{"version":"1.0.0","segments":[{"startTime":0.3,"endTime":1.86,"body":"- Today, I'm going to tell you"},{"startTime":1.86,"endTime":3.93,"body":"about The Mathematics of Game Theory."},{"startTime":3.93,"endTime":7.98,"body":"When we buy and sell,"},{"startTime":3.93,"endTime":7.98,"body":"barter, bid at auctions,"},{"startTime":7.98,"endTime":10.83,"body":"and compete for resources more widely,"},{"startTime":10.83,"endTime":14.13,"body":"we want to make sure that"},{"startTime":10.83,"endTime":14.13,"body":"we're using the best strategy"},{"startTime":14.13,"endTime":16.41,"body":"that will get us the best deal,"},{"startTime":16.41,"endTime":18.63,"body":"and mathematics can help us with this."},{"startTime":18.63,"endTime":19.463,"body":"In particular,"},{"startTime":19.463,"endTime":21.9,"body":"the field of mathematics"},{"startTime":19.463,"endTime":21.9,"body":"known as game theory."},{"startTime":21.9,"endTime":23.82,"body":"And in fact the Nobel Prize for Economics"},{"startTime":23.82,"endTime":27.06,"body":"has been won several"},{"startTime":23.82,"endTime":27.06,"body":"times by mathematicians"},{"startTime":27.06,"endTime":30.333,"body":"who were using game theory to"},{"startTime":27.06,"endTime":30.333,"body":"analyze economic questions."},{"startTime":31.26,"endTime":33.39,"body":"So we're going to talk"},{"startTime":31.26,"endTime":33.39,"body":"through some of those,"},{"startTime":33.39,"endTime":34.59,"body":"so is questions today."},{"startTime":34.59,"endTime":36.06,"body":"On the way, we're going to learn"},{"startTime":36.06,"endTime":38.22,"body":"how much to charge for a cup of coffee,"},{"startTime":38.22,"endTime":40.26,"body":"the best way to fight a duel,"},{"startTime":40.26,"endTime":42.423,"body":"and when you should buy a house."},{"startTime":43.41,"endTime":47.4,"body":"So I'm going to begin just by"},{"startTime":43.41,"endTime":47.4,"body":"giving a couple of examples"},{"startTime":47.4,"endTime":51.09,"body":"of the sorts of questions that"},{"startTime":47.4,"endTime":51.09,"body":"mathematics can help us with."},{"startTime":51.09,"endTime":52.7,"body":"So you've got three things there."},{"startTime":52.7,"endTime":54.87,"body":"On the top left, you've got price war."},{"startTime":54.87,"endTime":57.87,"body":"So imagine you've got something to sell,"},{"startTime":57.87,"endTime":59.13,"body":"you've got a crowded market"},{"startTime":59.13,"endTime":61.86,"body":"that you're competing for market share in."},{"startTime":61.86,"endTime":64.35,"body":"One possibility is that"},{"startTime":61.86,"endTime":64.35,"body":"you could lower your prices"},{"startTime":64.35,"endTime":66.06,"body":"to try and increase your market share,"},{"startTime":66.06,"endTime":68.04,"body":"your profit per item will go down"},{"startTime":68.04,"endTime":69.87,"body":"but hopefully you sell more items."},{"startTime":69.87,"endTime":72.75,"body":"What should you do? How"},{"startTime":69.87,"endTime":72.75,"body":"do you set your prices?"},{"startTime":72.75,"endTime":75.36,"body":"Below that, a sort of"},{"startTime":72.75,"endTime":75.36,"body":"slightly similar thing"},{"startTime":75.36,"endTime":77.22,"body":"is the problem of the world's best coffee."},{"startTime":77.22,"endTime":79.53,"body":"Now as you know, whenever"},{"startTime":77.22,"endTime":79.53,"body":"you travel anywhere"},{"startTime":79.53,"endTime":82.47,"body":"and you see some cafes"},{"startTime":79.53,"endTime":82.47,"body":"and all of them claim"},{"startTime":82.47,"endTime":84.42,"body":"that they sell the world's best coffee."},{"startTime":84.42,"endTime":87.06,"body":"So if you're a tourist, you"},{"startTime":84.42,"endTime":87.06,"body":"don't know which cafe to go in,"},{"startTime":87.06,"endTime":89.16,"body":"you just sort of pick randomly,"},{"startTime":89.16,"endTime":91.41,"body":"and so you are not very"},{"startTime":89.16,"endTime":91.41,"body":"price sensitive there"},{"startTime":91.41,"endTime":93.66,"body":"and the cafe owners perhaps know that."},{"startTime":93.66,"endTime":96.18,"body":"So they also have to bear in mind,"},{"startTime":96.18,"endTime":98.16,"body":"if you are setting the"},{"startTime":96.18,"endTime":98.16,"body":"price of your coffee"},{"startTime":98.16,"endTime":101.64,"body":"that the locals get"},{"startTime":98.16,"endTime":101.64,"body":"disgruntled if you massively,"},{"startTime":101.64,"endTime":103.92,"body":"you know, charge make"},{"startTime":101.64,"endTime":103.92,"body":"pounds for a cappuccino."},{"startTime":103.92,"endTime":106.23,"body":"So there the question is,"},{"startTime":106.23,"endTime":110.167,"body":"how much is the price"},{"startTime":106.23,"endTime":110.167,"body":"sensitivity in that market"},{"startTime":110.167,"endTime":112.14,"body":"comparing with what tourists will"},{"startTime":112.14,"endTime":113.88,"body":"sort of end up grudgingly paying"},{"startTime":113.88,"endTime":117.0,"body":"versus the locals that"},{"startTime":113.88,"endTime":117.0,"body":"you need to keep on side."},{"startTime":117.0,"endTime":118.62,"body":"Again, mathematical models and ideas"},{"startTime":118.62,"endTime":121.05,"body":"can help us make those sorts of decisions."},{"startTime":121.05,"endTime":122.58,"body":"And the other picture got up there"},{"startTime":122.58,"endTime":124.77,"body":"is clearly some very, very educated,"},{"startTime":124.77,"endTime":128.31,"body":"knowledgeable art buyers at an auction."},{"startTime":128.31,"endTime":131.52,"body":"This is a fun old cartoon,"},{"startTime":128.31,"endTime":131.52,"body":"modern connoisseurs it's called,"},{"startTime":131.52,"endTime":133.203,"body":"so the picture's upside"},{"startTime":131.52,"endTime":133.203,"body":"down there when you are."},{"startTime":133.203,"endTime":135.54,"body":"When you're bidding for things"},{"startTime":135.54,"endTime":138.63,"body":"and it need not be at a"},{"startTime":135.54,"endTime":138.63,"body":"Christie's or Sotheby's,"},{"startTime":138.63,"endTime":140.79,"body":"but on eBay or any other sort of place"},{"startTime":140.79,"endTime":144.717,"body":"where you're offering a price,"},{"startTime":140.79,"endTime":144.717,"body":"what's your best strategy?"},{"startTime":144.717,"endTime":146.82,"body":"And from the other side,"},{"startTime":144.717,"endTime":146.82,"body":"if you are an auctioneer,"},{"startTime":146.82,"endTime":149.883,"body":"what kind of auction"},{"startTime":146.82,"endTime":149.883,"body":"structure should you use?"},{"startTime":150.81,"endTime":154.17,"body":"Well, a lecture with the word"},{"startTime":150.81,"endTime":154.17,"body":"game theory in the title,"},{"startTime":154.17,"endTime":157.23,"body":"that's two words, with the"},{"startTime":154.17,"endTime":157.23,"body":"phrase game theory in the title"},{"startTime":157.23,"endTime":162.23,"body":"almost legally has to mention"},{"startTime":157.23,"endTime":162.23,"body":"one very, very famous game"},{"startTime":162.24,"endTime":165.96,"body":"or situation where you are"},{"startTime":162.24,"endTime":165.96,"body":"trying to think of a strategy."},{"startTime":165.96,"endTime":167.67,"body":"So I thought, I should mention this,"},{"startTime":167.67,"endTime":169.5,"body":"even though probably a"},{"startTime":167.67,"endTime":169.5,"body":"few of you in the audience"},{"startTime":169.5,"endTime":172.08,"body":"will have heard of this before,"},{"startTime":169.5,"endTime":172.08,"body":"but let's talk about it,"},{"startTime":172.08,"endTime":173.73,"body":"the prisoner's dilemma."},{"startTime":173.73,"endTime":175.62,"body":"And this is a famous example"},{"startTime":175.62,"endTime":180.62,"body":"of where you can use a"},{"startTime":175.62,"endTime":180.62,"body":"bit of reasoning to decide"},{"startTime":180.78,"endTime":183.42,"body":"there's sort of a dilemma"},{"startTime":180.78,"endTime":183.42,"body":"that happens in a situation."},{"startTime":183.42,"endTime":184.86,"body":"And the situation is this,"},{"startTime":184.86,"endTime":186.81,"body":"you and your criminal accomplice"},{"startTime":186.81,"endTime":188.85,"body":"have been caught by the police"},{"startTime":188.85,"endTime":192.0,"body":"and you're suspected of some crime,"},{"startTime":192.0,"endTime":194.04,"body":"maybe it's burglary or something."},{"startTime":194.04,"endTime":196.05,"body":"Now they take you off to separate places"},{"startTime":196.05,"endTime":197.15,"body":"and they question you."},{"startTime":198.12,"endTime":200.73,"body":"If both of you don't say anything,"},{"startTime":200.73,"endTime":202.23,"body":"then they haven't got enough evidence"},{"startTime":202.23,"endTime":203.97,"body":"to charge you with the big thing,"},{"startTime":203.97,"endTime":206.52,"body":"so maybe they can get"},{"startTime":203.97,"endTime":206.52,"body":"you on a lesser charge."},{"startTime":206.52,"endTime":209.19,"body":"So you would each serve one year in prison"},{"startTime":209.19,"endTime":213.21,"body":"because they can't get"},{"startTime":209.19,"endTime":213.21,"body":"you for the big crime."},{"startTime":213.21,"endTime":216.24,"body":"But if one of you talks"},{"startTime":216.24,"endTime":219.6,"body":"and you can implicate your accomplice,"},{"startTime":219.6,"endTime":222.45,"body":"frowned upon in some circles,"},{"startTime":219.6,"endTime":222.45,"body":"but, you know, let's..."},{"startTime":222.45,"endTime":224.19,"body":"It's a theoretical game."},{"startTime":224.19,"endTime":226.11,"body":"You implicate your accomplice."},{"startTime":226.11,"endTime":228.27,"body":"If they don't talk,"},{"startTime":228.27,"endTime":230.64,"body":"then there's not enough"},{"startTime":228.27,"endTime":230.64,"body":"evidence against you,"},{"startTime":230.64,"endTime":232.86,"body":"so you get to go free because"},{"startTime":230.64,"endTime":232.86,"body":"you've helped the police"},{"startTime":232.86,"endTime":235.47,"body":"and you can't yourself be convicted"},{"startTime":235.47,"endTime":238.2,"body":"and then you are unfortunate acquaintance"},{"startTime":238.2,"endTime":240.75,"body":"will get the full sentence."},{"startTime":240.75,"endTime":241.86,"body":"If both of you talk,"},{"startTime":241.86,"endTime":244.23,"body":"and we can put this in"},{"startTime":241.86,"endTime":244.23,"body":"a table of outcomes."},{"startTime":244.23,"endTime":247.08,"body":"If both of you talk"},{"startTime":244.23,"endTime":247.08,"body":"then you've cooperated,"},{"startTime":247.08,"endTime":248.37,"body":"but you're both implicated"},{"startTime":248.37,"endTime":250.68,"body":"so maybe you get three"},{"startTime":248.37,"endTime":250.68,"body":"years in prison each."},{"startTime":250.68,"endTime":253.29,"body":"So you can write down all of the outcomes"},{"startTime":253.29,"endTime":254.97,"body":"in a little table like this."},{"startTime":254.97,"endTime":258.21,"body":"And the question is,"},{"startTime":254.97,"endTime":258.21,"body":"what strategy should I,"},{"startTime":258.21,"endTime":261.183,"body":"if I'm prison A, let's"},{"startTime":258.21,"endTime":261.183,"body":"say, what should I do?"},{"startTime":262.26,"endTime":263.28,"body":"First of all, let's think,"},{"startTime":263.28,"endTime":265.48,"body":"if we're acting purely"},{"startTime":263.28,"endTime":265.48,"body":"out of self-interest"},{"startTime":266.7,"endTime":268.68,"body":"and let's say there's no repercussions,"},{"startTime":268.68,"endTime":270.39,"body":"which may not be very realistic."},{"startTime":270.39,"endTime":272.73,"body":"But if you are prisoner A, what do you do?"},{"startTime":272.73,"endTime":276.9,"body":"You don't know what the"},{"startTime":272.73,"endTime":276.9,"body":"other person is going to do."},{"startTime":276.9,"endTime":281.13,"body":"They're over there, they've"},{"startTime":276.9,"endTime":281.13,"body":"separated you, you don't know,"},{"startTime":281.13,"endTime":283.14,"body":"are they going to talk,"},{"startTime":281.13,"endTime":283.14,"body":"are they not going to talk?"},{"startTime":283.14,"endTime":286.8,"body":"So you have to plan for"},{"startTime":283.14,"endTime":286.8,"body":"both possible outcomes"},{"startTime":286.8,"endTime":289.14,"body":"for what they will do, you don't know."},{"startTime":289.14,"endTime":291.72,"body":"So what we can see is"},{"startTime":289.14,"endTime":291.72,"body":"that that the column there"},{"startTime":291.72,"endTime":293.73,"body":"that has prisoner B talks,"},{"startTime":293.73,"endTime":295.47,"body":"those are the outcomes if prisoner B talks"},{"startTime":295.47,"endTime":297.48,"body":"depending on whether you talk or not."},{"startTime":297.48,"endTime":301.86,"body":"So you can see if you talk,"},{"startTime":297.48,"endTime":301.86,"body":"in that column, prisoner B,"},{"startTime":301.86,"endTime":304.86,"body":"you as prisoner A will"},{"startTime":301.86,"endTime":304.86,"body":"either get three years"},{"startTime":304.86,"endTime":306.66,"body":"or five years in prison."},{"startTime":306.66,"endTime":309.57,"body":"And of course, you want"},{"startTime":306.66,"endTime":309.57,"body":"the least of time possible."},{"startTime":309.57,"endTime":311.28,"body":"So the best thing for you to do,"},{"startTime":311.28,"endTime":312.9,"body":"if prisoner B is going to talk,"},{"startTime":312.9,"endTime":315.33,"body":"the best thing for you to"},{"startTime":312.9,"endTime":315.33,"body":"do is also talk, right?"},{"startTime":315.33,"endTime":316.53,"body":"So that's that."},{"startTime":316.53,"endTime":319.95,"body":"What about if prisoner B"},{"startTime":316.53,"endTime":319.95,"body":"doesn't talk, they stay silent?"},{"startTime":319.95,"endTime":323.76,"body":"Well, that's the column on the"},{"startTime":319.95,"endTime":323.76,"body":"right, second column there."},{"startTime":323.76,"endTime":326.37,"body":"But again look, for you, your outcomes,"},{"startTime":326.37,"endTime":329.22,"body":"depending on your decision,"},{"startTime":326.37,"endTime":329.22,"body":"you either get no years,"},{"startTime":329.22,"endTime":331.35,"body":"you go free or you get one year."},{"startTime":331.35,"endTime":335.94,"body":"So again, in that case,"},{"startTime":331.35,"endTime":335.94,"body":"it's better for you to talk."},{"startTime":335.94,"endTime":337.62,"body":"So this is sort of a bit of paradox"},{"startTime":337.62,"endTime":341.46,"body":"because if you're just acting"},{"startTime":337.62,"endTime":341.46,"body":"purely from self-interest,"},{"startTime":341.46,"endTime":343.32,"body":"this kind of decision making would say"},{"startTime":343.32,"endTime":347.34,"body":"you should always talk,"},{"startTime":343.32,"endTime":347.34,"body":"because whatever B does,"},{"startTime":347.34,"endTime":350.67,"body":"the least worst outcome"},{"startTime":347.34,"endTime":350.67,"body":"for you comes from talking."},{"startTime":350.67,"endTime":351.69,"body":"The problem is"},{"startTime":351.69,"endTime":354.93,"body":"prisoner B is making"},{"startTime":351.69,"endTime":354.93,"body":"exactly that calculation,"},{"startTime":354.93,"endTime":357.0,"body":"so they're going to talk as well."},{"startTime":357.0,"endTime":358.587,"body":"So you'll be in the"},{"startTime":357.0,"endTime":358.587,"body":"situation where you both talk"},{"startTime":358.587,"endTime":360.27,"body":"and you both get three years."},{"startTime":360.27,"endTime":364.11,"body":"However, if you had both stayed silent,"},{"startTime":364.11,"endTime":365.61,"body":"it would be a better outcome."},{"startTime":365.61,"endTime":367.14,"body":"But the problem with this better outcome"},{"startTime":367.14,"endTime":370.2,"body":"is you can improve it if"},{"startTime":367.14,"endTime":370.2,"body":"you unilaterally decide,"},{"startTime":370.2,"endTime":371.85,"body":"actually I'm going to talk."},{"startTime":371.85,"endTime":375.3,"body":"So it's not a kind of stable equilibrium,"},{"startTime":375.3,"endTime":377.19,"body":"the one where you get one year each."},{"startTime":377.19,"endTime":381.03,"body":"But colluding in this case"},{"startTime":377.19,"endTime":381.03,"body":"or agreeing to cooperate"},{"startTime":381.03,"endTime":383.04,"body":"and both not saying anything"},{"startTime":383.04,"endTime":385.71,"body":"would get you a total of two prison years,"},{"startTime":385.71,"endTime":390.21,"body":"but it's unfortunately changing"},{"startTime":385.71,"endTime":390.21,"body":"and betraying your friend,"},{"startTime":390.21,"endTime":391.38,"body":"you may be able to go free."},{"startTime":391.38,"endTime":393.9,"body":"So that's a sort of a moral thing"},{"startTime":393.9,"endTime":395.97,"body":"mixed in with the mathematical thing."},{"startTime":395.97,"endTime":399.63,"body":"But this idea that you can make"},{"startTime":395.97,"endTime":399.63,"body":"a kind of table of outcomes"},{"startTime":399.63,"endTime":401.37,"body":"and look at the payoffs for each strategy"},{"startTime":401.37,"endTime":404.16,"body":"and make decisions based on those numbers"},{"startTime":404.16,"endTime":407.34,"body":"is where we start to do some mathematics."},{"startTime":407.34,"endTime":411.42,"body":"So our first putative"},{"startTime":407.34,"endTime":411.42,"body":"financial situation, price war,"},{"startTime":411.42,"endTime":415.11,"body":"is exactly the prisoner's"},{"startTime":411.42,"endTime":415.11,"body":"dilemma just in disguise."},{"startTime":415.11,"endTime":417.78,"body":"So here I've got two companies,"},{"startTime":417.78,"endTime":422.04,"body":"Acme Widgets and BestCo"},{"startTime":417.78,"endTime":422.04,"body":"who also make widgets"},{"startTime":422.04,"endTime":425.79,"body":"and they are competing for"},{"startTime":422.04,"endTime":425.79,"body":"the market share of widgets,"},{"startTime":425.79,"endTime":428.07,"body":"which I still don't really"},{"startTime":425.79,"endTime":428.07,"body":"know what a widget is,"},{"startTime":428.07,"endTime":429.81,"body":"but all the examples always use widgets."},{"startTime":429.81,"endTime":432.72,"body":"Whatever they are,"},{"startTime":429.81,"endTime":432.72,"body":"they're both selling them,"},{"startTime":432.72,"endTime":436.47,"body":"and there's a total market for"},{"startTime":432.72,"endTime":436.47,"body":"let's say a hundred widgets."},{"startTime":436.47,"endTime":440.34,"body":"If they both charge the"},{"startTime":436.47,"endTime":440.34,"body":"same price for each widget,"},{"startTime":440.34,"endTime":443.7,"body":"then they will get equal"},{"startTime":440.34,"endTime":443.7,"body":"market share, 50 each."},{"startTime":443.7,"endTime":445.83,"body":"But if one price is lower,"},{"startTime":445.83,"endTime":448.02,"body":"then there's a bit of laziness"},{"startTime":448.02,"endTime":450.21,"body":"about swapping over to"},{"startTime":448.02,"endTime":450.21,"body":"a different company."},{"startTime":450.21,"endTime":452.91,"body":"But let's say that 80% of people"},{"startTime":452.91,"endTime":456.21,"body":"will then switch to the"},{"startTime":452.91,"endTime":456.21,"body":"lower priced provider,"},{"startTime":456.21,"endTime":460.62,"body":"and so the lower price will"},{"startTime":456.21,"endTime":460.62,"body":"get an 80-20 market split."},{"startTime":460.62,"endTime":463.41,"body":"And so they'll sell 80"},{"startTime":460.62,"endTime":463.41,"body":"widgets rather than..."},{"startTime":463.41,"endTime":466.02,"body":"The more expensive one will only sell 20."},{"startTime":466.02,"endTime":467.28,"body":"So you could, you know,"},{"startTime":467.28,"endTime":469.77,"body":"put in some numbers in"},{"startTime":467.28,"endTime":469.77,"body":"and see what will happen."},{"startTime":469.77,"endTime":472.14,"body":"So let's say, if you"},{"startTime":469.77,"endTime":472.14,"body":"charge a higher price,"},{"startTime":472.14,"endTime":475.44,"body":"you make five pounds profit"},{"startTime":472.14,"endTime":475.44,"body":"on every widget that you sell,"},{"startTime":475.44,"endTime":476.28,"body":"and at lower price,"},{"startTime":476.28,"endTime":477.51,"body":"you'll make four pounds profit"},{"startTime":477.51,"endTime":480.09,"body":"and you could construct a"},{"startTime":477.51,"endTime":480.09,"body":"table of all the outcomes."},{"startTime":480.09,"endTime":483.81,"body":"And so this is now Acme versus BestCo."},{"startTime":483.81,"endTime":485.55,"body":"So A for Acme, of course."},{"startTime":485.55,"endTime":487.77,"body":"So here you've got in the first row there,"},{"startTime":487.77,"endTime":489.99,"body":"what happens if A cuts their price"},{"startTime":489.99,"endTime":492.364,"body":"and you can work out"},{"startTime":489.99,"endTime":492.364,"body":"what profit will happen"},{"startTime":492.364,"endTime":495.06,"body":"for each scenario."},{"startTime":495.06,"endTime":498.75,"body":"So in the top they've got, if"},{"startTime":495.06,"endTime":498.75,"body":"A and B both cut their prices,"},{"startTime":498.75,"endTime":500.52,"body":"then their price will"},{"startTime":498.75,"endTime":500.52,"body":"end up being the same,"},{"startTime":500.52,"endTime":502.92,"body":"and so they'll both make the same profit"},{"startTime":502.92,"endTime":506.67,"body":"which is four pounds each for every 50,"},{"startTime":506.67,"endTime":509.37,"body":"and you sell 50 widgets, so 200 pounds."},{"startTime":509.37,"endTime":512.28,"body":"If A cuts the price but B doesn't."},{"startTime":512.28,"endTime":513.33,"body":"So we are starting off here,"},{"startTime":513.33,"endTime":515.82,"body":"they're both charging this higher price"},{"startTime":515.82,"endTime":517.17,"body":"and they're making five"},{"startTime":515.82,"endTime":517.17,"body":"pound profits each."},{"startTime":517.17,"endTime":519.75,"body":"So we're start off in the"},{"startTime":517.17,"endTime":519.75,"body":"bottom right hand corner"},{"startTime":519.75,"endTime":522.27,"body":"where they're selling 50"},{"startTime":519.75,"endTime":522.27,"body":"widgets for five pounds profit,"},{"startTime":522.27,"endTime":524.43,"body":"they make 250 pounds profit."},{"startTime":524.43,"endTime":529.43,"body":"So what's the incentive"},{"startTime":524.43,"endTime":529.43,"body":"here for company A or Acme,"},{"startTime":529.44,"endTime":532.26,"body":"they, if they cut their price,"},{"startTime":532.26,"endTime":535.35,"body":"which would be the top entry here,"},{"startTime":535.35,"endTime":537.33,"body":"if they cut their price and B doesn't,"},{"startTime":537.33,"endTime":538.5,"body":"their profit shoots up"},{"startTime":538.5,"endTime":540.45,"body":"'cause they suddenly now sell 80 widgets"},{"startTime":540.45,"endTime":543.24,"body":"for four pounds profit, 320 pounds profit."},{"startTime":543.24,"endTime":544.893,"body":"But poor old B, BestCo,"},{"startTime":545.88,"endTime":547.56,"body":"they are only selling 20 widgets now."},{"startTime":547.56,"endTime":549.87,"body":"So even though they're"},{"startTime":547.56,"endTime":549.87,"body":"making five pounds on each,"},{"startTime":549.87,"endTime":551.73,"body":"their profit has gone right down."},{"startTime":551.73,"endTime":555.27,"body":"So if A cuts their price,"},{"startTime":555.27,"endTime":558.9,"body":"they increase their profit"},{"startTime":555.27,"endTime":558.9,"body":"at the expense of company B,"},{"startTime":558.9,"endTime":562.32,"body":"and now BestCo, what's it going to do?"},{"startTime":562.32,"endTime":565.11,"body":"Well, it's got to cut"},{"startTime":562.32,"endTime":565.11,"body":"it's prices as well now,"},{"startTime":565.11,"endTime":567.39,"body":"otherwise it's really in a bad place."},{"startTime":567.39,"endTime":568.92,"body":"So maybe it does that."},{"startTime":568.92,"endTime":571.74,"body":"And now you're in this"},{"startTime":568.92,"endTime":571.74,"body":"position in the top left"},{"startTime":571.74,"endTime":573.63,"body":"where they've both cut their prices,"},{"startTime":573.63,"endTime":576.93,"body":"there's a new position"},{"startTime":576.93,"endTime":579.45,"body":"where they both have"},{"startTime":576.93,"endTime":579.45,"body":"equal market share again,"},{"startTime":579.45,"endTime":581.07,"body":"but their profits are lower."},{"startTime":581.07,"endTime":583.74,"body":"And again, now whats to stop company A"},{"startTime":583.74,"endTime":584.94,"body":"cutting its prices again,"},{"startTime":584.94,"endTime":586.95,"body":"and increasing its profits temporarily"},{"startTime":586.95,"endTime":589.08,"body":"until the other one cuts the price."},{"startTime":589.08,"endTime":591.3,"body":"So this is why governments"},{"startTime":589.08,"endTime":591.3,"body":"have to legislate"},{"startTime":591.3,"endTime":593.55,"body":"against collusion because, of course,"},{"startTime":593.55,"endTime":596.13,"body":"what you'd really like to do"},{"startTime":593.55,"endTime":596.13,"body":"if you are these companies,"},{"startTime":596.13,"endTime":598.17,"body":"we should want to keep our profits high"},{"startTime":598.17,"endTime":599.94,"body":"and that means we have to"},{"startTime":598.17,"endTime":599.94,"body":"know what the other guy,"},{"startTime":599.94,"endTime":603.75,"body":"we have to agree between"},{"startTime":599.94,"endTime":603.75,"body":"us to have a higher price."},{"startTime":603.75,"endTime":606.21,"body":"But if one cuts, then"},{"startTime":603.75,"endTime":606.21,"body":"they all have to cut."},{"startTime":606.21,"endTime":609.33,"body":"And so governments have to"},{"startTime":606.21,"endTime":609.33,"body":"legislate against these kind of,"},{"startTime":609.33,"endTime":613.5,"body":"let's all agree to inflate"},{"startTime":609.33,"endTime":613.5,"body":"our prices and keep them high"},{"startTime":613.5,"endTime":615.18,"body":"because it's a tempting strategy,"},{"startTime":615.18,"endTime":617.613,"body":"but it's just the prison's"},{"startTime":615.18,"endTime":617.613,"body":"dilemma in disguise."},{"startTime":618.48,"endTime":622.05,"body":"Now there's a game theorist"},{"startTime":618.48,"endTime":622.05,"body":"called David Blackwell"},{"startTime":622.05,"endTime":625.143,"body":"who had an interesting comment"},{"startTime":622.05,"endTime":625.143,"body":"about the prisoner's dilemma."},{"startTime":626.613,"endTime":627.446,"body":"Here he is."},{"startTime":627.446,"endTime":630.42,"body":"He was working in the 40s"},{"startTime":627.446,"endTime":630.42,"body":"and 50s on game theory,"},{"startTime":630.42,"endTime":633.12,"body":"and we'll talk about one of"},{"startTime":630.42,"endTime":633.12,"body":"the things he did in a moment."},{"startTime":633.12,"endTime":638.12,"body":"And he said, you know, a lot"},{"startTime":633.12,"endTime":638.12,"body":"of these perhaps toy problems,"},{"startTime":638.64,"endTime":640.92,"body":"there are real world"},{"startTime":638.64,"endTime":640.92,"body":"applications in economics,"},{"startTime":640.92,"endTime":645.36,"body":"but also this time it was at"},{"startTime":640.92,"endTime":645.36,"body":"the height of the Cold War,"},{"startTime":645.36,"endTime":647.31,"body":"and there are lots of scenarios in that"},{"startTime":647.31,"endTime":650.01,"body":"that mimic situations like"},{"startTime":647.31,"endTime":650.01,"body":"the prisoner's dilemma."},{"startTime":650.01,"endTime":651.63,"body":"So he remarked about this"},{"startTime":651.63,"endTime":656.16,"body":"that it was a bit similar"},{"startTime":651.63,"endTime":656.16,"body":"to the Cold War situation"},{"startTime":656.16,"endTime":659.58,"body":"with sort of America or the"},{"startTime":656.16,"endTime":659.58,"body":"West and the Soviet Union"},{"startTime":659.58,"endTime":662.85,"body":"and they kept sort of escalating"},{"startTime":659.58,"endTime":662.85,"body":"how many weapons they have."},{"startTime":662.85,"endTime":664.23,"body":"And, of course, the best thing really"},{"startTime":664.23,"endTime":667.77,"body":"is for everyone to disarm and"},{"startTime":664.23,"endTime":667.77,"body":"get rid of all their weapons."},{"startTime":667.77,"endTime":671.82,"body":"But then you worry that your"},{"startTime":667.77,"endTime":671.82,"body":"opponent is not doing that"},{"startTime":671.82,"endTime":673.86,"body":"and then they keep their weapons."},{"startTime":673.86,"endTime":676.08,"body":"So unilateral disarmament"},{"startTime":676.08,"endTime":678.96,"body":"was felt to be like"},{"startTime":676.08,"endTime":678.96,"body":"too risky of a strategy"},{"startTime":678.96,"endTime":681.15,"body":"because then you are sort"},{"startTime":678.96,"endTime":681.15,"body":"of at a disadvantage."},{"startTime":681.15,"endTime":683.64,"body":"So it's a bit like the prisoner's dilemma"},{"startTime":683.64,"endTime":685.62,"body":"where actually it would be nice"},{"startTime":685.62,"endTime":687.54,"body":"if you would cooperate and disarm,"},{"startTime":687.54,"endTime":691.05,"body":"but that was too risky"},{"startTime":691.05,"endTime":693.537,"body":"because what if the other"},{"startTime":691.05,"endTime":693.537,"body":"person doesn't do that,"},{"startTime":693.537,"endTime":695.04,"body":"and then they've got all the weapons"},{"startTime":695.04,"endTime":697.11,"body":"and you haven't got any, was the thinking."},{"startTime":697.11,"endTime":701.343,"body":"So again, it's the prisoner's"},{"startTime":697.11,"endTime":701.343,"body":"dilemma in another guise."},{"startTime":702.66,"endTime":705.093,"body":"So David Blackwell, I mean, what a guy,"},{"startTime":706.29,"endTime":707.76,"body":"we don't hear enough about him actually."},{"startTime":707.76,"endTime":711.03,"body":"He was a real trailblazer."},{"startTime":711.03,"endTime":714.33,"body":"When he went to Princeton University,"},{"startTime":714.33,"endTime":718.14,"body":"the Institute of Advanced"},{"startTime":714.33,"endTime":718.14,"body":"Study at Princeton,"},{"startTime":718.14,"endTime":721.83,"body":"he was not only the first"},{"startTime":718.14,"endTime":721.83,"body":"black faculty member there,"},{"startTime":721.83,"endTime":722.76,"body":"but it was at a time"},{"startTime":722.76,"endTime":724.98,"body":"when they'd never even"},{"startTime":722.76,"endTime":724.98,"body":"had a black student."},{"startTime":724.98,"endTime":727.53,"body":"So like real, real trailblazer."},{"startTime":727.53,"endTime":730.23,"body":"I think he was only the"},{"startTime":727.53,"endTime":730.23,"body":"seventh African-American"},{"startTime":730.23,"endTime":734.46,"body":"to get a PhD, and he was"},{"startTime":730.23,"endTime":734.46,"body":"absolutely top of his field,"},{"startTime":734.46,"endTime":736.59,"body":"was the Head of Department"},{"startTime":734.46,"endTime":736.59,"body":"of Statistics, I think,"},{"startTime":736.59,"endTime":738.57,"body":"at Berkeley for 30 years"},{"startTime":738.57,"endTime":740.67,"body":"and some amazing work in game theory,"},{"startTime":740.67,"endTime":744.27,"body":"which I will just give"},{"startTime":740.67,"endTime":744.27,"body":"you a little idea of."},{"startTime":744.27,"endTime":747.24,"body":"And it's about duels and you could see"},{"startTime":747.24,"endTime":752.24,"body":"as a kind of cold war aspect"},{"startTime":747.24,"endTime":752.24,"body":"if you think. So, yes."},{"startTime":752.417,"endTime":754.8,"body":"This is the picture, Pistols At Dawn,"},{"startTime":754.8,"endTime":758.4,"body":"but there are more real world scenarios"},{"startTime":758.4,"endTime":760.32,"body":"where you might be in"},{"startTime":758.4,"endTime":760.32,"body":"a situation like this."},{"startTime":760.32,"endTime":762.75,"body":"For example, in a warfare type situation"},{"startTime":762.75,"endTime":765.0,"body":"or some kinds of auction as well."},{"startTime":765.0,"endTime":769.59,"body":"And the question here is"},{"startTime":765.0,"endTime":769.59,"body":"if you are fighting a duel,"},{"startTime":769.59,"endTime":773.01,"body":"so I really like these, these"},{"startTime":769.59,"endTime":773.01,"body":"are the petticoat duelists,"},{"startTime":773.01,"endTime":776.13,"body":"which is an old picture from 1792."},{"startTime":776.13,"endTime":778.32,"body":"So if you're fighting a duel"},{"startTime":778.32,"endTime":781.403,"body":"kind of in the traditional"},{"startTime":778.32,"endTime":781.403,"body":"setting, what you do is,"},{"startTime":781.403,"endTime":783.45,"body":"I know, you meet up at dawn"},{"startTime":783.45,"endTime":784.95,"body":"and then you walk away from each other,"},{"startTime":784.95,"endTime":787.53,"body":"you turn around to face"},{"startTime":784.95,"endTime":787.53,"body":"each other, pistols raised,"},{"startTime":787.53,"endTime":789.96,"body":"and then you start"},{"startTime":787.53,"endTime":789.96,"body":"walking towards each other"},{"startTime":789.96,"endTime":793.05,"body":"and at some point you"},{"startTime":789.96,"endTime":793.05,"body":"fire, you have one bullet,"},{"startTime":793.05,"endTime":796.86,"body":"you could fire your bullet and"},{"startTime":793.05,"endTime":796.86,"body":"if you hit the other person"},{"startTime":796.86,"endTime":799.53,"body":"then you know you've won, and if you miss,"},{"startTime":799.53,"endTime":803.19,"body":"then they can come up and"},{"startTime":799.53,"endTime":803.19,"body":"let's say finish things off"},{"startTime":803.19,"endTime":804.393,"body":"and then they've won."},{"startTime":805.95,"endTime":809.76,"body":"So the question here is"},{"startTime":805.95,"endTime":809.76,"body":"when should you shoot?"},{"startTime":809.76,"endTime":812.016,"body":"When should you fire your pistol?"},{"startTime":812.016,"endTime":814.05,"body":"If you think this is too frivolous,"},{"startTime":814.05,"endTime":816.54,"body":"you could imagine two nuclear submarines"},{"startTime":816.54,"endTime":818.92,"body":"deciding whether or when"},{"startTime":816.54,"endTime":818.92,"body":"to fire their missiles"},{"startTime":819.829,"endTime":821.64,"body":"if the stakes aren't high enough."},{"startTime":821.64,"endTime":822.75,"body":"When should you fire?"},{"startTime":822.75,"endTime":827.49,"body":"And there's a interplay"},{"startTime":822.75,"endTime":827.49,"body":"between two things."},{"startTime":827.49,"endTime":829.17,"body":"If you fire very early,"},{"startTime":829.17,"endTime":831.51,"body":"then you're really far"},{"startTime":829.17,"endTime":831.51,"body":"away from your opponent"},{"startTime":831.51,"endTime":834.24,"body":"and so your chance of actually"},{"startTime":831.51,"endTime":834.24,"body":"hitting them is lower."},{"startTime":834.24,"endTime":837.03,"body":"But if you wait until you're very close"},{"startTime":837.03,"endTime":839.007,"body":"then there's a risk that"},{"startTime":837.03,"endTime":839.007,"body":"they already fired first,"},{"startTime":839.007,"endTime":841.74,"body":"and so you are in no position to fire."},{"startTime":841.74,"endTime":844.32,"body":"So the question is what's the right place?"},{"startTime":844.32,"endTime":846.81,"body":"What's the sweet spot?"},{"startTime":844.32,"endTime":846.81,"body":"When should you fire?"},{"startTime":846.81,"endTime":849.27,"body":"And you can model this mathematically."},{"startTime":849.27,"endTime":852.09,"body":"So you can decide what rules"},{"startTime":849.27,"endTime":852.09,"body":"would work best for you,"},{"startTime":852.09,"endTime":855.18,"body":"you can vary these settings,"},{"startTime":852.09,"endTime":855.18,"body":"you can explore all of this,"},{"startTime":855.18,"endTime":858.06,"body":"and I encourage you to do"},{"startTime":855.18,"endTime":858.06,"body":"that 'cause playing is fun"},{"startTime":858.06,"endTime":859.56,"body":"and that's what we like doing."},{"startTime":860.73,"endTime":864.48,"body":"Don't re-enact it in real life"},{"startTime":860.73,"endTime":864.48,"body":"though, health and safety."},{"startTime":864.48,"endTime":867.0,"body":"So let's say, we'll make"},{"startTime":864.48,"endTime":867.0,"body":"some rules of engagement."},{"startTime":867.0,"endTime":869.703,"body":"So you're going to have"},{"startTime":867.0,"endTime":869.703,"body":"one chance each to fire."},{"startTime":870.69,"endTime":873.03,"body":"You initially are standing 20 paces apart"},{"startTime":873.03,"endTime":876.097,"body":"and then you start walking"},{"startTime":873.03,"endTime":876.097,"body":"towards each other,"},{"startTime":876.097,"endTime":877.68,"body":"and so after you've walked 10 paces,"},{"startTime":877.68,"endTime":878.513,"body":"if you get to that point,"},{"startTime":878.513,"endTime":880.83,"body":"you'll be right next to each other."},{"startTime":880.83,"endTime":883.14,"body":"We'll assume that the"},{"startTime":880.83,"endTime":883.14,"body":"opponent's equally skilled"},{"startTime":883.14,"endTime":885.36,"body":"so they have an equal chance"},{"startTime":883.14,"endTime":885.36,"body":"at any particular stage"},{"startTime":885.36,"endTime":888.66,"body":"of successfully hitting another person."},{"startTime":888.66,"endTime":891.03,"body":"And that the probability"},{"startTime":888.66,"endTime":891.03,"body":"of hitting your opponent"},{"startTime":891.03,"endTime":892.65,"body":"after p paces,"},{"startTime":892.65,"endTime":894.66,"body":"and I'll assume you have"},{"startTime":892.65,"endTime":894.66,"body":"a whole number of paces"},{"startTime":894.66,"endTime":897.9,"body":"if you want to do this, you"},{"startTime":894.66,"endTime":897.9,"body":"know, do it kind of continuous."},{"startTime":897.9,"endTime":899.97,"body":"You can have a continuous"},{"startTime":897.9,"endTime":899.97,"body":"idea of this, instead of this,"},{"startTime":899.97,"endTime":902.49,"body":"which is discrete a whole number of paces."},{"startTime":902.49,"endTime":906.138,"body":"The probability is we're"},{"startTime":902.49,"endTime":906.138,"body":"going to say is p over 10."},{"startTime":906.138,"endTime":910.47,"body":"After 10 paces, when you're"},{"startTime":906.138,"endTime":910.47,"body":"right next to each other,"},{"startTime":910.47,"endTime":911.91,"body":"that'll be a 10 out of 10,"},{"startTime":911.91,"endTime":913.2,"body":"a hundred percent probability"},{"startTime":913.2,"endTime":915.18,"body":"of actually successfully hitting them."},{"startTime":915.18,"endTime":916.86,"body":"But if you fire after zero paces,"},{"startTime":916.86,"endTime":918.6,"body":"you're so far away, you're"},{"startTime":916.86,"endTime":918.6,"body":"not going to hit them,"},{"startTime":918.6,"endTime":922.233,"body":"and then it's one in 10"},{"startTime":918.6,"endTime":922.233,"body":"probability for one pace and so on."},{"startTime":923.1,"endTime":925.95,"body":"So how do we decide the outcome?"},{"startTime":925.95,"endTime":930.95,"body":"Well, if I fire first and"},{"startTime":925.95,"endTime":930.95,"body":"I hit you, then I've won."},{"startTime":932.37,"endTime":935.13,"body":"And if I fire and I miss,"},{"startTime":935.13,"endTime":937.08,"body":"then that's bad for me because you,"},{"startTime":937.08,"endTime":939.45,"body":"wherever you were planning to fire from,"},{"startTime":939.45,"endTime":941.13,"body":"you can then just actually walk up to me"},{"startTime":941.13,"endTime":944.31,"body":"and take your time and"},{"startTime":941.13,"endTime":944.31,"body":"be guaranteed to win."},{"startTime":944.31,"endTime":947.37,"body":"So if I fire first and hit you, I win."},{"startTime":947.37,"endTime":950.67,"body":"If I fire first and miss you, I lose."},{"startTime":950.67,"endTime":952.98,"body":"If we both fire at the same time,"},{"startTime":952.98,"endTime":956.493,"body":"and both hit or both"},{"startTime":952.98,"endTime":956.493,"body":"miss, then it's a draw."},{"startTime":957.96,"endTime":959.82,"body":"So that's the kind of what the idea is."},{"startTime":959.82,"endTime":961.5,"body":"Now how are we going to"},{"startTime":959.82,"endTime":961.5,"body":"put some numbers to this?"},{"startTime":961.5,"endTime":964.2,"body":"It's not about profits or"},{"startTime":961.5,"endTime":964.2,"body":"time in prison or whatever,"},{"startTime":964.2,"endTime":966.66,"body":"well, it might be"},{"startTime":964.2,"endTime":966.66,"body":"depending on the outcome."},{"startTime":966.66,"endTime":967.8,"body":"It's not about those things,"},{"startTime":967.8,"endTime":971.46,"body":"so what we can do is just"},{"startTime":967.8,"endTime":971.46,"body":"assign a value to winning."},{"startTime":971.46,"endTime":972.87,"body":"And the simplest one is..."},{"startTime":972.87,"endTime":974.31,"body":"So the two people fighting"},{"startTime":974.31,"endTime":975.417,"body":"are going to be called Colin and Rose."},{"startTime":975.417,"endTime":977.67,"body":"And it's 'cause I've got a table,"},{"startTime":977.67,"endTime":979.86,"body":"and Colin is going to have the columns,"},{"startTime":979.86,"endTime":983.1,"body":"and Rose is going to"},{"startTime":979.86,"endTime":983.1,"body":"have the rows. (laughs)"},{"startTime":983.1,"endTime":986.073,"body":"So if you like that humor,"},{"startTime":983.1,"endTime":986.073,"body":"watch my other Gresham lectures."},{"startTime":987.432,"endTime":989.7,"body":"So the outcome is going to be,"},{"startTime":989.7,"endTime":994.29,"body":"Rose gets kind of plus one"},{"startTime":989.7,"endTime":994.29,"body":"for a win, if she wins,"},{"startTime":994.29,"endTime":998.37,"body":"she gets zero if it's a draw,"},{"startTime":994.29,"endTime":998.37,"body":"and minus one if she loses."},{"startTime":998.37,"endTime":1001.91,"body":"And this type of game is"},{"startTime":998.37,"endTime":1001.91,"body":"what we call a zero sum game"},{"startTime":1001.91,"endTime":1004.28,"body":"because Colin, if Rose gets plus one,"},{"startTime":1004.28,"endTime":1006.11,"body":"then, of course, that means Colin's lost"},{"startTime":1006.11,"endTime":1007.49,"body":"and will get minus one."},{"startTime":1007.49,"endTime":1011.0,"body":"So the sum of the outcomes is zero always."},{"startTime":1011.0,"endTime":1013.55,"body":"So actually when we do this little table,"},{"startTime":1013.55,"endTime":1016.49,"body":"there's no point including"},{"startTime":1013.55,"endTime":1016.49,"body":"the outcomes for Colin"},{"startTime":1016.49,"endTime":1018.74,"body":"from his point of view because"},{"startTime":1016.49,"endTime":1018.74,"body":"we can just deduce them,"},{"startTime":1018.74,"endTime":1021.05,"body":"it's minus whatever the"},{"startTime":1018.74,"endTime":1021.05,"body":"outcome was for Rose."},{"startTime":1021.05,"endTime":1024.623,"body":"So that simplifies things"},{"startTime":1021.05,"endTime":1024.623,"body":"a bit in the zero sum game."},{"startTime":1026.72,"endTime":1028.88,"body":"So how do we actually work out"},{"startTime":1028.88,"endTime":1031.85,"body":"what the payoff is for"},{"startTime":1028.88,"endTime":1031.85,"body":"a particular strategy?"},{"startTime":1031.85,"endTime":1034.159,"body":"Well, you can work out the probabilities"},{"startTime":1034.159,"endTime":1036.41,"body":"and then do the calculation"},{"startTime":1034.159,"endTime":1036.41,"body":"with these outcomes."},{"startTime":1036.41,"endTime":1038.03,"body":"I'll show you."},{"startTime":1038.03,"endTime":1042.74,"body":"Example, Rose decides she's"},{"startTime":1038.03,"endTime":1042.74,"body":"going to fire after four paces"},{"startTime":1042.74,"endTime":1045.59,"body":"and Colin decides he will fire after six"},{"startTime":1045.59,"endTime":1047.27,"body":"if he gets the chance."},{"startTime":1047.27,"endTime":1048.98,"body":"So obviously they don't know"},{"startTime":1047.27,"endTime":1048.98,"body":"what the other one's doing."},{"startTime":1048.98,"endTime":1053.12,"body":"So after four paces, that"},{"startTime":1048.98,"endTime":1053.12,"body":"means Rose will fire first,"},{"startTime":1053.12,"endTime":1055.4,"body":"fires after four paces,"},{"startTime":1055.4,"endTime":1058.49,"body":"she has a four in 10 'cause"},{"startTime":1055.4,"endTime":1058.49,"body":"that was our little rule,"},{"startTime":1058.49,"endTime":1062.78,"body":"a four in 10 chance of"},{"startTime":1058.49,"endTime":1062.78,"body":"successfully hitting Colin"},{"startTime":1062.78,"endTime":1064.1,"body":"and therefore winning."},{"startTime":1064.1,"endTime":1069.1,"body":"So with a four in 10 chance"},{"startTime":1064.1,"endTime":1069.1,"body":"she'll have plus one value,"},{"startTime":1069.77,"endTime":1073.07,"body":"but six times out of 10 she'll miss"},{"startTime":1073.07,"endTime":1074.9,"body":"and that means Colin will win."},{"startTime":1074.9,"endTime":1078.17,"body":"So then the outcome will"},{"startTime":1074.9,"endTime":1078.17,"body":"be minus one for Rose."},{"startTime":1078.17,"endTime":1079.37,"body":"So you can just work, you know,"},{"startTime":1079.37,"endTime":1083.45,"body":"add up those probabilities"},{"startTime":1079.37,"endTime":1083.45,"body":"and find the expected outcome,"},{"startTime":1083.45,"endTime":1085.7,"body":"the expected value of this strategy"},{"startTime":1085.7,"endTime":1087.563,"body":"and you get minus 0.2."},{"startTime":1089.06,"endTime":1090.5,"body":"So that's a negative number."},{"startTime":1090.5,"endTime":1092.54,"body":"So it looks like it's perhaps, you know,"},{"startTime":1092.54,"endTime":1094.88,"body":"zero would be kind of neutral for Rose"},{"startTime":1094.88,"endTime":1098.72,"body":"and anything negative"},{"startTime":1094.88,"endTime":1098.72,"body":"means it's likely that Rose"},{"startTime":1098.72,"endTime":1101.483,"body":"is going to come off"},{"startTime":1098.72,"endTime":1101.483,"body":"worse with this strategy."},{"startTime":1102.32,"endTime":1103.76,"body":"Okay, so you can do some other ones."},{"startTime":1103.76,"endTime":1106.01,"body":"If Rose says I'll fire after four paces"},{"startTime":1106.01,"endTime":1110.06,"body":"but Colin is firing after two,"},{"startTime":1106.01,"endTime":1110.06,"body":"then Colin will fire first,"},{"startTime":1110.06,"endTime":1113.329,"body":"so Rose will lose with the"},{"startTime":1110.06,"endTime":1113.329,"body":"probability of two in 10"},{"startTime":1113.329,"endTime":1115.37,"body":"'cause two tenths of the time Colin hits,"},{"startTime":1115.37,"endTime":1118.16,"body":"and she'll win with the"},{"startTime":1115.37,"endTime":1118.16,"body":"probability of eight in 10."},{"startTime":1118.16,"endTime":1121.34,"body":"So you can add those up"},{"startTime":1118.16,"endTime":1121.34,"body":"and you get plus 0.6,"},{"startTime":1121.34,"endTime":1123.833,"body":"so that strategy looks"},{"startTime":1121.34,"endTime":1123.833,"body":"pretty good for Rose."},{"startTime":1124.82,"endTime":1128.36,"body":"And if they fire at the same"},{"startTime":1124.82,"endTime":1128.36,"body":"time, both after four paces,"},{"startTime":1128.36,"endTime":1131.54,"body":"then if they both hit or"},{"startTime":1128.36,"endTime":1131.54,"body":"they both miss, it's a draw."},{"startTime":1131.54,"endTime":1132.74,"body":"So we don't need to worry about those"},{"startTime":1132.74,"endTime":1134.72,"body":"'cause that'll be multiplying"},{"startTime":1132.74,"endTime":1134.72,"body":"by zero at some point"},{"startTime":1134.72,"endTime":1136.16,"body":"it won't affect the outcome."},{"startTime":1136.16,"endTime":1140.57,"body":"So the only thing we have to"},{"startTime":1136.16,"endTime":1140.57,"body":"account is Rose could win by,"},{"startTime":1140.57,"endTime":1143.45,"body":"she hits with a probability for four in 10"},{"startTime":1143.45,"endTime":1146.99,"body":"and Colin misses with a"},{"startTime":1143.45,"endTime":1146.99,"body":"probability six in 10,"},{"startTime":1146.99,"endTime":1148.67,"body":"so that would give her a plus one."},{"startTime":1148.67,"endTime":1153.44,"body":"And then a kind of symmetrical"},{"startTime":1148.67,"endTime":1153.44,"body":"calculation for Rose missing,"},{"startTime":1153.44,"endTime":1156.35,"body":"which she does six tenths of"},{"startTime":1153.44,"endTime":1156.35,"body":"a time, and Colin hitting,"},{"startTime":1156.35,"endTime":1158.48,"body":"which he does four tenths of a time"},{"startTime":1158.48,"endTime":1160.94,"body":"will give you a minus"},{"startTime":1158.48,"endTime":1160.94,"body":"one, a loss for Rose."},{"startTime":1160.94,"endTime":1162.71,"body":"So you just sort of average these out"},{"startTime":1162.71,"endTime":1165.71,"body":"to get the expected value"},{"startTime":1162.71,"endTime":1165.71,"body":"and actually it's zero,"},{"startTime":1165.71,"endTime":1167.36,"body":"they cancel each other"},{"startTime":1165.71,"endTime":1167.36,"body":"out which makes sense"},{"startTime":1167.36,"endTime":1170.03,"body":"because these are"},{"startTime":1167.36,"endTime":1170.03,"body":"equally matched opponents"},{"startTime":1170.03,"endTime":1171.2,"body":"firing at the same moment."},{"startTime":1171.2,"endTime":1173.12,"body":"So you would hope that this would be..."},{"startTime":1173.12,"endTime":1174.59,"body":"That they'd have the same chance"},{"startTime":1174.59,"endTime":1176.09,"body":"with that particular strategy."},{"startTime":1176.96,"endTime":1179.99,"body":"So okay, you can do all these"},{"startTime":1176.96,"endTime":1179.99,"body":"individual calculations."},{"startTime":1179.99,"endTime":1183.53,"body":"I'm going to put a big table"},{"startTime":1179.99,"endTime":1183.53,"body":"of numbers up, don't worry,"},{"startTime":1183.53,"endTime":1184.97,"body":"don't expect to read every one of them,"},{"startTime":1184.97,"endTime":1186.17,"body":"I just want to show you it."},{"startTime":1186.17,"endTime":1188.81,"body":"You can do these calculations"},{"startTime":1186.17,"endTime":1188.81,"body":"for all the possibilities."},{"startTime":1188.81,"endTime":1192.23,"body":"So Rose gets the rows and"},{"startTime":1188.81,"endTime":1192.23,"body":"Colin gets the columns."},{"startTime":1192.23,"endTime":1194.5,"body":"So you can see like that R4, for example,"},{"startTime":1194.5,"endTime":1196.82,"body":"in the fourth row down,"},{"startTime":1196.82,"endTime":1200.87,"body":"that is what happens when"},{"startTime":1196.82,"endTime":1200.87,"body":"Rose fires after four paces."},{"startTime":1200.87,"endTime":1201.74,"body":"And you can see that,"},{"startTime":1201.74,"endTime":1204.92,"body":"so in the first column it's"},{"startTime":1201.74,"endTime":1204.92,"body":"Colin firing after one pace"},{"startTime":1204.92,"endTime":1207.77,"body":"that's good for Rose, it's an"},{"startTime":1204.92,"endTime":1207.77,"body":"expected value of plus 0.8."},{"startTime":1208.85,"endTime":1210.95,"body":"So you get all these numbers, okay."},{"startTime":1210.95,"endTime":1211.783,"body":"How does that help?"},{"startTime":1211.783,"endTime":1214.37,"body":"How are we going to decide"},{"startTime":1211.783,"endTime":1214.37,"body":"what the best strategy is?"},{"startTime":1214.37,"endTime":1218.45,"body":"So one that you could various"},{"startTime":1214.37,"endTime":1218.45,"body":"ways of attacking this,"},{"startTime":1218.45,"endTime":1219.68,"body":"you don't know, if you are Rose,"},{"startTime":1219.68,"endTime":1221.27,"body":"you don't know what Colin's going to do,"},{"startTime":1221.27,"endTime":1222.95,"body":"and you don't know kind of the probability"},{"startTime":1222.95,"endTime":1226.34,"body":"of which strategy Colin might pick."},{"startTime":1226.34,"endTime":1229.37,"body":"So you probably don't want to"},{"startTime":1226.34,"endTime":1229.37,"body":"just add up all these numbers"},{"startTime":1229.37,"endTime":1231.59,"body":"in each row and say which one's the best."},{"startTime":1231.59,"endTime":1235.76,"body":"So one strategy you could use"},{"startTime":1231.59,"endTime":1235.76,"body":"or one decision making process"},{"startTime":1235.76,"endTime":1238.4,"body":"for what's the best strategy"},{"startTime":1235.76,"endTime":1238.4,"body":"is to do with this column"},{"startTime":1238.4,"endTime":1240.23,"body":"on the far right hand side."},{"startTime":1240.23,"endTime":1243.98,"body":"What you think is, in"},{"startTime":1240.23,"endTime":1243.98,"body":"this method, you say,"},{"startTime":1243.98,"endTime":1247.25,"body":"okay, if I fire after four paces"},{"startTime":1247.25,"endTime":1248.54,"body":"I'm going to be pessimistic and think,"},{"startTime":1248.54,"endTime":1250.97,"body":"well, Colin, obviously,"},{"startTime":1248.54,"endTime":1250.97,"body":"Colin's my opponent,"},{"startTime":1250.97,"endTime":1254.54,"body":"Colin wants to do the best"},{"startTime":1250.97,"endTime":1254.54,"body":"for Colin, not for me."},{"startTime":1254.54,"endTime":1255.8,"body":"So I'm going to be pessimistic"},{"startTime":1255.8,"endTime":1259.49,"body":"and say what's the worst"},{"startTime":1255.8,"endTime":1259.49,"body":"case scenario here?"},{"startTime":1259.49,"endTime":1263.21,"body":"And after four paces,"},{"startTime":1259.49,"endTime":1263.21,"body":"the worst case scenario"},{"startTime":1263.21,"endTime":1266.54,"body":"is that Colin decides to"},{"startTime":1263.21,"endTime":1266.54,"body":"fire after five or more paces"},{"startTime":1266.54,"endTime":1270.11,"body":"because then the expected"},{"startTime":1266.54,"endTime":1270.11,"body":"kind of payoff for Rose"},{"startTime":1270.11,"endTime":1271.103,"body":"is this minus 0.2."},{"startTime":1272.0,"endTime":1275.21,"body":"So for Rose, kind of"},{"startTime":1272.0,"endTime":1275.21,"body":"the worst case scenario,"},{"startTime":1275.21,"endTime":1280.13,"body":"the minimum expected"},{"startTime":1275.21,"endTime":1280.13,"body":"outcome is this minus 0.2."},{"startTime":1280.13,"endTime":1281.63,"body":"So you can put that on"},{"startTime":1280.13,"endTime":1281.63,"body":"the right hand side,"},{"startTime":1281.63,"endTime":1283.04,"body":"this is what this column is measuring,"},{"startTime":1283.04,"endTime":1286.64,"body":"the sort of the worst"},{"startTime":1283.04,"endTime":1286.64,"body":"case in each row for Rose."},{"startTime":1286.64,"endTime":1289.16,"body":"And so you could then"},{"startTime":1286.64,"endTime":1289.16,"body":"perhaps judge the strategies"},{"startTime":1289.16,"endTime":1289.993,"body":"by saying, all right,"},{"startTime":1289.993,"endTime":1294.32,"body":"I want to kind of have"},{"startTime":1289.993,"endTime":1294.32,"body":"the least worst outcome"},{"startTime":1294.32,"endTime":1296.75,"body":"so I'm going to maximize"},{"startTime":1294.32,"endTime":1296.75,"body":"this minimum thing."},{"startTime":1296.75,"endTime":1298.7,"body":"You can call it a maximin strategy."},{"startTime":1298.7,"endTime":1300.35,"body":"So she looks down all"},{"startTime":1298.7,"endTime":1300.35,"body":"of these things, says,"},{"startTime":1300.35,"endTime":1302.63,"body":"well, I definitely don't"},{"startTime":1300.35,"endTime":1302.63,"body":"want to fire after one pace"},{"startTime":1302.63,"endTime":1306.56,"body":"'cause then worst case"},{"startTime":1302.63,"endTime":1306.56,"body":"scenario is a minus 0.8,"},{"startTime":1306.56,"endTime":1309.533,"body":"that's awfully close to Colin"},{"startTime":1306.56,"endTime":1309.533,"body":"being guaranteed to win."},{"startTime":1310.46,"endTime":1314.63,"body":"It looks like the middle two"},{"startTime":1310.46,"endTime":1314.63,"body":"firing after five or six paces"},{"startTime":1314.63,"endTime":1318.29,"body":"give you the least worst"},{"startTime":1314.63,"endTime":1318.29,"body":"outcome here of zero."},{"startTime":1318.29,"endTime":1321.593,"body":"And so, Rose will probably"},{"startTime":1318.29,"endTime":1321.593,"body":"pick one of those."},{"startTime":1322.64,"endTime":1325.01,"body":"Meanwhile, Colin, all of these values"},{"startTime":1325.01,"endTime":1326.84,"body":"are giving Rose's pay off"},{"startTime":1326.84,"endTime":1329.12,"body":"but remember Colin's are just minus that."},{"startTime":1329.12,"endTime":1330.83,"body":"So Colin here will,"},{"startTime":1330.83,"endTime":1333.86,"body":"instead of wanting to"},{"startTime":1330.83,"endTime":1333.86,"body":"maximize the minimum values,"},{"startTime":1333.86,"endTime":1335.96,"body":"he'll want to minimize the maximum values"},{"startTime":1335.96,"endTime":1338.78,"body":"'cause the maximum values are"},{"startTime":1335.96,"endTime":1338.78,"body":"when Rose does really well."},{"startTime":1338.78,"endTime":1340.88,"body":"And so Colin will also end up thinking,"},{"startTime":1340.88,"endTime":1344.54,"body":"well, I'll go in the middle"},{"startTime":1340.88,"endTime":1344.54,"body":"somewhere, five or six paces."},{"startTime":1344.54,"endTime":1346.64,"body":"And if you compare rows five and six,"},{"startTime":1346.64,"endTime":1350.33,"body":"six appears to be sort of a"},{"startTime":1346.64,"endTime":1350.33,"body":"dominant strategy you call it"},{"startTime":1350.33,"endTime":1353.45,"body":"because they're the same"},{"startTime":1350.33,"endTime":1353.45,"body":"except in some places"},{"startTime":1353.45,"endTime":1354.83,"body":"the six row numbers are higher."},{"startTime":1354.83,"endTime":1358.1,"body":"So if I were Rose, I would"},{"startTime":1354.83,"endTime":1358.1,"body":"fire after six paces,"},{"startTime":1358.1,"endTime":1359.837,"body":"and Colin probably will as well."},{"startTime":1359.837,"endTime":1362.0,"body":"And actually that outcome means,"},{"startTime":1362.0,"endTime":1364.64,"body":"since they're fairly"},{"startTime":1362.0,"endTime":1364.64,"body":"matched this makes sense"},{"startTime":1364.64,"endTime":1366.68,"body":"that there's basically an even chance"},{"startTime":1366.68,"endTime":1368.66,"body":"of each of them winning."},{"startTime":1368.66,"endTime":1371.93,"body":"So it's basically going"},{"startTime":1368.66,"endTime":1371.93,"body":"to be a draw probably."},{"startTime":1371.93,"endTime":1373.64,"body":"Of course, it's all probability,"},{"startTime":1373.64,"endTime":1375.41,"body":"so you don't know what's"},{"startTime":1373.64,"endTime":1375.41,"body":"definitely going to happen"},{"startTime":1375.41,"endTime":1377.93,"body":"but this is about choosing a strategy."},{"startTime":1377.93,"endTime":1382.52,"body":"So okay, now you know how to fight a duel."},{"startTime":1382.52,"endTime":1384.38,"body":"Can we extend these sorts of ideas?"},{"startTime":1384.38,"endTime":1385.91,"body":"And in particular this idea"},{"startTime":1385.91,"endTime":1389.42,"body":"of minimizing our worst case scenario"},{"startTime":1389.42,"endTime":1392.03,"body":"and choosing it through that way."},{"startTime":1392.03,"endTime":1395.63,"body":"Well, I want to tell you"},{"startTime":1392.03,"endTime":1395.63,"body":"about one of the first bits"},{"startTime":1395.63,"endTime":1398.54,"body":"of mathematic around game"},{"startTime":1395.63,"endTime":1398.54,"body":"theory, and to do that,"},{"startTime":1398.54,"endTime":1400.43,"body":"I want to give you a"},{"startTime":1398.54,"endTime":1400.43,"body":"slightly smaller example"},{"startTime":1400.43,"endTime":1401.45,"body":"than this giant one."},{"startTime":1401.45,"endTime":1405.95,"body":"But also, duel's are"},{"startTime":1401.45,"endTime":1405.95,"body":"pretty much, you know,"},{"startTime":1405.95,"endTime":1407.75,"body":"one off things probably,"},{"startTime":1407.75,"endTime":1410.39,"body":"you're not going to repeatedly"},{"startTime":1407.75,"endTime":1410.39,"body":"fight duels, I hope."},{"startTime":1410.39,"endTime":1414.35,"body":"Whereas you might"},{"startTime":1410.39,"endTime":1414.35,"body":"repeatedly take penalties"},{"startTime":1414.35,"endTime":1415.79,"body":"in a football game,"},{"startTime":1415.79,"endTime":1417.95,"body":"or at least over your footballing career."},{"startTime":1417.95,"endTime":1419.93,"body":"I'm aware there's maybe"},{"startTime":1417.95,"endTime":1419.93,"body":"some sort of competition"},{"startTime":1419.93,"endTime":1423.08,"body":"in football going on at the moment."},{"startTime":1423.08,"endTime":1426.173,"body":"So maybe this will be useful"},{"startTime":1423.08,"endTime":1426.173,"body":"to the players, I don't know."},{"startTime":1427.19,"endTime":1428.93,"body":"The question is here,"},{"startTime":1428.93,"endTime":1431.36,"body":"and this is an example of"},{"startTime":1428.93,"endTime":1431.36,"body":"what we're going to see."},{"startTime":1431.36,"endTime":1433.28,"body":"Can you find some sort"},{"startTime":1431.36,"endTime":1433.28,"body":"of equilibrium point"},{"startTime":1433.28,"endTime":1436.64,"body":"where both of you have got strategies"},{"startTime":1436.64,"endTime":1441.64,"body":"that are optimal in the sense"},{"startTime":1436.64,"endTime":1441.64,"body":"that deviating unilaterally"},{"startTime":1441.83,"endTime":1445.223,"body":"from that strategy makes"},{"startTime":1441.83,"endTime":1445.223,"body":"your predicted outcome worse."},{"startTime":1446.96,"endTime":1448.49,"body":"I suppose you call it a local maximum"},{"startTime":1448.49,"endTime":1451.37,"body":"where you've got something"},{"startTime":1448.49,"endTime":1451.37,"body":"that you both have developed"},{"startTime":1451.37,"endTime":1453.14,"body":"and if you change what you do,"},{"startTime":1453.14,"endTime":1455.36,"body":"it can't improve your your situation."},{"startTime":1455.36,"endTime":1458.84,"body":"So the game here I'm talking about,"},{"startTime":1458.84,"endTime":1460.79,"body":"for our American friends, soccer,"},{"startTime":1460.79,"endTime":1462.74,"body":"for everybody else, football."},{"startTime":1462.74,"endTime":1466.07,"body":"It's when you have a penalty."},{"startTime":1466.07,"endTime":1469.67,"body":"And this is where, so you've got a striker"},{"startTime":1469.67,"endTime":1471.44,"body":"who's trying to kick"},{"startTime":1469.67,"endTime":1471.44,"body":"the ball into the goal,"},{"startTime":1471.44,"endTime":1472.28,"body":"and you've got the keeper"},{"startTime":1472.28,"endTime":1474.44,"body":"who's trying to stop the"},{"startTime":1472.28,"endTime":1474.44,"body":"ball going into the goal."},{"startTime":1474.44,"endTime":1476.03,"body":"So this is a zero sum game"},{"startTime":1476.03,"endTime":1478.76,"body":"because either a goal"},{"startTime":1476.03,"endTime":1478.76,"body":"is scored or it isn't,"},{"startTime":1478.76,"endTime":1482.267,"body":"and the one is a good"},{"startTime":1478.76,"endTime":1482.267,"body":"outcome for the striker,"},{"startTime":1482.267,"endTime":1485.54,"body":"and the other one is a good"},{"startTime":1482.267,"endTime":1485.54,"body":"outcome for the keeper."},{"startTime":1485.54,"endTime":1489.023,"body":"So it's a two player zero sum"},{"startTime":1485.54,"endTime":1489.023,"body":"game like the dueling problem."},{"startTime":1489.92,"endTime":1493.823,"body":"And in this example,"},{"startTime":1494.9,"endTime":1498.44,"body":"picking a single strategy"},{"startTime":1494.9,"endTime":1498.44,"body":"is a terrible idea."},{"startTime":1498.44,"endTime":1500.45,"body":"So that's called a pure strategy."},{"startTime":1500.45,"endTime":1502.898,"body":"For instance, if you"},{"startTime":1500.45,"endTime":1502.898,"body":"said, well, I'm actually,"},{"startTime":1502.898,"endTime":1504.05,"body":"you know, I'm right footed"},{"startTime":1504.05,"endTime":1506.72,"body":"so I'm always going to aim to the right."},{"startTime":1506.72,"endTime":1508.58,"body":"That might work first time,"},{"startTime":1508.58,"endTime":1510.26,"body":"but if you always aim to the right,"},{"startTime":1510.26,"endTime":1512.57,"body":"then, of course, the keeper will know that"},{"startTime":1512.57,"endTime":1515.753,"body":"and then they'll always move"},{"startTime":1512.57,"endTime":1515.753,"body":"to the right, your right,"},{"startTime":1517.25,"endTime":1520.88,"body":"your right, and so then that's"},{"startTime":1517.25,"endTime":1520.88,"body":"going to lessen your chances"},{"startTime":1520.88,"endTime":1521.99,"body":"of actually scoring."},{"startTime":1521.99,"endTime":1524.69,"body":"So you can draw up a sort of..."},{"startTime":1524.69,"endTime":1527.24,"body":"You know, you can watch lots"},{"startTime":1524.69,"endTime":1527.24,"body":"of penalties being taken"},{"startTime":1527.24,"endTime":1530.81,"body":"and you can see what the"},{"startTime":1527.24,"endTime":1530.81,"body":"odds are of different..."},{"startTime":1530.81,"endTime":1532.46,"body":"What proportion of penalties will go in"},{"startTime":1532.46,"endTime":1534.71,"body":"under different circumstances."},{"startTime":1534.71,"endTime":1536.307,"body":"So I've basically simplified it to say,"},{"startTime":1536.307,"endTime":1539.09,"body":"you know, you either, you go right,"},{"startTime":1539.09,"endTime":1541.79,"body":"you stay where you are, or you go left,"},{"startTime":1541.79,"endTime":1546.05,"body":"that's essentially your choice"},{"startTime":1541.79,"endTime":1546.05,"body":"as both striker and keeper."},{"startTime":1546.05,"endTime":1549.44,"body":"So here, I've exclusively"},{"startTime":1546.05,"endTime":1549.44,"body":"talked about you're the striker,"},{"startTime":1549.44,"endTime":1552.02,"body":"I'm the keeper, I've talked"},{"startTime":1549.44,"endTime":1552.02,"body":"about the striker's right."},{"startTime":1552.02,"endTime":1554.69,"body":"So in that top left hand, that 50% thing,"},{"startTime":1554.69,"endTime":1557.36,"body":"that's what happens if both of you"},{"startTime":1557.36,"endTime":1560.9,"body":"go in the same direction"},{"startTime":1557.36,"endTime":1560.9,"body":"to the striker's right"},{"startTime":1560.9,"endTime":1563.42,"body":"then about 50% of the goals will go in"},{"startTime":1563.42,"endTime":1564.98,"body":"'cause you might not get there in time"},{"startTime":1564.98,"endTime":1566.72,"body":"or, you know, something like this."},{"startTime":1566.72,"endTime":1568.85,"body":"If the striker is going to the right"},{"startTime":1568.85,"endTime":1572.21,"body":"and you stay in the"},{"startTime":1568.85,"endTime":1572.21,"body":"center, 80% will go in."},{"startTime":1572.21,"endTime":1575.39,"body":"So some, you know, will"},{"startTime":1572.21,"endTime":1575.39,"body":"be the striker is wide"},{"startTime":1575.39,"endTime":1577.19,"body":"or goes over the goal post or something,"},{"startTime":1577.19,"endTime":1579.29,"body":"and some you might just get there in time."},{"startTime":1580.43,"endTime":1581.33,"body":"Even if you stay there,"},{"startTime":1581.33,"endTime":1583.61,"body":"you might just get to with fingers on it."},{"startTime":1583.61,"endTime":1586.55,"body":"And then the right hand"},{"startTime":1583.61,"endTime":1586.55,"body":"side is where the striker"},{"startTime":1586.55,"endTime":1589.19,"body":"is kicking to the right"},{"startTime":1586.55,"endTime":1589.19,"body":"but the keeper moves left"},{"startTime":1589.19,"endTime":1592.52,"body":"and then you can imagine 90%"},{"startTime":1589.19,"endTime":1592.52,"body":"of those are going to go in,"},{"startTime":1592.52,"endTime":1594.8,"body":"it'll be the striker's"},{"startTime":1592.52,"endTime":1594.8,"body":"error if they don't."},{"startTime":1594.8,"endTime":1597.201,"body":"So you can look at these"},{"startTime":1594.8,"endTime":1597.201,"body":"statistics and you can say,"},{"startTime":1597.201,"endTime":1598.034,"body":"you know, you could say,"},{"startTime":1598.034,"endTime":1600.83,"body":"well, I should always go"},{"startTime":1598.034,"endTime":1600.83,"body":"right then or always go left."},{"startTime":1600.83,"endTime":1602.48,"body":"But if you always do that"},{"startTime":1602.48,"endTime":1605.21,"body":"then people will get wise to the strategy."},{"startTime":1605.21,"endTime":1607.91,"body":"So a pure strategy like"},{"startTime":1605.21,"endTime":1607.91,"body":"that is not going to work,"},{"startTime":1607.91,"endTime":1609.59,"body":"you're going to need a mixed strategy"},{"startTime":1609.59,"endTime":1612.56,"body":"where you just mix it up a bit"},{"startTime":1609.59,"endTime":1612.56,"body":"enough to create uncertainty"},{"startTime":1612.56,"endTime":1614.72,"body":"so that they are not going to, you know,"},{"startTime":1614.72,"endTime":1616.61,"body":"always go to the right or left."},{"startTime":1616.61,"endTime":1619.55,"body":"For example, if both"},{"startTime":1616.61,"endTime":1619.55,"body":"players adopted a strategy"},{"startTime":1619.55,"endTime":1622.64,"body":"of going just randomly one"},{"startTime":1619.55,"endTime":1622.64,"body":"third of the time going right,"},{"startTime":1622.64,"endTime":1624.35,"body":"left, or staying in the middle,"},{"startTime":1624.35,"endTime":1627.68,"body":"then you could work out"},{"startTime":1624.35,"endTime":1627.68,"body":"the expected payoff of that"},{"startTime":1627.68,"endTime":1629.48,"body":"and it would be so, you know,"},{"startTime":1629.48,"endTime":1630.92,"body":"if they're doing this randomly,"},{"startTime":1630.92,"endTime":1633.2,"body":"there'll be one ninth of the time"},{"startTime":1633.2,"endTime":1635.45,"body":"for each of those nine boxes in the table."},{"startTime":1635.45,"endTime":1637.46,"body":"So you can just add all"},{"startTime":1635.45,"endTime":1637.46,"body":"of those up, so 50%,"},{"startTime":1637.46,"endTime":1640.67,"body":"that's a probability of 0.5, for instance,"},{"startTime":1640.67,"endTime":1644.003,"body":"add all 'em up, divide by nine"},{"startTime":1640.67,"endTime":1644.003,"body":"and the outcome would be 0.7."},{"startTime":1645.02,"endTime":1647.543,"body":"Otherwise 70% of the time there's a goal."},{"startTime":1648.798,"endTime":1652.94,"body":"But actually I looked it up,"},{"startTime":1648.798,"endTime":1652.94,"body":"in international competition,"},{"startTime":1652.94,"endTime":1655.31,"body":"many, many penalties have"},{"startTime":1652.94,"endTime":1655.31,"body":"been taken over the years,"},{"startTime":1655.31,"endTime":1660.31,"body":"about 75% of penalties result"},{"startTime":1655.31,"endTime":1660.31,"body":"in a goal, are converted."},{"startTime":1660.74,"endTime":1664.94,"body":"And so this is less than what"},{"startTime":1660.74,"endTime":1664.94,"body":"we know the actual outcome is."},{"startTime":1664.94,"endTime":1666.47,"body":"Can you improve this?"},{"startTime":1666.47,"endTime":1667.58,"body":"Maybe you can tweak it a bit,"},{"startTime":1667.58,"endTime":1669.02,"body":"maybe you can go right a little bit more"},{"startTime":1669.02,"endTime":1670.91,"body":"than you go to the middle."},{"startTime":1670.91,"endTime":1674.09,"body":"And the question is, is there"},{"startTime":1670.91,"endTime":1674.09,"body":"ever an equilibrium point?"},{"startTime":1674.09,"endTime":1677.3,"body":"Is there a time when you've"},{"startTime":1674.09,"endTime":1677.3,"body":"refined your strategy,"},{"startTime":1677.3,"endTime":1678.89,"body":"you've got the percentages just right"},{"startTime":1678.89,"endTime":1681.35,"body":"so that if you were to change that"},{"startTime":1681.35,"endTime":1684.29,"body":"you can't improve what your outcome is?"},{"startTime":1684.29,"endTime":1687.62,"body":"And here's where John"},{"startTime":1684.29,"endTime":1687.62,"body":"von Neumann comes in,"},{"startTime":1687.62,"endTime":1690.8,"body":"and he proves something"},{"startTime":1687.62,"endTime":1690.8,"body":"called the Minimax Theorem"},{"startTime":1690.8,"endTime":1694.31,"body":"which says in a two player zero sum game,"},{"startTime":1694.31,"endTime":1697.82,"body":"there is always going to"},{"startTime":1694.31,"endTime":1697.82,"body":"be an equilibrium strategy"},{"startTime":1697.82,"endTime":1700.43,"body":"that's minimax one player"},{"startTime":1697.82,"endTime":1700.43,"body":"and maximin for the other."},{"startTime":1700.43,"endTime":1701.263,"body":"In other words,"},{"startTime":1701.263,"endTime":1705.41,"body":"you're getting the best of"},{"startTime":1701.263,"endTime":1705.41,"body":"the worst case scenarios"},{"startTime":1705.41,"endTime":1706.55,"body":"if you're one player,"},{"startTime":1706.55,"endTime":1709.1,"body":"and the reverse of that if"},{"startTime":1706.55,"endTime":1709.1,"body":"you're the other player."},{"startTime":1710.886,"endTime":1713.27,"body":"Part of the reason for this"},{"startTime":1710.886,"endTime":1713.27,"body":"is, this is zero sum game,"},{"startTime":1713.27,"endTime":1715.79,"body":"so this concept of minimax and maximin"},{"startTime":1715.79,"endTime":1718.19,"body":"can relate to each other."},{"startTime":1718.19,"endTime":1719.543,"body":"So he proved that,"},{"startTime":1720.38,"endTime":1725.38,"body":"and you'll want to know what"},{"startTime":1720.38,"endTime":1725.38,"body":"the footballers should do."},{"startTime":1725.93,"endTime":1727.67,"body":"So there's a paper by,"},{"startTime":1727.67,"endTime":1730.34,"body":"I put a link in the"},{"startTime":1727.67,"endTime":1730.34,"body":"transcript, Ferenc Forgó,"},{"startTime":1730.34,"endTime":1733.31,"body":"which solves this for the football example"},{"startTime":1733.31,"endTime":1735.5,"body":"using something called linear programming."},{"startTime":1735.5,"endTime":1737.12,"body":"And actually both players"},{"startTime":1737.12,"endTime":1739.94,"body":"should go to the right 42% of the time,"},{"startTime":1739.94,"endTime":1741.83,"body":"to the left 42% of the time,"},{"startTime":1741.83,"endTime":1744.11,"body":"and in the center 16% of the time."},{"startTime":1744.11,"endTime":1745.49,"body":"If you never go in the center,"},{"startTime":1745.49,"endTime":1748.49,"body":"then, of course, people will"},{"startTime":1745.49,"endTime":1748.49,"body":"learn, go to the center."},{"startTime":1748.49,"endTime":1751.19,"body":"So that's the equilibrium"},{"startTime":1748.49,"endTime":1751.19,"body":"for this particular example."},{"startTime":1752.12,"endTime":1755.21,"body":"Now, I'm aging myself now,"},{"startTime":1755.21,"endTime":1757.73,"body":"but when I was young in the olden days,"},{"startTime":1757.73,"endTime":1761.69,"body":"there was an advert for a trainer company,"},{"startTime":1761.69,"endTime":1765.77,"body":"let's say a sports sort of"},{"startTime":1761.69,"endTime":1765.77,"body":"brand, which I won't name,"},{"startTime":1765.77,"endTime":1770.6,"body":"and it said 1966 was a good"},{"startTime":1765.77,"endTime":1770.6,"body":"year for English football."},{"startTime":1770.6,"endTime":1773.21,"body":"If anyone remembers that campaign,"},{"startTime":1773.21,"endTime":1776.81,"body":"of course, in 1966,"},{"startTime":1773.21,"endTime":1776.81,"body":"England won the World Cup."},{"startTime":1776.81,"endTime":1779.06,"body":"But also that the pure"},{"startTime":1776.81,"endTime":1779.06,"body":"point of this advert"},{"startTime":1779.06,"endTime":1781.43,"body":"was that also in 1966,"},{"startTime":1781.43,"endTime":1783.38,"body":"a footballer called Eric Cantona was born"},{"startTime":1783.38,"endTime":1784.73,"body":"and he played for Manchester United"},{"startTime":1784.73,"endTime":1786.05,"body":"and he was a good footballer."},{"startTime":1786.05,"endTime":1787.85,"body":"So he was sort of playing within England."},{"startTime":1787.85,"endTime":1789.26,"body":"So they sort of steered"},{"startTime":1787.85,"endTime":1789.26,"body":"you in the wrong way."},{"startTime":1789.26,"endTime":1793.58,"body":"So I will now say, 1928 was"},{"startTime":1789.26,"endTime":1793.58,"body":"a good year for game theory"},{"startTime":1793.58,"endTime":1798.143,"body":"because John Nash was born, and John Nash,"},{"startTime":1799.85,"endTime":1801.59,"body":"perhaps the most famous game theorist"},{"startTime":1801.59,"endTime":1802.58,"body":"because there was a film"},{"startTime":1802.58,"endTime":1804.77,"body":"called \"A Beautiful Mind\" about him."},{"startTime":1804.77,"endTime":1808.22,"body":"This is what he looked like"},{"startTime":1804.77,"endTime":1808.22,"body":"in I guess the early 90s."},{"startTime":1808.22,"endTime":1810.74,"body":"If you watch the film, he's"},{"startTime":1808.22,"endTime":1810.74,"body":"played by Russell Crowe,"},{"startTime":1810.74,"endTime":1813.44,"body":"which is perhaps not an"},{"startTime":1810.74,"endTime":1813.44,"body":"absolutely accurate depiction,"},{"startTime":1813.44,"endTime":1815.24,"body":"but anyway, it doesn't"},{"startTime":1813.44,"endTime":1815.24,"body":"matter. (audience laughing)"},{"startTime":1815.24,"endTime":1817.34,"body":"It's a great film, it's a great film,"},{"startTime":1817.34,"endTime":1821.007,"body":"and it's based on the equally"},{"startTime":1817.34,"endTime":1821.007,"body":"great book by Sylvia Nasar,"},{"startTime":1821.007,"endTime":1822.02,"body":"\"A Beautiful Mind.\""},{"startTime":1822.02,"endTime":1824.63,"body":"And he struggled with mental"},{"startTime":1822.02,"endTime":1824.63,"body":"illness through his life"},{"startTime":1824.63,"endTime":1825.86,"body":"and the film, you know,"},{"startTime":1825.86,"endTime":1829.49,"body":"addresses and discusses those challenges."},{"startTime":1829.49,"endTime":1833.21,"body":"But he managed to extend"},{"startTime":1829.49,"endTime":1833.21,"body":"the idea of an equilibrium"},{"startTime":1833.21,"endTime":1834.56,"body":"so that it would hold"},{"startTime":1834.56,"endTime":1836.39,"body":"not just in two player zero some games,"},{"startTime":1836.39,"endTime":1839.63,"body":"but in multiplayer non-zero sum games."},{"startTime":1839.63,"endTime":1842.0,"body":"So the prison's dilemma"},{"startTime":1839.63,"endTime":1842.0,"body":"is a non-zero sum game"},{"startTime":1842.0,"endTime":1845.12,"body":"because the sum of the"},{"startTime":1842.0,"endTime":1845.12,"body":"outcomes is not always zero"},{"startTime":1845.12,"endTime":1847.61,"body":"and it varies according to the strategies."},{"startTime":1847.61,"endTime":1848.93,"body":"And he showed that"},{"startTime":1847.61,"endTime":1848.93,"body":"there's always something"},{"startTime":1848.93,"endTime":1852.38,"body":"which is been come to be"},{"startTime":1848.93,"endTime":1852.38,"body":"known as the Nash equilibrium."},{"startTime":1852.38,"endTime":1854.84,"body":"There's always an equilibrium position"},{"startTime":1854.84,"endTime":1858.23,"body":"where no one player"},{"startTime":1854.84,"endTime":1858.23,"body":"can improve the outcome"},{"startTime":1858.23,"endTime":1860.03,"body":"by unilaterally changing strategy."},{"startTime":1860.03,"endTime":1862.04,"body":"If a whole bunch of people"},{"startTime":1860.03,"endTime":1862.04,"body":"change their strategy,"},{"startTime":1862.04,"endTime":1863.06,"body":"maybe you can,"},{"startTime":1863.06,"endTime":1865.82,"body":"but nobody can unilaterally"},{"startTime":1863.06,"endTime":1865.82,"body":"change their strategy"},{"startTime":1865.82,"endTime":1868.7,"body":"and improve what their outcome will be."},{"startTime":1868.7,"endTime":1871.07,"body":"So that's a really huge generalization"},{"startTime":1871.07,"endTime":1873.86,"body":"and a really important"},{"startTime":1871.07,"endTime":1873.86,"body":"result in game theory."},{"startTime":1873.86,"endTime":1877.64,"body":"So this now brings us to"},{"startTime":1873.86,"endTime":1877.64,"body":"our second little puzzle"},{"startTime":1877.64,"endTime":1879.32,"body":"of the best coffee in town."},{"startTime":1879.32,"endTime":1882.02,"body":"And again, you have to put"},{"startTime":1879.32,"endTime":1882.02,"body":"some numbers into these things,"},{"startTime":1882.02,"endTime":1883.55,"body":"but suppose you have."},{"startTime":1883.55,"endTime":1885.59,"body":"In this little town,"},{"startTime":1883.55,"endTime":1885.59,"body":"there are two coffee shops"},{"startTime":1885.59,"endTime":1887.75,"body":"that both sell the best coffee in town,"},{"startTime":1887.75,"endTime":1890.48,"body":"and there are 400 locals who will go in"},{"startTime":1890.48,"endTime":1892.19,"body":"and who will know what"},{"startTime":1890.48,"endTime":1892.19,"body":"price is being charged"},{"startTime":1892.19,"endTime":1894.98,"body":"and will remember, and if"},{"startTime":1892.19,"endTime":1894.98,"body":"someone's charging too much,"},{"startTime":1894.98,"endTime":1896.39,"body":"they'll go to the other cafe."},{"startTime":1896.39,"endTime":1899.87,"body":"But there are also 200"},{"startTime":1896.39,"endTime":1899.87,"body":"tourists who, you know,"},{"startTime":1899.87,"endTime":1902.42,"body":"they visit once, they pick at random"},{"startTime":1902.42,"endTime":1905.03,"body":"to have the best coffee"},{"startTime":1902.42,"endTime":1905.03,"body":"in town here or here."},{"startTime":1905.03,"endTime":1907.07,"body":"And so they're not going to be..."},{"startTime":1907.07,"endTime":1908.72,"body":"You know, they might grumble"},{"startTime":1907.07,"endTime":1908.72,"body":"if it's a high price,"},{"startTime":1908.72,"endTime":1909.65,"body":"but they'll have the coffee,"},{"startTime":1909.65,"endTime":1910.61,"body":"but then you'll never see 'em again."},{"startTime":1910.61,"endTime":1915.23,"body":"So those, that you can"},{"startTime":1910.61,"endTime":1915.23,"body":"perhaps weather a higher price"},{"startTime":1915.23,"endTime":1916.79,"body":"for those people but not for the locals."},{"startTime":1916.79,"endTime":1919.7,"body":"And you can work out a table,"},{"startTime":1919.7,"endTime":1921.29,"body":"just like for the prisoner's dilemma,"},{"startTime":1921.29,"endTime":1922.25,"body":"only this time it's coffee."},{"startTime":1922.25,"endTime":1927.25,"body":"So here I've got, you"},{"startTime":1922.25,"endTime":1927.25,"body":"know, cafe A is charging..."},{"startTime":1927.26,"endTime":1930.08,"body":"Could they charge three"},{"startTime":1927.26,"endTime":1930.08,"body":"pounds or four pounds?"},{"startTime":1930.08,"endTime":1931.79,"body":"Cafe B, same thing."},{"startTime":1931.79,"endTime":1934.25,"body":"So that decision making process,"},{"startTime":1934.25,"endTime":1936.08,"body":"I'm not going into the"},{"startTime":1934.25,"endTime":1936.08,"body":"details, but, you know,"},{"startTime":1936.08,"endTime":1938.48,"body":"all of the locals will"},{"startTime":1936.08,"endTime":1938.48,"body":"go, if somebody's cheaper,"},{"startTime":1938.48,"endTime":1940.76,"body":"the locals will gravitate there."},{"startTime":1940.76,"endTime":1944.09,"body":"So if they're both charging"},{"startTime":1940.76,"endTime":1944.09,"body":"the same price, three pounds,"},{"startTime":1944.09,"endTime":1947.93,"body":"then there are 600 people,"},{"startTime":1944.09,"endTime":1947.93,"body":"they've got 300 each,"},{"startTime":1947.93,"endTime":1950.3,"body":"so they each make 900 pounds."},{"startTime":1950.3,"endTime":1953.57,"body":"If they both charge four"},{"startTime":1950.3,"endTime":1953.57,"body":"pounds, again, captive audience,"},{"startTime":1953.57,"endTime":1958.52,"body":"they're both going to make"},{"startTime":1953.57,"endTime":1958.52,"body":"four lots of 300, so 1200."},{"startTime":1958.52,"endTime":1961.28,"body":"But if they've got this higher price"},{"startTime":1961.28,"endTime":1965.45,"body":"and if somebody lowers their"},{"startTime":1961.28,"endTime":1965.45,"body":"price, then suddenly they get,"},{"startTime":1965.45,"endTime":1966.8,"body":"all the locals will switch to them"},{"startTime":1966.8,"endTime":1968.63,"body":"so their profit will go up."},{"startTime":1968.63,"endTime":1971.3,"body":"So you can think about, you"},{"startTime":1968.63,"endTime":1971.3,"body":"know, the various examples,"},{"startTime":1971.3,"endTime":1974.69,"body":"you could have multiple"},{"startTime":1971.3,"endTime":1974.69,"body":"different prices, many cafes."},{"startTime":1974.69,"endTime":1977.12,"body":"But this example here"},{"startTime":1974.69,"endTime":1977.12,"body":"is not a zero sum game"},{"startTime":1977.12,"endTime":1981.833,"body":"because the total amount"},{"startTime":1977.12,"endTime":1981.833,"body":"made is not a constant thing."},{"startTime":1983.42,"endTime":1985.487,"body":"But there is an equilibrium here"},{"startTime":1985.487,"endTime":1989.12,"body":"and the equilibrium is that..."},{"startTime":1989.12,"endTime":1990.383,"body":"It's this top left."},{"startTime":1991.37,"endTime":1993.02,"body":"So because in this situation,"},{"startTime":1993.02,"endTime":1995.93,"body":"if A for instance changes their strategy,"},{"startTime":1995.93,"endTime":1998.6,"body":"so they move to this, the higher price,"},{"startTime":1998.6,"endTime":2002.5,"body":"their profit goes down to"},{"startTime":1998.6,"endTime":2002.5,"body":"400 and profit for B goes up"},{"startTime":2002.5,"endTime":2003.73,"body":"and the same for B."},{"startTime":2003.73,"endTime":2007.09,"body":"So neither of them can raise"},{"startTime":2003.73,"endTime":2007.09,"body":"their prices unilaterally"},{"startTime":2007.09,"endTime":2009.55,"body":"without having a worse outcome."},{"startTime":2009.55,"endTime":2010.487,"body":"But if they both were to do it,"},{"startTime":2010.487,"endTime":2013.15,"body":"then they both improve their outcome."},{"startTime":2013.15,"endTime":2017.05,"body":"So again, cooperation,"},{"startTime":2013.15,"endTime":2017.05,"body":"collusion to think about."},{"startTime":2017.05,"endTime":2019.3,"body":"So this work was so influential,"},{"startTime":2019.3,"endTime":2024.3,"body":"the work of Nash and others"},{"startTime":2019.3,"endTime":2024.3,"body":"who have extended his work,"},{"startTime":2024.79,"endTime":2027.49,"body":"was so influential that"},{"startTime":2024.79,"endTime":2027.49,"body":"three game theorists"},{"startTime":2027.49,"endTime":2031.66,"body":"shared the Nobel Prize,"},{"startTime":2027.49,"endTime":2031.66,"body":"including Nash in 1994."},{"startTime":2031.66,"endTime":2034.54,"body":"And there have been other"},{"startTime":2031.66,"endTime":2034.54,"body":"Nobel prizes for economics"},{"startTime":2034.54,"endTime":2036.1,"body":"that have been won by game theorists."},{"startTime":2036.1,"endTime":2038.02,"body":"And in particular, in 2020,"},{"startTime":2038.02,"endTime":2043.02,"body":"very recently the prize was"},{"startTime":2038.02,"endTime":2043.02,"body":"shared by two game theorists"},{"startTime":2043.18,"endTime":2044.68,"body":"who worked on auctions."},{"startTime":2044.68,"endTime":2046.43,"body":"So I want to talk about auctions now."},{"startTime":2047.5,"endTime":2049.894,"body":"And auctions, the first thing to say"},{"startTime":2049.894,"endTime":2051.64,"body":"is there are lots of"},{"startTime":2049.894,"endTime":2051.64,"body":"different kinds of auction,"},{"startTime":2051.64,"endTime":2054.245,"body":"and the one we perhaps"},{"startTime":2051.64,"endTime":2054.245,"body":"have in our mind with a,"},{"startTime":2054.245,"endTime":2057.82,"body":"you know, a very expensive"},{"startTime":2054.245,"endTime":2057.82,"body":"painting being sold,"},{"startTime":2057.82,"endTime":2062.23,"body":"that's called an English open auction."},{"startTime":2062.23,"endTime":2064.27,"body":"So this style of auction"},{"startTime":2062.23,"endTime":2064.27,"body":"is called English."},{"startTime":2064.27,"endTime":2066.34,"body":"So it's an open auction,"},{"startTime":2064.27,"endTime":2066.34,"body":"so everyone's there,"},{"startTime":2066.34,"endTime":2069.07,"body":"they know what the bids"},{"startTime":2066.34,"endTime":2069.07,"body":"are and it's ascending,"},{"startTime":2069.07,"endTime":2071.56,"body":"the prices are going up and"},{"startTime":2069.07,"endTime":2071.56,"body":"they go up and up and up"},{"startTime":2071.56,"endTime":2073.81,"body":"until all the other bidders drop out"},{"startTime":2073.81,"endTime":2075.37,"body":"and there's just one person left,"},{"startTime":2075.37,"endTime":2078.43,"body":"and then that person will win the auction,"},{"startTime":2078.43,"endTime":2079.543,"body":"they'll get the item."},{"startTime":2080.65,"endTime":2082.09,"body":"There's another kind of open auction"},{"startTime":2082.09,"endTime":2083.98,"body":"which is often called a Dutch auction."},{"startTime":2083.98,"endTime":2088.45,"body":"It's because it's often used"},{"startTime":2083.98,"endTime":2088.45,"body":"Dutch flower selling auctions,"},{"startTime":2088.45,"endTime":2091.0,"body":"hence the tulips, and that one,"},{"startTime":2091.0,"endTime":2094.42,"body":"the price is set at the"},{"startTime":2091.0,"endTime":2094.42,"body":"beginning extremely high,"},{"startTime":2094.42,"endTime":2095.89,"body":"higher than anyone would ever pay."},{"startTime":2095.89,"endTime":2097.33,"body":"A million pounds, whatever."},{"startTime":2097.33,"endTime":2099.43,"body":"And then it comes down incrementally"},{"startTime":2099.43,"endTime":2102.88,"body":"until somebody bites and"},{"startTime":2099.43,"endTime":2102.88,"body":"says, I'll have that thing."},{"startTime":2102.88,"endTime":2105.283,"body":"So that's a descending open auction."},{"startTime":2106.9,"endTime":2108.52,"body":"Now these envelopes here"},{"startTime":2108.52,"endTime":2110.74,"body":"are representing sealed bid auctions."},{"startTime":2110.74,"endTime":2115.3,"body":"So these happen, for instance,"},{"startTime":2110.74,"endTime":2115.3,"body":"quite often in Scotland,"},{"startTime":2115.3,"endTime":2117.46,"body":"houses are sold this way, a sealed bid."},{"startTime":2117.46,"endTime":2120.16,"body":"So everyone puts in their"},{"startTime":2117.46,"endTime":2120.16,"body":"bid for what they think"},{"startTime":2120.16,"endTime":2122.83,"body":"they will pay for this"},{"startTime":2120.16,"endTime":2122.83,"body":"house, and then, you know,"},{"startTime":2122.83,"endTime":2125.02,"body":"when that process finishes,"},{"startTime":2125.02,"endTime":2126.4,"body":"all the envelopes are opened"},{"startTime":2126.4,"endTime":2128.11,"body":"and whatever the highest offer is,"},{"startTime":2128.11,"endTime":2130.45,"body":"they get the house and"},{"startTime":2128.11,"endTime":2130.45,"body":"they pay that amount"},{"startTime":2130.45,"endTime":2131.93,"body":"that they've bid."},{"startTime":2131.93,"endTime":2134.38,"body":"There is a variant of a sealed bid auction"},{"startTime":2134.38,"endTime":2137.413,"body":"which is called a second"},{"startTime":2134.38,"endTime":2137.413,"body":"price sealed bid auction."},{"startTime":2139.18,"endTime":2142.09,"body":"It feels counterintuitive"},{"startTime":2139.18,"endTime":2142.09,"body":"that this is a good idea,"},{"startTime":2142.09,"endTime":2143.53,"body":"but I think it is."},{"startTime":2143.53,"endTime":2146.02,"body":"In this type of auction,"},{"startTime":2143.53,"endTime":2146.02,"body":"it's a sealed bid,"},{"startTime":2146.02,"endTime":2147.61,"body":"nobody knows what anyone else is bidding."},{"startTime":2147.61,"endTime":2148.99,"body":"You'll put your bids in,"},{"startTime":2148.99,"endTime":2152.26,"body":"and the item goes to the highest bidder,"},{"startTime":2152.26,"endTime":2154.33,"body":"but the price they pay for it"},{"startTime":2154.33,"endTime":2157.03,"body":"is the amount of the second highest bid."},{"startTime":2157.03,"endTime":2159.58,"body":"So that feels odd. (chuckles)"},{"startTime":2159.58,"endTime":2163.78,"body":"We'll discuss about why"},{"startTime":2159.58,"endTime":2163.78,"body":"that might be a good thing,"},{"startTime":2163.78,"endTime":2165.43,"body":"you know, in a little bit."},{"startTime":2165.43,"endTime":2168.4,"body":"So there are loads of applications"},{"startTime":2168.4,"endTime":2171.46,"body":"of the mathematics of game"},{"startTime":2168.4,"endTime":2171.46,"body":"theory to this kind of situation,"},{"startTime":2171.46,"endTime":2173.38,"body":"loads of strategies you"},{"startTime":2171.46,"endTime":2173.38,"body":"might want to know about."},{"startTime":2173.38,"endTime":2174.58,"body":"What should I bid?"},{"startTime":2174.58,"endTime":2175.9,"body":"How should I design my auction"},{"startTime":2175.9,"endTime":2178.0,"body":"to sell my thing for the highest price?"},{"startTime":2178.0,"endTime":2180.7,"body":"If I want to get an item in an auction,"},{"startTime":2180.7,"endTime":2182.863,"body":"what should my bidding strategy be?"},{"startTime":2183.97,"endTime":2186.55,"body":"So just, I'll give you a"},{"startTime":2183.97,"endTime":2186.55,"body":"couple of examples of these,"},{"startTime":2186.55,"endTime":2190.06,"body":"and the first one is"},{"startTime":2186.55,"endTime":2190.06,"body":"setting a reserve price."},{"startTime":2190.06,"endTime":2194.92,"body":"So in, say, an English"},{"startTime":2190.06,"endTime":2194.92,"body":"auction that the typical one"},{"startTime":2194.92,"endTime":2196.78,"body":"that we sort of feel that we know."},{"startTime":2196.78,"endTime":2198.97,"body":"Sometimes there'll be a reserve price,"},{"startTime":2198.97,"endTime":2203.38,"body":"and that's where the person"},{"startTime":2198.97,"endTime":2203.38,"body":"selling the object says,"},{"startTime":2203.38,"endTime":2204.58,"body":"okay, I'm not going to sell it"},{"startTime":2204.58,"endTime":2206.92,"body":"if I don't make at least this much money."},{"startTime":2206.92,"endTime":2208.3,"body":"So, you know, I've got this thing"},{"startTime":2208.3,"endTime":2211.45,"body":"and I think it's worth a million pounds"},{"startTime":2211.45,"endTime":2213.52,"body":"but I won't sell for under"},{"startTime":2211.45,"endTime":2213.52,"body":"half a million pounds."},{"startTime":2213.52,"endTime":2215.83,"body":"So if the bidding does not get that high,"},{"startTime":2215.83,"endTime":2220.21,"body":"then the item is not sold,"},{"startTime":2215.83,"endTime":2220.21,"body":"and that's the end of it."},{"startTime":2220.21,"endTime":2222.34,"body":"And this is to prevent a situation"},{"startTime":2222.34,"endTime":2225.25,"body":"where maybe there's very"},{"startTime":2222.34,"endTime":2225.25,"body":"few bidders on the day"},{"startTime":2225.25,"endTime":2228.19,"body":"and so, you know, worst case"},{"startTime":2225.25,"endTime":2228.19,"body":"maybe you just have one bidder"},{"startTime":2228.19,"endTime":2230.29,"body":"and then they could buy it for a pound"},{"startTime":2230.29,"endTime":2232.66,"body":"and you don't want to do that."},{"startTime":2232.66,"endTime":2235.33,"body":"So it's quite often"},{"startTime":2232.66,"endTime":2235.33,"body":"there'll be a reserve price."},{"startTime":2235.33,"endTime":2238.3,"body":"So what we want to know is what"},{"startTime":2235.33,"endTime":2238.3,"body":"should we set that price at?"},{"startTime":2238.3,"endTime":2239.89,"body":"What's the best reserve price"},{"startTime":2239.89,"endTime":2241.51,"body":"in order to maximize the revenue"},{"startTime":2241.51,"endTime":2244.33,"body":"that you are going to get"},{"startTime":2241.51,"endTime":2244.33,"body":"from selling your thing?"},{"startTime":2244.33,"endTime":2248.77,"body":"So to analyze this problem,"},{"startTime":2244.33,"endTime":2248.77,"body":"the first step is to say,"},{"startTime":2248.77,"endTime":2251.26,"body":"actually we can reduce this"},{"startTime":2251.26,"endTime":2253.45,"body":"to assuming there's just a single bidder,"},{"startTime":2253.45,"endTime":2256.36,"body":"which, again, why would that be okay?"},{"startTime":2256.36,"endTime":2258.34,"body":"Intuitively the reason is that"},{"startTime":2258.34,"endTime":2262.75,"body":"you can kind of coalesce"},{"startTime":2258.34,"endTime":2262.75,"body":"all the bidders into one"},{"startTime":2262.75,"endTime":2264.13,"body":"and really the thing you care about"},{"startTime":2264.13,"endTime":2267.46,"body":"is the highest bid amongst"},{"startTime":2264.13,"endTime":2267.46,"body":"that collection of people"},{"startTime":2267.46,"endTime":2269.44,"body":"that would be made,"},{"startTime":2269.44,"endTime":2270.273,"body":"'cause that's the thing"},{"startTime":2270.273,"endTime":2273.34,"body":"that ultimately is going"},{"startTime":2270.273,"endTime":2273.34,"body":"to govern what happens."},{"startTime":2273.34,"endTime":2275.53,"body":"So we'll say, we'll just"},{"startTime":2273.34,"endTime":2275.53,"body":"pretend we have a single bidder"},{"startTime":2275.53,"endTime":2277.3,"body":"that's really like, you know,"},{"startTime":2277.3,"endTime":2279.04,"body":"an amalgamated collection of bidders."},{"startTime":2279.04,"endTime":2281.2,"body":"So now we'll say we have a single bidder"},{"startTime":2281.2,"endTime":2285.507,"body":"and we don't know what"},{"startTime":2281.2,"endTime":2285.507,"body":"they've got in their mind"},{"startTime":2285.507,"endTime":2289.39,"body":"as the maximum price that"},{"startTime":2285.507,"endTime":2289.39,"body":"they will be prepared to bid."},{"startTime":2289.39,"endTime":2291.61,"body":"Let's say, we've sort of"},{"startTime":2289.39,"endTime":2291.61,"body":"understood previous auctions"},{"startTime":2291.61,"endTime":2294.7,"body":"and we think that the"},{"startTime":2291.61,"endTime":2294.7,"body":"highest price you could get"},{"startTime":2294.7,"endTime":2296.41,"body":"for something like this"},{"startTime":2294.7,"endTime":2296.41,"body":"is a hundred pounds,"},{"startTime":2296.41,"endTime":2298.27,"body":"whatever this object is."},{"startTime":2298.27,"endTime":2302.17,"body":"So we are going to say"},{"startTime":2298.27,"endTime":2302.17,"body":"that their maximum bid"},{"startTime":2302.17,"endTime":2305.68,"body":"is going to be somewhere in"},{"startTime":2302.17,"endTime":2305.68,"body":"the range zero to a hundred."},{"startTime":2305.68,"endTime":2307.51,"body":"Now have to think about"},{"startTime":2307.51,"endTime":2308.89,"body":"what money are we actually going to get,"},{"startTime":2308.89,"endTime":2310.51,"body":"what's our revenue going to be?"},{"startTime":2310.51,"endTime":2313.87,"body":"Well, they're not going to pay"},{"startTime":2310.51,"endTime":2313.87,"body":"more than they have to pay."},{"startTime":2313.87,"endTime":2316.81,"body":"So if their maximum bid"},{"startTime":2313.87,"endTime":2316.81,"body":"is a hundred pounds,"},{"startTime":2316.81,"endTime":2318.64,"body":"but your reserve price is 10 pounds,"},{"startTime":2318.64,"endTime":2320.29,"body":"they're not going to bid a hundred pounds,"},{"startTime":2320.29,"endTime":2321.73,"body":"they're going to bid 10 pounds."},{"startTime":2321.73,"endTime":2324.01,"body":"They'll get it for the"},{"startTime":2321.73,"endTime":2324.01,"body":"smallest amount they can."},{"startTime":2324.01,"endTime":2326.77,"body":"So if that x, which we don't know,"},{"startTime":2326.77,"endTime":2331.45,"body":"is equal or higher than the"},{"startTime":2326.77,"endTime":2331.45,"body":"reserve price that you have set,"},{"startTime":2331.45,"endTime":2334.18,"body":"then you will sell your"},{"startTime":2331.45,"endTime":2334.18,"body":"item for the reserve price"},{"startTime":2334.18,"endTime":2338.5,"body":"'cause they're not going to"},{"startTime":2334.18,"endTime":2338.5,"body":"go above that unnecessarily."},{"startTime":2338.5,"endTime":2341.41,"body":"If this maximum bid is lower"},{"startTime":2338.5,"endTime":2341.41,"body":"than your reserve price,"},{"startTime":2341.41,"endTime":2342.88,"body":"then your revenue is zero"},{"startTime":2342.88,"endTime":2344.59,"body":"'cause you're not going to sell the thing."},{"startTime":2344.59,"endTime":2347.29,"body":"So you can work out, again,"},{"startTime":2344.59,"endTime":2347.29,"body":"it's sort of probabilistic,"},{"startTime":2347.29,"endTime":2350.838,"body":"but you are expected"},{"startTime":2347.29,"endTime":2350.838,"body":"revenue based on this,"},{"startTime":2350.838,"endTime":2353.023,"body":"and, you know, a very simple graph there."},{"startTime":2354.88,"endTime":2357.79,"body":"It's bidding on the x-axis, your bid is x,"},{"startTime":2357.79,"endTime":2362.74,"body":"and the revenue that the seller"},{"startTime":2357.79,"endTime":2362.74,"body":"will make is on the y-axis."},{"startTime":2362.74,"endTime":2364.93,"body":"So if the maximum bid"},{"startTime":2362.74,"endTime":2364.93,"body":"anyone's prepared to make"},{"startTime":2364.93,"endTime":2368.11,"body":"is less than the reserve,"},{"startTime":2364.93,"endTime":2368.11,"body":"your revenue is zero,"},{"startTime":2368.11,"endTime":2369.55,"body":"if it's bigger than the reserve,"},{"startTime":2369.55,"endTime":2373.0,"body":"then your revenue will"},{"startTime":2369.55,"endTime":2373.0,"body":"be R, the reserve price."},{"startTime":2373.0,"endTime":2376.24,"body":"And to work out the expected revenue then,"},{"startTime":2376.24,"endTime":2379.0,"body":"basically you want to average"},{"startTime":2376.24,"endTime":2379.0,"body":"all of the revenues you get"},{"startTime":2379.0,"endTime":2381.88,"body":"for these different values over a hundred,"},{"startTime":2381.88,"endTime":2382.93,"body":"which is a hundred, you know,"},{"startTime":2382.93,"endTime":2384.73,"body":"you're going from zero to a hundred."},{"startTime":2386.83,"endTime":2387.94,"body":"How do you do this?"},{"startTime":2387.94,"endTime":2390.28,"body":"Well, it's basically going to"},{"startTime":2387.94,"endTime":2390.28,"body":"be the area under the graph here"},{"startTime":2390.28,"endTime":2391.81,"body":"divided by a hundred."},{"startTime":2391.81,"endTime":2393.04,"body":"So what is the area?"},{"startTime":2393.04,"endTime":2394.69,"body":"Well, you've got this rectangle,"},{"startTime":2394.69,"endTime":2397.567,"body":"and the width of the"},{"startTime":2394.69,"endTime":2397.567,"body":"rectangle is a hundred minus R"},{"startTime":2397.567,"endTime":2399.1,"body":"and the height is R."},{"startTime":2399.1,"endTime":2402.1,"body":"So your expected revenue,"},{"startTime":2402.1,"endTime":2404.59,"body":"your sort of on average"},{"startTime":2402.1,"endTime":2404.59,"body":"revenue will be this amount,"},{"startTime":2404.59,"endTime":2405.55,"body":"it's R times a hundred minus R,"},{"startTime":2405.55,"endTime":2407.77,"body":"which is the area of that rectangle,"},{"startTime":2407.77,"endTime":2410.89,"body":"and then you're averaging"},{"startTime":2407.77,"endTime":2410.89,"body":"out so over a hundred."},{"startTime":2410.89,"endTime":2412.3,"body":"So that's the expected revenue"},{"startTime":2412.3,"endTime":2414.1,"body":"for that particular reserve price."},{"startTime":2414.1,"endTime":2415.36,"body":"And what we want to know is,"},{"startTime":2415.36,"endTime":2418.06,"body":"what should I set that"},{"startTime":2415.36,"endTime":2418.06,"body":"reserve price to be at"},{"startTime":2418.06,"endTime":2420.67,"body":"to maximize this number?"},{"startTime":2420.67,"endTime":2422.62,"body":"So there are various ways to attack this"},{"startTime":2422.62,"endTime":2424.69,"body":"and depending on kind of"},{"startTime":2422.62,"endTime":2424.69,"body":"what stage you are at,"},{"startTime":2424.69,"endTime":2427.345,"body":"so you can do it with"},{"startTime":2424.69,"endTime":2427.345,"body":"calculus, you can, you know,"},{"startTime":2427.345,"endTime":2428.49,"body":"you can differentiate respect to R,"},{"startTime":2428.49,"endTime":2431.53,"body":"if that means anything to"},{"startTime":2428.49,"endTime":2431.53,"body":"you, then good do that,"},{"startTime":2431.53,"endTime":2434.2,"body":"if that doesn't, then there"},{"startTime":2431.53,"endTime":2434.2,"body":"are other ways of doing it."},{"startTime":2434.2,"endTime":2437.14,"body":"You can complete the square"},{"startTime":2434.2,"endTime":2437.14,"body":"or you can draw a picture."},{"startTime":2437.14,"endTime":2439.27,"body":"So here's a picture, in this plotting,"},{"startTime":2439.27,"endTime":2443.26,"body":"the value of R, the revenue,"},{"startTime":2439.27,"endTime":2443.26,"body":"against the expected, sorry,"},{"startTime":2443.26,"endTime":2444.85,"body":"the value of the reserve,"},{"startTime":2444.85,"endTime":2447.31,"body":"both things begin with"},{"startTime":2444.85,"endTime":2447.31,"body":"R that's very annoying."},{"startTime":2447.31,"endTime":2449.05,"body":"The value of the reserve R"},{"startTime":2449.05,"endTime":2451.63,"body":"and then against the expected revenue."},{"startTime":2451.63,"endTime":2453.436,"body":"And you can see very clearly that,"},{"startTime":2453.436,"endTime":2454.9,"body":"I mean, this is an upturn parabola"},{"startTime":2454.9,"endTime":2456.4,"body":"and the highest value is attained"},{"startTime":2456.4,"endTime":2458.83,"body":"when the reserve price is 50."},{"startTime":2458.83,"endTime":2461.38,"body":"So that's what mathematics"},{"startTime":2458.83,"endTime":2461.38,"body":"tells you, you should do,"},{"startTime":2461.38,"endTime":2462.58,"body":"you should set the reserve price"},{"startTime":2462.58,"endTime":2466.123,"body":"to be half the maximum"},{"startTime":2462.58,"endTime":2466.123,"body":"possible value you might get."},{"startTime":2467.14,"endTime":2468.763,"body":"So that's one example."},{"startTime":2469.6,"endTime":2472.6,"body":"From the other viewpoint, if"},{"startTime":2469.6,"endTime":2472.6,"body":"we are bidding at an auction,"},{"startTime":2472.6,"endTime":2475.66,"body":"we can ask what's the best bid?"},{"startTime":2475.66,"endTime":2478.312,"body":"Oh, I've said that there."},{"startTime":2475.66,"endTime":2478.312,"body":"What's the best bid?"},{"startTime":2478.312,"endTime":2479.8,"body":"And there have been"},{"startTime":2478.312,"endTime":2479.8,"body":"different types of auctions,"},{"startTime":2479.8,"endTime":2482.35,"body":"there'll be a different calculation to do."},{"startTime":2482.35,"endTime":2483.43,"body":"I won't go into the details,"},{"startTime":2483.43,"endTime":2485.98,"body":"but the mathematics around the best bid"},{"startTime":2485.98,"endTime":2489.22,"body":"for a first price sealed bid auction,"},{"startTime":2489.22,"endTime":2492.88,"body":"you all put in a bid secretly"},{"startTime":2489.22,"endTime":2492.88,"body":"and the highest one wins it"},{"startTime":2492.88,"endTime":2494.653,"body":"and pays out whatever they bid."},{"startTime":2495.73,"endTime":2499.75,"body":"A similar kind of calculation"},{"startTime":2495.73,"endTime":2499.75,"body":"like the reserve price says,"},{"startTime":2499.75,"endTime":2502.75,"body":"what you should do is bid half"},{"startTime":2502.75,"endTime":2506.713,"body":"what your maximum value that"},{"startTime":2502.75,"endTime":2506.713,"body":"you think that the object has."},{"startTime":2508.03,"endTime":2510.1,"body":"But there's a risk with"},{"startTime":2508.03,"endTime":2510.1,"body":"these kinds of auctions,"},{"startTime":2510.1,"endTime":2512.77,"body":"with the first price auction,"},{"startTime":2512.77,"endTime":2516.88,"body":"and it's to do with how we value things."},{"startTime":2516.88,"endTime":2520.39,"body":"Now some things, everyone"},{"startTime":2516.88,"endTime":2520.39,"body":"could legitimately"},{"startTime":2520.39,"endTime":2523.21,"body":"have a different feeling"},{"startTime":2520.39,"endTime":2523.21,"body":"of what it's worth."},{"startTime":2523.21,"endTime":2525.67,"body":"If you are a collector of"},{"startTime":2523.21,"endTime":2525.67,"body":"something, and I don't know,"},{"startTime":2525.67,"endTime":2530.23,"body":"there are 200 different"},{"startTime":2525.67,"endTime":2530.23,"body":"types of, I don't know,"},{"startTime":2530.23,"endTime":2531.58,"body":"this stamp or whatever,"},{"startTime":2531.58,"endTime":2535.18,"body":"and you've got 199 of them"},{"startTime":2531.58,"endTime":2535.18,"body":"and there's the 200th,"},{"startTime":2535.18,"endTime":2536.983,"body":"you will put a very high value on that."},{"startTime":2536.983,"endTime":2539.47,"body":"Your own private valuation"},{"startTime":2536.983,"endTime":2539.47,"body":"of that will be very high,"},{"startTime":2539.47,"endTime":2540.88,"body":"much higher than someone"},{"startTime":2540.88,"endTime":2543.7,"body":"who's just started collecting"},{"startTime":2540.88,"endTime":2543.7,"body":"stamps, doesn't sort of care,"},{"startTime":2543.7,"endTime":2545.41,"body":"they just want a stamp."},{"startTime":2545.41,"endTime":2547.75,"body":"They're not going to value"},{"startTime":2545.41,"endTime":2547.75,"body":"that particular one very high."},{"startTime":2547.75,"endTime":2551.44,"body":"So in that case, people"},{"startTime":2547.75,"endTime":2551.44,"body":"might legitimately bid"},{"startTime":2551.44,"endTime":2552.31,"body":"very different amounts"},{"startTime":2552.31,"endTime":2555.88,"body":"and still be very happy to"},{"startTime":2552.31,"endTime":2555.88,"body":"win the auction if it comes in"},{"startTime":2555.88,"endTime":2559.15,"body":"at something that's like"},{"startTime":2555.88,"endTime":2559.15,"body":"the value they put on it."},{"startTime":2559.15,"endTime":2563.02,"body":"Another kind of item or"},{"startTime":2559.15,"endTime":2563.02,"body":"something that can be auctioned"},{"startTime":2563.02,"endTime":2568.02,"body":"will have a value that"},{"startTime":2563.02,"endTime":2568.02,"body":"everyone will agree in common."},{"startTime":2568.09,"endTime":2571.48,"body":"But we don't necessarily"},{"startTime":2568.09,"endTime":2571.48,"body":"know what that value is,"},{"startTime":2571.48,"endTime":2573.94,"body":"a common value, but we"},{"startTime":2571.48,"endTime":2573.94,"body":"don't know what it is."},{"startTime":2573.94,"endTime":2577.03,"body":"For example, drilling rights"},{"startTime":2573.94,"endTime":2577.03,"body":"or something like this,"},{"startTime":2577.03,"endTime":2580.18,"body":"you know, rights to drill"},{"startTime":2577.03,"endTime":2580.18,"body":"on a particular oil field,"},{"startTime":2580.18,"endTime":2581.86,"body":"everyone will have a..."},{"startTime":2581.86,"endTime":2583.12,"body":"There will be a value for that"},{"startTime":2583.12,"endTime":2584.77,"body":"depending on how much oil is there,"},{"startTime":2584.77,"endTime":2587.11,"body":"and you can try and estimate"},{"startTime":2584.77,"endTime":2587.11,"body":"that with geological surveys"},{"startTime":2587.11,"endTime":2590.05,"body":"or something like this and"},{"startTime":2587.11,"endTime":2590.05,"body":"everyone will do their own survey"},{"startTime":2590.05,"endTime":2591.75,"body":"and you're all making an estimate."},{"startTime":2593.47,"endTime":2595.96,"body":"There's uncertainty about"},{"startTime":2593.47,"endTime":2595.96,"body":"what the true value is,"},{"startTime":2595.96,"endTime":2597.64,"body":"but there is a true value."},{"startTime":2597.64,"endTime":2599.5,"body":"And the problem there is"},{"startTime":2599.5,"endTime":2601.21,"body":"that you are at the risk"},{"startTime":2599.5,"endTime":2601.21,"body":"of the winner's curse,"},{"startTime":2601.21,"endTime":2604.307,"body":"which is if you win against 50"},{"startTime":2601.21,"endTime":2604.307,"body":"other people and you're like,"},{"startTime":2604.307,"endTime":2608.17,"body":"\"Oh, all their valuations must"},{"startTime":2604.307,"endTime":2608.17,"body":"have come in less than mine."},{"startTime":2608.17,"endTime":2611.68,"body":"So, you know, I must have overbid,"},{"startTime":2611.68,"endTime":2613.3,"body":"my valuation must have been wrong.\""},{"startTime":2613.3,"endTime":2614.62,"body":"So even if you win,"},{"startTime":2614.62,"endTime":2617.07,"body":"you then doubt yourself"},{"startTime":2614.62,"endTime":2617.07,"body":"and perhaps aren't happy."},{"startTime":2617.98,"endTime":2622.72,"body":"So in this situation, it's"},{"startTime":2617.98,"endTime":2622.72,"body":"really tricky, like if you win,"},{"startTime":2622.72,"endTime":2625.09,"body":"there's always that"},{"startTime":2622.72,"endTime":2625.09,"body":"risk of not being happy."},{"startTime":2625.09,"endTime":2627.25,"body":"If everyone has a private value,"},{"startTime":2627.25,"endTime":2629.89,"body":"then you don't know what others"},{"startTime":2627.25,"endTime":2629.89,"body":"are valuing something at."},{"startTime":2629.89,"endTime":2632.95,"body":"So example, suppose you"},{"startTime":2629.89,"endTime":2632.95,"body":"are a furniture dealer,"},{"startTime":2632.95,"endTime":2634.48,"body":"you see there's a broken lamp,"},{"startTime":2634.48,"endTime":2635.607,"body":"you know that you can fix it"},{"startTime":2635.607,"endTime":2637.587,"body":"'cause you've got the skills to do it"},{"startTime":2637.587,"endTime":2638.95,"body":"and then you'd be able to"},{"startTime":2637.587,"endTime":2638.95,"body":"sell it for a hundred pounds."},{"startTime":2638.95,"endTime":2640.24,"body":"So your private value"},{"startTime":2640.24,"endTime":2642.04,"body":"would be a hundred pounds for that thing."},{"startTime":2642.04,"endTime":2643.21,"body":"What should you bid?"},{"startTime":2643.21,"endTime":2645.28,"body":"You shouldn't bid a hundred pounds"},{"startTime":2645.28,"endTime":2647.11,"body":"because then you make no profit on it."},{"startTime":2647.11,"endTime":2648.16,"body":"But what should you bid?"},{"startTime":2648.16,"endTime":2650.35,"body":"And the same sort of"},{"startTime":2648.16,"endTime":2650.35,"body":"mathematics as we just used"},{"startTime":2650.35,"endTime":2652.39,"body":"for the reserve price"},{"startTime":2652.39,"endTime":2654.76,"body":"tells you that actually"},{"startTime":2652.39,"endTime":2654.76,"body":"you should bid 50 pounds"},{"startTime":2654.76,"endTime":2657.67,"body":"and that balances out, I"},{"startTime":2654.76,"endTime":2657.67,"body":"won't go into the details,"},{"startTime":2657.67,"endTime":2659.05,"body":"but it's a very similar calculation,"},{"startTime":2659.05,"endTime":2663.46,"body":"it balances out the kind of the risk"},{"startTime":2663.46,"endTime":2665.05,"body":"between if you bid too low,"},{"startTime":2665.05,"endTime":2666.76,"body":"you're unlikely to actually get it,"},{"startTime":2666.76,"endTime":2670.03,"body":"if you bid too high, you"},{"startTime":2666.76,"endTime":2670.03,"body":"don't get very much profit."},{"startTime":2670.03,"endTime":2672.46,"body":"So in that scenario you should bid half"},{"startTime":2672.46,"endTime":2675.37,"body":"what you actually value the thing at."},{"startTime":2675.37,"endTime":2676.42,"body":"And sometimes you won't get it,"},{"startTime":2676.42,"endTime":2678.16,"body":"but at least you won't lose out."},{"startTime":2678.16,"endTime":2682.45,"body":"Now the other kind of auction,"},{"startTime":2678.16,"endTime":2682.45,"body":"the sealed bid auction,"},{"startTime":2682.45,"endTime":2684.04,"body":"which is sometimes called Vickrey auctions"},{"startTime":2684.04,"endTime":2687.04,"body":"because the economist, William Vickrey,"},{"startTime":2687.04,"endTime":2689.2,"body":"studied them and wrote about them."},{"startTime":2689.2,"endTime":2692.71,"body":"In this case, your winning"},{"startTime":2689.2,"endTime":2692.71,"body":"bid gets you the item,"},{"startTime":2692.71,"endTime":2694.21,"body":"but the price you pay"},{"startTime":2694.21,"endTime":2697.45,"body":"is whatever the second"},{"startTime":2694.21,"endTime":2697.45,"body":"highest bidder contributed."},{"startTime":2697.45,"endTime":2699.55,"body":"Now these feel a bit odd,"},{"startTime":2699.55,"endTime":2701.35,"body":"but they are actually pretty close"},{"startTime":2701.35,"endTime":2704.89,"body":"to what would happen in an open auction"},{"startTime":2704.89,"endTime":2706.6,"body":"because in an open auction,"},{"startTime":2706.6,"endTime":2707.433,"body":"if you think we get down to a point"},{"startTime":2707.433,"endTime":2709.18,"body":"where there's two bidders left,"},{"startTime":2709.18,"endTime":2711.19,"body":"and at some point, you"},{"startTime":2709.18,"endTime":2711.19,"body":"know, one of them said,"},{"startTime":2711.19,"endTime":2712.99,"body":"okay, I'm not going any higher."},{"startTime":2712.99,"endTime":2714.28,"body":"If we're at that point,"},{"startTime":2714.28,"endTime":2716.62,"body":"then you can just add like one penny"},{"startTime":2716.62,"endTime":2719.65,"body":"to their maximum bid and win the item."},{"startTime":2719.65,"endTime":2724.03,"body":"So what you end up paying"},{"startTime":2719.65,"endTime":2724.03,"body":"there is not your maximum bid,"},{"startTime":2724.03,"endTime":2727.15,"body":"but their maximum bid"},{"startTime":2724.03,"endTime":2727.15,"body":"plus a tiny teeny bit."},{"startTime":2727.15,"endTime":2728.98,"body":"So that's exactly what's mimicked really"},{"startTime":2728.98,"endTime":2730.93,"body":"in this second price seal bid."},{"startTime":2730.93,"endTime":2733.24,"body":"You can put your true"},{"startTime":2730.93,"endTime":2733.24,"body":"valuation, in that case,"},{"startTime":2733.24,"endTime":2734.94,"body":"that's actually the best strategy."},{"startTime":2737.069,"endTime":2738.55,"body":"You should bid your valuation"},{"startTime":2738.55,"endTime":2740.86,"body":"because the amount you"},{"startTime":2738.55,"endTime":2740.86,"body":"end up actually paying"},{"startTime":2740.86,"endTime":2743.47,"body":"will be the second bidders maximum,"},{"startTime":2743.47,"endTime":2746.08,"body":"so you're not sort of going"},{"startTime":2743.47,"endTime":2746.08,"body":"to pay more than you need to"},{"startTime":2746.08,"endTime":2747.103,"body":"in that situation."},{"startTime":2748.458,"endTime":2753.458,"body":"And in this case then you"},{"startTime":2748.458,"endTime":2753.458,"body":"should bid your valuation"},{"startTime":2753.46,"endTime":2757.21,"body":"because yes, if you bid"},{"startTime":2753.46,"endTime":2757.21,"body":"more than your valuation,"},{"startTime":2757.21,"endTime":2759.193,"body":"there's a risk, someone comes"},{"startTime":2757.21,"endTime":2759.193,"body":"in just like less than that,"},{"startTime":2759.193,"endTime":2761.14,"body":"then you have to pay that amount,"},{"startTime":2761.14,"endTime":2762.61,"body":"which is more than you think."},{"startTime":2762.61,"endTime":2764.89,"body":"So you'd be annoyed, you've overpaid."},{"startTime":2764.89,"endTime":2767.65,"body":"If you bid less than your"},{"startTime":2764.89,"endTime":2767.65,"body":"valuation, maybe bid 90 pounds,"},{"startTime":2767.65,"endTime":2770.44,"body":"then there's a risk someone"},{"startTime":2767.65,"endTime":2770.44,"body":"comes in and bids 95 pounds,"},{"startTime":2770.44,"endTime":2772.96,"body":"then you don't get the"},{"startTime":2770.44,"endTime":2772.96,"body":"item and you wish you had."},{"startTime":2772.96,"endTime":2774.37,"body":"So you should bid your exact valuation,"},{"startTime":2774.37,"endTime":2776.5,"body":"but the amount you are going to pay,"},{"startTime":2776.5,"endTime":2778.96,"body":"if you think we don't know what"},{"startTime":2776.5,"endTime":2778.96,"body":"any of these other people..."},{"startTime":2778.96,"endTime":2781.257,"body":"We have no idea what the"},{"startTime":2778.96,"endTime":2781.257,"body":"other people are bidding,"},{"startTime":2781.257,"endTime":2783.82,"body":"it's a silent sealed bid auction,"},{"startTime":2783.82,"endTime":2785.95,"body":"so without any further information,"},{"startTime":2785.95,"endTime":2787.96,"body":"you know, the other person,"},{"startTime":2787.96,"endTime":2790.21,"body":"if your valuation is a hundred pounds,"},{"startTime":2790.21,"endTime":2793.06,"body":"the expected value that you"},{"startTime":2790.21,"endTime":2793.06,"body":"would have to put on that"},{"startTime":2793.06,"endTime":2794.8,"body":"is kind of a randomly chosen number"},{"startTime":2794.8,"endTime":2796.39,"body":"between zero and a hundred,"},{"startTime":2796.39,"endTime":2799.837,"body":"and so the expected value"},{"startTime":2796.39,"endTime":2799.837,"body":"of that number is 50,"},{"startTime":2799.837,"endTime":2802.45,"body":"and so they would, kind of on average."},{"startTime":2802.45,"endTime":2804.94,"body":"So what you actually end up paying,"},{"startTime":2804.94,"endTime":2807.28,"body":"the expected amount would be 50 pounds."},{"startTime":2807.28,"endTime":2809.56,"body":"So actually in both of these cases"},{"startTime":2809.56,"endTime":2811.6,"body":"you ended up paying 50 pounds,"},{"startTime":2811.6,"endTime":2813.31,"body":"you got to it from a different route."},{"startTime":2813.31,"endTime":2816.91,"body":"And it's actually a curious"},{"startTime":2813.31,"endTime":2816.91,"body":"fact of auction theory"},{"startTime":2816.91,"endTime":2819.31,"body":"that we have the revenue"},{"startTime":2816.91,"endTime":2819.31,"body":"equivalence theorem that says,"},{"startTime":2819.31,"endTime":2822.22,"body":"the theoretical expected"},{"startTime":2819.31,"endTime":2822.22,"body":"revenue, so not the maximum bid,"},{"startTime":2822.22,"endTime":2823.6,"body":"but the actual amount paid,"},{"startTime":2823.6,"endTime":2826.36,"body":"is the same in all of these auction types."},{"startTime":2826.36,"endTime":2829.21,"body":"Theoretical is an important word there."},{"startTime":2829.21,"endTime":2831.76,"body":"It turns out to be half"},{"startTime":2829.21,"endTime":2831.76,"body":"the highest valuation"},{"startTime":2831.76,"endTime":2834.19,"body":"which we've seen in those two examples."},{"startTime":2834.19,"endTime":2836.92,"body":"Of course, real life has complications."},{"startTime":2836.92,"endTime":2839.95,"body":"For example, the biggest one"},{"startTime":2836.92,"endTime":2839.95,"body":"is people tend to have budgets."},{"startTime":2839.95,"endTime":2842.08,"body":"So even if I think this thing"},{"startTime":2839.95,"endTime":2842.08,"body":"is worth a million pounds,"},{"startTime":2842.08,"endTime":2844.36,"body":"I may not have a million pounds to pay."},{"startTime":2844.36,"endTime":2847.57,"body":"So that will obviously affect strategies."},{"startTime":2847.57,"endTime":2850.57,"body":"But in principle these are"},{"startTime":2847.57,"endTime":2850.57,"body":"the same kind of thing."},{"startTime":2850.57,"endTime":2854.92,"body":"Now designing auctions"},{"startTime":2850.57,"endTime":2854.92,"body":"and the right strategies"},{"startTime":2854.92,"endTime":2857.503,"body":"has been very, very"},{"startTime":2854.92,"endTime":2857.503,"body":"fruitful for governments"},{"startTime":2857.503,"endTime":2859.64,"body":"that have asked mathematicians"},{"startTime":2860.59,"endTime":2862.45,"body":"and game theorists for help with this."},{"startTime":2862.45,"endTime":2863.98,"body":"And there's just one example."},{"startTime":2863.98,"endTime":2866.47,"body":"So I put this very old"},{"startTime":2863.98,"endTime":2866.47,"body":"phone to illustrate the fact"},{"startTime":2866.47,"endTime":2868.36,"body":"that this happened in the year 2000"},{"startTime":2868.36,"endTime":2870.82,"body":"when phones barely existed."},{"startTime":2870.82,"endTime":2872.92,"body":"So at this point, so we're on 5G now,"},{"startTime":2872.92,"endTime":2876.31,"body":"but at this point it was 3G"},{"startTime":2872.92,"endTime":2876.31,"body":"was the exciting new thing,"},{"startTime":2876.31,"endTime":2877.143,"body":"and the government,"},{"startTime":2877.143,"endTime":2880.33,"body":"the UK government was"},{"startTime":2877.143,"endTime":2880.33,"body":"auctioning off 3G airspace."},{"startTime":2880.33,"endTime":2882.13,"body":"I guess, for want a better word."},{"startTime":2882.13,"endTime":2884.56,"body":"And they've had five licenses for it,"},{"startTime":2884.56,"endTime":2886.96,"body":"and they wanted to have"},{"startTime":2884.56,"endTime":2886.96,"body":"five different people"},{"startTime":2886.96,"endTime":2887.83,"body":"having each license."},{"startTime":2887.83,"endTime":2890.89,"body":"So they did this auction"},{"startTime":2887.83,"endTime":2890.89,"body":"with series of bids"},{"startTime":2890.89,"endTime":2893.02,"body":"and the first round"},{"startTime":2890.89,"endTime":2893.02,"body":"everyone could make a bid"},{"startTime":2893.02,"endTime":2894.673,"body":"on exactly one license."},{"startTime":2895.51,"endTime":2898.9,"body":"And then they looked at"},{"startTime":2895.51,"endTime":2898.9,"body":"which ones were the top bids,"},{"startTime":2898.9,"endTime":2902.47,"body":"and for those five companies"},{"startTime":2898.9,"endTime":2902.47,"body":"could not then make another bid"},{"startTime":2902.47,"endTime":2905.2,"body":"in the second round, they"},{"startTime":2902.47,"endTime":2905.2,"body":"had to just stick with that."},{"startTime":2905.2,"endTime":2908.14,"body":"Everyone else could"},{"startTime":2905.2,"endTime":2908.14,"body":"choose either to drop out"},{"startTime":2908.14,"endTime":2911.92,"body":"or to increase exactly"},{"startTime":2908.14,"endTime":2911.92,"body":"one of those top bids"},{"startTime":2911.92,"endTime":2913.36,"body":"and then repeat."},{"startTime":2913.36,"endTime":2914.86,"body":"You know, people would drop out"},{"startTime":2914.86,"endTime":2917.41,"body":"and you carry on until you"},{"startTime":2914.86,"endTime":2917.41,"body":"have exactly five people left,"},{"startTime":2917.41,"endTime":2918.73,"body":"five bidders left,"},{"startTime":2918.73,"endTime":2923.11,"body":"and by construction they"},{"startTime":2918.73,"endTime":2923.11,"body":"currently hold the top bids"},{"startTime":2923.11,"endTime":2925.42,"body":"on five different licenses."},{"startTime":2925.42,"endTime":2927.82,"body":"So this was designed very carefully,"},{"startTime":2927.82,"endTime":2929.29,"body":"they had did a lot of"},{"startTime":2927.82,"endTime":2929.29,"body":"mathematical modeling"},{"startTime":2929.29,"endTime":2930.64,"body":"on the possible things,"},{"startTime":2930.64,"endTime":2932.38,"body":"and it raised gigantically more"},{"startTime":2932.38,"endTime":2934.84,"body":"than any other telecom's"},{"startTime":2932.38,"endTime":2934.84,"body":"auction before it."},{"startTime":2934.84,"endTime":2937.78,"body":"It raised 22 and a half billion pounds,"},{"startTime":2937.78,"endTime":2939.93,"body":"which you know, is good"},{"startTime":2937.78,"endTime":2939.93,"body":"for the ex-checker,"},{"startTime":2941.08,"endTime":2942.52,"body":"and this is kind of the power"},{"startTime":2942.52,"endTime":2944.77,"body":"of using mathematics to"},{"startTime":2942.52,"endTime":2944.77,"body":"design your auctions."},{"startTime":2945.61,"endTime":2948.52,"body":"So in the few minutes we have remaining,"},{"startTime":2948.52,"endTime":2953.52,"body":"I want to talk about a final"},{"startTime":2948.52,"endTime":2953.52,"body":"question and it's about houses."},{"startTime":2954.01,"endTime":2955.66,"body":"Well, it's sort of, it"},{"startTime":2954.01,"endTime":2955.66,"body":"could be about houses."},{"startTime":2955.66,"endTime":2957.04,"body":"This is one of the scenarios"},{"startTime":2957.04,"endTime":2959.62,"body":"in which what I'm about"},{"startTime":2957.04,"endTime":2959.62,"body":"to say can be relevant."},{"startTime":2959.62,"endTime":2963.58,"body":"So if you're thinking about"},{"startTime":2959.62,"endTime":2963.58,"body":"in various times in our life,"},{"startTime":2963.58,"endTime":2966.13,"body":"maybe one day we are lucky"},{"startTime":2963.58,"endTime":2966.13,"body":"enough to be able to buy a house."},{"startTime":2966.13,"endTime":2969.22,"body":"But actually a situation"},{"startTime":2966.13,"endTime":2969.22,"body":"that's upmost in my mind is,"},{"startTime":2969.22,"endTime":2970.78,"body":"for example, if you're a student,"},{"startTime":2970.78,"endTime":2974.92,"body":"you go to university and you"},{"startTime":2970.78,"endTime":2974.92,"body":"have to rent accommodation,"},{"startTime":2974.92,"endTime":2977.41,"body":"you have to rent a house"},{"startTime":2974.92,"endTime":2977.41,"body":"with some of your friends."},{"startTime":2977.41,"endTime":2980.92,"body":"The market for that is"},{"startTime":2977.41,"endTime":2980.92,"body":"very rapidly moving."},{"startTime":2980.92,"endTime":2983.47,"body":"You go round, you look at places,"},{"startTime":2983.47,"endTime":2986.41,"body":"everyone else is also going"},{"startTime":2983.47,"endTime":2986.41,"body":"around and looking at places."},{"startTime":2986.41,"endTime":2988.75,"body":"So if you see a place and you like it,"},{"startTime":2988.75,"endTime":2991.42,"body":"but you don't make a decision quickly,"},{"startTime":2991.42,"endTime":2993.867,"body":"the thing is, if you go"},{"startTime":2991.42,"endTime":2993.867,"body":"back next day and say,"},{"startTime":2993.867,"endTime":2996.1,"body":"well, actually I did like"},{"startTime":2993.867,"endTime":2996.1,"body":"that, it'll already be gone."},{"startTime":2996.1,"endTime":2998.77,"body":"So in that kind of market,"},{"startTime":2998.77,"endTime":3000.873,"body":"and renting houses is this one example."},{"startTime":3001.92,"endTime":3003.42,"body":"Really, you see something"},{"startTime":3003.42,"endTime":3005.19,"body":"and you have to decide yes or no to it,"},{"startTime":3005.19,"endTime":3006.09,"body":"kind of there and then,"},{"startTime":3006.09,"endTime":3010.32,"body":"and you can't go back and"},{"startTime":3006.09,"endTime":3010.32,"body":"change your mind subsequently."},{"startTime":3010.32,"endTime":3013.47,"body":"And so you can think,"},{"startTime":3010.32,"endTime":3013.47,"body":"what's my strategy here?"},{"startTime":3013.47,"endTime":3015.54,"body":"I'll see some houses and at"},{"startTime":3013.47,"endTime":3015.54,"body":"some point I have to say,"},{"startTime":3015.54,"endTime":3019.41,"body":"okay, this one is okay, I'm"},{"startTime":3015.54,"endTime":3019.41,"body":"going to choose this one."},{"startTime":3019.41,"endTime":3022.5,"body":"Even though you haven't"},{"startTime":3019.41,"endTime":3022.5,"body":"seen all the possible houses"},{"startTime":3022.5,"endTime":3025.68,"body":"because, you know, there's"},{"startTime":3022.5,"endTime":3025.68,"body":"more that you could see."},{"startTime":3025.68,"endTime":3027.45,"body":"So you think about, okay,"},{"startTime":3025.68,"endTime":3027.45,"body":"maybe I'm going to see,"},{"startTime":3027.45,"endTime":3030.42,"body":"you know, three houses"},{"startTime":3027.45,"endTime":3030.42,"body":"every day for up to a week,"},{"startTime":3030.42,"endTime":3033.21,"body":"that's how long I've got"},{"startTime":3030.42,"endTime":3033.21,"body":"to find this house to rent."},{"startTime":3033.21,"endTime":3036.24,"body":"So maybe I'm going to"},{"startTime":3033.21,"endTime":3036.24,"body":"be able to see up to,"},{"startTime":3036.24,"endTime":3038.46,"body":"let's say, 20 houses,"},{"startTime":3038.46,"endTime":3041.79,"body":"I'll see some and then I'm going"},{"startTime":3038.46,"endTime":3041.79,"body":"to have to make a decision,"},{"startTime":3041.79,"endTime":3043.8,"body":"and when should I do that?"},{"startTime":3043.8,"endTime":3046.29,"body":"So the idea is if you don't see any at all"},{"startTime":3046.29,"endTime":3048.33,"body":"and then just take the first house,"},{"startTime":3048.33,"endTime":3051.33,"body":"that's unlikely, you're not"},{"startTime":3048.33,"endTime":3051.33,"body":"going to know what's out there."},{"startTime":3051.33,"endTime":3056.25,"body":"So a good strategy will be to"},{"startTime":3051.33,"endTime":3056.25,"body":"view some houses, a sample,"},{"startTime":3056.25,"endTime":3060.54,"body":"let's say, s houses, out of"},{"startTime":3056.25,"endTime":3060.54,"body":"the potential, we said 20,"},{"startTime":3060.54,"endTime":3063.09,"body":"but let's say n, you know,"},{"startTime":3060.54,"endTime":3063.09,"body":"we're mathematicians."},{"startTime":3063.09,"endTime":3065.28,"body":"And then, so you're looking at some"},{"startTime":3065.28,"endTime":3066.51,"body":"just to gauge what's out there,"},{"startTime":3066.51,"endTime":3068.34,"body":"you're not going to take any of them."},{"startTime":3068.34,"endTime":3071.16,"body":"And then after that point,"},{"startTime":3068.34,"endTime":3071.16,"body":"after your initial sample,"},{"startTime":3071.16,"endTime":3073.41,"body":"you are then going to take the next house"},{"startTime":3073.41,"endTime":3076.41,"body":"that's the best so far, if that happens."},{"startTime":3076.41,"endTime":3077.243,"body":"I mean, there's a chance"},{"startTime":3077.243,"endTime":3078.36,"body":"you'll already have seen the best house,"},{"startTime":3078.36,"endTime":3080.97,"body":"it was your first house, then"},{"startTime":3078.36,"endTime":3080.97,"body":"this strategy won't work."},{"startTime":3080.97,"endTime":3082.59,"body":"But that's what the"},{"startTime":3080.97,"endTime":3082.59,"body":"strategy's going to be."},{"startTime":3082.59,"endTime":3085.98,"body":"So you view some sample number of houses,"},{"startTime":3085.98,"endTime":3087.72,"body":"then beyond that point,"},{"startTime":3087.72,"endTime":3088.91,"body":"if the house you're looking at right now"},{"startTime":3088.91,"endTime":3091.83,"body":"is the best one so far, you"},{"startTime":3088.91,"endTime":3091.83,"body":"take it, that's the strategy,"},{"startTime":3091.83,"endTime":3093.03,"body":"at least for that point,"},{"startTime":3093.03,"endTime":3095.03,"body":"you are happy that you've"},{"startTime":3093.03,"endTime":3095.03,"body":"got the best one so far."},{"startTime":3095.03,"endTime":3096.87,"body":"If you took while, and"},{"startTime":3095.03,"endTime":3096.87,"body":"take the second best,"},{"startTime":3096.87,"endTime":3099.39,"body":"you're always a bit sad, you"},{"startTime":3096.87,"endTime":3099.39,"body":"dunno what's out there left."},{"startTime":3099.39,"endTime":3102.48,"body":"So it's a tricky challenge."},{"startTime":3102.48,"endTime":3103.62,"body":"So let's try this out"},{"startTime":3103.62,"endTime":3105.627,"body":"with a very small total number of houses."},{"startTime":3105.627,"endTime":3108.54,"body":"Now you've only got one day,"},{"startTime":3105.627,"endTime":3108.54,"body":"you can only see three houses."},{"startTime":3108.54,"endTime":3110.52,"body":"So what sample size should you have?"},{"startTime":3110.52,"endTime":3112.95,"body":"How many should you view"},{"startTime":3110.52,"endTime":3112.95,"body":"and reject automatically"},{"startTime":3112.95,"endTime":3115.71,"body":"before you start really making a decision?"},{"startTime":3115.71,"endTime":3117.33,"body":"So we'll have three houses."},{"startTime":3117.33,"endTime":3119.61,"body":"So you could have a sample size of zero"},{"startTime":3119.61,"endTime":3120.93,"body":"or you're not looking at any,"},{"startTime":3120.93,"endTime":3123.78,"body":"and then you start doing"},{"startTime":3120.93,"endTime":3123.78,"body":"the strategy, which is,"},{"startTime":3123.78,"endTime":3124.98,"body":"for any house,"},{"startTime":3124.98,"endTime":3127.65,"body":"we're going to take it if it's"},{"startTime":3124.98,"endTime":3127.65,"body":"the best we've seen so far."},{"startTime":3127.65,"endTime":3129.36,"body":"Well, if you don't look at any,"},{"startTime":3129.36,"endTime":3131.61,"body":"then the first house"},{"startTime":3129.36,"endTime":3131.61,"body":"is going to be the best"},{"startTime":3131.61,"endTime":3134.28,"body":"you've seen so far, so you'll take it."},{"startTime":3134.28,"endTime":3137.1,"body":"But I mean, I guess we"},{"startTime":3134.28,"endTime":3137.1,"body":"assume that these houses,"},{"startTime":3137.1,"endTime":3139.14,"body":"these n houses do have..."},{"startTime":3139.14,"endTime":3141.99,"body":"We could all put them in"},{"startTime":3139.14,"endTime":3141.99,"body":"an order best to worst"},{"startTime":3141.99,"endTime":3146.01,"body":"and that they're randomly"},{"startTime":3141.99,"endTime":3146.01,"body":"seen in random order."},{"startTime":3146.01,"endTime":3147.686,"body":"So you don't know that any one house"},{"startTime":3147.686,"endTime":3150.09,"body":"has a kind of one in n"},{"startTime":3147.686,"endTime":3150.09,"body":"or one in three chance"},{"startTime":3150.09,"endTime":3151.19,"body":"of being the best one."},{"startTime":3152.94,"endTime":3155.82,"body":"If you didn't have any"},{"startTime":3152.94,"endTime":3155.82,"body":"in your sample size,"},{"startTime":3155.82,"endTime":3157.23,"body":"you're going to choose the first house,"},{"startTime":3157.23,"endTime":3158.91,"body":"success rate one in three,"},{"startTime":3158.91,"endTime":3162.453,"body":"33% chance of being the"},{"startTime":3158.91,"endTime":3162.453,"body":"actually best house."},{"startTime":3163.32,"endTime":3165.06,"body":"If you have a sample size of one"},{"startTime":3165.06,"endTime":3166.59,"body":"that means you view the first house,"},{"startTime":3166.59,"endTime":3167.97,"body":"reject it automatically."},{"startTime":3167.97,"endTime":3169.5,"body":"And then you look at house number two,"},{"startTime":3169.5,"endTime":3170.79,"body":"is it better than house number one?"},{"startTime":3170.79,"endTime":3174.99,"body":"If so, I take it, if not, reject"},{"startTime":3170.79,"endTime":3174.99,"body":"it, move on to house three."},{"startTime":3174.99,"endTime":3178.62,"body":"Now, each possible way in which"},{"startTime":3174.99,"endTime":3178.62,"body":"the houses could be arranged"},{"startTime":3178.62,"endTime":3180.21,"body":"and we'll have to look at each one,"},{"startTime":3180.21,"endTime":3181.95,"body":"if we're just assuming it's all random,"},{"startTime":3181.95,"endTime":3184.83,"body":"has a one in six chance of"},{"startTime":3181.95,"endTime":3184.83,"body":"occurring each ordering."},{"startTime":3184.83,"endTime":3186.247,"body":"So if house one is the best,"},{"startTime":3186.247,"endTime":3189.57,"body":"it could go one, two,"},{"startTime":3186.247,"endTime":3189.57,"body":"three, or one, three, two,"},{"startTime":3189.57,"endTime":3191.37,"body":"if house one is the best,"},{"startTime":3191.37,"endTime":3193.68,"body":"that's bad news for us cause"},{"startTime":3191.37,"endTime":3193.68,"body":"we reject it automatically."},{"startTime":3193.68,"endTime":3195.6,"body":"So our strategy fails in that case"},{"startTime":3195.6,"endTime":3198.0,"body":"to choose the best house for us."},{"startTime":3198.0,"endTime":3200.79,"body":"If house two is the best, that's great."},{"startTime":3200.79,"endTime":3202.83,"body":"We've rejected house one automatically,"},{"startTime":3202.83,"endTime":3204.48,"body":"house two is the next one we see,"},{"startTime":3204.48,"endTime":3206.46,"body":"it's definitely better than house one"},{"startTime":3206.46,"endTime":3207.36,"body":"and it's the best overall."},{"startTime":3207.36,"endTime":3209.943,"body":"So we choose it and"},{"startTime":3207.36,"endTime":3209.943,"body":"it's the best, success."},{"startTime":3211.17,"endTime":3214.17,"body":"House three, if house three is the best,"},{"startTime":3214.17,"endTime":3219.06,"body":"then if the ordering goes"},{"startTime":3214.17,"endTime":3219.06,"body":"like this, best is three,"},{"startTime":3219.06,"endTime":3221.73,"body":"worst is two, then that's good"},{"startTime":3221.73,"endTime":3223.29,"body":"because we see house one reject it,"},{"startTime":3223.29,"endTime":3226.08,"body":"house two is worse than house"},{"startTime":3223.29,"endTime":3226.08,"body":"one, we reject it as well,"},{"startTime":3226.08,"endTime":3227.34,"body":"and then we see house three"},{"startTime":3227.34,"endTime":3229.56,"body":"and it's the best so we pick it, success."},{"startTime":3229.56,"endTime":3232.5,"body":"But if house two is better than house one,"},{"startTime":3232.5,"endTime":3234.99,"body":"then we would take it,"},{"startTime":3232.5,"endTime":3234.99,"body":"we'd sort of peak too soon,"},{"startTime":3234.99,"endTime":3236.28,"body":"we wouldn't even look at house three."},{"startTime":3236.28,"endTime":3237.63,"body":"So that would be a failure."},{"startTime":3237.63,"endTime":3240.06,"body":"So of those six possibilities,"},{"startTime":3240.06,"endTime":3241.95,"body":"three of them result in success."},{"startTime":3241.95,"endTime":3244.29,"body":"So it's a 50% success rate."},{"startTime":3244.29,"endTime":3246.12,"body":"So a sample size of one is looking better"},{"startTime":3246.12,"endTime":3247.98,"body":"than a sample size of zero,"},{"startTime":3247.98,"endTime":3250.32,"body":"and a sample size of two adds too many"},{"startTime":3250.32,"endTime":3253.68,"body":"'cause you reject house"},{"startTime":3250.32,"endTime":3253.68,"body":"one and two automatically."},{"startTime":3253.68,"endTime":3256.11,"body":"And then you move on to house three,"},{"startTime":3256.11,"endTime":3257.73,"body":"you're only going to be able to choose it"},{"startTime":3257.73,"endTime":3259.41,"body":"if it's better than both of those things"},{"startTime":3259.41,"endTime":3261.42,"body":"where there's a one in"},{"startTime":3259.41,"endTime":3261.42,"body":"three chance of that."},{"startTime":3261.42,"endTime":3263.73,"body":"So in this case, with just three houses,"},{"startTime":3263.73,"endTime":3268.08,"body":"the best strategy is to sample"},{"startTime":3263.73,"endTime":3268.08,"body":"a third of them, one house,"},{"startTime":3268.08,"endTime":3270.48,"body":"and then you choose after that point,"},{"startTime":3270.48,"endTime":3272.973,"body":"the best one so far, if that happens."},{"startTime":3273.99,"endTime":3275.73,"body":"So you can try this for bigger numbers."},{"startTime":3275.73,"endTime":3277.44,"body":"So if you have five houses,"},{"startTime":3277.44,"endTime":3279.72,"body":"it turns out the best sample size is two,"},{"startTime":3279.72,"endTime":3282.0,"body":"which is 40% of the sample."},{"startTime":3282.0,"endTime":3283.62,"body":"If there are eight houses,"},{"startTime":3283.62,"endTime":3285.78,"body":"you move up to looking at three houses."},{"startTime":3285.78,"endTime":3287.58,"body":"You can just do all these calculations,"},{"startTime":3287.58,"endTime":3290.79,"body":"and that's 37.5% of sample size."},{"startTime":3290.79,"endTime":3292.5,"body":"I mean, these numbers are sort"},{"startTime":3290.79,"endTime":3292.5,"body":"of moving up and down a bit,"},{"startTime":3292.5,"endTime":3293.58,"body":"these are small numbers."},{"startTime":3293.58,"endTime":3296.52,"body":"As you increase the"},{"startTime":3293.58,"endTime":3296.52,"body":"total number of houses,"},{"startTime":3296.52,"endTime":3299.37,"body":"the best sample size as a percentage"},{"startTime":3299.37,"endTime":3301.47,"body":"seems to settle on a particular number."},{"startTime":3301.47,"endTime":3304.86,"body":"And so, I've done this for a large number,"},{"startTime":3304.86,"endTime":3305.94,"body":"here's another graph."},{"startTime":3305.94,"endTime":3309.09,"body":"So this is if we have a"},{"startTime":3305.94,"endTime":3309.09,"body":"thousand houses to look at,"},{"startTime":3309.09,"endTime":3311.31,"body":"hopefully that would never happen,"},{"startTime":3311.31,"endTime":3313.59,"body":"and the sample size is on the x-axis"},{"startTime":3313.59,"endTime":3315.69,"body":"going from zero to a thousand,"},{"startTime":3315.69,"endTime":3320.52,"body":"and then the likelihood of"},{"startTime":3315.69,"endTime":3320.52,"body":"success is on the y-axis."},{"startTime":3320.52,"endTime":3322.65,"body":"And you can see that there's a maximum"},{"startTime":3322.65,"endTime":3326.04,"body":"and it's about sort of 370 houses"},{"startTime":3326.04,"endTime":3328.383,"body":"somewhere around that point."},{"startTime":3329.79,"endTime":3331.71,"body":"And this is sort of generally"},{"startTime":3329.79,"endTime":3331.71,"body":"it's tending to some,"},{"startTime":3331.71,"endTime":3332.82,"body":"converging on some number,"},{"startTime":3332.82,"endTime":3336.469,"body":"and that number is viewing 36.79%"},{"startTime":3336.469,"endTime":3339.93,"body":"of the total number of objects."},{"startTime":3339.93,"endTime":3343.68,"body":"Now, what on earth, that's"},{"startTime":3339.93,"endTime":3343.68,"body":"not a very nice number, 36.79."},{"startTime":3343.68,"endTime":3346.26,"body":"If I express it in a"},{"startTime":3343.68,"endTime":3346.26,"body":"slightly different way,"},{"startTime":3346.26,"endTime":3347.73,"body":"you can look at the fraction."},{"startTime":3347.73,"endTime":3349.92,"body":"So the best sample fraction"},{"startTime":3347.73,"endTime":3349.92,"body":"of these three things,"},{"startTime":3349.92,"endTime":3354.92,"body":"little low numbers, one in"},{"startTime":3349.92,"endTime":3354.92,"body":"three, one in 2.5, one in 2.7,"},{"startTime":3355.89,"endTime":3360.12,"body":"for large n, one in 2.718."},{"startTime":3360.12,"endTime":3363.0,"body":"Now there may be, I think that's..."},{"startTime":3363.0,"endTime":3364.65,"body":"Again, depending on what stage"},{"startTime":3364.65,"endTime":3365.97,"body":"of your mathematical knowledge you're at,"},{"startTime":3365.97,"endTime":3368.34,"body":"you might recognize 2.718."},{"startTime":3368.34,"endTime":3369.39,"body":"It's an approximation"},{"startTime":3369.39,"endTime":3373.35,"body":"to the very, very important"},{"startTime":3369.39,"endTime":3373.35,"body":"mathematical constant called E,"},{"startTime":3373.35,"endTime":3374.82,"body":"which you will encounter at some point"},{"startTime":3374.82,"endTime":3376.83,"body":"in your mathematical career"},{"startTime":3374.82,"endTime":3376.83,"body":"if you haven't already."},{"startTime":3376.83,"endTime":3379.09,"body":"The best sample size turns out to be"},{"startTime":3380.04,"endTime":3382.86,"body":"one over E of the total."},{"startTime":3382.86,"endTime":3387.273,"body":"Now this is a nice little"},{"startTime":3382.86,"endTime":3387.273,"body":"fact, it's a lovely fact."},{"startTime":3388.44,"endTime":3392.49,"body":"Also, this sort of idea is"},{"startTime":3388.44,"endTime":3392.49,"body":"not just for house buying,"},{"startTime":3392.49,"endTime":3395.463,"body":"you can also use it for love."},{"startTime":3396.72,"endTime":3399.84,"body":"Think about your dating strategy,"},{"startTime":3399.84,"endTime":3401.19,"body":"I'm sure you will have one."},{"startTime":3403.32,"endTime":3404.79,"body":"You might think, okay,"},{"startTime":3404.79,"endTime":3408.0,"body":"I'm going to want to"},{"startTime":3404.79,"endTime":3408.0,"body":"settle down at some point,"},{"startTime":3408.0,"endTime":3409.92,"body":"you might work out how many people"},{"startTime":3409.92,"endTime":3413.13,"body":"you might potentially"},{"startTime":3409.92,"endTime":3413.13,"body":"date in a life of dating."},{"startTime":3413.13,"endTime":3416.34,"body":"Well, if I want to be settled"},{"startTime":3413.13,"endTime":3416.34,"body":"down by the age of 30,"},{"startTime":3416.34,"endTime":3419.25,"body":"let's say, and maybe I'll"},{"startTime":3416.34,"endTime":3419.25,"body":"have 10 years of dating"},{"startTime":3419.25,"endTime":3422.103,"body":"and maybe I'll have, I don't know,"},{"startTime":3422.94,"endTime":3425.97,"body":"two people I might date in a"},{"startTime":3422.94,"endTime":3425.97,"body":"year or something like this."},{"startTime":3425.97,"endTime":3430.97,"body":"So maybe you have up to"},{"startTime":3425.97,"endTime":3430.97,"body":"20 candidates for the one"},{"startTime":3431.4,"endTime":3433.02,"body":"during a potential dating life,"},{"startTime":3433.02,"endTime":3434.79,"body":"just to pick some random numbers."},{"startTime":3434.79,"endTime":3437.58,"body":"And so, now you know what you should do,"},{"startTime":3437.58,"endTime":3439.59,"body":"is you should date seven people"},{"startTime":3439.59,"endTime":3441.75,"body":"with absolutely no intention"},{"startTime":3439.59,"endTime":3441.75,"body":"of settling down with them"},{"startTime":3441.75,"endTime":3444.42,"body":"just to see what the market's like."},{"startTime":3444.42,"endTime":3446.43,"body":"And then after that point,"},{"startTime":3446.43,"endTime":3448.71,"body":"you should settle down"},{"startTime":3446.43,"endTime":3448.71,"body":"with the best ones so far."},{"startTime":3448.71,"endTime":3452.463,"body":"Okay, so dating advice from"},{"startTime":3448.71,"endTime":3452.463,"body":"your Gresham professor."},{"startTime":3453.57,"endTime":3454.92,"body":"So lots of these things can appear"},{"startTime":3454.92,"endTime":3456.81,"body":"in lots of different situations."},{"startTime":3456.81,"endTime":3459.69,"body":"I hope you've enjoyed a"},{"startTime":3456.81,"endTime":3459.69,"body":"little mathematical tour"},{"startTime":3459.69,"endTime":3461.28,"body":"of the mathematics of games,"},{"startTime":3461.28,"endTime":3462.78,"body":"and you could see these ideas"},{"startTime":3462.78,"endTime":3465.84,"body":"can be relevant in many, many places."},{"startTime":3465.84,"endTime":3467.94,"body":"I will just flag up my next lecture"},{"startTime":3467.94,"endTime":3469.65,"body":"which is on the 31st of January,"},{"startTime":3469.65,"endTime":3471.54,"body":"lottery winning mathematics."},{"startTime":3471.54,"endTime":3472.92,"body":"So I hope to see you all there,"},{"startTime":3472.92,"endTime":3475.284,"body":"but for the moment, thank you very much."},{"startTime":3475.284,"endTime":3478.451,"body":"(audience applauding)"},{"startTime":3484.503,"endTime":3485.79,"body":"- Thank you so much, Professor Hart."},{"startTime":3485.79,"endTime":3487.32,"body":"We don't have a lot of time for questions,"},{"startTime":3487.32,"endTime":3488.91,"body":"so can we start with one online"},{"startTime":3488.91,"endTime":3490.89,"body":"and then take one in the room?"},{"startTime":3490.89,"endTime":3492.09,"body":"We've got a question here"},{"startTime":3492.09,"endTime":3496.743,"body":"about your auction slides from earlier."},{"startTime":3497.94,"endTime":3500.88,"body":"And he says, this might"},{"startTime":3497.94,"endTime":3500.88,"body":"be answered in due course,"},{"startTime":3500.88,"endTime":3503.61,"body":"but would second bid"},{"startTime":3500.88,"endTime":3503.61,"body":"plus one or some value"},{"startTime":3503.61,"endTime":3505.92,"body":"between the top two be better?"},{"startTime":3505.92,"endTime":3509.67,"body":"Surely, yes, for the seller,"},{"startTime":3505.92,"endTime":3509.67,"body":"and not so bad for the buyer?"},{"startTime":3509.67,"endTime":3510.503,"body":"- Well, yes."},{"startTime":3510.503,"endTime":3511.74,"body":"So this kind of thing"},{"startTime":3511.74,"endTime":3513.78,"body":"where I said it's quite"},{"startTime":3511.74,"endTime":3513.78,"body":"like the open auction"},{"startTime":3513.78,"endTime":3515.73,"body":"because you can bid a tiny little bit more"},{"startTime":3515.73,"endTime":3519.87,"body":"than the second person who"},{"startTime":3515.73,"endTime":3519.87,"body":"drops out at the last minute."},{"startTime":3519.87,"endTime":3522.27,"body":"So potentially you could"},{"startTime":3519.87,"endTime":3522.27,"body":"tweak that and say, yeah,"},{"startTime":3522.27,"endTime":3525.78,"body":"actually we'll take the"},{"startTime":3522.27,"endTime":3525.78,"body":"average or something like this."},{"startTime":3525.78,"endTime":3528.24,"body":"But that, yeah, I don't think"},{"startTime":3528.24,"endTime":3531.0,"body":"it would make the underlying"},{"startTime":3528.24,"endTime":3531.0,"body":"mathematics very different."},{"startTime":3531.0,"endTime":3532.47,"body":"It might be like, well, you know,"},{"startTime":3532.47,"endTime":3535.26,"body":"maybe 1% or something 10% of the value"},{"startTime":3535.26,"endTime":3538.29,"body":"would please the seller and"},{"startTime":3535.26,"endTime":3538.29,"body":"not be so bad for the buyer."},{"startTime":3538.29,"endTime":3539.31,"body":"Yeah."},{"startTime":3539.31,"endTime":3540.69,"body":"- [Audience Member] Is the calculation"},{"startTime":3540.69,"endTime":3542.37,"body":"for the Colin and Rose example"},{"startTime":3542.37,"endTime":3545.4,"body":"suitable for large number of outcomes?"},{"startTime":3545.4,"endTime":3547.8,"body":"Or is it limited to"},{"startTime":3545.4,"endTime":3547.8,"body":"only the three outcomes"},{"startTime":3547.8,"endTime":3549.27,"body":"of win, draw, or lose?"},{"startTime":3549.27,"endTime":3552.84,"body":"And does the zero sum game"},{"startTime":3549.27,"endTime":3552.84,"body":"only include numbers one,"},{"startTime":3552.84,"endTime":3554.34,"body":"plus one, or minus one,"},{"startTime":3554.34,"endTime":3558.27,"body":"or is there ever a plus two, minus two?"},{"startTime":3558.27,"endTime":3560.25,"body":"Or if there were separate outcomes,"},{"startTime":3560.25,"endTime":3563.04,"body":"do they count within the total possibility"},{"startTime":3563.04,"endTime":3565.23,"body":"of plus one and minus one?"},{"startTime":3565.23,"endTime":3568.56,"body":"- So you could have"},{"startTime":3565.23,"endTime":3568.56,"body":"more complicated setups,"},{"startTime":3568.56,"endTime":3570.42,"body":"you could have a range"},{"startTime":3568.56,"endTime":3570.42,"body":"of different outcomes."},{"startTime":3570.42,"endTime":3572.85,"body":"And in fact, yeah,"},{"startTime":3572.85,"endTime":3575.7,"body":"the goal scoring was kind"},{"startTime":3572.85,"endTime":3575.7,"body":"of really plus one and zero,"},{"startTime":3575.7,"endTime":3577.02,"body":"you could argue."},{"startTime":3577.02,"endTime":3578.64,"body":"As long as it adds up to zero"},{"startTime":3578.64,"endTime":3580.89,"body":"or even if it adds up"},{"startTime":3578.64,"endTime":3580.89,"body":"to a constant amount,"},{"startTime":3580.89,"endTime":3582.48,"body":"then that's okay."},{"startTime":3582.48,"endTime":3584.61,"body":"So you could have plus two, minus two,"},{"startTime":3584.61,"endTime":3588.51,"body":"as long as if I get plus"},{"startTime":3584.61,"endTime":3588.51,"body":"two you get minus two."},{"startTime":3588.51,"endTime":3591.09,"body":"So as long as it, you know,"},{"startTime":3588.51,"endTime":3591.09,"body":"the sum is zero still,"},{"startTime":3591.09,"endTime":3593.43,"body":"then it still can be amenable"},{"startTime":3591.09,"endTime":3593.43,"body":"to that kind of analysis."},{"startTime":3593.43,"endTime":3594.263,"body":"But yeah,"},{"startTime":3594.263,"endTime":3595.65,"body":"there are much more complicated"},{"startTime":3594.263,"endTime":3595.65,"body":"examples you could use."},{"startTime":3595.65,"endTime":3600.33,"body":"But I didn't want to put a"},{"startTime":3595.65,"endTime":3600.33,"body":"hundred by hundred diagram"},{"startTime":3600.33,"endTime":3601.2,"body":"on a slide."},{"startTime":3601.2,"endTime":3604.44,"body":"So yes, yeah, you can extend"},{"startTime":3601.2,"endTime":3604.44,"body":"these in many ways, yeah."},{"startTime":3604.44,"endTime":3606.056,"body":"- Thanks so much."},{"startTime":3606.056,"endTime":3607.819,"body":"(audience applauding)"}]}