Preference orderings represented by coherent upper and lower conditional previsions

Modelling human decisions under uncertainty has become a crucial issue in the field of Artificial Intelligence over recent years. Mathematical models of decision making under risk provide the user with an ‘optimal’ solution. These rational decision models, however, are not always able to describe the typical human approach to making decisions.

Dr Serena Doria, from The Gabriele d'Annunzio University in Italy, presents a new mathematical  updating model that can represent the awareness process of the unconscious and conscious thought.

Read her paper here:  https://doi.org/10.1007/s11238-019-09699-3

Transcript

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Modelling human decisions under uncertainty has become a crucial issue in the field of Artificial Intelligence over recent years. Mathematical models of decision making under risk provide the user with an ‘optimal’ solution. These rational decision models, however, are not always able to describe the typical human approach to making decisions. In particular, preference ordering, where items are ranked according to their preference, cannot always be modelled by a clear, linear prevision, or additive probability, especially when the nature of the preferences is determined by each individual personal tastes.

In the 1950s, Gustave Choquet, a French mathematician, started using nonadditive probabilities to help make sense of choices in what is known as Choquet utility theory. Later, in the 1970s, Israeli psychologists Daniel Kahneman and Amos Tversky developed their Prospect theory. They performed a variety of experiments in their investigation of human reasoning decision making in order to describe the actual behaviour and find out what kind of evidence biases human intuition and choice. 

They uncovered two processes of brain activity that are regulated by two different ways of thinking: fast and slow thought. They refer to these dual processes as System 1 and System 2.

System 1 is fast thinking which regulates intuitive, involuntary, unconscious and effortless activities, while System 2 is the slow process, the conscious part of the brain that is in charge of logical reasoning. 

Around the same time the Chilean psychiatrist and psychoanalyst, Matte Blanco, was developing a logic best explanation for the operation of the unconscious. He found that conscious and unconscious activities were in fact two different modes of being.  In particular, he drew a distinction between logical conscious thought that is structured on the categories of time and space and ruled by Aristotle's principle of non-contradiction, and unconscious thought, which is based upon the principles of symmetry and generalization. Both types of thoughts are supposed to combine in different human thinking experiences since they yield a bi-logic asset. Emotions function in the same way and offer ways to reach and decode the unconscious. Moreover, both scenarios stress the importance of emotion-driven, unconscious thinking in human decision-making.  It is therefore natural to wonder if mathematical models are able to describe and reproduce these different aspects of human mind.

In her paper ‘Preference orderings represented by coherent upper and lower conditional previsions’, Dr Serena Doria, from The Gabriele d'Annunzio University in Italy, presents a new mathematical model that describes this dual aspect of human brain activity. Her approach is based on the notion of  non-linear uncertainty measures, namely coherent upper and lower conditional probabilities, that  may actually preclude the possibility of an optimal choice. 

The concept of coherence was proposed by the Italian applied mathematician Bruno de Finetti, when his research into betting and probability led him to develop his ideas on subjective probability. Coherent upper and lower conditional previsions are non-linear operators that satisfy the axioms of coherence. They are obtained when only indicator functions are considered. Indicator functions are used in probability theory to simplify notation and to prove theorems. They are assigned the value 1 when an event happens and the value 0 when the event does not happen. 

Walley defined them in 1990 as "lower prevision can be regarded as a supremum buying price for the random variable X, and the upper prevision is an infimum selling price."

In  the subjective approach, the probability of an event is the amount we are willing to pay for the event in a fair bet - where no-one is sure to win or lose. In a fair bet, the gain is a random variable, as its value depends on the result. It follows that it cannot always be positive, or always be negative, because this would imply that the gambler either always wins or always loses. 

Coherent upper and lower conditional previsions were later developed by Peter Walley. 

Dr Doria proposes  a new model of coherent upper and lower conditional previsions, based on Hausdorff outer and inner measure, that assures the optimal decision can be made if measurable sets are considered.

Her approach suggests that in order to obtain an optimal choice, both the conscious and the unconscious thought arrive at the same decision. Her model of coherent upper and lower conditional previsions, based on Hausdorff outer and inner measures, assures that the optimal decision can be made if measurable sets are considered.

The German mathematician, Felix Hausdorff is considered to be one of the founders of modern topology. He developed Hausdorff measures as a generalisation the traditional notions of area and volume to non-integer dimensions, initially for use with fractals and their Hausdorff dimensions. The outer and inner  measures are functions defined on all subsets of a given set satisfying some technical conditions. 

Basically, the inner measure of a set is a lower bound of the size of the set.

This proposed model of coherent lower and upper conditional previsions  represents the preference orderings, as well as the equivalences, that are assigned respectively by the conscious and unconscious thought in human decision making under uncertainty. In fact preference orderings, represented by the coherent lower previsions, satisfy the antisymmetric property which is not satisfied by the binary relation  represented by their conjugate  coherent upper conditional previsions. For this reason the two binary relations can  describe, respectively, the activity of the conscious human thought, ruled by the antisymmetric property, and of the unconscious human thought, which is governed by the symmetric principle and the generalization principle according to the theory developed by Matte-Blanco.

 In addition, the model highlights the role of unexpected events, those with zero probability, in the updating of knowledge and awareness of both the conscious and unconscious thought. The complexity of partial information is represented by the Hausdorff dimension of the conditioning event.  

When the events that describe the decision problem, are measurable in terms of  Hausdorff outer measure, an optimal decision can be reached and it occurs, according to the model,  when both the conscious and the unconscious thought arrive at the same decision.

Dr Doria applies her model to resolve Tversky’s and Kahneman’s conjunction fallacy, the Linda′s Problem: 

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which current event about Linda is more likely?

Option 1: Linda is a bank teller.

Option 2 Linda is a bank teller and is active in the feminist movement.

Tversky and Kahneman found that contrary to the axioms of probability, 85 percent of participants chose option 2, the conjunction, regardless of whether they were expert statisticians or not. The conjunction rule states, however, that the probability of two independent events occurring conjointly cannot be higher than the probability of either event occurring separately. The probability of two events occurring in conjunction, therefore, is always less than or equal to the probability of either event occurring on its own.

Dr Dorias’ approach proposes that the conjunction fallacy can be dissolved once the preference ordering is represented by conditional probability which is defined, precisely, on the basis of the complexity of the given information measured in terms of Hausdorff dimension. 

In the original experiment, the information given about Linda, produces, in the majority of the participants, the idea that it is more likely that Linda is a feminist than she is not. This result can be described applying the conditional probability concept, which depends on the complexity of the information so that the level of knowledge about Linda is updated. Thus, the resulting preference ordering does not incur in the fallacy.  

The model of coherent upper and lower conditional previsions also  assures that the optimal decision can be made if measurable sets are considered as she demonstrates for Linda′s Problem. According to this interpretation, we could conclude that if two random variables are indifferent with respect to the unconscious thought, then one of them cannot be preferable to the other with respect to the conscious thought. 

It follows that if a random variable is preferable to another one with respect to the conscious mind, then they cannot be indifferent with respect to the unconscious thought.  

 The model also describes how pathological or counterintuitive situations can be obtained when two random variables are indifferent with respect to the unconscious thought, but one of them is preferable to the other one with respect to the conscious thought, or when a random variable is preferable to another one with respect to the conscious thought, but the two random variables are indifferent with respect to the unconscious thought.

Dr Dorias’ model can capture these situations as it recognises that the lower conditional prevision should be greater than the upper conditional prevision. Moreover, it can represent the awareness processes of both the unconscious and conscious thoughts which depend, according to the model,  on unexpected events in all cases. 

This research has revealed that the conjunction fallacy can also be solved if the preference ordering of the events is represented by the linear model of coherent conditional probability as defined by Hausdorff measures. Dr Doria has demonstrated this by representing the considered events involved in Linda’s Problem with sets that are measurable with respect to the Hausdorff measure of orders. Her updating model can represent the awareness process of the unconscious and conscious thought, which depends on unexpected events in all cases.

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