The Society for Imprecise Probabilities:
Theories and Applications

# Arianna Casanova's PhD thesis "Rationality and Desirability - A foundational study"

Posted on February 28, 2024 by Arianna Casanova Flores (edited by H. Diego Estrada-Lugo & Alexander Erreygers)
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In July 2023, I successfully defended my PhD thesis entitled “Rationality and desirability - A foundational study”. This work reflects five years of research at the Dalle Molle Institute for Artificial Intelligence USI-SUPSI (IDSIA) in Lugano, under the supervision of Prof. Marco Zaffalon and Prof. Luca Maria Gambardella. Alongside Prof. Zaffalon and Prof. Gambardella, my committee included Prof. Fabio Crestani, Prof. Ernst C. Wit, Prof. Matthias C.M. Troffaes and Dr. Nic Wilson.

## Introduction

The past century has witnessed major achievements in scientific fields that have come to be regarded as classical by now: among them, we can mention decision theory, social choice theory, and, of course, the Bayesian theory of probability. Despite the differences, all those theories have been founded on the same idea of rationality, that of consistency: in order to be rational, a calculus needs to be consistent, i.e., deliver coherent inferences, or a subject needs to maintain coherent preferences among options. Bayesian probability is the arena where this idea of rationality has been faced more vividly, starting with the work of de Finetti [1]. Already then, de Finetti clearly saw that rationality, in the mentioned interpretation of coherence, and probability, were just the same thing. Another arena where coherence has been identified with a science field is logic: this should not be surprising, given that consistency is substantially the subject matter of logic.

A turning point in the interplay of logic, probability, and coherence, has been Peter Williams’ definition of desirability [2], an extension of de Finetti’s theory of probability crafted to address imprecision, as originated by incompleteness or other reasons. The main tools of desirability, namely the coherent sets of desirable gambles (or, for brevity, coherent sets of gambles) correspond to sets of bets, also called gambles, that are acceptable for an agent satisfying a set of rationality axioms. Coherence underlying these tools can also be expressed using elements from logic and probability. Coherent sets of gambles can be interpreted as closed (with respect to an algebraic closure operator) and consistent subsets of a language, thereby placing desirability in the realm of logic. Moreover, echoing de Finetti’s main ideas, it is possible to show that coherent sets of gambles encompass most generalizations of probability that have been proposed in the literature: lower and upper probabilities and previsions, convex sets of distributions, belief functions, possibility measures, etcetera [3, 4].

Rationality axioms of desirability are grounded on the assumption that rewards of gambles are expressed in a linear utility scale. This creates some difficulties in interpreting gambles as proper monetary bets, i.e., bets that return money. For example, for a large positive scalar $$\lambda$$ it can be hard to accept $$\lambda f$$ for every acceptable gamble $$f$$, because of lack of market liquidity at some degree. The recent works of Zaffalon and Miranda [5, 6], however, show that enriching the space where gambles are defined in a suitable way, desirability can be generalized to deal with nonlinear utility considerations. In particular, in this context, coherent sets of gambles result to be in a one-to-one correspondence with the (incomplete) coherent preference relations used in the context of traditional rational decision making. Desirability therefore turns out to be not only a very powerful theory for uncertainty but also a very general framework for decision making. In more recent work, Miranda and Zaffalon [7] address the challenge of modeling general rational decision-making using desirability by proposing a more direct approach based on generalizations of its foundational axioms. This refined version of desirability offers solutions to issues related to the traditional axiomatization of preferences, such as those highlighted by the well-known Allais paradox [8].

## Contributions

In the thesis, we aim to further exploit and generalize desirability to create a more powerful and comprehensive framework for addressing issues in probability, decision-making, logic, and potentially other research fields. In particular, we pursue this goal by following three main research lines.

The first one is devoted to illustrating how desirability can offer a unified framework for disciplines grounded in the concept of rationality mentioned earlier. In particular, we show how desirability can be used to unify (and treat at a very general level) some issues related to social choice [9, 10] and probabilistic opinion pooling [11, 12], two research fields dealing with the problem of aggregation of opinions but historically based on different formalisms: (coherent) preference relations over alternatives (social choice) and probability distributions (probabilistic opinion pooling). This leads, on the one hand, to a new perspective on traditional results (particularly Arrow’s theorem and conditions for the existence of oligarchy and democracy in social choice), and on the other hand, to the application of the same framework in analyzing probabilistic opinion pooling. Further exploration of these topics may also involve taking a more application-oriented approach. It would be interesting, indeed, to move our results closer to computational social choice, an active research area focusing on the algorithmic tasks in social choice and their complexity.

In the second and the third research lines we focus on more abstract considerations.

Information algebras are general algebraic structures for managing knowledge or information [13]. They abstract away the most important features that appear in almost all representation of information and, at this level, they provide general formulations of architectures for inference. They encompass diverse formalisms from both logic and probability domains. Moreover, they can be represented using systems consisting of a language and a closure operator. For this reason, they can be fully considered as part of the realm of logic. Given the structural similarities between desirability and information algebras, we proceed by verifying that the former formally induces an instance of the latter. This result enriches the view of desirability as an algebraic structure and provides us with a useful machinery for inference, usable to simplify the treatment of some related problems such as, for example, the marginal problem, i.e., the problem of checking the compatibility of a number of marginal assessments with a global model. Further, the modeling capability of desirability makes it a very general instance of information algebras able to manipulate very different forms of information. Our treatment however, is limited to the unconditional case. It would be interesting in the future to pursue further research on the conditional case, focusing in particular on conditional independence structures and their use for efficient local computation schema.

We conclude the work by following the same idea as explored more recently in [7], i.e., that of relaxing desirability’s foundational axioms to allow for a more realistic interpretation of gambles. While their approach is more general, ours is closer to applications. Specifically, we proceed by analyzing two other axiomatic definitions of desirability which relax the linearity assumption on gambles rewards. Moving to the application side, we provide an alternative interpretation of desirability and its generalisations as binary (usually nonlinear) classification problems, where in particular the family of classifiers involved changes with the axioms assumed. By borrowing ideas from machine learning, we also demonstrate the possibility of defining feature mappings that allow us to reformulate the aforementioned nonlinear classification problems as linear ones in higher-dimensional spaces. It would be interesting in the future to attempt to combine the approach mentioned here with the one proposed in [7]. In particular, a detailed analysis of how integrating our feature mappings within the framework of [7] could prove beneficial in bringing it even closer to the application side. Another promising avenue for future research stemming from these studies involves analyzing well-known examples of non-expected utility theories in terms of desirability. Notable works in [14, 15] present promising opportunities for exploration in this regard.

## References

[1] De Finetti, B.. La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut Henri Poincaŕe, Vol. 7, pp. 1–68, 1937. http://www.numdam.org/item/AIHP_1937__7_1_1_0/

[2] Williams, P. M. Notes on conditional previsions, Technical report, School of Mathematical and Physical Science, University of Sussex, UK, 1975. doi:10.1016/j.ijar.2006.07.019

[3] Quaeghebeur, E.. Desirability, in T. Augustin, F. Coolen, G. de Cooman and M. Troffaes (eds), Introduction to Imprecise Probabilities, Wiley, pp. 1–27, 2014. doi:10.1002/9781118763117.ch1

[4] Walley, P. Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991.

[5] Zaffalon, M. and Miranda, E. Axiomatising incomplete preferences through sets of desirable gambles, Journal of Artificial Intelligence Research 60: 1057–1126, 2017. doi:10.1613/jair.5230

[6] Zaffalon, M. and Miranda, E. Desirability foundations of robust rational decision making, Synthese 198(27): 6529–6570, 2021. doi:10.1007/s11229-018-02010-x

[7] Miranda, E. and Zaffalon, M.. Nonlinear desirability theory, International Journal of Approximate Reasoning 154: 176–199, 2023. doi:10.1016/j.ijar.2022.12.015

[8] Allais, M. Le comportment de l‘homme rationnel devant le risque: critique des postulats et axiomes de l‘école americaine, Econometrica 22, 1953. doi:10.2307/1907921

[9] Arrow, K. Social Choice and Individual Values, John Wiley & Sons, 1951.

[10] Weymark, J. A. Arrow’s theorem with social quasi-orderings, Public Choice 42(3): 235–246, 1984. doi:10.1007/BF00124943

[11] Lindley, D. V., Tversky, A. and Brown, R. V. On the reconciliation of probability assessments, Journal of the Royal Statistical Society, Series A 142(2): 146–162, 1979. doi:10.2307/2345078

[12] Stewart, R. and Quintana, I. O. Probabilistic opinion pooling with imprecise probabilities, Journal of Philosophical Logic 47(1): 17–45, 2018. doi:10.1007/s10992-016-9415-9

[13] Kohlas, J. Information Algebras: Generic Structures for Inference, Springer-Verlag, 2003. doi:10.1007/978-1-4471-0009-6

[14] Kahneman, D. and Tversky, A. Prospect theory: an analysis of decision under risk, Handbook of the Fundamentals of Financial Decision Making: Part I, World Scientific, pp. 99–127, 2013. doi:10.1142/9789814417358_0006

[15] Quiggin, J. A theory of anticipated utility, Journal of Economic Behavior & Organization 3(4): 323–343, 1982. doi:10.1016/0167-2681(82)90008-7