Rationality and desirability

You are cordially invited to attend the PhD Dissertation Defence of Arianna Casanova Flores on Wednesday 19 July 2023 at 14:30 online (ID: 765 5306 4388, Passcode: 5HKsy0)

Abstract:
In the past century, significant achievements have been made in classical scientific fields, including decision theory, social choice theory, and Bayesian probability. Despite of 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 deliver coherent inferences, or a subject needs to maintain coherent preferences among options. Bayesian probability, particularly highlighted in the work of de Finetti, vividly exploits this notion of rationality, revealing the intrinsic connection between rationality, in the mentioned interpretation of coherence, and probability. 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. Williams’ definition of desirability marks a significant milestone in the intersection of logic, probability, and coherence. Desirability extends de Finetti’s theory of probability to address imprecision. Its primary tools, known as coherent sets of desirable gambles, represent sets of bets acceptable to a rational agent and embody a notion of coherence, which can be explicitly formalized using elements of logic. On the other hand, they also encompass most generalisations of probability that have been proposed in the literature: lower and upper probabilities and previsions, convex sets of distributions, belief functions, possibility measures, etcetera. Moreover, recent works show that if we enrich the space where gambles are defined in a suitable way, desirability can also become a powerful framework for decision making. This thesis further enhances and broadens this perspective by exploring additional developments and applications of desirability. Specifically, we follow three primary research lines. Considering the powerful modeling capabilities of desirability and its underlying idea of rationality, we first explore its application as a formalism for aggregating opinions of different forms. In particular, we propose a formulation of the problem where both individual and group opinions are represented by coherent sets of desirable gambles. We then undertake an analysis to determine whether Arrowian-like limitations are still present in this context. Lastly, we proceed by examining the minimum requirements necessary for the group opinion to exhibit some degree of consistency with the individual ones, thereby uncovering the fundamental role played by the unanimity requirement. We then enrich the view of desirability as an algebraic (logic-like) structure encapsulating it into the formalism of information algebras. These are general algebraic structures introduced to manage information at an abstract level, providing basic operations and architectures for inference. Information algebras, however, can also be alternatively represented by means of systems composed of a language and a closure operator. We demonstrate, in particular, that the main tools of desirability induce instances of information algebras. Then, we illustrate how properties of information algebras can be used to more directly derive some results in the context of desirability. An underlying assumption behind the rationality concept expressed by desirability is that rewards of gambles are measured in units of linear utility. This requirement, however, is not so realistic, especially when one needs to consider monetary gambles. Therefore, we conclude the thesis by discussing some possible generalisations of this formalism that may enable us to also directly deal with non-linear utility considerations.

Dissertation Committee:
- Prof. Luca Maria Gambardella, Università della Svizzera italiana, Switzerland (Research Advisor)
- Prof. Marco Zaffalon, IDSIA USI-SUPSI, Switzerland (Research co-Advisor)
- Prof. Fabio Crestani, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Ernst Wit, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Matthias Troffaes, Durham University, UK (External Member)
- Prof. Nic Wilson, University College Cork, Ireland (External Member)