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Categorical Decision Theory
October 27, 2015 @ 2:00 pm - 4:00 pm
Marcus Pivato (Université de Cergy-Pontoise, France)
(Joint work with Vassili Vergopoulos of the Paris School of Economics)
Individuals and societies must often make difficult decisions, which are fraught with uncertainty. How should an agent decide when faced with such uncertainty? This is the subject of a branch of theoretical economics called Decision Theory.
Bernoulli (1738) claimed that we should choose the alternative which yields the highest expected utility. But what justifies this methodology? Savage (1954) showed that, if our decision-making process satisfies certain axioms (encoding basic properties of "consistency" and "rationality"), then it must maximize expected utility. Savage's Theorem is considered the foundational result of modern Decision Theory.
Savage posited a set S of possible "states of nature" and a set X of possible "outcomes". He supposed that each alternative defined a function (an "act") mapping states to outcomes. His theorem constructs a probability measure on S and a utility function on X. However, this approach raises at least three issues.
1. Savage assumed that S and X were arbitrary sets, and acts were arbitrary functions. But what if S and X are topological spaces, and acts must be continuous? What if S and X are differentiable manifolds, and acts must be differentiable? We would like a single theory which works in all of these environments (and others).
2. In many applications, it is unrealistic to suppose that we can enumerate all possible states of nature or all possible outcomes "in advance". Thus, there is growing interest in developing decision theory without an explicit specification of S or X.
3. At different times, the same agent might be faced with many different sources of uncertainty (i.e. different instances of S) and many different menus of outcomes (different instances of X), in different combinations. We would like a single holistic description of the agent's decisions over all of these possible decision problems.
In this talk, we will reformulate decision theory using the tools of category theory, and derive a version of Savage's theorem which addresses all three of these issues.