Ulric B. and Evelyn L. Bray Seminar

We consider an environment that is complex in the sense a Decision Maker (DM) finds it impossible to describe the state state space and is thus prevented from an application of standard Bayesian methods. This paper presents an axiomatic theory of choice in such environments. The DM chooses from a finite set of actions that result in random rewards at discrete moments in time. The DM has a well-defined preference relation over the actions that depends on history of observed rewards. We present three simple axioms --- two of them highlight the bounded rationality of DM and the third describes how these history contingent preferences are updated, loosely based on the ``the principle of insufficient reason''. These are then shown to be necessary and sufficient to characterize (a generalized) fictitious play. We also show that if rewards follow an exchangeable stochastic process, a DM that obeys these axioms (almost surely) cannot be distinguished from one that is fully cognizant of the environment and satisfies EUH. The main result requires proving a representation result on multisets which may be of independent interest.