Limited Computational Resources Favor Rationality

A choice function is a rule that chooses a single alternative from every set of alternatives drawn from a finite ground set. A rationalizable choice function satisfies the consistency condition; i.e., if an alternative is chosen from a set A, then the same alternative is also chosen from every subset of A that contains it. In this paper we study computational aspects of choice, through choice functions. We explore two simple models that demonstrate two important aspects of choice procedures: the ability to remember the past and the ability to perform complex computations. We show that a choice function is optimal in terms of the amount of memory and the computational power required for its computation if and only if the function is rationalizable. We also show that the computation of most other choice functions, including some “natural” ones, requires much more memory and computational power. ∗I am indebted to Gil Kalai for his devoted guidance, constant support, and most valuable suggestions. I am grateful to Ariel Rubinstein and Ran Spiegler for the encouragement, productive discussions, and important comments. I thank Elchanan Ben-Porath, Michael Borns, Michal Maimaran, Zvika Neeman, Abraham Neyman, Motty Perry, Ron Siegel, and Menahem Yaari for fruitful conversations and helpful remarks. †Email: [email protected].