Bounding Rationality by Discounting Time

TL;DR

This paper introduces a new bounded rationality model where players' payoffs are discounted by computation time, establishing a link between game equilibria and the hardness of factoring, with implications for computational game theory.

Contribution

It defines a novel bounded rationality framework based on time-discounted payoffs and connects equilibrium existence to the computational hardness of factoring.

Findings

Equilibria exist in the model even with countable actions and computable payoffs.

A correspondence between equilibria with Alice's winning strategy and the hardness of factoring.

The Largest Integer game has an equilibrium in this model despite lacking Nash equilibria.

Abstract

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or epsilon-Nash equilibria.