# Bounding Rationality by Discounting Time - Computer Science > Computer Science and Game Theory

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Abstract: Consider a game where Alice generates an integer and Bob wins if he canfactor that integer. Traditional game theory tells us that Bob will always winthis game even though in practice Alice will win given our usual assumptionsabout the hardness of factoring.We define a new notion of bounded rationality, where the payoffs of playersare discounted by the computation time they take to produce their actions. Weuse this notion to give a direct correspondence between the existence ofequilibria where Alice has a winning strategy and the hardness of factoring.Namely, under a natural assumption on the discount rates, there is anequilibriumwhere Alice has a winning strategy iff there is a linear-timesamplable distribution with respect to which Factoring is hard on average.We also give general results for discounted games over countable actionspaces, including showing that any game with bounded and computable payoffs hasan equilibrium in our model, even if each player is allowed a countable numberof actions. It follows, for example, that the Largest Integer game has anequilibrium in our model though it has no Nash equilibria or epsilon-Nashequilibria.

Author: ** Lance Fortnow, Rahul Santhanam**

Source: https://arxiv.org/