Abstract
We explore how an incremental change in complexity of strategies (“an inch of memory”) in repeated interactions influences the sets of Nash equilibrium (NE) strategy and payoff profiles. For this, we introduce the two most basic setups of repeated games, where players are allowed to use only reactive strategies for which a probability of players’ actions depends only on the opponent’s preceding move. The first game is trivial and inherits equilibria of the stage game since players have only unconditional (memory-less) Reactive Strategies (RSs); in the second one, players also have conditional stochastic RSs. This extension of the strategy sets can be understood as a result of evolution or learning that increases the complexity of strategies. For the game with conditional RSs, we characterize all possible NE profiles in stochastic RSs and find all possible symmetric games admitting these equilibria. By setting the unconditional benchmark as the least symmetric equilibrium payoff profile in memory-less RSs, we demonstrate that for most classes of symmetric stage games, infinitely many equilibria in conditional stochastic RSs (“a mile of equilibria”) Pareto dominate the benchmark. Since there is no folk theorem for RSs, Pareto improvement over the benchmark is the best one can gain with an inch of memory.
Highlights
In the theory of repeated games, restrictions on players’ strategies are not usually imposed
We show analytically that even an incremental change of complexity of strategies in repeated interactions has a huge influence on the sets of equilibrium payoff profiles for some classes of stage games that are circumscribed by Theorem 2
The first thing to note is that except for some trivial cases, payoff-relevant indeterminacy holds true [17]; i.e., there exists a continuum of equilibria in “new” conditional strategies with distinct payoff profiles
Summary
In the theory of repeated games, restrictions on players’ strategies are not usually imposed (see, for example, [1]). This paper falls into the strategy-restrictions framework: we make an extensive study of the Nash equilibrium in infinitely repeated 2 × 2 games where payoffs are determined by Reactive Strategies (RSs) and players’ payoffs are evaluated according to the limit of means. Players may ignore the available information about the action of the opponent in the previous round; we distinguish between unconditional and conditional RSs. The second aspect is related to predictability of actions. By omitting the issue of opening moves, one derives 1-memory form for the strategies that were found to be the most popular in experimental study [3] These strategies are Tit-for-Tat, (1, 0, 1, 0); Always Defect, (0, 0, 0, 0); both are reactive ones, while Grim, (1, 0, 0, 0), is not
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