Nikoleta Glynatsi (LEG Seminar 2025/3/3)

Best responses in repeated games: Reactive strategies with longer memory.

Repeated games have been central to studying the evolution of human cooperation. In these games, strategies define the rules players follow when choosing actions based on past interactions. Traditional theoretical models often assume that players can remember only the preceding turn. This assumption arises because, as memory length increases, the strategy space expands exponentially, making analytical results more challenging to derive.

We explore strategies that extend beyond one-round memory by focusing on reactive-n strategies, which respond solely to the co-player’s actions in the previous n rounds. We develop an algorithm to determine whether a given reactive-n strategy qualifies as a partner strategy, a Nash strategy that ensures mutual cooperation. We characterize all partner strategies among reactive-2 and reactive-3 strategies. Additionally, we analyze reactive counting strategies, a specific subset that only considers the number of cooperative moves in the last n turns, and we characterize partner strategies for all memory lengths in this class.

Our results build on a central finding: if a player employs a reactive strategy, then a co-player using a memory-n strategy can switch to a self-reactive-n strategy without altering the resulting payoffs. However, for a given subclass of games, we show an even stronger result: the co-player can afford to remember even less without any change to their payoff, switching to a self-reactive-(n-1) strategy while maintaining the same payoffs.