Can I Afford to Remember Less Than You? Best Responses in Repeated Additive Games

A new study from the Max Planck Institute for Evolutionary Biology uncovers surprising conditions under which remembering less than your opponent in repeated games can still lead to optimal outcomes.

April 14, 2025

Repeated games assume two players interact over many rounds, and at each round, players make a decision. They are sometimes assumed to have finite memory (i.e., they can remember the outcomes of the last rounds). If one player hasfinite memory, an interesting question arises: how much memory does one need to remember to play optimally? Researchers have found conditions under which remembering less than your opponent in repeated games can still lead to optimal outcomes.

© Nikoleta E. Glynatsi

Repeated games assume two players interact over many rounds, and at each round, players make a decision. They are sometimes assumed to have finite memory (i.e., they can remember the outcomes of the last rounds). If one player hasfinite memory, an interesting question arises: how much memory does one need to remember to play optimally? Researchers have found conditions under which remembering less than your opponent in repeated games can still lead to optimal outcomes.

© Nikoleta E. Glynatsi

To the point

Less memory required: Under certain conditions, players can remember less than their opponent.
Optimal play still achievable: Reduced-memory strategies can perform just as well.
Easier equilibrium analysis: Helps identify stable strategies in complex repeated games.

Repeated games are a core concept in game theory, modeling how players adjust their strategies over time based on past outcomes. A key question is: How much memory does a player truly need to succeed? Famously, Press and Dyson (2012) showed that having more memory than your opponent doesn’t offer a strategic edge — in fact, the player with less memory can determine the optimal complexity of the game.

This new study takes it a step further: If one player uses a strategy that recalls the last nn rounds, could their co-player perform just as well with even less memory — say, m<nm<n rounds?

The researchers answer: yes, under specific conditions. For reactive strategies — where a player’s move depends only on the opponent’s most recent choices — and within a class of games called additive games, players can perform optimally even with reduced memory. That is, the co-player can ignore the distant past, rely only on recent rounds, and still play optimally.

The findings are conceptually striking and practically useful. They enable researchers to identify equilibrium strategies more efficiently — a task known to be computationally difficult. As memory increases, so does the number of possible outcomes, making traditional equilibrium calculations infeasible. With this result, researchers can now explore stable strategies that were previously out of reach due to their complexity.