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Beyond Symmetry in Repeated Games with Restarts

Henry Fleischmann, Kiriaki Fragkia, Ratip Emin Berker

TL;DR

The paper addresses cooperation in anonymous pairwise interactions by extending repeated games with restarts to asymmetric settings, where agents may rematch with new partners after deviations. It introduces hazing periods and goal sequences to sustain stable equilibria, and develops a formal framework to compare these sequences via Pareto-optimality, welfare maximization, and limit-utility fairness. The authors prove that computing minimum hazing sequences is (weakly) NP-hard, but show a pseudo-polynomial-time dynamic programming algorithm when the goal sequence maximizes social welfare; they also extend the model to random role reassignment and discuss existence results in various game classes. This work advances the design and analysis of cooperative protocols under anonymity, revealing both the potential gains from asymmetric strategies and the computational barriers to achieving them.

Abstract

Infinitely repeated games support equilibrium concepts beyond those present in one-shot games (e.g., cooperation in the prisoner's dilemma). Nonetheless, repeated games fail to capture our real-world intuition for settings with many anonymous agents interacting in pairs. Repeated games with restarts, introduced by Berker and Conitzer [IJCAI '24], address this concern by giving players the option to restart the game with someone new whenever their partner deviates from an agreed-upon sequence of actions. In their work, they studied symmetric games with symmetric strategies. We significantly extend these results, introducing and analyzing more general notions of equilibria in asymmetric games with restarts. We characterize which goal strategies players can be incentivized to play in equilibrium, and we consider the computational problem of finding such sequences of actions with minimal cost for the agents. We show that this problem is NP-hard in general. However, when the goal sequence maximizes social welfare, we give a pseudo-polynomial time algorithm.

Beyond Symmetry in Repeated Games with Restarts

TL;DR

The paper addresses cooperation in anonymous pairwise interactions by extending repeated games with restarts to asymmetric settings, where agents may rematch with new partners after deviations. It introduces hazing periods and goal sequences to sustain stable equilibria, and develops a formal framework to compare these sequences via Pareto-optimality, welfare maximization, and limit-utility fairness. The authors prove that computing minimum hazing sequences is (weakly) NP-hard, but show a pseudo-polynomial-time dynamic programming algorithm when the goal sequence maximizes social welfare; they also extend the model to random role reassignment and discuss existence results in various game classes. This work advances the design and analysis of cooperative protocols under anonymity, revealing both the potential gains from asymmetric strategies and the computational barriers to achieving them.

Abstract

Infinitely repeated games support equilibrium concepts beyond those present in one-shot games (e.g., cooperation in the prisoner's dilemma). Nonetheless, repeated games fail to capture our real-world intuition for settings with many anonymous agents interacting in pairs. Repeated games with restarts, introduced by Berker and Conitzer [IJCAI '24], address this concern by giving players the option to restart the game with someone new whenever their partner deviates from an agreed-upon sequence of actions. In their work, they studied symmetric games with symmetric strategies. We significantly extend these results, introducing and analyzing more general notions of equilibria in asymmetric games with restarts. We characterize which goal strategies players can be incentivized to play in equilibrium, and we consider the computational problem of finding such sequences of actions with minimal cost for the agents. We show that this problem is NP-hard in general. However, when the goal sequence maximizes social welfare, we give a pseudo-polynomial time algorithm.

Paper Structure

This paper contains 14 sections, 19 theorems, 18 equations, 1 figure, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Motivating Examples
  4. Symmetric games with asymmetric strategies.
  5. Some (even symmetric) games are unfair.
  6. Organization of the Paper
  7. Preliminaries
  8. Equilibrium Sequences
  9. Existence Results
  10. Computing Minimum Hazing Sequences
  11. Random Player Reassignment
  12. Directions for Future Work
  13. Existence of Stable sequences
  14. Omitted proofs

Key Result

Theorem 1

(Informal version of Proposition 1, Lemma 1, and Lemma 2 of berker2024computing) In repeated symmetric games $\Gamma$ with restarts and discount factor $\beta$, where all agents follow an identical sequence of actions (i.e., the strategy is symmetric), we have each of the following:

Figures (1)

  • Figure 1: Pictorial representation of $\texttt{memo}$. The green shading denotes the bounds of hazing value pairs that Algorithm \ref{['alg:dp']} will search according to Lemma \ref{['lem: hazing per struct']}.

Theorems & Definitions (49)

  • Theorem 1
  • Example 1: Restarting without Deviating
  • Definition 2: Stable Sequences
  • Remark 3
  • Proposition 3
  • Remark 4
  • Definition 5: Pareto-Optimal Sequence
  • Definition 6: Welfare maximization
  • Definition 7: Limit-utility fairness
  • Remark 8
  • ...and 39 more