Swap Regret and Correlated Equilibria Beyond Normal-Form Games (2502.20229v1)

Abstract: Swap regret is a notion that has proven itself to be central to the study of general-sum normal-form games, with swap-regret minimization leading to convergence to the set of correlated equilibria and guaranteeing non-manipulability against a self-interested opponent. However, the situation for more general classes of games -- such as Bayesian games and extensive-form games -- is less clear-cut, with multiple candidate definitions for swap-regret but no known efficiently minimizable variant of swap regret that implies analogous non-manipulability guarantees. In this paper, we present a new variant of swap regret for polytope games that we call ``profile swap regret'', with the property that obtaining sublinear profile swap regret is both necessary and sufficient for any learning algorithm to be non-manipulable by an opponent (resolving an open problem of Mansour et al., 2022). Although we show profile swap regret is NP-hard to compute given a transcript of play, we show it is nonetheless possible to design efficient learning algorithms that guarantee at most $O(\sqrt{T})$ profile swap regret. Finally, we explore the correlated equilibrium notion induced by low-profile-swap-regret play, and demonstrate a gap between the set of outcomes that can be implemented by this learning process and the set of outcomes that can be implemented by a third-party mediator (in contrast to the situation in normal-form games).

Insights into Swap Regret and Correlated Equilibria in Complex Game Structures

The paper presents a comprehensive paper of swap regret in polytope games, which expands on the foundational theories in normal-form games. The concept of swap regret, which originally focused on normal-form games, is innovatively extended to polytope games, a category encompassing Bayesian games and extensive-form games. The exploration leads to introducing "profile swap regret," a variant specifically designed for polytope games that proves both necessary and sufficient for ensuring non-manipulability — the ability for a learning strategy within a game not to be exploited by an opponent.

Key Contributions

1. Profile Swap Regret: The researchers propose a new variant of swap regret, termed "profile swap regret," that is relevant in polytope games. Sublinear profile swap regret emerges as both necessary and sufficient for ensuring non-manipulability in any learning strategy executed against an adversary, as sought in earlier conjectures by Mansour et al.
2. Computational Complexity: Despite being NP-hard to compute from a direct transcript of play, the authors propose strategies to ensure a learning algorithm can achieve at most $O(\sqrt{T})$ profile swap regret. These algorithms present computational feasibility through Blackwell's approachability theorem, refined through techniques such as semi-separation oracles.
3. Equilibrium Dynamics: The dynamics induced by minimizing profile swap regret lead naturally to a type of correlated equilibrium (PCE), distinct from normal-form correlated equilibria. Interestingly, the paper notes a gap between these outcomes and those achievable via a third-party mediation, highlighting differences in outcome sets in normal-form and polytope gaming structures.

Theoretical and Practical Implications

• Strategic Learning and Manipulability: By proving that profile swap regret characterizes non-manipulability, this research ensures that learning agents cannot be exploited, thereby providing robust strategies in environments where opponent manipulation is probable.
• Algorithmic Efficiency: The results bridge the gap between theoretical possible outcomes in game theory and practical learning algorithms within reasonable time frames, promoting the application of these enhanced strategic models in real-world scenarios such as auctions or any structured strategic decision processes.
• Future Research Directions: The framework and methodology expand the potential for researching equilibria in multi-agent systems, where understanding the full spectrum of possible game outcomes in intricate action spaces remains critical. The exploration of decompositions into convex sets also suggests further evolutionary paths for examining game-theoretical solutions that operate under more efficient and computationally feasible manners.

Conclusion

The paper enriches the understanding of how swap regret can be adapted to more complex game structures, offering significant improvements in strategic robustness through the introduction of profile swap regret. This advancement ensures that the strategies derived are not only theoretically sound but also computationally tractable, opening up expansive avenues in fields where strategic interactions are key. This is particularly relevant in areas such as economic modeling and competitive strategy environments, where understanding and predicting opponent actions are crucial. The intricate balance achieved between theoretical exploration and practical application serves as a testament to the efficacy of sophisticated learning and adaptation in strategic gaming scenarios.