Swap Regret and Correlated Equilibria Beyond Normal-Form Games

TL;DR

This paper introduces a new variant of swap regret called profile swap regret for complex games, establishing its importance for non-manipulability, designing efficient algorithms despite NP-hardness, and exploring its implications for correlated equilibria beyond normal-form games.

Contribution

The paper proposes profile swap regret for polytope games, proving its necessity and sufficiency for non-manipulability, and develops efficient algorithms despite computational hardness.

Findings

Profile swap regret is NP-hard to compute from play transcripts.

Efficient algorithms can guarantee $O( oot T)$ profile swap regret.

Low-profile-swap-regret play leads to a different set of outcomes than mediator-based correlated equilibria.

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).