Symmetric solution of the Bellman optimality equation for repeated harmony game

In this paper we study team-symmetric games with $m\ge 2$ teams. Players within a team have symmetric identity and have a common payoff function. We show that team-symmetric games always have a team-symmetric Nash equilibrium. We develop and solve a linear complementarity problem of team-symmetric Nash equilibria. We propose an actor-critic based multi-agent reinforcement learning algorithm for team-symmetric games. Through simulations, we show that this multi-agent reinforcement learning algorithm performs much better than many existing algorithms.

Reinforcement learning agents trained to maximize their own reward in repeated interactions can converge to supra-competitive outcomes resembling explicit collusion, without communication or shared design. Existing mitigation approaches are largely tied to specific economic settings, like two-sided platforms and auctions, leaving open how to design interventions for general repeated games. We address this gap by formalizing the connection between empirical observations from prior work on Q-learning collusion and classical theory of Simple Penal Codes (SPCs). We show any non-trivial SPC induces a quantifiable conditional dependence in agents' policies, detectable via the total variation distance between an agent's action distributions across cooperation and defection histories. Building on this connection, we propose CURB (Collusion Unwinding via Reward shaping and Belief injection), a reward-shaping framework that penalizes this Total Variation (TV) distance signal during Q-learning and is guaranteed to convert any SPC fixed point of the dynamics into a trivial one, thus precluding collusive equilibria sustained by punishment threats. Empirically, CURB substantially reduces collusion by Q-learning agents in both Bertrand and Cournot Competition Repeated Games. We further demonstrate that CURB extends to deep Q-network agents in Bertrand competition, suggesting the mechanism generalizes beyond tabular Q-learning.

We extend the study of learning in games to dynamics that exhibit non-asymptotic stability. We do so through the notion of uniform stability, which is concerned with equilibria of individually utility-seeking dynamics. Perhaps surprisingly, it turns out to be closely connected to economic properties of collective rationality. Up to strategic equivalence, if a mixed equilibrium is uniformly stable, then it is weakly Pareto optimal; there is no way for all players to improve by jointly deviating from the equilibrium. This is a form of collective rationality that rules out the types of behaviors in the prisoner's dilemma or the tragedy of the commons. Moreover, we show that uniform stability determines the last-iterate convergence behavior for the family of incremental smoothed best-response dynamics, used to model individual and corporate behaviors in the markets. Unlike dynamics around strict equilibria, which can stabilize to socially-inefficient solutions, individually utility-seeking behaviors near mixed Nash equilibria lead to collective rationality.