Regret Minimization with Adaptive Opponents in Repeated Games

Problem
The paper addresses the inadequacy of traditional external regret metrics in online learning when applied to repeated games with adaptive opponents. Existing regret definitions do not account for the dynamic nature of opponents who adjust their strategies based on historical play. This work is particularly relevant as it proposes a new metric, Repeated Policy Regret (RP-Regret), which is tailored for this context. The authors highlight that their approach is necessary for understanding regret minimization in scenarios where players can respond to the history of play, a gap in the current literature that remains unaddressed in prior works.

Method
The authors introduce RP-Regret as a game-theoretic metric that quantifies the difference between the accumulated utility achieved by a player and the best possible utility that could have been achieved in hindsight, considering the adaptive nature of opponents. They establish necessary conditions for achieving sublinear RP-Regret over time, which depend on the variability of comparator strategies and the memory of both the player and the opponents. To minimize RP-Regret, the authors propose three algorithms:

1. An optimization oracle-based algorithm, leveraging assumptions from previous online non-convex learning research.
2. A method that minimizes a convex and linearized surrogate of RP-Regret iteratively.
3. A direct minimization approach for RP-Regret when opponents’ strategies change gradually. The algorithms are designed to facilitate learning subgame perfect equilibria in repeated games when all players adopt RP-Regret minimization strategies.

Results
The paper presents experimental results demonstrating that minimizing RP-Regret can lead to more cooperative outcomes in repeated games, specifically in the Stag-Hunt game scenario. The authors report that their algorithms outperform traditional regret minimization techniques, although specific numerical results and comparisons to baseline methods are not detailed in the abstract. The experiments indicate a significant improvement in utility, suggesting that the proposed RP-Regret framework can enhance strategic interactions among adaptive players.

Limitations
The authors acknowledge that their approach is inherently non-convex, which complicates the optimization process. They also note that the effectiveness of their algorithms may depend on the specific dynamics of the opponents’ strategies and the assumptions made regarding their adaptivity. Additionally, the paper does not explore the scalability of the proposed algorithms in larger game settings or with more complex opponent behaviors, which could limit practical applicability.

Why it matters
This work has significant implications for the design of algorithms in multi-agent systems where adaptive behavior is prevalent. By providing a robust framework for regret minimization in repeated games, the authors pave the way for future research on cooperative strategies and equilibrium learning in dynamic environments. The introduction of RP-Regret could influence various applications, including economic modeling, reinforcement learning, and strategic decision-making in competitive settings, as discussed in related literature. For further details, see the full paper available on arXiv.