Instance-dependent Sample Complexity Bounds for Zero-sum Matrix Games
This work addresses the sample complexity of learning equilibria in game theory, providing foundational insights for algorithm design in multi-agent systems, though it is incremental as it focuses on a specific game class.
The paper tackles the problem of determining how many rounds are needed for players to reach an approximate equilibrium in two-player zero-sum matrix games, deriving instance-dependent bounds that depend on the game matrix and desired accuracy, and proving that these bounds are achievable with specific strategies.
We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum $n\times 2$ matrix games. That is, in a sequence of repeated game plays, how many rounds must the two players play before reaching an approximate equilibrium (e.g., Nash)? We derive instance-dependent bounds that define an ordering over game matrices that captures the intuition that the dynamics of some games converge faster than others. Specifically, we consider a stochastic observation model such that when the two players choose actions $i$ and $j$, respectively, they both observe each other's played actions and a stochastic observation $X_{ij}$ such that $\mathbb E[ X_{ij}] = A_{ij}$. To our knowledge, our work is the first case of instance-dependent lower bounds on the number of rounds the players must play before reaching an approximate equilibrium in the sense that the number of rounds depends on the specific properties of the game matrix $A$ as well as the desired accuracy. We also prove a converse statement: there exist player strategies that achieve this lower bound.