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We study the query complexity of finding the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)in two-player zero-sum matrix games. Fearnley and Savani [18] showed that for any randomized algorithm, there exists ann×ninput matrix where it needs to queryΩ(n²) entries in expectation to compute asingleNash equilibrium. On the other hand, Bienstock et al. [5] showed that there is a special class of matrices for which one can queryO(n) entries and compute its set of all Nash equilibria. However, these results do not fully characterize the query complexity of finding the set of all Nash equilibria in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)in terms of the number of rowsnof the input matrix\(A \in \mathbb {R}^{n \times n} \), row support size\(k_1 := |\bigcup \limits _{x \in \mathcal {X}_\ast } \text{supp}(x)| \), and column support size\(k_2 := |\bigcup \limits _{y \in \mathcal {Y}_\ast } \text{supp}(y)| \). We design a simple yet non-trivial randomized algorithm that returns the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \)by querying at mostO(nk⁵· polylog(n)) entries of the input matrix\(A \in \mathbb {R}^{n \times n} \)in expectation, wherek≔ max{k₁,k₂}. This upper bound is tight up to a factor of poly(k), as we show that for any randomized algorithm, there exists ann×ninput matrix with min {k₁,k₂} = 1, for which it needs to queryΩ(nk) entries in expectation in order to find the set of all Nash equilibria\(\mathcal {X}_\ast \times \mathcal {Y}_\ast \). 

We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum n × 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 Xij such that E [Xij ] = Aij . 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.