Query Complexity of Approximate Nash Equilibria

We study the query complexity of approximate notions of Nash equilibrium in games with a large number of players $n$. Our main result states that for $n$-player binary-action games and for constant $\varepsilon$, the query complexity of an $\varepsilon$-well-supported Nash equilibrium is exponential in $n$. One of the consequences of this result is an exponential lower bound on the rate of convergence of adaptive dynamics to approxiamte Nash equilibrium

Query Complexity of Approximate Equilibria in Anonymous Games

We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is \emph{query complexity}, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by \cite{bfh09, dp14}. Our main result is a new randomized query-efficient algorithm that finds a $O(n^{-1/4})$-approximate Nash equilibrium querying $\tilde{O}(n^{3/2})$ payoffs and runs in time $\tilde{O}(n^{3/2})$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that $\Omega(n \log{n})$ payoffs must be queried in order to find any $\epsilon$-well-supported Nash equilibrium, even by randomized algorithms

Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω(k2), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O(k.log k/ϵ2) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O(k.log k/ϵ4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O(k.log k/ϵ4) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria

Distributed Methods for Computing Approximate Equilibria

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing \emph{approximate well-supported Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring $O(n \log n)$ payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing \emph{approximate Nash equilibria}: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium

Lower bounds for the query complexity of equilibria in Lipschitz games

Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player's payoff function is λ-Lipschitz with respect to the actions of the other players. They showed that such games admit ϵ-approximate pure Nash equilibria for certain settings of ϵ and λ. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding ϵ-approximate pure Nash equilibria of n-player, binary-action, λ-Lipschitz games that is exponential in nλ/ϵ. In addition, we introduce “Multi-Lipschitz games,” a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the average of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding ϵ-approximate Nash equilibria of n-player, m-action, λ-Lipschitz games for strong values of ϵ, motivating the consideration of explicitly randomized algorithms in the above results

The complexity of solution concepts in Lipschitz games

Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player’s payoff function is λ-Lipschitz with respect to the actions of the other players. They showed via the probabilistic method that n-player Lipschitz games with m strategies per player have pure -approximate Nash equilibria, for ≥ λ√8n log(2mn). They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop an efficient reduction from more general games to Lipschitz games. We use this reduction to study both the query and computational complexity of algorithms finding λ-approximate pure Nash equilibria of λ-Lipschitz games and related classes. We show a query lower bound exponential in nλ/ against randomized algorithms finding - approximatepure Nash equilibria of n-player, λ-Lipschitz games. We additionally present the first PPAD-completeness result for finding pure Nash equilibria in a class of finite, non-Bayesian games (we show this for λ-Lipschitz polymatrix games for suitable pairs of values and λ) in which both the proof of PPAD-hardness and the proof of containment in PPAD require novel approaches (in fact, our approach implies containment in PPAD for any class of Lipschitz games in which payoffs from mixed-strategy profiles can be deterministically computed), and present a definition of “randomized PPAD”. We define and subsequently analyze the class of “Multi-Lipschitz games”, a generalization of Lipschitz games involving player-specific Lipschitz parameters in which the value of interest appears to be the average of the individual Lipschitz parameters. We discuss a dichotomy of the deterministic query complexity of finding -approximate Nash equilibria of general games and, subsequently, a query lower bound for λ-Lipschitz games in which any non-trivial value of requires exponentially-many queries to achieve. We examine which parts of this extend to the concepts of approximate correlated and coarse correlated equilibria, and in the process generalize the edge-isoperimetric inequalities to generalizations of the hypercube. Finally, we improve the block update algorithm presented by Goldberg and Marmolejo to break the potential boundary of a 0.75-approximation factor, presenting a randomized algorithm achieving a 0.7368-approximate Nash equilibrium making polynomially-many profile queries of an n-player 1/n−1 -Lipschitz game with an unbounded number of actions

Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games

In this paper we present a generic reduction from the problem of finding an epsilon-well-supported Nash equilibrium (WSNE) to that of finding an Theta(epsilon)-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player. Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE. This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions. As an application we prove a 2^{Omega(n/log n)} lower bound on the randomized query complexity of finding an epsilon-ANE in binary-action n-player games, for some constant epsilon>0. This answers an open problem posed by Hart and Nisan and Babichenko, and is very close to the trivial upper bound of 2^n. Previously for WSNE, Babichenko showed a 2^{Omega(n)} lower bound on the randomized query complexity of finding an epsilon-WSNE for some constant epsilon>0. Our result follows directly from combining Babichenko\u27s result and our new reduction from WSNE to ANE

Query-Efficient Algorithms to Find the Unique Nash Equilibrium in a Two-Player Zero-Sum Matrix Game

We study the query complexity of identifying Nash equilibria in two-player zero-sum matrix games. Grigoriadis and Khachiyan (1995) showed that any deterministic algorithm needs to query $\Omega(n^2)$ entries in worst case from an $n\times n$ input matrix in order to compute an $\varepsilon$-approximate Nash equilibrium, where $\varepsilon<\frac{1}{2}$. Moreover, they designed a randomized algorithm that queries $\mathcal O(\frac{n\log n}{\varepsilon^2})$ entries from the input matrix in expectation and returns an $\varepsilon$-approximate Nash equilibrium when the entries of the matrix are bounded between $-1$ and $1$. However, these two results do not completely characterize the query complexity of finding an exact Nash equilibrium in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding an exact Nash equilibrium for two-player zero-sum matrix games that have a unique Nash equilibrium $(x_\star,y_\star)$. We first show that any randomized algorithm needs to query $\Omega(nk)$ entries of the input matrix $A\in\mathbb{R}^{n\times n}$ in expectation in order to find the unique Nash equilibrium where $k=|\text{supp}(x_\star)|$. We complement this lower bound by presenting a simple randomized algorithm that, with probability $1-\delta$, returns the unique Nash equilibrium by querying at most $\mathcal O(nk^4\cdot \text{polylog}(\frac{n}{\delta}))$ entries of the input matrix $A\in\mathbb{R}^{n\times n}$. In the special case when the unique Nash Equilibrium is a pure-strategy Nash equilibrium (PSNE), we design a simple deterministic algorithm that finds the PSNE by querying at most $\mathcal O(n)$ entries of the input matrix.Comment: 17 page