Suppose that an @math -simplex is partitioned into @math convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance @math from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant @math uses @math queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant @math uses @math queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.
The Nash equilibrium (NE) is a fundamental concept in game theory @cite_19 . They are guaranteed to exist in finite games, yet computational challenges in finding one abound, most notably, the PPAD-completeness of computing an exact equilibrium even for two-player normal form games @cite_12 @cite_24 . For this reason, query complexity has been extensively used as a tool to differentiate hardness of equilibrium concepts in games. For payoff queries, some notable examples include: exponential lower bounds for randomised computation of exact Nash equilibria and exact correlated equilibria via communication complexity lower bounds in multiplayer games @cite_14 @cite_3 ; exponential lower bounds for randomised computation of approximate well-supported equilibria and general approximate equilibria for a small enough approximation factor in multiplayer games @cite_8 ; upper and lower bounds for equilibrium computation in bimatrix games, congestion games @cite_5 and anonymous games @cite_6 ; upper and lower bounds for randomised algorithms computing approximate correlated equilibria @cite_10 . have also proved lower bounds in communication complexity for computing @math -WSNE for small enough @math in both bimatrix and multiplayer games @cite_21 .
@cite_14 @cite_8 @cite_21 @cite_3 @cite_6 @cite_24 @cite_19 @cite_5 @cite_10 @cite_12
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We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players. 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 e, the query complexity of an e-well-supported Nash equilibrium is exponential in n. As a consequence of this result, we get an exponential lower bound on the rate of convergence of adaptive dynamics to approximate Nash equilibria. For a constant ϵ, we prove a (N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N x N games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1-ϵ)-fraction of the players are ϵ-best replying. Abstract We study the computation of Nash equilibria of anonymous games, via algorithms that use adaptive queries to a game's payoff function. We show that exact equilibria cannot be found via query-efficient algorithms, and exhibit a two-strategy, 3-player anonymous game whose exact equilibria require irrational numbers. We obtain positive results for known sub-classes of anonymous games. Our main result is a new randomized query-efficient algorithm for approximate equilibria of two-strategy anonymous games that improves on the running time of previous algorithms. It is the first to obtain an inverse polynomial approximation in poly-time, and yields an efficient polynomial-time approximation scheme. A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic results for exact equilibria of bimatrix and graphical games. We then study the query complexity of approximate equilibria in bimatrix games. Finally, we study the query complexity of exact equilibria in symmetric network congestion games. For directed acyclic networks, we can learn the cost functions (and hence compute an equilibrium) while querying just a small fraction of pure-strategy profiles. For the special case of parallel links, we have the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values. We analyze the number of payoff queries needed to compute approximate equilibria of multi-player games. We find that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and we develop new techniques for upper- and lower bounding the query complexity. For binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated equilibrium (a restriction where a player’s behavior must always be approximately optimal, in the worst case over draws from the distribution) we show a linear lower bound, thus separating the query complexity of well supported approximate correlated equilibrium from the standard notion of approximate correlated equilibrium. Finally, we give a query-efficient reduction from the problem of computing an approximate well-supported Nash equilibrium to the problem of verifying a well supported Nash equilibrium, where the additional query overhead is proportional to the description length of the game. This gives a polynomial-query algorithm for computing well supported approximate Nash equilibria (and hence correlated equilibria) in concisely represented games. We identify a class of games (which includes congestion games) in which the reduction can be made not only query efficient, but also computationally efficient. In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof.