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The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
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Cited by 324 (23 self)
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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
Graphical Models for Game Theory
, 2001
"... We introduce a compact graphtheoretic representation for multiparty game theory. Our main result is a provably correct and efficient algorithm for computing approximate Nash equilibria in onestage games represented by trees or sparse graphs. ..."
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Cited by 283 (22 self)
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We introduce a compact graphtheoretic representation for multiparty game theory. Our main result is a provably correct and efficient algorithm for computing approximate Nash equilibria in onestage games represented by trees or sparse graphs.
Computing the optimal strategy to commit to
 IN PROCEEDINGS OF THE 7TH ACM CONFERENCE ON ELECTRONIC COMMERCE (ACMEC
, 2006
"... In multiagent systems, strategic settings are often analyzed under the assumption that the players choose their strategies simultaneously. However, this model is not always realistic. In many settings, one player is able to commit to a strategy before the other player makes a decision. Such models a ..."
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Cited by 144 (22 self)
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In multiagent systems, strategic settings are often analyzed under the assumption that the players choose their strategies simultaneously. However, this model is not always realistic. In many settings, one player is able to commit to a strategy before the other player makes a decision. Such models are synonymously referred to as leadership, commitment, or Stackelberg models, and optimal play in such models is often significantly different from optimal play in the model where strategies are selected simultaneously. The recent surge in interest in computing gametheoretic solutions has so far ignored leadership models (with the exception of the interest in mechanism design, where the designer is implicitly in a leadership position). In this paper, we study how to compute optimal strategies to commit to under both commitment to pure strategies and commitment to mixed strategies, in both normalform and Bayesian games. We give both positive results (efficient algorithms) and negative results (NPhardness results).
Complexity Results about Nash Equilibria
, 2002
"... Noncooperative game theory provides a normative framework for analyzing strategic interactions. ..."
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Cited by 136 (11 self)
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Noncooperative game theory provides a normative framework for analyzing strategic interactions.
Playing large games using simple strategies
 IN: PROC. OF THE 4TH ACM CONF. ON EL. COMMERCE (EC ’03). ASSOC. OF COMP. MACH
, 2003
"... We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be approximated by the payoffs to the players in some such logarithmic support Nash equilibrium. These ..."
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Cited by 123 (4 self)
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We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be approximated by the payoffs to the players in some such logarithmic support Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasipolynomial algorithm for computing an Nash equilibrium. To our knowledge this is the rst subexponential algorithm for finding an Nash equilibrium. Our results hold for any multipleplayer game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a xed number of players m, the payos to all players in any mtuple of mixed strategies can be approximated by the payos in some mtuple of constant support strategies. We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an equilibrium with small support. It also implies that if the payo matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.
The Structure and Complexity of Nash Equilibria for a Selfish Routing Game
, 2002
"... In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribu ..."
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Cited by 122 (28 self)
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In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
Efficient Computation of Equilibria for Extensive Twoperson Games
, 1996
"... The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, howev ..."
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Cited by 121 (8 self)
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The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, however, is often exponentially large in the size of the game tree. If the game has perfect recall, a linearsized strategic description is the sequence form. For the resulting small LCP, we show that an equilibrium is found efficiently by Lemke’s algorithm, a generalization of the Lemke–Howson method.
Fast Algorithms for Finding Randomized Strategies in Game Trees
, 1994
"... Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computation ..."
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Cited by 119 (11 self)
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Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of twoplayer games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.
Intrinsic Robustness of the Price of Anarchy
"... The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium ..."
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Cited by 100 (12 self)
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The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regretminimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regretminimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and BayesianNash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomiallength bestresponse sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worstcase upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with nonpolynomial cost functions, and the first
AWESOME: A general multiagent learning algorithm that converges in selfplay and learns a best response against stationary opponents
, 2003
"... A satisfactory multiagent learning algorithm should, at a minimum, learn to play optimally against stationary opponents and converge to a Nash equilibrium in selfplay. The algorithm that has come closest, WoLFIGA, has been proven to have these two properties in 2player 2action repeated games— as ..."
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Cited by 98 (5 self)
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A satisfactory multiagent learning algorithm should, at a minimum, learn to play optimally against stationary opponents and converge to a Nash equilibrium in selfplay. The algorithm that has come closest, WoLFIGA, has been proven to have these two properties in 2player 2action repeated games— assuming that the opponent’s (mixed) strategy is observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have these two properties in all repeated (finite) games. It requires only that the other players ’ actual actions (not their strategies) can be observed at each step. It also learns to play optimally against opponents that eventually become stationary. The basic idea behind AWESOME (Adapt When Everybody is Stationary, Otherwise Move to Equilibrium) is to try to adapt to the others’ strategies when they appear stationary, but otherwise to retreat to a precomputed equilibrium strategy. The techniques used to prove the properties of AWESOME are fundamentally different from those used for previous algorithms, and may help in analyzing other multiagent learning algorithms also.