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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 222 (14 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
Equilibria and Efficiency Loss in Games on Networks
"... Abstract—Social networks are the substrate upon which we make and evaluate many of our daily decisions: our costs and benefits depend on whether—or how many of, or which of—our friends are willing to go to that restaurant, choose that cellular provider, already own that gaming platform. Much of the ..."
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Cited by 3 (0 self)
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Abstract—Social networks are the substrate upon which we make and evaluate many of our daily decisions: our costs and benefits depend on whether—or how many of, or which of—our friends are willing to go to that restaurant, choose that cellular provider, already own that gaming platform. Much of the research on the “diffusion of innovation,” for example, takes a gametheoretic perspective on strategic decisions made by people embedded in a social context. Indeed, multiplayer games played on social networks, where the network’s nodes correspond to the game’s players, have proven to be fruitful models of many natural scenarios involving strategic interaction. In this paper, we embark on a mathematical and general exploration of the relationship between 2person strategic interactions (a “base game”) and a “networked ” version
Equilibria of Graphical Games with Symmetries
, 2007
"... We study graphical games where the payoff function of each player satisfies one of four types of symmetries in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NPhard in graphical games with each of the four types of symmetry. Using a characteriza ..."
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Cited by 3 (1 self)
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We study graphical games where the payoff function of each player satisfies one of four types of symmetries in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NPhard in graphical games with each of the four types of symmetry. Using a characterization of games with pure equilibria in terms of even cycles in the neighborhood graph, as well as a connection to a generalized satisfiability problem, we identify tractable subclasses of the games satisfying the most restrictive type of symmetry. In the process, we characterize a satisfiability problem that remains NPhard in the presence of a matching, a result that may be of independent interest. Finally, games with symmetries of two of the four types are shown to possess a symmetric mixed equilibrium which can be computed in polynomial time. We have thus identified a class of games where the pure equilibrium problem is computationally harder than the mixed equilibrium problem, unless P=NP.
Connectivity and Equilibrium in Random Games
, 2007
"... We study how the structure of the interaction graph affects the Nash equilibria of the resulting game. In particular, for a fixed interaction graph, we are interested if there exist Nash equilibria which arise when random utility tables are assigned to the players. We provide conditions for the stru ..."
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We study how the structure of the interaction graph affects the Nash equilibria of the resulting game. In particular, for a fixed interaction graph, we are interested if there exist Nash equilibria which arise when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementary conditions which make the existence of equilibria highly unlikely. Our results have immediate implications for many deterministic graphs and generalize known results for games on the complete graph. In particular, our results imply that the probability that bounded degree graphs have Nash equilibria is exponentially small in the size of the graph and yield a simple algorithm that finds small nonexistence certificates for a large family of graphs. In order to obtained a refined characterization of the degree of connectivity associated with the existence of equilibria, we study the model in the random graph setting. In particular, we look at the case where the interaction graph is drawn from the ErdősRényi, G(n, p), where each edge is present independently with probability p. For this model we establish a double phase transition for the existence of pure Nash equilibria as a function of the average degree pn consistent with the nonmonotone behavior of the model. We show that when the average degree satisfies np> (2+Ω(1))log n, the number of pure Nash equilibria follows a Poisson distribution with parameter 1. When 1/n << np < (0.5 −Ω(1))logn pure Nash equilibria fail to exist with high probability. Finally, when np << 1/n a pure Nash equilibrium exists with high probability. 1