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 recently-established equivalence between polynomialtime solvability of normal-form games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPAD-complete class of Brouwer functions. 1

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 NP-hard 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 NP-hard 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.

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 game-theoretic 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 2-person strategic interactions (a “base game”) and a “networked ” version

c ○ Xin Jiang, 2011itriou and Roughgarden described a polynomial-time algorithm (”Ellipsoid Against Hope”) for computing sample correlated equilibria of compactlyrepresented games. Recently, Stein, Parrilo and Ozdaglar showed that this algorithm can fail to find an exact correlated equilibrium. We present a variant of the Ellipsoid Against Hope algorithm that guarantees the polynomialtime identification of exact correlated equilibrium. Efficient computation of optimal correlated equilibria. We show that the polynomialtime solvability of what we call the deviation-adjusted social welfare problem is a sufficient condition for the tractability of the optimal correlated equilibrium problem. iii Preface Certain chapters of this thesis are based on publications (or submissions to publications) by my collaborators and me (under the name Albert Xin Jiang). Per requirement of UBC Faculty of Graduate Studies, I describe here the relative contributions of all collaborators. Chapter 3 is based on the article Action-Graph Games by Albert Xin Jiang, Kevin Leyton-Brown and Navin Bhat, published in Games and Economic Behavior,

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