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Register a new userConstrained Pure Nash Equilibria in Polymatrix Games
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Added by
Dominik Wojtczak
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consistent with some pure Nash equilibrium or, alternatively, with all pure Nash equilibria. We identify the most natural tractable cases and show NP or coNP-completness of these problems already for unweighted DAGs.
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We consider polymatrix coordination games with individual preferences where every player corresponds to a node in a graph who plays with each neighbor a separate bimatrix game with non-negative symmetric payoffs. In this paper, we study $alpha$-approximate $k$-equilibria of these games, i.e., outcomes where no group of at most $k$ players can deviate such that each member increases his payoff by at least a factor $alpha$. We prove that for $alpha ge 2$ these games have the finite coalitional improvement property (and thus $alpha$-approximate $k$-equilibria exist), while for $alpha < 2$ this property does not hold. Further, we derive an almost tight bound of $2alpha(n-1)/(k-1)$ on the price of anarchy, where $n$ is the number of players; in particular, it scales from unbounded for pure Nash equilibria ($k = 1)$ to $2alpha$ for strong equilibria ($k = n$). We also settle the complexity of several problems related to the verification and existence of these equilibria. Finally, we investigate natural means to reduce the inefficiency of Nash equilibria. Most promisingly, we show that by fixing the strategies of $k$ players the price of anarchy can be reduced to $n/k$ (and this bound is tight).
We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)
Graphical games are a useful framework for modeling the interactions of (selfish) agents who are connected via an underlying topology and whose behaviors influence each other. They have wide applications ranging from computer science to economics and biology. Yet, even though a players payoff only depends on the actions of their direct neighbors in graphical games, computing the Nash equilibria and making statements about the convergence time of natural local dynamics in particular can be highly challenging. In this work, we present a novel approach for classifying complexity of Nash equilibria in graphical games by establishing a connection to local graph algorithms, a subfield of distributed computing. In particular, we make the observation that the equilibria of graphical games are equivalent to locally verifiable labelings (LVL) in graphs; vertex labelings which are verifiable with a constant-round local algorithm. This connection allows us to derive novel lower bounds on the convergence time to equilibrium of best-response dynamics in graphical games. Since we establish that distributed convergence can sometimes be provably slow, we also introduce and give bounds on an intuitive notion of time-constrained inefficiency of best responses. We exemplify how our results can be used in the implementation of mechanisms that ensure convergence of best responses to a Nash equilibrium. Our results thus also give insight into the convergence of strategy-proof algorithms for graphical games, which is still not well understood.
Extensive study on the complexity of computing Nash Equilibrium has resulted in the definition of the complexity class PPAD by Papadimitriou cite{Papa2}, Subsequently shown to be PPAD-complete, first by Daskalakis, Goldberg, and Papadimitriou cite{Papa} for $3$ or more and even for the bimatrix case by Chen and Deng cite{Chen}. On the other hand, it is well known that Nash equilibria of games with smooth payoff functions are generally Pareto-inefficient cite{Dubey} In the spirit of Von Neumanns Minimax Theorem and its polynomial-time solvability via Linear Programming, Kalantari cite{Kalantari} has described a multilinear minimax relaxation (MMR) that provides an approximation to a convex combination of expected payoffs in any Nash Equilibrium via LP. In this paper, we study this relaxation for the bimatrix game, solving its corresponding LP formulation and comparing its solution to the solution computed by the Lemke-Howson algorithm. We also give a game theoretic interpretation of MMR for the bimatrix game involving a meta-player. Our relaxation has the following theoretical advantages: (1) It can be computed in polynomial time; (2) For at least one player, the computed MMR payoff is at least as good any Nash Equilibrium payoff; (3) There exists a convex scaling of the payoff matrices giving equal payoffs. Such a solution is a satisfactory compromise. Computationally, we have compared our approach with the state-of-the-art implementation of the Lemke-Howson algorithm cite{Lemke}. We have observed the following advantages: (i) MMR outperformed Lemke-Howson in time complexity; (ii) In about $80%$ of the cases the MMR payoffs for both players are better than any Nash Equilibria; (iii) in the remaining $20%$, while one players payoff is better than any Nash Equilibrium payoff, the other players payoff is only within a relative error of $17%$.
This paper shows the existence of $mathcal{O}(frac{1}{n^gamma})$-Nash equilibria in $n$-player noncooperative aggregative games where the players cost functions depend only on their own action and the average of all the players actions, and is lower semicontinuous in the former while $gamma$-H{o}lder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes congestion games with $gamma$ being 1, a proximal best-reply algorithm is used to construct an $mathcal{O}(frac{1}{n})$-Nash equilibria with at most $mathcal{O}(n^3)$ iterations. These results are applied in a numerical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when $n$ tends to infinity.
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