Anti-coordination Games and Stable Graph Colorings

Abstract

Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We characterize the cases when such colorings exist and when the decision problem is NP-hard. These correspond to finding pure strategy equilibria in the anti-coordination games, whose price of anarchy we also analyze. We further consider the directed case, a generalization that captures both coordination and anti-coordination. We prove the decision problem for non-strict equilibria in directed graphs is NP-hard. Our notions also have multiple connections to other combinatorial questions, and our results resolve some open problems in these areas, most notably the complexity of the strictly unfriendly partition problem.

Coalition Resilient Outcomes in Max k-Cut Games

“…We conjecture that it is indeed always an SE, even if this seems to be challenging to prove in general. A natural approach could be that of using the size of the cut as a strong potential function, that is Φ S (σ) = S(σ), as it has already been done for proving that max k-cut games admit a Nash equilibrium [15] [18]. However, it can be argued that this approach cannot work in general, since a profitable coalition deviation could sometimes result in a cutvalue decrease.…”

“…Further related work. The max k-cut game has been first investigated in [15,18], where the authors show that, when the graph is unweighted and undirected, it is possible to compute a Nash Equilibrium in polynomial time by exploiting the potential function method. When the graph is weighted undirected, even if the potential function ensures the existence of NE the problem of computing an equilibrium is PLS-complete even for k = 2 [22].…”

“…When the graph is directed, the max k-cut game in general does not admit a potential function. Indeed, in this case, even the problem of understanding whether they admit a Nash equilibrium is NP-complete for any fixed k ≥ 2 [18]. In [7], the authors present a randomized polynomial time algorithm that computes a constant approximate Nash equilibrium for a large class of directed unweighted graphs.…”

Computing and Optimizing Over All Fixed-Points of Discrete Systems on Large Networks

“…Special cases of finding all fixed points of system (1) such as deciding on the existence of a Nash equilibrium for certain systems [5] and calculating the minimum-energy state of ferromagnetic spin models [4] are proven to be NP-hard. This generally means that the time required by any existing algorithm to compute a solution is exponential in the size of the problem (e.g., number of nodes in the network).…”

Price of anarchy for graph coloring games with concave payoff

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We study the price of anarchy in a class of graph coloring games (a subclass of polymatrix common-payoff games). In those games, players are vertices of an undirected, simple graph, and the strategy space of each player is the set of colors from 1 to k. A tight bound on the price of anarchy of k k´1 is known (Hoefer 2007, Kun et al 2013, for the case that each player's payoff is the number of her neighbors with different color than herself. The study of more complex payoff functions was left as an open problem.We compute payoff for a player by determining the distance of her color to the color of each of her neighbors, applying a non-negative, real-valued, concave function f to each of those distances, and then summing up the resulting values.

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