Abstract
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 \(2\alpha (n-1)/(k-1)\) 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 \(2\alpha \) 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).
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Notes
- 1.
In [9] the bimatrix games on the edges may have negative payoffs and this is exploited in the PLS-completeness proof. However, we can accommodate this in our model by adding a sufficiently large constant to each payoff.
- 2.
A graph is a pseudoforest if each of its connected components has at most one cycle.
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Rahn, M., Schäfer, G. (2015). Efficient Equilibria in Polymatrix Coordination Games. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_44
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