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Pure Nash Equilibria in Graphical Games and Treewidth

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Abstract

We treat PNE-GG, the problem of deciding the existence of a Pure Nash Equilibrium in a graphical game, and the role of treewidth in this problem. PNE-GG is known to be \(NP\)-complete in general, but polynomially solvable for graphical games of bounded treewidth. We prove that PNE-GG is \(W[1]\)-Hard when parameterized by treewidth. On the other hand, we give a dynamic programming approach that solves the problem in \(O^*(\alpha ^w)\) time, where \(\alpha \) is the cardinality of the largest strategy set and \(w\) is the treewidth of the input graph (and \(O^*\) hides polynomial factors). This proves that PNE-GG is in \(FPT\) for the combined parameter \((\alpha ,w)\). Moreover, we prove that there is no algorithm that solves PNE-GG in \(O^*((\alpha -\epsilon )^w)\) time for any \(\epsilon > 0\), unless the Strong Exponential Time Hypothesis fails. Our lower bounds implicitly assume that \(\alpha \ge 3\); we show that for \(\alpha =2\) the problem can be solved in polynomial time. Finally, we discuss the implication for computing pure Nash equilibria in graphical games (PNE-GG) of \(O(\log n)\) treewidth, the existence of polynomial kernels for PNE-GG parameterized by treewidth, and the construction of a sample and maximum-payoff pure Nash equilibrium.

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Notes

  1. Here we give the payoffs as a utility function over the global configurations. Of course, the only relevant part of the global configuration is the one referring to \(\mathcal {N}(p)\) for a player \(p\in P\). The only reason we use \(St(P)\) as the domain of the function is to simplify the presentation. The same convention will be used in the proof of Theorem 4.

  2. Observe that in the case of a clique all the nodes of \(\mathcal {T}\) but one are introduce nodes. The treewidth of an \(n\)-clique is \(n-1\).

  3. Note that if the degree of the graph is bounded, then the description of the graphical game is polynomial in the number of players.

  4. The actual size of the matrices are \(|M_{w_i^k}| \le 3^3\alpha ^p\) and \(|M_{v_k}| \le \alpha ^3 3^{\alpha ^p}\).

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Antonis Thomas

  2. Department of Information and Computing Sciences, Utrecht University, 3584 CC, Utrecht, The Netherlands

    Jan van Leeuwen

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Thomas, A., van Leeuwen, J. Pure Nash Equilibria in Graphical Games and Treewidth. Algorithmica 71, 581–604 (2015). https://doi.org/10.1007/s00453-014-9923-3

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