Abstract
We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between \(2(n-1)/(k-1) - 1\) and \(2(n-1)/(k-1)\). The latter upper bound is tight for \(k=n\) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases.
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Notes
Recall that in a pseudoforest each connected component has at most one cycle.
The k-price of anarchy is also commonly known as the k-strong price of anarchy.
Recall that a feedback edge set is a set of edges whose removal makes the graph acyclic.
In the case of division by zero, we define the outcome as \(\infty \).
Observe that we do not enforce that the endpoints of the removed edge \(\{i_j, i_{j+1}\}\) obtain different colors in the optimal solution. In fact, subsequently it will become clear that we do not have to do so.
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Acknowledgements
The fact that for the case of a ring the coordination game has the c-FIP was first observed by Dariusz Leniowski. We thank José Correa for allowing us to use his lower bound in Theorem 9. It improves on our original one by a factor of 2. We thank the anonymous reviewers for their valuable comments. The first author is also a Visiting Professor at the University of Warsaw. He was partially supported by the NCN Grant No. 2014/13/B/ST6/01807.
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An extended abstract of this paper appeared as Apt et al. (2014). Part of this research has been carried out while the second author was a post-doctoral researcher at Sapienza University of Rome, Italy.
Appendix: c-FIP and generalized ordinal c-potentials
Appendix: c-FIP and generalized ordinal c-potentials
Theorem 14
A finite game has the c-FIP iff a generalized ordinal c-potential for it exists.
Proof
\((\Rightarrow )\) We use here the argument given in the proof of Milchtaich (1996) of the fact that every finite game that has the FIP (finite improvement property) has a generalized ordinal potential.
Consider a branching tree of which the root has all joint strategies as successors, of which the non-root elements are joint strategies, and of which the branches are the c-improvement paths. Because the game is finite, this tree is finitely branching.
König’s Lemma of König (1927) states that any finitely branching tree is either finite or it has an infinite path. So by the assumption, the considered tree is finite. Hence the number of c-improvement paths is finite. Given a joint strategy s, define P(s) to be the number of prefixes of the c-improvement paths that terminate in s. Then P is a generalized ordinal c-potential, where we use the strict linear ordering on the natural numbers.
\((\Leftarrow )\) Immediate, as already noted in Holzman and Law-Yone (1997). \(\square \)
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Apt, K.R., de Keijzer, B., Rahn, M. et al. Coordination games on graphs. Int J Game Theory 46, 851–877 (2017). https://doi.org/10.1007/s00182-016-0560-8
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DOI: https://doi.org/10.1007/s00182-016-0560-8