International Journal of Game Theory

, Volume 46, Issue 3, pp 851–877 | Cite as

Coordination games on graphs

  • Krzysztof R. Apt
  • Bart de Keijzer
  • Mona Rahn
  • Guido Schäfer
  • Sunil Simon
Original Paper
First Online:
Accepted:

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.

Keywords

Coordination games Graphs Nash equilibria Strong equilibria Price of anarchy Price of stability 
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Notes

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.

References

  1. Andelman N, Feldman M, Mansour Y (2009) Strong price of anarchy. Games Econ Behav 65(2):289–317CrossRefGoogle Scholar
  2. Apt KR, Markakis E (2011) Diffusion in social networks with competing products. Proceedings of the 4th International Symposium on Algorithmic Game Theory (SAGT). Lecture Notes in Computer Science, vol 6982. Springer, New York, pp 212–223Google Scholar
  3. Apt KR, Rahn M, Schäfer G, Simon S (2014) Coordination games on graphs (extended abstract). Proceedings of the 10th Conference on Web and Internet Economics (WINE). Lecture Notes in Computer Science, vol 8877. Springer, New York, pp 441–446Google Scholar
  4. Apt KR, Simon S (2013) Social network games with obligatory product selection. In: Proceedings 8th International Symposium on Games, Automata, Logics and Formal Verification (GandALF), vol 119. Electronic Proceedings in Theoretical Computer Science, pp 180–193Google Scholar
  5. Apt KR, Simon S, Wojtczak D (2015) Coordination games on directed graphs. In: Proc. of the 12th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2015)Google Scholar
  6. Aumann RJ (1959) Acceptable points in general cooperative n-person games. In: Luce RD, Tucker AW (eds) Contribution to the theory of game IV, Annals of Mathematical Study, vol 40. University Press, pp 287–324Google Scholar
  7. Aziz H, Brandt F (2012) Existence of stability in hedonic coalition formation games. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp 763–770Google Scholar
  8. Aziz H, Brandt F, Seedig HG (2010) Optimal partitions in additively separable hedonic games. In: Proceedings of the 3rd International Workshop on Computational Social Choice (COMSOC), pp 271–282Google Scholar
  9. Aziz H, Brandt F, Seedig HG (2011) Stable partitions in additively separable hedonic games. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp 183–190Google Scholar
  10. Banerjee Konishi S (2001) Core in a simple coalition formation game. Soc Choice Welf 18:135–153CrossRefGoogle Scholar
  11. Bilò V, Fanelli A, Flammini M, Moscardelli L (2011) Graphical congestion games. Algorithmica 61(2):274–297CrossRefGoogle Scholar
  12. Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38(2):201230CrossRefGoogle Scholar
  13. Cai Y, Daskalakis C (2011) On minmax theorems for multiplayer games. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pp 217–234Google Scholar
  14. Chatzigiannakis I, Koninis C, Panagopoulou PN, Spirakis PG (2010) Distributed game-theoretic vertex coloring. In: Proceedings of 14th International Conference on Principles of Distributed Systems (OPODIS), Lecture Notes in Computer Science, vol 6490. Springer, New YorkGoogle Scholar
  15. Escoffier B, Gourvès L, Monnot J (2012) Strategic coloring of a graph. Internet Math 8(4):424–455CrossRefGoogle Scholar
  16. Feldman M, Friedler O (2015) A unified framework for strong price of anarchy in clustering games. Automata, Languages, and Programming. Lecture Notes in Computer Science, vol 9135. Springer, New York, pp 601–613Google Scholar
  17. Feldman M, Lewin-Eytan L, Naor JS (2015) Hedonic clustering games. ACM Trans Parallel Comput 2(1):4:1–48Google Scholar
  18. Gairing M, Savani R (2010) Computing stable outcomes in hedonic games. In: Proceedings of the 3rd International Symposium on Algorithmic Game Theory (SAGT), pp 174–185Google Scholar
  19. Gourvès L, Monnot J (2009) On strong equilibria in the max cut game. In: Proc. 5th International Workshop on Internet and Network Economics, WINE, Lecture Notes in Computer Science, vol 5929. Springer, New York, pp 608–615Google Scholar
  20. Gourvès L, Monnot J (2010) The max k-cut game and its strong equilibria. Proceedings of the, 7th Annual Conference on the Theory and Applications of Models of Computation TAMC. Lecture Notes in Computer Science, vol 6108. Springer, New York, pp 234–246Google Scholar
  21. Harks T, Klimm M, Möhring R (2013) Strong equilibria in games with the lexicographical improvement property. Int J Game Theory 42(2):461–482CrossRefGoogle Scholar
  22. Hoefer M (2007) Cost sharing and clustering under distributed competition. Ph.D. Thesis, University of Konstanz. http://www.mpiinf.mpg.de/~mhoefer/05-07/diss.pdf
  23. Holzman R, Law-Yone N (1997) Strong equilibrium in congestion games. Games Econ Behav 21(1–2):85–101CrossRefGoogle Scholar
  24. Howson J (1972) Equilibria of polymatrix games. Manag Sci 18(5):312–318CrossRefGoogle Scholar
  25. Jackson M, Zenou Y (2012) Games on networks. Centre for Economic Policy Research Discussion Paper No. 9127:86Google Scholar
  26. Janovskaya E (1968) Equilibrium points in polymatrix games. Litovskii Mat Sbornik 8:381–384Google Scholar
  27. König D (1927) Über eine Schlußweise aus dem Endlichen ins Unendliche. Acta Litt Ac Sci 3:121–130Google Scholar
  28. Konishi H, Le Breton M, Weber S (1997) Pure strategy Nash equilibrium in a group formation game with positive externalities. Games Econ Behav 21:161–182CrossRefGoogle Scholar
  29. Milchtaich I (1996) Congestion games with player-specific payoff functions. Games Econ Behav 13:111–124CrossRefGoogle Scholar
  30. Panagopoulou PN, Spirakis PG (2008) A game theoretic approach for efficient graph coloring. Proceedings of the 19th International Symposium on Algorithms and Computation, (ISAAC). Lecture Notes in Computer Science, vol 5369. Springer, New York, pp 183–195Google Scholar
  31. Rahn M, Schäfer G (2015) Efficient equilibria in polymatrix coordination games. In: Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS), Lecture Notes in Computer Science, vol 9235, pp 529–541Google Scholar
  32. Rosenthal RW (1973) A class of games possessing pure-strategy Nash equilibria. Int J Game Theory 2(1):65–67CrossRefGoogle Scholar
  33. Rozenfeld O, Tennenholtz M (2006) Strong and correlated strong equilibria in monotone congestion games. Proceedings of the 2nd International Workshop on Internet and Network Economics (WINE). Lecture Notes in Computer Science, vol 4286. Springer, New York, pp 74–86Google Scholar
  34. Simon S, Apt KR (2015) Social network games. J Logic Comput 25(1):207–242CrossRefGoogle Scholar
  35. Simon S, Wojtczak D (2016) Efficient local search in coordination games on graphs. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI). AAAI Press, USA, pp 482–488Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Krzysztof R. Apt
    • 1
  • Bart de Keijzer
    • 1
  • Mona Rahn
    • 1
  • Guido Schäfer
    • 1
    • 2
  • Sunil Simon
    • 3
  1. 1.Centrum Wiskunde & Informatica (CWI), Networks and Optimization GroupAmsterdamThe Netherlands
  2. 2.Department of Econometrics and Operations ResearchVrije Universiteit AmsterdamAmsterdamThe Netherlands
  3. 3.Department of CSEIIT KanpurKanpurIndia

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