Potential games, path independence and Poisson’s binomial distribution

Original Article
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Abstract

This paper provides a simple characterization of potential games in terms of path independence. Using this characterization we propose an algorithm to determine if a finite game is potential or not. We define the storage requirement for our algorithm and provide its upper bound. The number of equations required in this algorithm is lower or equal to the number obtained in the algorithms proposed in the recent literature. We also show that for games with same numbers of players and strategy profiles, the number of equations for our algorithm is maximum when all players have the same number of strategies. The key technique of this paper is to identify an associated Poisson’s binomial distribution with any finite game. This distribution enables us to derive explicit forms of the number of equations, storage requirement and related aspects.

Keywords

Potential games Zero strategy Path independence Poisson’s binomial distribution Storage requirement 

Notes

Acknowledgements

I express my sincere gratitude to the editor and two anonymous reviewers for their helpful comments and suggestions. I am also grateful to the seminar participants of the 22nd International Conference on Game Theory at Stony Brook for their comments on an earlier version of the paper. Research support by the Faculty of Arts, Ryerson University is gratefully acknowledged.

Supplementary material

186_2018_631_MOESM1_ESM.docx (626 kb)
Supplementary material 1 (docx 626 KB)

References

  1. Anderson SP, Goeree JP, Holt CA (2001) Minimum-effort coordination games: stochastic potential and logit equilibrium. Games Econ Behav 34:177–199MathSciNetCrossRefMATHGoogle Scholar
  2. Bramoullé Y (2007) Anti-coordination and social interactions. Games Econ Behav 58:30–49MathSciNetCrossRefMATHGoogle Scholar
  3. Bramoullé Y, Kranton R, D’Amours M (2014) Strategic interaction and networks. Am Econ Rev 104:898–930CrossRefGoogle Scholar
  4. Cheng D, Liu T, Zhang K, Qi H (2016) On decomposed subspaces of finite games. IEEE Trans Autom Control 61:3651–3656MathSciNetCrossRefMATHGoogle Scholar
  5. Chien S, Sinclair A (2011) Convergence to approximate Nash equilibria in congestion games. Games Econ Behav 71:315–327MathSciNetCrossRefMATHGoogle Scholar
  6. Darroch JN (1964) On the distribution of the number of successes in independent trials. Ann Math Stat 35:1317–1321MathSciNetCrossRefMATHGoogle Scholar
  7. Ellingsæter B, Skjegstad M, Maseng T (2012) A potential game for power and frequency allocation in large-scale wireless networks, pp 1–10 arXiv preprint arXiv:1212.0724
  8. Hino Y (2011) An improved algorithm for detecting potential games. Int J Game Theory 40:199–205MathSciNetCrossRefMATHGoogle Scholar
  9. Hoeffding W (1956) On the distribution of the number of successes in independent trials. Ann Math Stat 27:713–721MathSciNetCrossRefMATHGoogle Scholar
  10. Marden JR, Arslan G, Shamma JS (2009) Cooperative control and potential games. IEEE Trans Syst Man Cybern Part B 39:1393–1407CrossRefGoogle Scholar
  11. Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14:124–143MathSciNetCrossRefMATHGoogle Scholar
  12. Neel JO, Reed JH, Gilles RP (2004) Convergence of cognitive radio networks. In: Proceedings of IEEE wireless communications network conference (WCNC), Atlanta, GA, Mar. 2004, pp 2250–2255Google Scholar
  13. Pitman J (1997) Probabilistic bounds on the coefficients of polynomials with only real zeros. J Comb Theory Ser A 77:279–303MathSciNetCrossRefMATHGoogle Scholar
  14. Roughgarden T, Tardos E (2002) How bad is selfish routing? J ACM 49:236–259MathSciNetCrossRefMATHGoogle Scholar
  15. Sandholm WH (2010) Decompositions and potentials for normal form games. Games Econ Behav 70:446–456MathSciNetCrossRefMATHGoogle Scholar
  16. Todd MJ (2016) Computation, multiplicity, and comparative statics of Cournot equilibria in integers. Math Oper Res 41:1125–1134MathSciNetCrossRefMATHGoogle Scholar
  17. Wang YH (1993) On the number of successes in independent trials. Stat Sin 3:295–312MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ryerson UniversityTorontoCanada

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