Advertisement

Many-Candidate Nash Equilibria for Elections Involving Random Selection

  • Jeffrey S. RosenthalEmail author
Article
  • 15 Downloads

Abstract

We consider various voter game theory models which involve some form of random selection, including random tie-breaking, and single elimination, and runoff of the top two candidates. Under certain rules for resolving ties, we prove that with any number of candidates, each such model has a Nash equilibrium in which all candidates attempt to contest the election at the median policy. For models which do not permit ties, we prove that each such model has a Nash equilibrium in which the number of candidates contesting the election is essentially equal to the ratio of the positive payoff for winning divided by the negative of the payoff for losing. All of these model variations thus predict lots of candidates. This result contrasts with Duverger’s Law, which asserts that only two (major) candidates will contest the election at all, and which has been confirmed in some other voter game theory models. However, it is consistent with recent primary and leadership and runoff elections where the number of major candidates reached two figures. We close with a simulation study showing that, through repeated elections and averaging and tweaking, candidates’ actions will sometimes converge to their predicted equilibrium behaviour.

Keywords

Voter model Game theory Nash equilibrium Multiple candidates Runoff election Random selection 

Mathematics Subject Classification (2010)

Primary 91A06 Secondary 60E15, 91F10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

I thank Martin J. Osborne for introducing me to this topic and for many helpful discussions. This research was partially supported by NSERC of Canada.

References

  1. Andrieu C, Roberts GO (2009) The pseudo-marginal approach for efficient Monte Carlo computations. Ann Stat 37(2):697–725MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bouton L, Gratton G (2015) Majority runoff elections: strategic voting and Duverger’s hypothesis. Theor Econ 10:283–314MathSciNetCrossRefzbMATHGoogle Scholar
  3. de Vries J-P (2015) Duverger’s (f)law: counterproof to the Osborne Conjecture. MSc thesis. Available at: https://thesis.eur.nl/pub/17645/
  4. de Vries J-P, Kamphorst JJA, Osborne MJ, Rosenthal JS (2016) On a conjecture about the sequential positioning of political candidates. Work in progressGoogle Scholar
  5. Downs A (1957) An economic theory of democracy. Harper & Row, New YorkGoogle Scholar
  6. Duverger M (1951) Les Partis Politiques. Armand Colin, ParisGoogle Scholar
  7. Eaton BC, Lipsey RG (1975) The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Rev Econ Stud 42:27–49CrossRefzbMATHGoogle Scholar
  8. Hotelling H (1929) Stability in competition. Econ J 39:41–57CrossRefGoogle Scholar
  9. Hug S (1995) Third parties in equilibrium. Public Choice 82:159–180CrossRefGoogle Scholar
  10. Nash J (1951) Non-cooperative games. Ann Math 52(2):286–295MathSciNetCrossRefzbMATHGoogle Scholar
  11. Osborne MJ (1993) Candidate positioning and entry in a political competition. Games Econom Behav 5:133–151MathSciNetCrossRefzbMATHGoogle Scholar
  12. Osborne MJ (1996) A conjecture about the subgame perfect equilibria of a model of sequential location. Available at: https://www.economics.utoronto.ca/osborne/research/conjecture.html
  13. Osborne MJ (2003) An introduction to game theory. Oxford University Press, OxfordGoogle Scholar
  14. Riker W (1982) The two-party system and Duverger’s law: an essay on the history of political science. Am Polit Sci Rev 76(4):753–766CrossRefGoogle Scholar
  15. Rosenthal JS (2016a) Stochastic simulation of sequential game-theory voting models. Commun Stat Simul Comput, to appearGoogle Scholar
  16. Rosenthal JS (2016b) Nash equilibria for voter models with randomly perceived positions. Submitted for publicationGoogle Scholar
  17. Schlesinger JA, Schesinger MS (2006) Maurice Duverger and the study of political parties. Fr Polit 4:58–68CrossRefGoogle Scholar
  18. Shaked A (1982) Existence and computation of mixed strategy Nash equilibrium for 3-firms location problem. J Ind Econ 31:93–96CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada

Personalised recommendations