Abstract

We establish the existence of a Nash equilibrium in a class of stochastic games representing electoral competitions between two candidates facing a continuum of voters. The underlying uncertainty in such games is due to the fact that candidates are uncertain about the fraction of the voters that will actually vote for them on election day. The payoff functions of the candidates are neither continuous nor concave. Therefore, we transform a game in this class into a new game with nicer geometric and continuity properties. We establish the existence of an equilibrium for the new game using a standard fixed point argument, and we finally show that an equilibrium of the new games is also an equilibrium of the original game. We further show the equilibria we found can be approximated statistically using non-parametric maximum likelihood estimators of the distribution of the random variable representing the uncertainty in the game.

Author information

Contact details are reproduced from the original publication and may be historical.

Adib Bagh

Depts. of Mathematics and Economics, University of Kentucky, Lexington, U.S.A.

adib.bagh@uky.edu

Keywords

  • Stochastic voting models
  • Nash Equilibrium
  • log-concave random variables
  • non-parametric maximum likelihood estimators

Mathematics Subject Classification

91A80, 90C15, 62F12

References

  1. A. Alesina: Credibility and policy convergence in a two-party system with rational voters, Amer. Econ. Review 78 (1988) 796–805.Google Scholar
  2. A. Bagh: Existence of equilibria in constrained discontinuous games, Int. J. Game Theory 45 (2016) 769–793.https://doi.org/10.1007/s00182-015-0480-zGoogle Scholar
  3. M. Bagnoli, T. Bergstorm: Log-concave probability and its applications, Econ. Theory 26 (2005) 445–469.https://doi.org/10.1007/s00199-004-0514-4Google Scholar
  4. R. Ball: Discontinuity and non-existence of equilibirum in the probabilistic spatial voting model, Social Choice Welfare 16 (1999) 533–555.https://doi.org/10.1007/s003550050159Google Scholar
  5. G. R. M. Borzadaran, H. A. M. Borzadaran: Log-concavity properties for some well-known distributions, Surveys Math. Appl. 6 (2011) 203–219.Google Scholar
  6. S. Boyd, L. Vandenberghe: Convex Optimization, Cambridge University Press, Cambridge (2009).Google Scholar
  7. B. C. Burden: Deterministic and probabilistic voting models, Amer. J. Polit. Science 41 (1997) 1150–1169.https://doi.org/10.2307/2960485Google Scholar
  8. R. L. Calvert: Robustness of the multidimensional voting model: Candidate motivation, uncertainty, and convergence, Amer. J. Polit. Science 3 (1985) 69–95.https://doi.org/10.2307/2111212Google Scholar
  9. P. J. Coughlin: Candidate uncertainty and electoral equilibria, in: Advances in the Spatial Theory of Voting, J. M. Enelow et al. (eds.), Cambridge University Press, Cambridge (1990) 145–166.https://doi.org/10.1017/CBO9780511896606.008Google Scholar
  10. M. X. Dong, R. J.-B. Wets: Estimating density functions: A constrained maximum likelihood approach, Nonparametric Statistics 12 (2000) 549–595.https://doi.org/10.1080/10485250008832822Google Scholar
  11. M. Droubvelis, A. Saporiti, N. J. Vriend: Political motivations and electoral competition: Equilibrium analysis and experimental evidence, Games Econ. Behavior 183 (2014) 86–115.https://doi.org/10.1016/j.geb.2013.10.004Google Scholar
  12. L. Dumbgen, K. Rufibach: Maximium likelihood estimation of a long concave density and its distribution function: Basic properties and uniform consistency, Bernoulli 15 (2009) 40–68.https://doi.org/10.3150/08-BEJ141Google Scholar
  13. J. Dupačová, R. J.-B. Wets: Asymptotic behavior of statistical estimators and of optimal solutions for stochastic optimization problems, Annals of Statistics 16 (1988) 1517–1549.https://doi.org/10.1214/aos/1176351052Google Scholar
  14. J. X. Eguia: Citizen candidates under uncertainty, Social Choice Welfare 29 (2007) 317–331.https://doi.org/10.1007/s00355-006-0207-yGoogle Scholar
  15. R. S. Erikson, D. W. Roomero: Candidate equilibrium and the behavioral model of the vote, Amer. J. Polit. Sci. Review 84 (1990) 1103–1126.https://doi.org/10.2307/1963255Google Scholar
  16. J. K. Goeree, J. Grosser: Welfare reducing polls, Econ. Theory 31 (2007) 51–68.https://doi.org/10.1007/s00199-006-0082-xGoogle Scholar
  17. G. Gomez, T. G. Hansford, G. A. Krause: The republicans should pray for rain: Weather turn out and voting in U.S. presidential elections, J. Politics 3 (2007) 649–663.https://doi.org/10.1111/j.1468-2508.2007.00565.xGoogle Scholar
  18. I. Hansson, C. Stuart: Voting competitions with interested politicians: Platforms do not converge to the prefrences of the median voter, Public Choice 44 (1984) 431–441.https://doi.org/10.1007/BF00119691Google Scholar
  19. A. King, R. J.-B. Wets: Epi-consistency of convex stochastic programs, Stochastics and Stochastics Reports 34 (1990) 83–92.https://doi.org/10.1080/17442509108833676Google Scholar
  20. M. Persson, A. Sundell, R. Ohrvall: Does election day weather affect voter turn out? Evidence for Swedish elections, Elections Studies 33 (2014) 335–342.https://doi.org/10.1016/j.electstud.2013.07.021Google Scholar
  21. P. Reny: On the existence of equilibrium in discontinuous games, Econometrica 167 (1999) 1029–1056.https://doi.org/10.1111/1468-0262.00069Google Scholar
  22. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (2009).Google Scholar
  23. J. E. Roemer: Political-economic equilibrium when parties represent constituents: The undimensional case, Social Choice Welfare 14 (1997) 479–502.https://doi.org/10.1007/s003550050082Google Scholar
  24. J. E. Roemer: Political Competition, Harvard University Press, Cambridge (200).Google Scholar
  25. J. O. Royset, R. J.-B. Wets: Fusion of hard and soft information in nonparametric density estimation, Eur. J. Oper. Research 247 (2015) 532–547.https://doi.org/10.1016/j.ejor.2015.06.034Google Scholar
  26. A. Saporiti: Existence and uniqueness of Nash equilibrium in electoral competition game: The hybrid case, J. Public Econ. Theory 10 (2008) 827–857.https://doi.org/10.1111/j.1467-9779.2008.00388.xGoogle Scholar
  27. B. W. Silverman: On the estimation of a probability density function by the maximum penalized likelihood method, Ann. Statistics 10 (1982) 795–810.https://doi.org/10.1214/aos/1176345872Google Scholar
  28. C. R. Taylor, H. Yildirim: Public information and electoral bias, mimeo (2009).Google Scholar
  29. G. Walther: Inference and modeling with log-concavity distributions, Statist. Science 24 (2009) 319–327.https://doi.org/10.1214/09-STS303Google Scholar
  30. R. J.-B. Wets: Stochastic programming, in: Handbook for Operations Research and Mangement Science, G. Nemhauser et al. (eds.), Elsevier, Amsterdam (1989) 573–629.https://doi.org/10.1016/S0927-0507(89)01009-1Google Scholar

Suggested citation

A. Bagh. “Existence and Statistical Estimation of Equilibria in Stochastic Electoral Competitions.” Journal of Convex Analysis 30 (2023), No. 2, 481–498. https://doi.org/10.68381/jca30028

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press.