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Stochastic Contests with Linex Utility Functions

Year 2019, Volume: 7 Issue: 1, 113 - 128, 30.06.2019
https://doi.org/10.17093/alphanumeric.456860

Abstract

In this paper, we analyze the Nash equilibrium in a class of winner-takes-all stochastic contests among players with linear-exponential (linex) utility functions. In this contest, players are required to make upfront investments, which collectively determine their winning probabilities. We first show that a Nash equilibrium for such a contest exists and is unique, then set the equilibrium conditions, and study the properties of these conditions to gain insights into the structure of equilibrium. We show that the total equilibrium investment is bounded below and above, that the equilibrium has a cut characterization with respect to wealth, and that wealthier players invest more. The latter implies that richer is likely to get richer. For the special case with identical players, we show that an increase in the wealth or a decrease in the weight on the nonlinear component of the linex utility function results in an increase in the equilibrium investment.

References

  • Arrow, Kenneth J. Aspects of the theory of risk-bearing. Yrjo Jahnssonin Saatio, Helsinki, 1965.
  • Bell, David E. "One-switch utility functions and a measure of risk." Management Science 34 (1988): 1416-1424.
  • Canbolat, Pelin G., Boaz Golany, Inbal Mund, and Uriel G. Rothblum. "A stochastic competitive R&D race where ``winner takes all."" Operations Research 60 (2012): 700-715.
  • Canbolat, Pelin G. and Uriel G. Rothblum. "Constant risk aversion in stochastic contests with exponential completion times." Naval Research Logistics (2017).
  • Clark, Derek J. and Christian Riis. "Contest success functions: An extension." Economic Theory 11 (1998): 201-204.
  • Cornes, Richard and Roger Hartley. "Risk aversion, heterogeneity and contests." Public Choice 117 (2003): 1-25.
  • Cornes, Richard and Roger Hartley. "Risk aversion in symmetric and asymmetric contests." Economic Theory 51 (2012): 247-275.
  • Denuit, Michael M., Louis Eeckhoudt, and Harris Schlesinger. "When Ross meets Bell: The linex utility function." Journal of Mathematical Economics 49 (2013): 177-182.
  • Pratt, John W. "Risk aversion in the small and in the large." Econometrica 32 (1964): 122–136.
  • Ross, Stephen A. "Some stronger measures of risk aversion in the small and in the large with applications." Econometrica 49 (1981): 621-638.
  • Skaperdas, Stergios. "Contest success functions." Economic Theory 7 (1996): 283-290.
  • Treich, Nicolas. "Risk-aversion and prudence in rent-seeking games." Public Choice 145 (2010): 339-349.
  • Tullock, Gordon. Efficient rent seeking. In Buchanan, James M., Robert D. Tollison, and Gordon Tullock (Eds.), Toward a theory of the rent-seeking society 4, Texas A & M University Press, 1980.
  • Yamazaki, Takeshi. "The uniqueness of pure-strategy Nash equilibrium in rent-seeking games with risk-averse players." Public Choice 139 (2009): 335-342.

Doğrusal-Üstel (Linex) Yarar Fonksiyonları ile Rassal Yarışmalar

Year 2019, Volume: 7 Issue: 1, 113 - 128, 30.06.2019
https://doi.org/10.17093/alphanumeric.456860

Abstract

Bu çalışmada tercihleri doğrusal-üstel yarar fonksiyonları ile belirlenen oyuncular arasında gerçekleşen ve tek kazananın olduğu yarışmaları incelenmektedir. Modelde oyuncuların yarışmanın başında yatırımlarını belirledikleri ve bütün bu yatırımların beraberce her bir oyuncunun kazanma olasılığını belirlediği varsayılmaktadır. Öncelikle bu modelin tek bir Nash dengesinin olduğunu gösterilmekte, ardından denge koşullarını belirlenip bu koşulların özellikleri incelenerek dengenin yapısı hakkında çıkarımlarda bulunulmaktadır. Bu çıkarımlar arasında dengede toplam yatırımın alt ve üst sınırlarının bulunduğu, yatırımların oyuncuların başlangıçtaki zenginliklerine göre bir kesi şeklinde gösterilebileceği, zengin oyuncuların daha fazla yatırım yaptıkları, dolayısıyla zenginlerin yarışmanın sonunda büyük olasılıkla daha da zengin olacakları yer almaktadır. Bunlar dışında, oyuncuların eşit zenginliğe sahip olduğu özel durumlarda zenginlikte bir artışın veya yarar fonksiyonunun doğrusal olmayan kısmının ağırlığında bir azalmanın yatırımlarda artışa neden olduğu kanıtlanmaktadır.

References

  • Arrow, Kenneth J. Aspects of the theory of risk-bearing. Yrjo Jahnssonin Saatio, Helsinki, 1965.
  • Bell, David E. "One-switch utility functions and a measure of risk." Management Science 34 (1988): 1416-1424.
  • Canbolat, Pelin G., Boaz Golany, Inbal Mund, and Uriel G. Rothblum. "A stochastic competitive R&D race where ``winner takes all."" Operations Research 60 (2012): 700-715.
  • Canbolat, Pelin G. and Uriel G. Rothblum. "Constant risk aversion in stochastic contests with exponential completion times." Naval Research Logistics (2017).
  • Clark, Derek J. and Christian Riis. "Contest success functions: An extension." Economic Theory 11 (1998): 201-204.
  • Cornes, Richard and Roger Hartley. "Risk aversion, heterogeneity and contests." Public Choice 117 (2003): 1-25.
  • Cornes, Richard and Roger Hartley. "Risk aversion in symmetric and asymmetric contests." Economic Theory 51 (2012): 247-275.
  • Denuit, Michael M., Louis Eeckhoudt, and Harris Schlesinger. "When Ross meets Bell: The linex utility function." Journal of Mathematical Economics 49 (2013): 177-182.
  • Pratt, John W. "Risk aversion in the small and in the large." Econometrica 32 (1964): 122–136.
  • Ross, Stephen A. "Some stronger measures of risk aversion in the small and in the large with applications." Econometrica 49 (1981): 621-638.
  • Skaperdas, Stergios. "Contest success functions." Economic Theory 7 (1996): 283-290.
  • Treich, Nicolas. "Risk-aversion and prudence in rent-seeking games." Public Choice 145 (2010): 339-349.
  • Tullock, Gordon. Efficient rent seeking. In Buchanan, James M., Robert D. Tollison, and Gordon Tullock (Eds.), Toward a theory of the rent-seeking society 4, Texas A & M University Press, 1980.
  • Yamazaki, Takeshi. "The uniqueness of pure-strategy Nash equilibrium in rent-seeking games with risk-averse players." Public Choice 139 (2009): 335-342.
There are 14 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Pelin G. Canbolat 0000-0003-0028-3285

Publication Date June 30, 2019
Submission Date September 3, 2018
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Canbolat, P. G. (2019). Stochastic Contests with Linex Utility Functions. Alphanumeric Journal, 7(1), 113-128. https://doi.org/10.17093/alphanumeric.456860

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