American Institute of Mathematical Sciences

Advanced
April  2020, 7(2): 97-104. doi: 10.3934/jdg.2020006

On the uniqueness of Nash equilibrium in strategic-form games

Junichi Minagawa

Faculty of Economics, Chuo University, 742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan

Received  June 2019 Revised  February 2020 Published  April 2020

We consider a sufficient condition for the uniqueness of a Nash equilibrium in strategic-form games: for any two distinct strategy profiles, there is a player who can obtain a higher payoff by unilaterally changing the strategy from one strategy profile to the other strategy profile. An example of a game that satisfies this condition is the prisoner's dilemma. Viewed as a solution concept, the Nash equilibrium satisfying the condition is stronger than strict Nash Equilibrium and weaker than strict dominant strategy equilibrium.

Keywords: Diagonal strict concavity, prisoner's dilemma, strict Nash equilibrium, strict dominant strategy equilibrium, competitive equilibrium.
Mathematics Subject Classification: 91A10.
Citation: Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
References:
[1]

H. Carlsson and E. van Damme, Equilibrium selection in stag hunt games, Frontiers of Game Theory, MIT Press, Cambridge, MA, (1993), 237–253.  Google Scholar

[2]

L. A. Chenault, On the uniqueness of Nash equilibria, Economics Letters, 20 (1986), 203-205.  doi: 10.1016/0165-1765(86)90023-6.  Google Scholar

[3]

A. A. Cournot, Researches into the Mathematical Principles of the Theory of Wealth, English edition of Recherches sur les Principes Mathématiques de la Théorie des Richesses, Kelley, New York, 1971. Google Scholar

[4] D. Fudenberg and J. Tirole, Game Theory, The MIT Press, Cambridge, MA, 1991.   Google Scholar
[5]

J. C. Harsanyi, Oddness of the number of equilibrium points: A new proof, International Journal of Game Theory, 2 (1973), 235-250.  doi: 10.1007/BF01737572.  Google Scholar

[6]

H. Hotelling, Stability in competition, Economic Journal, 39 (1929), 41-57.  doi: 10.1007/978-1-4613-8905-7_4.  Google Scholar

[7]

R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons, Inc., New York, N. Y., 1957.  Google Scholar

[8] A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, New York, 1995.   Google Scholar
[9] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511794216.  Google Scholar
[10]

A. Matsumoto and F. Szidarovszky, Game Theory and Its Applications, Springer, Tokyo, 2016. doi: 10.1007/978-4-431-54786-0.  Google Scholar

[11]

J. F. Jr Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[12]

J. Nash, Non-cooperative games, Annals of Mathematics (2), 54 (1951), 286-295. doi: 10.2307/1969529.  Google Scholar

[13]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.  Google Scholar

[14] J. Sutton, Technology and Market Structure: Theory and History, The MIT Press, Cambridge, MA, 1998.   Google Scholar
[15] A. Takayama, Mathematical Economics, Second edition, Cambridge University Press, Cambridge, 1985.   Google Scholar
[16]

H. Uzawa, Walras' existence theorem and Brouwer's fixed-point theorem, Economic Studies Quarterly, 13 (1962), 59-62.   Google Scholar

[17]

E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-58242-4.  Google Scholar

[18]

A. van den Nouweland, Rock-paper-scissors: A new and elegant proof, Economics Bulletin, 3 (2007), 1-6.   Google Scholar

[19]

A. Wald, Über einige Gleichungssysteme der mathematischen Ökonomie, Econometrica, 19 (1951), 368-403.   Google Scholar

show all references

References:
[1]

H. Carlsson and E. van Damme, Equilibrium selection in stag hunt games, Frontiers of Game Theory, MIT Press, Cambridge, MA, (1993), 237–253.  Google Scholar

[2]

L. A. Chenault, On the uniqueness of Nash equilibria, Economics Letters, 20 (1986), 203-205.  doi: 10.1016/0165-1765(86)90023-6.  Google Scholar

[3]

A. A. Cournot, Researches into the Mathematical Principles of the Theory of Wealth, English edition of Recherches sur les Principes Mathématiques de la Théorie des Richesses, Kelley, New York, 1971. Google Scholar

[4] D. Fudenberg and J. Tirole, Game Theory, The MIT Press, Cambridge, MA, 1991.   Google Scholar
[5]

J. C. Harsanyi, Oddness of the number of equilibrium points: A new proof, International Journal of Game Theory, 2 (1973), 235-250.  doi: 10.1007/BF01737572.  Google Scholar

[6]

H. Hotelling, Stability in competition, Economic Journal, 39 (1929), 41-57.  doi: 10.1007/978-1-4613-8905-7_4.  Google Scholar

[7]

R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons, Inc., New York, N. Y., 1957.  Google Scholar

[8] A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, New York, 1995.   Google Scholar
[9] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511794216.  Google Scholar
[10]

A. Matsumoto and F. Szidarovszky, Game Theory and Its Applications, Springer, Tokyo, 2016. doi: 10.1007/978-4-431-54786-0.  Google Scholar

[11]

J. F. Jr Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[12]

J. Nash, Non-cooperative games, Annals of Mathematics (2), 54 (1951), 286-295. doi: 10.2307/1969529.  Google Scholar

[13]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.  Google Scholar

[14] J. Sutton, Technology and Market Structure: Theory and History, The MIT Press, Cambridge, MA, 1998.   Google Scholar
[15] A. Takayama, Mathematical Economics, Second edition, Cambridge University Press, Cambridge, 1985.   Google Scholar
[16]

H. Uzawa, Walras' existence theorem and Brouwer's fixed-point theorem, Economic Studies Quarterly, 13 (1962), 59-62.   Google Scholar

[17]

E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-58242-4.  Google Scholar

[18]

A. van den Nouweland, Rock-paper-scissors: A new and elegant proof, Economics Bulletin, 3 (2007), 1-6.   Google Scholar

[19]

A. Wald, Über einige Gleichungssysteme der mathematischen Ökonomie, Econometrica, 19 (1951), 368-403.   Google Scholar

[1]

Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2203-2215. doi: 10.3934/jimo.2020065
[2]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1
[3]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[4]

Arrigo Cellina, Carlo Mariconda, Giulia Treu. Comparison results without strict convexity. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 57-65. doi: 10.3934/dcdsb.2009.11.57
[5]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[6]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[7]

Kashi Behrstock, Michel Benaïm, Morris W. Hirsch. Smale strategies for network prisoner's dilemma games. Journal of Dynamics & Games, 2015, 2 (2) : 141-155. doi: 10.3934/jdg.2015.2.141
[8]

Sharon M. Cameron, Ariel Cintrón-Arias. Prisoner's Dilemma on real social networks: Revisited. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1381-1398. doi: 10.3934/mbe.2013.10.1381
[9]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
[10]

María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1
[11]

P. Daniele, S. Giuffrè, S. Pia. Competitive financial equilibrium problems with policy interventions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 39-52. doi: 10.3934/jimo.2005.1.39
[12]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[13]

Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060
[14]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123
[15]

Ethan Akin. Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov. Journal of Dynamics & Games, 2017, 4 (3) : 217-253. doi: 10.3934/jdg.2017014
[16]

Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control & Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401
[17]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[18]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[19]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[20]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

 Impact Factor: 

Tools

Metrics

  • PDF downloads (577)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences