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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

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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
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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

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