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On the uniqueness of Nash equilibrium in strategic-form games
Faculty of Economics, Chuo University, 742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan |
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.
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. |
| [2] |
L. A. Chenault,
On the uniqueness of Nash equilibria, Economics Letters, 20 (1986), 203-205.
doi: 10.1016/0165-1765(86)90023-6. |
| [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. |
| [4] |
D. Fudenberg and J. Tirole, Game Theory, The MIT Press, Cambridge, MA, 1991.
|
| [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. |
| [6] |
H. Hotelling,
Stability in competition, Economic Journal, 39 (1929), 41-57.
doi: 10.1007/978-1-4613-8905-7_4. |
| [7] |
R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons, Inc., New York, N. Y., 1957. |
| [8] |
A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, New York, 1995.
|
| [9] |
M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9780511794216.
|
| [10] |
A. Matsumoto and F. Szidarovszky, Game Theory and Its Applications, Springer, Tokyo, 2016.
doi: 10.1007/978-4-431-54786-0. |
| [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. |
| [12] |
J. Nash, Non-cooperative games, Annals of Mathematics (2), 54 (1951), 286-295.
doi: 10.2307/1969529. |
| [13] |
J. B. Rosen,
Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.
doi: 10.2307/1911749. |
| [14] |
J. Sutton, Technology and Market Structure: Theory and History, The MIT Press, Cambridge, MA, 1998.
|
| [15] |
A. Takayama, Mathematical Economics, Second edition, Cambridge University Press, Cambridge, 1985.
|
| [16] |
H. Uzawa,
Walras' existence theorem and Brouwer's fixed-point theorem, Economic Studies Quarterly, 13 (1962), 59-62.
|
| [17] |
E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-58242-4. |
| [18] |
A. van den Nouweland,
Rock-paper-scissors: A new and elegant proof, Economics Bulletin, 3 (2007), 1-6.
|
| [19] |
A. Wald,
Über einige Gleichungssysteme der mathematischen Ökonomie, Econometrica, 19 (1951), 368-403.
|
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. |
| [2] |
L. A. Chenault,
On the uniqueness of Nash equilibria, Economics Letters, 20 (1986), 203-205.
doi: 10.1016/0165-1765(86)90023-6. |
| [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. |
| [4] |
D. Fudenberg and J. Tirole, Game Theory, The MIT Press, Cambridge, MA, 1991.
|
| [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. |
| [6] |
H. Hotelling,
Stability in competition, Economic Journal, 39 (1929), 41-57.
doi: 10.1007/978-1-4613-8905-7_4. |
| [7] |
R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey, John Wiley & Sons, Inc., New York, N. Y., 1957. |
| [8] |
A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, New York, 1995.
|
| [9] |
M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9780511794216.
|
| [10] |
A. Matsumoto and F. Szidarovszky, Game Theory and Its Applications, Springer, Tokyo, 2016.
doi: 10.1007/978-4-431-54786-0. |
| [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. |
| [12] |
J. Nash, Non-cooperative games, Annals of Mathematics (2), 54 (1951), 286-295.
doi: 10.2307/1969529. |
| [13] |
J. B. Rosen,
Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534.
doi: 10.2307/1911749. |
| [14] |
J. Sutton, Technology and Market Structure: Theory and History, The MIT Press, Cambridge, MA, 1998.
|
| [15] |
A. Takayama, Mathematical Economics, Second edition, Cambridge University Press, Cambridge, 1985.
|
| [16] |
H. Uzawa,
Walras' existence theorem and Brouwer's fixed-point theorem, Economic Studies Quarterly, 13 (1962), 59-62.
|
| [17] |
E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-3-642-58242-4. |
| [18] |
A. van den Nouweland,
Rock-paper-scissors: A new and elegant proof, Economics Bulletin, 3 (2007), 1-6.
|
| [19] |
A. Wald,
Über einige Gleichungssysteme der mathematischen Ökonomie, Econometrica, 19 (1951), 368-403.
|
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