### Abstract

In a recent article in this journal, Kandori, Mailath, and Rob (1993) (KMR) study the Darwinian dynamics of a 2x2 symmetric game, played repeatedly within a finite population. KMR first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2x2 game into three cases : dominant strategy (DS) games (e.g., prisoners' dilemma), coordination (C) games, and games with no symmetric pure strategy equilibrium ( NP) (e.g., battle of the sexes). In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases.

Original language | English (US) |
---|---|

Pages (from-to) | 443-449 |

Number of pages | 7 |

Journal | Econometrica |

Volume | 64 |

Issue number | 2 |

State | Published - 1996 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Economics and Econometrics
- Mathematics (miscellaneous)
- Statistics and Probability
- Social Sciences (miscellaneous)

### Cite this

*Econometrica*,

*64*(2), 443-449.

**A comment on : 'Learning, mutation, and long-run equilibria in games'.** / Rhode, P.; Stegeman, Mark W.

Research output: Contribution to journal › Article

*Econometrica*, vol. 64, no. 2, pp. 443-449.

}

TY - JOUR

T1 - A comment on

T2 - 'Learning, mutation, and long-run equilibria in games'

AU - Rhode, P.

AU - Stegeman, Mark W

PY - 1996

Y1 - 1996

N2 - In a recent article in this journal, Kandori, Mailath, and Rob (1993) (KMR) study the Darwinian dynamics of a 2x2 symmetric game, played repeatedly within a finite population. KMR first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2x2 game into three cases : dominant strategy (DS) games (e.g., prisoners' dilemma), coordination (C) games, and games with no symmetric pure strategy equilibrium ( NP) (e.g., battle of the sexes). In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases.

AB - In a recent article in this journal, Kandori, Mailath, and Rob (1993) (KMR) study the Darwinian dynamics of a 2x2 symmetric game, played repeatedly within a finite population. KMR first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2x2 game into three cases : dominant strategy (DS) games (e.g., prisoners' dilemma), coordination (C) games, and games with no symmetric pure strategy equilibrium ( NP) (e.g., battle of the sexes). In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases.

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UR - http://www.scopus.com/inward/citedby.url?scp=0030371559&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030371559

VL - 64

SP - 443

EP - 449

JO - Econometrica

JF - Econometrica

SN - 0012-9682

IS - 2

ER -