### Abstract

This paper attempts to explain the Nash equilibrium concept from the viewpoint of its one-shot play interpretation. We consider a final decision to be made by each player before the game is actually played. We formalize this game situation in terms of an infinitary first-order predicate logic. Then we give an axiom for final decisions-in the two-person case, the central requirement for this axiom is: for players i and j, if x is a possible final decision for player i, then (1) player i knows that x is his final decision; (2) there is a final decision y for player j; and (3) for any final decision y for j, x is a best response to y and player i knows that y is j's final decision. The entire axiom takes the form of the common knowledge of the above requirement by its very nature. We assume that the complete logical abilities of the players are common knowledge. Then we prove that for solvable games in Nash's sense, x is a final decision for player i iff it is common knowledge that x is a Nash strategy. A similar result will be obtained for unsolvable games.

Original language | English |
---|---|

Pages (from-to) | 229-255 |

Number of pages | 27 |

Journal | Mathematical Social Sciences |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

### Fingerprint

### Keywords

- Final decisions
- game logic
- Nash equilibrium
- solvability
- subsolution

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Economics and Econometrics

### Cite this

**Final decisions, the Nash equilibrium and solvability in games with common knowledge of logical abilities.** / Kaneko, Mamoru; Nagashima, Takashi.

Research output: Contribution to journal › Article

*Mathematical Social Sciences*, vol. 22, no. 3, pp. 229-255. https://doi.org/10.1016/0165-4896(91)90024-L

}

TY - JOUR

T1 - Final decisions, the Nash equilibrium and solvability in games with common knowledge of logical abilities

AU - Kaneko, Mamoru

AU - Nagashima, Takashi

PY - 1991

Y1 - 1991

N2 - This paper attempts to explain the Nash equilibrium concept from the viewpoint of its one-shot play interpretation. We consider a final decision to be made by each player before the game is actually played. We formalize this game situation in terms of an infinitary first-order predicate logic. Then we give an axiom for final decisions-in the two-person case, the central requirement for this axiom is: for players i and j, if x is a possible final decision for player i, then (1) player i knows that x is his final decision; (2) there is a final decision y for player j; and (3) for any final decision y for j, x is a best response to y and player i knows that y is j's final decision. The entire axiom takes the form of the common knowledge of the above requirement by its very nature. We assume that the complete logical abilities of the players are common knowledge. Then we prove that for solvable games in Nash's sense, x is a final decision for player i iff it is common knowledge that x is a Nash strategy. A similar result will be obtained for unsolvable games.

AB - This paper attempts to explain the Nash equilibrium concept from the viewpoint of its one-shot play interpretation. We consider a final decision to be made by each player before the game is actually played. We formalize this game situation in terms of an infinitary first-order predicate logic. Then we give an axiom for final decisions-in the two-person case, the central requirement for this axiom is: for players i and j, if x is a possible final decision for player i, then (1) player i knows that x is his final decision; (2) there is a final decision y for player j; and (3) for any final decision y for j, x is a best response to y and player i knows that y is j's final decision. The entire axiom takes the form of the common knowledge of the above requirement by its very nature. We assume that the complete logical abilities of the players are common knowledge. Then we prove that for solvable games in Nash's sense, x is a final decision for player i iff it is common knowledge that x is a Nash strategy. A similar result will be obtained for unsolvable games.

KW - Final decisions

KW - game logic

KW - Nash equilibrium

KW - solvability

KW - subsolution

UR - http://www.scopus.com/inward/record.url?scp=38149145818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149145818&partnerID=8YFLogxK

U2 - 10.1016/0165-4896(91)90024-L

DO - 10.1016/0165-4896(91)90024-L

M3 - Article

AN - SCOPUS:38149145818

VL - 22

SP - 229

EP - 255

JO - Mathematical Social Sciences

JF - Mathematical Social Sciences

SN - 0165-4896

IS - 3

ER -