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

Mamoru Kaneko, Takashi Nagashima

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

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 languageEnglish
Pages (from-to)229-255
Number of pages27
JournalMathematical Social Sciences
Volume22
Issue number3
DOIs
Publication statusPublished - 1991
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Economics and Econometrics

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