Notes
A (biased) sampling from my bookshelf: Shoenfield’s Mathematical Logic: “Logic is the study of reasoning; and mathematical logic is the study of the type of reasoning done by mathematicians”; Enderton’s A Mathematical Introduction of Logic: “Symbolic logic is a mathematical model of deductive thought”; and Chiswell and Hodges Mathematical Logic: “In this course we shall study some ways of proving statements.”
That is, if, according to player i’s beliefs, strategy s is optimal for player j, then i cannot rule out all states where player j follows strategy s.
I am assuming that the set of states is finite, so that individual states can be assigned a non-zero probability.
I assume that the reader is familiar with the standard formulation of common knowledge: An event E is commonly known provided everyone knows E, everyone knows that everyone knows E, everyone knows that everyone knows E, and so on ad infinitum.
Note that the framework used in [20] differs in small but important ways from the epistemic-probability models introduced in this paper. These technical details are not important for the main point I am making here, and, indeed, this argument can be made more formal using epistemic probability models.
This follows from the well-known fact that a strategy is weakly dominated iff it does not maximize expected utility with respect to any probability that assigns non-zero probability to all of the opponent’s choices.
Well, I do restrict attention to games with a finite set of players and and finite sets of actions for each player.
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Pacuit, E. On the use (and abuse) of Logic in Game Theory. J Philos Logic 44, 741–753 (2015). https://doi.org/10.1007/s10992-015-9356-8
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DOI: https://doi.org/10.1007/s10992-015-9356-8