On Generalizing a Temporal Formalism for Game Theory to the ...
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On Generalizing a Temporal Formalism for Game Theory to the ...
On Generalizing a Temporal Formalism for Game Theory to the Asymptotic Combinatorics of S5 Modal Frames Samuel Reid
arXiv:1305.0064v1 [math.LO] 1 May 2013
May 2, 2013 Abstract A temporal-theoretic formalism for understanding game theory is described where a strict ordering relation on a set of time points T defines a game on T . Using this formalism, a proof of Zermelo’s Theorem, which states that every finite 2-player zero-sum game is determined, is given and an exhaustive analysis of the game of Nim is presented. Furthermore, a combinatorial analysis of games on a set of arbitrary time points is given; in particular, it is proved that the number of distinct games on a set T with cardinality n is the number of partial orders on a set of n elements. By generalizing this theorem from temporal modal frames to S5 modal frames, it is proved that the number of isomorphism classes of S5 modal frames F = hW, Ri with |W | = n is equal to the partition function p(n). As a corollary of the fact that the partition function is asymptotic to the Hardy-Ramanujan number √ 1 √ eπ 2n/3 4 3n the number of isomorphism classes of S5 modal frames F = hW, Ri with |W | = n is asymptotically the Hardy-Ramanujan number. Lastly, we use these results to prove that an arbitrary modal frame is an S5 modal frame with probability zero.
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Temporal Syntax and Semantics
Temporal Logic provides a comprehensive framework for understanding time through the use of model theory and modalities. In particular, we can understand linear, branching, discrete, dense, or Dedekind Complete time flows by simple conditions on a linear ordering in a model. For applications to game theory, we will be interested in branching time flows. Definition 1. Let F = hT,
