On Generalizing a Temporal Formalism for Game Theory to the Asymptotic Combinatorics of S5 Modal Frames

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 $\mathcal{F} = \ < W, R \ >$ 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 $$\frac{1}{4\sqrt{3}n}e^{\pi \sqrt{2n/3}}$$ the number of isomorphism classes of S5 modal frames $\mathcal{F} = \ < W, R \ >$ 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.