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

TL;DR

This paper introduces a temporal formalism for game theory, proves key theorems including Zermelo's, analyzes Nim, and extends combinatorial results to S5 modal frames, showing their asymptotic growth and measure zero property.

Contribution

It generalizes the combinatorial analysis of game frames from temporal to S5 modal frames, linking to partition functions and asymptotic enumeration.

Findings

Number of S5 modal frames with n worlds equals the partition function p(n).

Asymptotic count of S5 modal frames grows like the Hardy-Ramanujan number.

Probability that an arbitrary modal frame is S5 is zero.

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.