$x$ Plays Pokemon, for Almost-Every $x$

TL;DR

This paper demonstrates that for any finite state game, a disjunctive number x will eventually win, using well-known proof techniques from graph theory and cellular automata, with Pokémon as an engaging context.

Contribution

It provides a clear exposition of a known theoretical result in game theory using accessible proof techniques and an engaging example.

Findings

Disjunctive numbers eventually win finite state games

Proof relies on established graph theory and cellular automata results

Uses Pokémon context to motivate the theoretical discussion

Abstract

This paper provides a brief write-up showing that for any finite state game, a disjunctive number $x$ will eventually win that game. The proof techniques here are well known and this result follows immediately from folklore results in graph theory and cellular automata. This short paper primarily serves as an expositional piece to collect this proof with the fun context of $\pi$ Plays Pok\'emon serving as motivation.