Infinite Separation between General and Chromatic Memory

TL;DR

This paper constructs a specific winning condition demonstrating that general memory strategies can be arbitrarily more efficient than chromatic memory strategies in finite arenas.

Contribution

It provides a formal example showing an infinite separation between general and chromatic memory in strategy optimality.

Findings

General memory strategies can be exponentially more efficient.

No finite chromatic memory size can match the optimality of general memory strategies.

The result highlights fundamental differences in strategy complexity.

Abstract

In this paper, we construct a winning condition $W$ over a finite set of colors such that, first, every finite arena has a strategy with 2 states of general memory which is optimal w.r.t.~$W$, and second, there exists no $k$ such that every finite arena has a strategy with $k$ states of chromatic memory which is optimal w.r.t.~$W$.