The Territorial Raider Game and Graph Derangements

TL;DR

This paper links graph derangements to Nash equilibria in a biological game, providing a new criterion for their existence based on strategic stability in spatial resource competition.

Contribution

It establishes an equivalence between graph derangements and strict Nash equilibria in the Territorial Raider Game, offering a novel criterion for derangement existence.

Findings

A graph admits a derangement if and only if a strict Nash equilibrium exists in the game.

Provides an alternative graph derangement criterion based on game theory.

Connects combinatorial graph properties with biological resource competition models.

Abstract

A derangement of a graph $G=(V,E)$ is an injective function $f:V\to V$ such that for all $v\in V$, $f(v)\neq v$ and $(v,f(v))\in E$. Not all graphs admit a derangement and previous results have characterized graphs with derangements using neighborhood conditions for subsets of $V$. We establish an alternative criterion for the existence of derangements on a graph. We analyze strict Nash equilibria of the biologically motivated Territorial Raider Game, a multi-player competition for resources in a spatially structured population based on animal raiding and defending behavior. We find that a graph $G$ admits a derangement if and only if there is a strict Nash equilibrium of the Territorial Raider game on $G$.