Cops vs. Gambler

TL;DR

This paper studies a pursuit-evasion game on graphs where the robber (gambler) moves independently according to a fixed probability distribution, and the cop aims to minimize expected capture time, providing exact and bounded results.

Contribution

It introduces a variation of the cop vs. robber game with a probabilistic robber and derives exact and bounded expected capture times under different information scenarios.

Findings

Expected capture time is exactly n when the distribution is known.

Bounds are provided for expected capture time when the distribution is unknown.

The game analysis applies to any connected n-vertex graph.

Abstract

We consider a variation of cop vs.\ robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on $V(G)$ and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected $n$-vertex graph is exactly $n$. We also give bounds on the (generally greater) expected capture time when the gambler's distribution is unknown to the cop.