Markov models for the tipsy cop and robber game on graphs

TL;DR

This paper models three variants of the tipsy cop and robber game on graphs using Markov chains, providing probabilistic analysis of game duration and persistence under different biological scenarios.

Contribution

It introduces Markov chain models for three new tipsy cop and robber game scenarios, analyzing game persistence and length based on tipsiness and distance.

Findings

Calculated probability of game persistence over multiple rounds.

Estimated expected game length for different initial conditions.

Provided insights into how tipsiness affects game outcomes.

Abstract

In this paper we analyze and model three open problems posed by Harris, Insko, Prieto-Langarica, Stoisavljevic, and Sullivan in 2020 concerning the tipsy cop and robber game on graphs. The three different scenarios we model account for different biological scenarios. The first scenario is when the cop and robber have a consistent tipsiness level though the duration of the game; the second is when the cop and robber sober up as a function of time; the third is when the cop and robber sober up as a function of the distance between them. Using Markov chains to model each scenario we calculate the probability of a game persisting through $\mathbf{M}$ rounds of the game and the expected game length given different starting positions and tipsiness levels for the cop and robber.