Markov models for the tipsy cop and robber game on graphs

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.