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Tipsy cop and tipsy robber: collisions of biased random walks on graphs

Pamela E. Harris, Erik Insko, Florian Lehner

TL;DR

This article adopts a slightly different interpretation of tipsiness of the cop and robber where it is assumed that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move,A tipsy (uniformly random) move by the cop, and a tipsy

Abstract

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the \emph{tipsy cop and drunken robber} is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including $δ$-regular trees %infinite binary trees with an infinite path rooted at every vertex, and $δ$-regular trees rooted to a $Δ$-regular tree, where $Δ\geq δ$. Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.

Tipsy cop and tipsy robber: collisions of biased random walks on graphs

TL;DR

This article adopts a slightly different interpretation of tipsiness of the cop and robber where it is assumed that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move,A tipsy (uniformly random) move by the cop, and a tipsy

Abstract

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the \emph{tipsy cop and drunken robber} is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including -regular trees %infinite binary trees with an infinite path rooted at every vertex, and -regular trees rooted to a -regular tree, where . Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.
Paper Structure (11 sections, 27 theorems, 80 equations, 3 figures)

This paper contains 11 sections, 27 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and general set up
  3. Markov chains and random walks
  4. The tipsy cop and robber game
  5. Optimal and less than optimal strategies on the infinite grid
  6. Cop playing an optimal strategy
  7. Cop playing foolish strategy
  8. Game on Trees
  9. Analysis of $S_r^\ast$ and $S_r^B$
  10. Comparison of the two strategies
  11. Future Directions and Open Problems

Key Result

Theorem 2.1

Let $X_{\mathbf{s}}$ be a time homogeneous, irreducible Markov chain with state space $V$, and let $v,x,y \in V$. The following (mutually exclusive) statements hold:

Figures (3)

  • Figure 1: Probability of each move based on a spinner model.
  • Figure 2: The trees $X(4,2)$ and $X(5,3)$.
  • Figure 3: Comparing the efficacy of the robber's dualing strategies on $X(\Delta, \delta)$ for $\Delta = 7$ and $\delta \in \{2,3,4,5\}$.

Theorems & Definitions (46)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: Gambler's ruin
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 36 more