[Paper Review] Cops and Robbers, Game Theory and Zermelo's Early Results

Athanasios Kehagias, Georgios Konstantinidis|arXiv (Cornell University)|Jul 7, 2014

Evolutionary Game Theory and Cooperation8 references5 citations

TL;DR

This paper formalizes the Cops and Robbers game within a rigorous game-theoretic framework, establishing the existence of optimal strategies and bounded capture times in cop-win graphs. It connects the game to Zermelo's early work on determinacy and shows that Cops and Robbers can be reduced to a reachability game, enabling memoryless optimal strategies via standard game-theoretic results.

ABSTRACT

We provide a game theoretic framework for the game of cops and robbers (CR). Within this framework we study certain assumptions which underlie the concepts of optimal strategies and capture time. We also point out a connection of these concepts to early work by Zermelo and D. Konig. Finally, we discuss the relationship between CR and related pursuit games to reachability games.

Motivation & Objective

• To provide a formal game-theoretic foundation for the Cops and Robbers game, addressing informal treatment of strategies and optimality in prior literature.
• To highlight the overlooked connection between Cops and Robbers and early work by Zermelo and D. König on determinacy and infinite games.
• To establish a formal link between Cops and Robbers and reachability games, showing that optimal strategies can be derived from reachability game theory.
• To prove that in cop-win graphs, the cop has a memoryless strategy ensuring bounded capture time regardless of the robber's actions.

Proposed method

• Models the Cops and Robbers game as a two-player, zero-sum, infinite-horizon game using game positions, histories, and strategies defined over finite and infinite sequences of moves.
• Defines legal and memoryless strategies for both players, where memoryless strategies depend only on current positions, not history.
• Introduces the capture time T(s_C, s_R) as the payoff, with the cop aiming to minimize it and the robber to maximize it.
• Applies game-theoretic concepts such as value of the game, optimal strategies, and the minimax equality to prove existence of optimal strategies under bounded capture time.
• Reduces the Cops and Robbers game to a reachability game by constructing a move digraph M_G and its expansion M̅_G, where winning conditions correspond to capture.
• Uses Theorem 5.1 on reachability games to show that memoryless winning strategies exist for both players, and proves that bounded capture time implies time-optimality.

Experimental results

Research questions

• RQ1 Does the Cops and Robbers game admit optimal strategies in the game-theoretic sense, and can such strategies be proven to exist?
• RQ2 How does the concept of capture time in Cops and Robbers relate to Zermelo's early results on determinacy in finite and infinite games?
• RQ3 Can the Cops and Robbers game be formally reduced to a reachability game, and what are the implications for strategy construction?
• RQ4 Under what conditions does the existence of a bounded capture time imply the existence of a time-optimal memoryless strategy?

Key findings

• For every cop-win graph G, there exists a finite upper bound T̄_G on capture time, independent of the robber's strategy, which ensures the existence of optimal strategies.
• The value of the game exists and equals the capture time T(s_C^*, s_R^*) when optimal strategies s_C^* and s_R^* exist, satisfying the minimax equality.
• Memoryless optimal strategies σ_C^* and σ_R^* exist for both players in cop-win graphs, derived from the reachability game framework.
• When G is robber-win, the capture time is infinite, and any cop strategy results in infinite time; thus, all memoryless cop strategies are time-optimal.
• The Cops and Robbers game can be reduced to a reachability game via the move digraph M̅_G, where the cop's target set F consists of positions where cop and robber are co-located.
• The cop number c(G) is the smallest K such that the initial state (∅, ∅, C) belongs to the winning region W_0 of the K-player reachability game.