Computing optimal strategy for cop in the game of Cop v.s. Gambler

TL;DR

This paper introduces two efficient algorithms for computing the optimal cop strategy in the game of Cop versus Gambler, leveraging shortest path algorithms adapted for different graph densities.

Contribution

It presents novel algorithms inspired by Bellman-Ford and Dijkstra's methods to efficiently determine optimal strategies in the game.

Findings

Algorithms run in $O(|V(G)||E(G)|)$ and $O(|E(G)|+|V(G)| ext{log}|V(G)|)$ time.

Algorithms are suitable for sparse and dense graphs respectively.

Provides practical methods for strategy computation in game theory.

Abstract

We present two efficient algorithms that compute the optimal strategy for cop in the game of Cop v.s. Gambler where the gambler's strategy is not optimal but known to the cop. The first algorithm is analogous to Bellman-Ford algorithm for single source shortest path problem and runs in $O(|V(G)||E(G)|)$ time. The second is analogous to Dijkstra's algorithm and runs in $O(|E(G)|+|V(G)|\log |V(G)|)$ time. Compared with each other, they are more suitable for sparse and dense graphs, respectively.