Two-person Positive Shortest Path Games Have Nash Equilibria in Pure Stationary Strategies
Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Mikhail Vyalyi
TL;DR
This work studies two-person shortest path games on directed graphs with positive local costs, proving that every finite positive SP game admits a Nash equilibrium in pure stationary strategies and that such NE can be computed in polynomial time. It further extends existence results to infinite graphs with finite out-degree and analyzes terminal versus cyclic outcomes, showing a terminal NE when at least one player can force reaching a terminal, and a cyclic NE when both can block terminals. The authors introduce short paths interdiction games, prove NE existence via reduction to SP games, and provide a polynomial-time algorithm under an independence-system oracle based on a Modified Dijkstra approach. By leveraging potential transformations, the paper connects SP games to deterministic stochastic games with perfect information and a 1-total effective cost, delivering a cohesive framework for NE analysis in both standard SP games and interdiction variants.
Abstract
We prove that every finite two-person shortest path game, where the local cost of every move is positive for each player, has a Nash equilibrium (NE) in pure stationary strategies, which can be computed in polynomial time. We also extend the existence result to infinite graphs with finite out-degrees. Moreover, our proof gives that a terminal NE (in which the play is a path from the initial position to a terminal) exists provided at least one of the two players can guarantee reaching a terminal. If none of the players can do it, in other words, if each of the two players has a strategy that separates all terminals from the initial position $s$, then, obviously, a cyclic NE exists, although its cost is infinite for both players, since we restrict ourselves to positive games. We conjecture that a terminal NE exists too, provided there exists a directed path from $s$ to a terminal. However, this is open. We extend our result to short paths interdiction games, where at each vertex, we allow one player to block some of the arcs and the other player to choose one of the non-blocked arcs. Assuming that blocking sets are chosen from an independence system given by an oracle, we give an algorithm for computing a NE in time $O(|E|(\log|V|+τ))$, where $V$ is the set of vertices, $E$ is the set of arcs, and $τ$ is the maximum time taken by the oracle on any input.