On Nash-solvability of finite $n$-person shortest path games; bi-shortest path conjecture

TL;DR

This paper explores a graph theory conjecture related to Nash-solvability in finite shortest path games, showing it holds for two players but not for three, highlighting differences in game complexity.

Contribution

It introduces a conjecture linking Nash-solvability to graph properties and demonstrates its validity for two-player games but failure for three-player cases.

Findings

Conjecture is equivalent to Nash-solvability in two-player shortest path games.

The conjecture does not hold for three-player shortest path games.

Nash-solvability behavior differs significantly between two and three players.

Abstract

We formulate a conjecture from graph theory that is equivalent to Nash-solvability of the finite two-person shortest path games with positive local costs. For the three-person games such conjecture fails.