The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values

We study equilibrium computation in discrete multiplayer Colonel Blotto games with player-specific battlefield values. In the two-player model with common battlefield values, equilibria can be computed in polynomial time. We show that this tractability breaks down in the multiplayer model with player-specific values under the standard uniform tie-breaking rule. In particular, computing a $(c\mathrm{/}n)$-approximate Nash equilibrium is PPAD-hard for some constant $c>0$, even when every player has three resources, where $n$ is the number of players. The main technical step is PPAD-hardness for computing a constant-approximate well-supported Nash equilibrium. In contrast, under uniform tie-breaking, a pure Nash equilibrium can be computed in polynomial time when every player has one resource. We also prove PPAD membership for computing $\epsilon$-approximate Nash equilibria for inverse-exponentially small $\epsilon$. Finally, for non-uniform monotone tie-breaking, we show PPAD-hardness even when every player has one resource and all players have identical battlefield values.