Problem
Research questions and friction points this paper is trying to address.
Colonel Blotto game
Nash equilibrium
computational complexity
PPAD-hardness
multiplayer games
The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values
🤖 AI Summary
This study addresses the open problem regarding the computational complexity of Nash equilibria in discrete multiplayer Blotto games with player-specific battlefield valuations. By leveraging computational complexity theory and PPAD reduction techniques, this work systematically analyzes the boundaries of equilibrium computation under varying resource constraints and tie-breaking rules. The primary contribution is the first proof establishing the PPAD-hardness of computing constant-factor approximate equilibria in the three-player setting, thereby revealing a fundamental complexity dichotomy between uniform and non-uniform tie-breaking rules. Furthermore, this research establishes the PPAD-completeness of the associated problems and delineates strict boundaries between polynomial-time solvable special cases and computationally hard instances.
📝 Abstract
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 $(c/n)$-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 $\varepsilon$-approximate Nash equilibria for inverse-exponentially small $\varepsilon$. 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Colonel Blotto game
PPAD-hardness
approximate Nash equilibrium
well-supported Nash equilibrium
tie-breaking rule
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