多人Colonel Blotto博弈中玩家特定价值的复杂性
The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values
- Technical University of Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究探讨多人Colonel Blotto博弈中玩家特定价值下的均衡计算复杂性，证明在均匀平局打破下计算近似纳什均衡为PPAD-hard，而在特定条件下可多项式求解，并扩展至非均匀平局打破的复杂性结果。
AI中文摘要：
我们研究了具有玩家特定战场价值的离散多人Colonel Blotto博弈中的均衡计算问题。在具有共同战场价值的双人模型中，均衡可以在多项式时间内计算。我们证明，在标准均匀平局打破规则下，这种可解性在具有玩家特定价值的多人模型中失效。特别地，计算一个$(c/n)$-近似纳什均衡是PPAD-hard的，对于某个常数$c>0$，即使每个玩家只有三个资源，其中$n$是玩家数量。主要技术步骤是计算常数近似良好支持的纳什均衡的PPAD-hard性。相比之下，在均匀平局打破下，当每个玩家只有一个资源时，纯纳什均衡可以在多项式时间内计算。我们还证明了对于逆指数小的$\varepsilon$，计算$\varepsilon$-近似纳什均衡的PPAD成员资格。最后，对于非均匀单调平局打破，我们证明即使每个玩家只有一个资源且所有玩家具有相同的战场价值，也是PPAD-hard的。
英文摘要：
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.
