计算Colonel Blotto博弈中纳什均衡的复杂性

中文摘要 AI 辅助

我们研究了离散Colonel Blotto博弈中纳什均衡计算的复杂性。对于具有单调分段常数战场收益的双人一般和Colonel Blotto博弈，我们证明即使只有两个战场且每个战场的分段数为常数，计算一个逆多项式近似纳什均衡也是PPAD难的，从而解决了[KPF+25]中提出的一个开放问题。允许更多战场时，当每个战场收益相对于任一玩家的分配都是2-分段常数时，困难性依然存在。一个适用于具有任意局部奖励函数的多玩家Blotto博弈的一般PPAD成员定理进而蕴含了这两个困难结果的PPAD完全性。受固定秩双矩阵博弈的启发，我们在双人一般和Colonel Blotto博弈中识别出一个可处理的边界。如果社会收益具有秩-r结构，那么对于每个固定的r，可以在关于B_1、B_2、k和1/ε的多项式时间内计算出一个ε-纳什均衡。因此，常数秩的双人Colonel Blotto博弈尽管其纯策略空间指数级大，仍然是可处理的。最后，我们研究了具有玩家特定战场价值的多玩家赢家通吃Colonel Blotto博弈，并揭示了一个尖锐的平局打破边界。如果每个最高出价者获得其完整的战场价值，那么对于任意二进制编码的预算，可以在多项式时间内计算出一个精确的纯纳什均衡。相比之下，在最高出价者之间进行普通等分的情况下，我们证明即使只有五个玩家，计算一个逆多项式精度的纳什均衡也是PPAD难的。这解决了Bichler和Ghosh [BG26]在并发工作中提出的一个开放问题，他们在玩家数量随实例增长且询问当玩家数量为常数时困难性是否持续的情况下，在相同的玩家特定等分设置中建立了PPAD困难性。

英文摘要

We study the complexity of Nash equilibrium computation in discrete Colonel Blotto games. For two-player general-sum Colonel Blotto with monotonic piecewise-constant battlefield payoffs, we show that computing an inverse-polynomial approximate Nash equilibrium is PPAD-hard, even with only two battlefields and a constant number of pieces per battlefield, thereby resolving an open question of [KPF+25]. Allowing more battlefields, the hardness persists when every battlefield payoff is $2$-piecewise constant with respect to either player's allocation. A general PPAD-membership theorem for multiplayer Blotto with arbitrary local reward functions then implies PPAD-completeness for both hardness results. Motivated by fixed-rank bimatrix games, we then identify a tractable frontier within two-player general-sum Colonel Blotto. If the social payoff has rank-$r$ structure, then, for every fixed $r$, an $\varepsilon$-Nash equilibrium can be computed in time polynomial in $B_1,B_2,k$, and $1/\varepsilon$. Thus constant-rank two-player Colonel Blotto remains tractable despite its exponentially large pure strategy spaces. Finally, we study multiplayer winner-takes-all Colonel Blotto with player-specific battlefield values and uncover a sharp tie-breaking frontier. If every highest bidder receives her full battlefield value, an exact pure Nash equilibrium can be computed in polynomial time for arbitrary binary-encoded budgets. In contrast, under ordinary equal splitting among highest bidders, we prove that computing an inverse-polynomial-accuracy Nash equilibrium is PPAD-hard even with only five players. This resolves an open question posed in concurrent work by Bichler and Ghosh [BG26], who established PPAD-hardness in the same player-specific equal-splitting setting when the number of players grows with the instance and asked whether hardness persists for a constant number of players.

发表机构

• National and Kapodistrian University of Athens（雅典国立卡波蒂斯坦大学）
• Archimedes, Athena Research Center（阿基米德雅典研究中心）
• National Technical University of Athens（雅典国立技术大学）

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