发表机构
Indian Institute of Technology Bombay（印度理工学院孟买分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文反驳了无权Max-k-Cut博弈中最优着色必为强均衡的猜想，构造反例并给出联盟收益恒等式，同时为一般纳什均衡的强偏离建立单色边数量的尖锐上界。
AI 中文摘要
在Max-$k$-Cut博弈中，最优着色最大化社会福利，但它们能否抵御使每个参与玩家严格受益的协调偏离？文献中一个反复出现的猜想断言，对于无权图，每个最优着色都具有这种稳定性性质，即强均衡。我们通过构造一个有限、简单、连通的无权图，其中存在一个不是强均衡的最优着色，从而反驳了对于每个$k\ge3$的该猜想。有利可图的联盟偏离导致另一个最优着色：每个联盟成员都获益，而联盟外的玩家承担相应的损失，社会福利保持不变。我们还给出了一个精确恒等式，将联盟收益、割值变化和内部冲突联系起来，当最大度至多为$2k-1$时，恢复了最优着色的已知稳定性保证。对于从一般纳什均衡的强偏离，我们建立了联盟内剩余单色边数量的尖锐上界，并表明该数量可以超过联盟规模。
英文摘要
In Max-$k$-Cut games, optimal colorings maximize social welfare, but are they stable against coordinated deviations that strictly benefit every participating player? A recurring conjecture in the literature asserts that, for unweighted graphs, every optimal coloring has this stability property, known as strong equilibrium. We disprove the conjecture for every $k\ge3$ by exhibiting a finite, simple, connected, unweighted graph with an optimal coloring that is not a strong equilibrium. The profitable coalitional deviation leads to another optimal coloring: every coalition member gains, while players outside the coalition bear the corresponding losses and social welfare remains unchanged. We also give an exact identity relating coalitional gains, changes in cut value, and internal conflicts, recovering the known stability guarantee for optimal colorings when the maximum degree is at most $2k-1$. For strong deviations from general Nash equilibria, we establish a sharp upper bound on the number of monochromatic edges remaining within the coalition and show that this number can exceed the coalition size.
Comments16 pages
