发表机构
Chang’an University; Grado Department of Industrial and Systems Engineering, Virginia Tech; Department of Industrial and Systems Engineering, University of Wisconsin–Madison（长安大学; 弗吉尼亚理工大学格雷多工业与系统工程系; 威斯康星大学麦迪逊分校工业与系统工程系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种割平面算法，通过条件均衡不等式刻画带共享约束整数博弈的纯纳什均衡，并嵌入分支割框架求解精确或近似均衡，在3300个实例上验证了其高效性。
AI 中文摘要
我们开发了一种割平面算法，用于计算具有共享约束的有限整数博弈中的纯纳什均衡，其中偏离（deviation）可能针对一个对手策略组合是可行的，而针对另一个对手策略组合则不可行。条件均衡不等式通过一个显式的激活项来捕捉这种依赖性。我们给出了成本和激活条件的仿射编码，并证明了由此产生的不等式能精确刻画均衡集合。将这些不等式作为惰性约束嵌入分支割（branch-and-cut）框架中，得到了广义零遗憾（Generalized Zero-Regret）算法，用于在精确或近似均衡上优化线性目标；一种二分法程序维持了关于可实现的最小近似因子的有界证明。我们还证明了，当条件多面体具有整数顶点时，凹成本在包含均衡策略的最小面上是常数，因此严格凹性迫使策略为顶点策略；对于均匀整数可分割装箱博弈，即使没有严格凹性，一种保成本变换也能产生顶点均衡。我们给出了带共享容量的装箱、网络形成和背包博弈的公式，并在3,300个实例上评估了该算法，除43个实例外均计算出了社会最优均衡，在16个实例上证明了不存在性，并在不存在精确均衡的实例上证明了近似因子在百分之几以内。
英文摘要
We develop a cutting-plane algorithm for computing pure Nash equilibria in finite integer games with shared constraints, where a deviation may be feasible against one opponent profile and infeasible against another. Conditional equilibrium inequalities capture this dependence through an explicit activation term. We give affine encodings of costs and activation conditions and prove that the resulting inequalities characterize the equilibrium set exactly. Embedded as lazy constraints in branch-and-cut, they yield the Generalized Zero-Regret algorithm for optimizing a linear objective over exact or approximate equilibria; a bisection procedure maintains certified bounds on the smallest achievable approximation factor. We also prove that, when the conditional polytopes have integral vertices, a concave cost is constant on the minimal face containing an equilibrium strategy, so strict concavity forces vertex strategies; for uniform integer-splittable bin packing, a cost-preserving transformation yields vertex equilibria even without strict concavity. We give formulations for bin packing, network formation, and knapsack games with shared capacities and evaluate the algorithm on 3,300 instances, computing a socially optimal equilibrium on all but 43, certifying nonexistence on 16, and certifying approximation factors within a few percent where no exact equilibrium exists.
