合作整数规划博弈：核稳定性与最优联盟结构
Cooperative Integer Programming Games: Core Stability and Optimal Coalition Structures
- Virginia Tech（弗吉尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出合作整数规划博弈，通过稳定性不等式和混合整数公式，实现最优及稳定联盟结构的求解，并在合作背包博弈中高效验证。
AI中文摘要：
我们引入了合作整数规划博弈（CIPGs），在该博弈中，智能体汇集预算约束以共同完成不可分割的任务，特征函数将每个联盟映射到一个汇集整数规划的最优值。我们的目标是识别最优联盟结构（OCS）和稳定联盟结构（OSCS）。我们推导出一个稳定性不等式，该不等式使每个形成的联盟相对于其自身保持在核（Core）内，并提出了两种混合整数OCS公式，即聚合公式和分解公式，证明了分解公式在整数等价的同时具有更紧的LP松弛。基于稳定性不等式，我们开发了提升稳定性割、切割平面算法内的多种分离策略、一个SCS可行的原始启发式算法，该算法构建具有保证稳定性的热启动，以及一个支付精化步骤，用于计算每个形成联盟的Shapley值和核仁。在基准合作背包博弈上，该方法在最多16个玩家时证明最优性，在30个玩家时达到低于1%的MIP间隙，同时评估了约10^9个联盟值中的766个。
英文摘要：
We introduce cooperative integer programming games (CIPGs), in which agents pool budget constraints to accomplish indivisible tasks jointly and the characteristic function maps every coalition to the optimal value of a pooled integer program. Our goal is to identify an optimal coalition structure (OCS) and a stable one (OSCS). We derive a stability inequality that keeps each formed coalition in the Core with respect to itself, and present two mixed-integer OCS formulations, aggregated and disaggregated, proving that the disaggregated formulation is integer-equivalent yet yields a tighter LP relaxation. Building on the stability inequality we develop lifted stability cuts, several separation strategies inside a cutting-plane algorithm, an SCS-feasible primal heuristic that constructs warm starts with guaranteed stability, and a payoff-refinement step computing the Shapley value and the nucleolus of every formed coalition. On benchmark cooperative knapsack games, the method certifies optimality with up to 16 players and reaches MIP gaps below 1% at 30 players while evaluating 766 of the roughly $10^9$ coalition values.
