发表机构
Ben-Gurion University of the Negev（内盖夫本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明在近似意义下，C-完备简单博弈蕴含加权性，给出ε*的上界，并表明当玩家类规模增大且类数有界时，误差ε*与误分类比例δ均趋于零。
AI 中文摘要
已知简单博弈中联盟（C-）合意关系的完备性并不蕴含加权多数表示的存在性。我们证明在近似意义下蕴含加权性。称一个n人简单博弈为ε-加权，若存在(n-1)-单纯形中的某个权重向量，使得任意输家联盟的权重超过任意赢家联盟的权重至多ε；令ε*为满足此性质的最小ε。我们在一般C-完备博弈中建立了ε*的上界，并由此推出：若所有等合意玩家类渐近地变大但类数有界，则对于最小玩家类规模n_min，有ε*=O(1/n_min)，且该界是紧的。重要的是，当n_min→∞且玩家类数有界时，由加权规则误分类的联盟的最小可达比例δ也趋于0，且δ=O(1/√n_min)。
英文摘要
It is known that completeness of the coalitional ($C$-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an $n$-player simple game $\varepsilon$-weighted if, for some weight vector in the $(n-1)$-simplex, any losing coalition weight exceeds any winning coalition weight by at most $\varepsilon$; let $\varepsilon^{\ast}$ be the smallest such $\varepsilon$. We establish an upper bound on $\varepsilon^{\ast}$ in a general $C$-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then $\varepsilon^{\ast}=O(1/n_{\min})$ for the minimal player class size $n_{\min}$, and the bound is tight. Importantly, the minimal attainable fraction $δ$ of coalitions mislabeled by a weighting rule also tends to $0$ when $n_{\min}\rightarrow\infty$ and the number of player classes is bounded, with $δ=O(1/\sqrt{n_{\min}})$.
