We study the design of voting rules for committees representing heterogeneous groups (countries, states, districts) when cooperation among groups is voluntary. While efficiency recommends weighting groups proportionally to their stakes, we show that accounting for participation constraints entails overweighting some groups, those for which the incentive to cooperate is the lowest. When collective decisions are not enforceable, cooperation induces more stringent constraints that may require granting veto power to certain groups. In the benchmark case where groups differ only in their population size (i.e., the apportionment problem), the model provides a rationale for setting a minimum representation for smaller groups. (JEL C73, D71, D72)

中文翻译：

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通过加权投票和否决权促进合作

当群体之间的合作是自愿的时，我们研究了代表不同群体（国家、州、地区）的委员会的投票规则设计。虽然效率建议按比例对群体进行加权，但我们表明，考虑参与限制需要过度加权某些群体，而这些群体的合作动机最低。当集体决定不可执行时，合作会引发更严格的限制，可能需要授予某些群体否决权。在群体仅在其人口规模上有所不同的基准案例中（即分配问题），该模型提供了为较小群体设置最小代表性的基本原理。(JEL C73, D71, D72)