We consider the fair division of a set of indivisible goods where each agent can receive more than one good, and monetary transfers are allowed. We show that if there are at least three goods to allocate, no efficient solution is population monotonic (PM) on the superadditive Cartesian product preference domain, and the Shapley solution is not PM even on the submodular domain. Moreover, the incompatibility between efficiency and PM prevails in the case of at least four goods on the subadditive Cartesian product domain. We also show that in case there are only two goods to allocate, the Shapley solution and the constrained egalitarian solution are PM on the subadditive preference domain but not on the full preference domain. For the two-good case, we provide a new tool (the hybrid solutions) to construct efficient solutions that are PM on the entire monotone preference domain. The hybrid Shapley solution and the hybrid constrained egalitarian solution are two important examples of such solutions.

我们考虑公平分配一组不可分割的商品，其中每个代理商可以收到多个商品，并且允许进行货币转移。我们表明，如果至少要分配三种商品，那么在超加性笛卡尔积偏好域上没有有效的解决方案是种群单调（PM），即使在亚模域上，Shapley解决方案也不是PM。此外，在亚加成笛卡尔积域上至少有四种商品的情况下，效率和PM之间的不相容性普遍存在。我们还表明，在仅分配两种商品的情况下，Shapley解和约束平均解在次加性偏好域上是PM，而在全偏好域上不是PM。对于两种情况，我们提供了一个新工具（混合解决方案）来构建有效的解决方案，这些解决方案在整个单调首选项域上都是PM。混合Shapley解和混合约束均等解是此类解决方案的两个重要示例。