In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

在本文中，我们考虑具有一致，匿名，对称和组策略证明的选择函数，并考虑在某些树上是单峰的域。在这种情况下，我们证明以下三个结果。首先，当且仅当该域在树上是单峰且代理数为奇数时，在路径连接的域上才存在一个一致，匿名，对称和组策略证明选择功能。其次，当且仅当它是成对的多数规则（也称为树中位数规则）和...的数目时，选择函数才能在树的单峰域上实现一致，匿名，对称和组策略证明。代理商很奇怪。第三，存在一个一致的，匿名的，对称的，并且仅当该域在树上为单峰且代理数为奇数时，才在强路径连接的域上提供具有策略保护的选择功能。作为这些结果的推论，如果代理数量为偶数，则我们得出在路径连接域上不存在一致，匿名，对称和组策略的选择功能。