At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.

在每个时刻，通过随机过程从有限集合中选择一个备选方案。玩家观察选定的选项并依次投赞成票或反对票。如果投赞成票的一组玩家对所选替代方案具有决定性意义，则接受并结束游戏。否则下一个时期开始。我们将这类问题称为停止游戏。集体选择博弈、退出博弈和联盟形成博弈是特殊的例子。当停止博弈的核心为非空时，纯平稳策略中的子博弈完美均衡被证明存在。但总的来说，即使是混合平稳策略中的子博弈完美均衡也可能不存在。我们表明，可以通过集体策略来总结聚合投票行为。我们坚持纯粹的策略，允许简单的惩罚形式，并提供一个建设性的证明来证明所谓的两步简单集体均衡总是存在的。这意味着纯策略子博弈完美均衡的存在。我们将我们的方法应用于具有孔多塞循环的三种替代方案和再分配政治模型的案例。