An extreme point characterization of strategy-proof and unanimous probabilistic rules over binary restricted domains

We show that every strategy-proof and unanimous probabilistic rule on a binary restricted domain has binary support, and is a probabilistic mixture of strategy-proof and unanimous deterministic rules. Examples of binary restricted domains include several types of single-dipped domains, the single-peaked domain where peaks are restricted to two adjacent alternatives, and the single-crossing domain with two tops. We also provide some extensions to infinitely many alternatives