The decomposition of strategy-proof random social choice functions on dichotomous domains

A feature of strategy-proof and efficient random social choice functions (RSCFs) defined over several important domains is that they are fixed probability distributions over deterministic strategy-proof and efficient social choice functions. We call such domains deterministic extreme point (DEP) domains. Examples of DEP domains are the domain of all strict preferences and the domain of single-peaked preferences. We show that the dichotomous domain introduced in Bogomolnaia et al. (2005) is not a DEP domain. We find a necessary condition for a strategy-proof RSCF to be written as a fixed probability distribution of deterministic strategy proof social choice functions. We show that this condition is compatible with efficiency. We also show that the condition is sufficient for decomposability in a special case.