A domain of preference orderings is a random dictatorship domain if every strategyproof random social choice function satisfying unanimity defined on the domain, is a random dictatorship. Gibbard (1977) showed that the universal domain is a random dictatorship domain. We investigate the relationship between dictatorial and random dictatorship domains. We show that there exist dictatorial domains that are not random dictatorship domains. We provide stronger versions of the linked domain condition (introduced in Aswal et al. (2003)) that guarantee that a domain is a random dictatorship domain. A key step in these arguments that is of independent interest, is a ramification result that shows that under certain assumptions, a domain that is a random dictatorship domain for two voters is also a random dictatorship domain for an arbitrary number of voters.

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This paper proves the following result: every path‐connected domain of preferences that admits a strategy‐proof, unanimous, tops‐only random social choice function satisfying a compromise property is single‐peaked. Conversely, every single‐peaked domain admits a random social choice function satisfying these properties. Single‐peakedness is defined with respect to arbitrary trees. The paper provides a justification of the salience of single‐peaked preferences and evidence in favor of the Gul conjecture ([Barberà, 2010] ).

The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose vertices are the possible types of the voter. Two vertices are connected by an edge in the graph if the associated types are “neighbors.” A social choice function is locally strategy‐proof if no type of a voter can gain by misrepresentation to a type that is a neighbor of her true type. A social choice function is strategy‐proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local‐global equivalence (LGE) is satisfied if local strategy‐proofness implies strategy‐proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of “localness” is perfectly general. We use this feature of our model to identify notions of localness according to which various models of multidimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.

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