Strategy-proof social choice on multiple and multi-dimensional single-peaked domains

We generalize the traditional concept of single-peaked preference domains in two ways. First, we introduce the concept of a multiple single-peaked domain, where the set of alternatives is equipped with several underlying orderings with respect to which a preference can be single-peaked, and we argue that these domains are appropriate to represent preferences over political parties. Second, we define a domain of multi-dimensional single-peaked preferences based on the condition of value-restricted preferences by Sen [34]. We provide complete characterizations of the strategy-proof social choice functions on both multiple and multi-dimensional single-peaked domains. For both domains, we also identify the subclasses consisting of all anonymous strategy-proof social choice functions.