Individual versus group strategy proofedness: when do they coincide?

A social choice function is group strategy-proof on a domain if no group of agents can manipulate its final outcome to their own benefit by declaring false preferences on that domain. Group strategy-proofness is a very attractive requirement of incentive compatibility. But in many cases it is hard or impossible to find nontrivial social choice functions satisfying even the weakest condition of individual strategy-proofness. However, there are a number of economically signicant domains where interesting rules satisfying individual strategy-proofness can be defi…ned, and for some of them, all these rules turn out to also satisfy the stronger requirement of group strategy-proofness. This is the case, for example, when preferences are single-peaked or single-dipped. In other cases, this equivalence does not hold. We provide sufficient conditions defining domains of preferences guaranteeing that individual and group strategy-proofness are equivalent for all rules defined on these domains. Our results extend to intermediate versions of strategy-proofness, defined to exclude manipulations by small group of agents. They also provide guidelines on how to restrict the ranges of functions defined on domains that only satisfy our condition partially. Finally, we provide a partial answer regarding the necessity of our conditions.