Two necessary conditions for strategy-proofness: On what domains are they also sufficient?

Abstract

A social choice function may or may not satisfy a desirable property depending on its domain of definition. For the same reason, different conditions may be equivalent for functions defined on some domains, while not in other cases. Understanding the role of domains is therefore a crucial issue in mechanism design. We illustrate this point by analyzing the role of two conditions, monotonicity and reshuffling invariance, that are always necessary, but not always sufficient for strategy-proofness. We identify domains, that we call intertwined, ensuring the equivalence between these two conditions and that of strategy-proofness for whatever functions are defined on them. We prove that some important domains are intertwined: those of single-peaked preferences, both with public and private goods, and also those arising in simple models of house allocation. We also discuss the relationship between our conditions on functions and domains with others considered in the literature.

Arrow on domain conditions: a fruitful road to travel

“…These two necessary conditions are, however, not always su¢ cient to precipitate strategy proofness, depending on the domain of de…nition of the social choice function where they apply. In Barberà, Berga, and Moreno (2012) we show that they do if the domain is intertwined, a condition that we now describe for strict preferences over consequences. For any i 2 N , let B i be a non-empty set of possible consequences for i, A i2N B i be the set of alternatives, and, abusing notation, let R i be the set of agent i's preferences on B i .…”

Restricted environments and incentive compatibility in interdependent values models

“…The theorem restricts attention to mechanisms that are respectful, but note that the latter requirement does not always have bite: it is irrelevant when the environment is strict, that is, when the preferences of all agents under all type pro…les are strict (see Corollary 1). 9 Respectfulness is an analogous condition to the one we use in Barberà, Berga, and Moreno (2016) but requiring here invariance in outcomes instead of indi¤erences in outcomes. Examples of mechanisms satisfying respectfulness are provided in Section 4.…”

“…Examples of mechanisms satisfying respectfulness are provided in Section 4. An example of a mechanism violating it is the Gale-Shapley mechanism (see Barberà, Berga, and Moreno, 2016). 10 We do not resort to the use of utility functions like other authors do, mostly because they also analyze other notions of incentive compatibility that require it.…”

“…The proof follows the same argument as the one in Proposition 2, given that agents have all possible strict preferences over individual assignments and preferences are sel…sh. As in Barberà, Berga, and Moreno (2016), just note that although preferences over individual assignments are strict, preferences over alternatives allow for in-di¤erences, by sel…shness: all alternatives with the same individual assignment are indi¤erent for such individual agent. Thus, in the case of housing C(R; z; x) C(R; z; x) holds and R z i are the preferences obtained by lifting z and also all alternatives with the same individual assignment z i to the …rst position and keep the relative position of all other alternatives.…”

“…Our purpose in the present paper is to study when it is possible to design satisfactory mechanisms even in the case where agent's values are interdependent. But before we introduce our present endeavor, it will be useful to refer to our previous and parallel work regarding mechanism design in private values environments (Barberà, Berga, and Moreno, 2016). There, we started from the observation that, under speci…c circumstances, it may be possible to de…ne mechanisms that are not only individually strategy-proof but also (weakly) group strategy-proof.…”