Dolors Berga

Individual versus group strategy-proofness: 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. There are a number of economically significant domains where interesting rules satisfying individual strategy-proofness can be defined, and for some of them, all these rules turn out to also satisfy the stronger requirement of group strategy-proofness. We provide conditions on domains guaranteeing that for all rules defined on them, individual and group strategy-proofness become equivalent. We also provide a partial answer regarding the necessity of our conditions.

Strategy-proofness and single-plateaued preferences

Abstract In the framework of the provision of one pure public good, we obtain a characterization of the class of strategy-proof voting schemes on single-plateaued preferences over a convex and closed subset of the real line (the set of feasible levels of the public good). Moulin [8] completely characterizes strategy-proof voting schemes on single-peaked preferences as the family of minmax rules. We obtain the result that any strategy-proof voting scheme on the domain of single-plateaued preferences can be viewed as a two-stage procedure. First, we choose a Moulins minmax rule. Then, in the tie-breaking stage, we select one representative alternative from each voters plateau using a strategy-proof scheme with respect to the other voters. The final choice is obtained by applying the minmax rule to the representative best alternatives. Similarly, we can also characterize the subclass of strategy-proof social choice functions satisfying uncompromisingness.

Domains, Ranges and Strategy-Proofness: The Case of Single-dipped Preferences

We characterize the set of all individual and group strategy-proof rules on the domain of all single-dipped preferences on a line. For rules defined on this domain, and on several of its subdomains, we explore the implications of these strategy-proofness requirements on the maximum size of the rules’ range. We show that when all single-dipped preferences are admissible, the range must contain two alternatives at most. But this bound changes as we consider different subclasses of single-dipped preferences: we provide examples of subdomains admitting strategy-proof rules with larger ranges. We establish exact bounds on the maximal size of strategy-proof functions on each of these domains, and prove that the relationship between the sizes of the subdomains and those of the ranges of strategy-proof functions on them need not be monotonic. Our results exhibit a sharp contrast between the structure of strategy-proof rules defined on subdomains of single-dipped preferences and those defined on subsets of single-peaked ones.

Strategic requirements with indifference: single-peaked versus single-plateaued preferences

We concentrate on the problem of the provision of one pure public good whenever agents that form the society have either single-plateaued preferences or single-peaked preferences over the set of alternatives. We are interested in comparing the relationships between different nonmanipulability notions under these two domains. On the single-peaked domain, under strategy-proofness, non-bossiness is equivalent to convexity of the range. Thus, minmax rules are the only strategy-proof non-bossy rules. On the single-plateaued domain, only constant rules are non-bossy or Maskin monotonic; but strategy-proofness and weak non-bossiness are equivalent to weak Maskin monotonicity. Moreover, strategy-proofness and plateau-invariance guarantee convexity of the range.

Stability and voting by committees with exit

Abstract.We study the problem of a society choosing a subset of new members from a finite set of candidates (as in Barberà et al. 1991). However, we explicitly consider the possibility that initial members of the society (founders) may want to leave it if they do not like the resulting new society. We show that, if founders have separable (or additive) preferences, the unique strategy-proof and stable social choice function satisfying founder’s sovereignty (on the set of candidates) is the one where candidates are chosen unanimously and no founder leaves the society.

On Exiting After Voting

We consider the problem of a society whose members must choose from a finite set of alternatives. After knowing the chosen alternative, members may reconsider their membership by either staying or exiting. In turn, and as a consequence of the exit of some of its members, other members might now find undesirable to belong to the society as well. For general exit procedures we analyze the exit behavior of members after knowing the chosen alternative. For the case of monotonic preferences we propose, for each chosen alternative, an unambiguous and meaningful prediction of the subset of members that will exit

Group Strategy-Proof Social Choice Functions with Binary Ranges and Arbitrary Domains: Characterization Results

We define different concepts of group strategy-proofness for social choice functions. We discuss the connections between the defined concepts under different assumptions on their domains of definition. We characterize the social choice functions that satisfy each one of them and whose ranges consist of two alternatives, in terms of two types of basic properties. Finally, we obtain the functional form of all rules satisfying our strongest version of group strategy-proofness.

Single-peakedness and strategy-proofness of generalized median voter schemes

Abstract. We identify, in a continuous multidimensional framework, a maximal domain of preferences compatible with strategy-proofness for a given generalized median voter scheme. It turns out that these domains are a variation of single-peakedness. A similar but stronger result for the discrete case and single-peakedness has been already obtained by Barberà et al. (1999). However, both results are independent and their proofs involve different arguments.

An undominated Nash equilibrium for voting by committees with exit

We consider the problem of a society whose members choose, with a voting by committees, a subset of new members from a given set of candidates. After knowing the elected candidates, former members may decide either stay or exit the society. We analyze the voting behavior of members who take into account the effect of their votes not only on the elected candidates, but also on the final composition of the society. For additive and monotonic preferences with dichotomous bads we construct a strategy profile that is an undominated pure strategy Nash equilibrium of the induced voting game.

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

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.