Resumen de Voting by Committees under Constraints

Jordi Massó , Salvador Barberá Fraguas, Alejandro Neme

• Many problems of social choice take the following form. There are n voters and a set K of objects. These objects may be bills considered by a legislature, candidates to some set of positions, or the collection of characteristics which distinguish a social alternative from another. The voters must choose a subset of the set of objects.

  Sometimes, any combination of objects is feasible: for example, if we consider the election of candidates to join a club which is ready to admit as many of them as the voters choose, or if we are modelling the global results of a legislature, which may pass or reject any number of bills. It is for these cases that Barberà, Sonnenschein, and Zhou (1991) provided characterizations of all voting procedures which are strategy-proof and respect voter�s sovereignty (all subsets of object may be chosen) when voters� preferences are additively representable, and also when these are separable. For both of these restricted domains, voting by committees turns out to be the family of all rules satisfying the above requirements. Rules in this class are defined by a collection of families of winning coalitions, one for each object; agents vote for sets of objects; to be elected, an object must get the vote of all members of some coalition among those that are winning for that object.

  Most often, though, some combinations of objects are not feasible, while others are: if there are more candidates than positions to be filled, only sets of size less than or equal to the available number of slots are feasible; if objects are the characteristics of an alternative, some collections of characteristics may be mutually incompatible, and others not. Our purpose in this paper is to characterize the families of strategy-proof voting procedures when not all possible subsets of objects are feasible, and voters� preferences are separable or additively representable. Our main conclusions are the following. First: rules that satisfy strategy-proofness are still voting by committees, with ballots indicating the best feasible set of objects. Second: the committees for different objects must be interrelated, in precise ways which depend on what families of sets of objects are feasible. Third: unlike in Barberà, Sonnenschein, and Zhou (1991), the class of strategy-proof rules when preferences are additively representable can be substantially larger that the set of rules satisfying the same requirement when voters� preferences are separable.

  Our characterization result for separable preferences is quite negative: infeasibilities quickly turn any non-dictatorial rule into a manipulable one, except for very limited cases. In contrast, our characterization result for additive preferences can be interpreted as either positive or negative, because it has different consequences depending on the exact shape of the range of feasible choices.

  The contrast between these two characterization results is a striking conclusion of our research, because until now the results regarding strategy-proof mechanisms for these two domains had gone hand to hand, even if they are, of course, logically independent.

  Notice that here, as in Barberà, Sonnenschein, and Zhou (1991), we could identify sets of objects with their characteristic function, and our objects of choice as (some of) the vertices of a |K|-dimensional hypercube. Barberà, Gul, and Stacchetti (1993) extended the analysis to cover situations where the objects of choice are Cartesian products of integer intervals, allowing for possibly more than two values on each dimension. In there and in other contexts of multidimensional choice where the range of the social choice rule is a Cartesian product, strategy-proof rules are necessarily decomposable into rules which independently choose a value for each dimension, and are themselves strategyproof (see Le Breton and Sen (1997) and (1999) for general expressions of this important result, which dates back to the pioneering work of Border and Jordan (1983)).

  In Barberà, Massó, and Neme (1997) we considered the consequences of introducing feasibility constraints in that larger framework. The range of feasible choices is no longer a Cartesian product and this requires a more complex and careful analysis. All strategy-proof rules are still decomposable, but choices in the different dimensions must now be coordinated in order to guarantee feasibility.

  While this previous paper makes an important step in understanding how this coordination is attained for each given shape of the range, it is marred by a strong assumption on the domain of admissible preferences. Specifically, we assume there that each agent� bliss point is feasible. This assumption is not always realistic. Moreover, it makes the domain of admissible preferences dependent on the range of feasible choices.

  In the present paper we study the question of voting under constraints for two rich and natural sets of admissible preferences: those that are additively representable or separable on the power set of K, regardless of the type of constraints faced by choosers.