Aggregation of Votes with Multiple Positions on Each Issue

Abstract

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e. for every issue j the corresponding component fj of every aggregator on every issue should satisfy fj (x1, , . . . , xn) ∈ {x1, , . . . , xn}. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of some size is possible if and only if either non-dictatorial aggregation is possible in a society of only two members or a ternary aggregator exists that either on every issue j is a majority operation, i.e. the corresponding component

On the Computational Complexity of Non-dictatorial Aggregation

“…As mentioned in the Introduction, the existence of a binary non-dictatorial aggregator on X is closely related to the total blockedness of X. More precisely, the following result was proved in [11].…”

“…We now turn our attention to a subclass of possibility domains, first defined in [11], called uniform possibility domains. Letf = (f 1 , .…”

“…A consequence of these results is that a set X of feasible evaluations is a possibility domain if and only if X admits a binary non-dictatorial aggregator or a ternary non-dictatorial aggregator; in other words, non-dictatorial aggregation is possible for a society of some size if and only if it is possible for a society with just two members or with just three members. This line of investigation was pursued further by Kirousis et al [11], who characterized possibility domains in terms of the existence of binary nondictatorial aggregators or ternary non-dictatorial aggregators of particular form.…”

“…Along the way, we also show that there is a polynomial-time algorithm for determining, given X, whether or not X is totally blocked. After this, we turn attention to uniform possibility domains, which were introduced in [11] and constitute a proper subclass of the class of possibility domains. Intuitively, uniform possibility domains are sets of feasible evaluations for which nondictatorial aggregation is possible, uniformly for all issues and when restricted to any two positions for each issue.…”

“…Intuitively, uniform possibility domains are sets of feasible evaluations for which nondictatorial aggregation is possible, uniformly for all issues and when restricted to any two positions for each issue. In [11], a tight connection was established between uniform possibility domains and constraint satisfaction by showing that multi-sorted conservative constraint satisfaction problems on uniform possibility domains are tractable, whereas such constraint satisfaction problems defined on all other domains are NP-complete. Here, using results of Carbonnel [6], we give a polynomial-time algorithm for the following decision problem: given a set X of feasible evaluations (be it in the Boolean or the non-Boolean framework), determine whether or not X is a uniform possibility domain; moreover, if X is a uniform possibility domain, then the algorithm can produce a suitable uniform non-dictatorial aggregator for X.…”