Implementation via approval mechanisms

Abstract

We focus on the single-peaked domain and study the class of Generalized Approval Mechanisms (GAMs): First, players simultaneously select subsets of the outcome space and scores are assigned to each alternative; and, then, a given quantile of the induced score distribution is implemented. Our main finding is that essentially for every Nash-implementable welfare optimum-including the Condorcet winner alternative-there exists a GAM that Nashimplements it. Importantly, the GAM that Nash-implements the Condorcet winner alternative is the first simple simultaneous game with this feature in the literature.

On the implementation of the median

“…Therefore, we show that one can design mechanisms whose unique equilibrium outcome coincides with the alternative indicated by the median rule. These positive results provide a novel intuition since, to the extent of our knowledge, little is known about (Bayes-) Nash equilibrium implementation of the median: it satisfies Maskin monotonicity (and hence is Nash implementable), and there are indirect rules that Nash implement it (see Sprumont [1995], Barberà and Jackson [1994], Berga and Moreno [2009] and Núñez and Xefteris [2017]).…”

“…Moreover, the designed mechanisms do not generate incentives for truth-telling, and for almost all admissible preference profiles, almost all agents largely misreport their preferences. The second approach (see Maskin [1999] and Núñez and Xefteris [2017]) has designed indirect mechanisms that Nash implement any generalized median rule such that its interior phantoms (if any) are all distinct. This allows, in particular, implementing the median rule.…”

“…Maskin [1999] builds the well-known integer games that apply to any setting. Núñez and Xefteris [2017] allow agents to approve of intervals of alternatives and the mechanism selects the median of these intervals. Finally, in an incomplete information setting, Gershkov et al [2017] succeeds in partially implementing the median by building dynamic voting games in which agents vote until a qualified majority is reached.…”

On strategy-proofness and single-peakedness: median-voting over intervals

“…Barberà and Coelho 2000), or situations where voters select subsets of the outcome space (e.g. Nuñez and Xefteris 2017). We assume that decisions are made under ignorance (Peterson 2009, p. 40) and that agents, when comparing sets, focus on their best (most favorite) point(s) and their worst (least favorite) point(s) in each set.…”

Piercing numbers in approval voting

“…We should mention that there is work on the social choice side that considers approval sets and their intersections (e.g., [15], [16] ) though these generally have not focused, as mathematicians have, on the geometric nature of the sets involved.…”