Nonmanipulability in two dimensions

doi:10.1016/0165-4896(84)90059-3

Mathematical Social Sciences

Volume 8, Issue 1, August 1984, Pages 29-43

https://doi.org/10.1016/0165-4896(84)90059-3 Get rights and content

Abstract

We prove that continuous, anonymous, strategyproof social choice functions in two dimensions forType I preferences are coordinatewise median functionsm(x₁,…,x₁,a₁,…,a_n−1). These are efficient only if thea₁ are ±∞ in equal numbers. For three or more dimensions, efficiency is not possible.

References (9)

ChichilniskyG. et al.
Necessary and sufficient conditions for a resolution of the social choice paradox
J. Econom. Theory
(1983)
BorderK. et al.
Straightforward elections, unanimity, and phantom voters
Rev. Econom. Studies
(1983)
KimK.H. et al.
Properties of consistent voting systems
Int. J. Game Theory
(1981)
KimK.H. et al.
Competitive Economics: Equilibrium and Arbitration
(1983)

There are more references available in the full text version of this article.

Cited by (23)

Truthful aggregation of budget proposals
2021, Journal of Economic Theory
We consider a participatory budgeting problem in which each voter submits a proposal for how to divide a single divisible resource (such as money or time) among several possible alternatives (such as public projects or activities) and these proposals must be aggregated into a single aggregate division. Under $ℓ_{1}$ preferences—for which a voter's disutility is given by the $ℓ_{1}$ distance between the aggregate division and the division he or she most prefers—the social welfare-maximizing mechanism, which minimizes the average $ℓ_{1}$ distance between the outcome and each voter's proposal, is incentive compatible (Goel et al., 2019). However, it fails to satisfy a natural fairness notion of proportionality, placing too much weight on majority preferences. Leveraging a connection between market prices and the generalized median rules of Moulin (1980), we introduce the independent markets mechanism, which is both incentive compatible and proportional. We unify the social welfare-maximizing mechanism and the independent markets mechanism by defining a broad class of moving phantom mechanisms that includes both. We show that every moving phantom mechanism is incentive compatible. Finally, we characterize the social welfare-maximizing mechanism as the unique Pareto-optimal mechanism in this class, suggesting an inherent tradeoff between Pareto optimality and proportionality.
Monotonic norms and orthogonal issues in multidimensional voting
2020, Journal of Economic Theory
Citation Excerpt :
We explain in Section 3.2 this discrepancy in terms of the different domains and focuses of the respective studies.15 For the Euclidean norm, Kim and Roush (1984) and Peters et al. (1992) connected the DIC property of marginal medians to orthogonal coordinate systems.16 An elegant result due to Peters et al. (1993) shows that marginal medians constitute DIC mechanisms for a general norm on the plane (i.e., when there are two dimensions) if and only if majority voting takes place along two directions that are BJ-mutually orthogonal (see Birkhoff, 1935, and James, 1947).17
We study issue-by-issue voting by majority and incentive compatibility in multidimensional frameworks where privately informed agents have preferences induced by general norms and where dimensions are endogenously chosen. We uncover the deep connections between dominant strategy incentive compatibility (DIC) on the one hand, and several geometric/functional analytic concepts on the other. Our main results are: 1) Marginal medians are DIC if and only if they are calculated with respect to coordinates defined by a basis such that the norm is orthant-monotonic in the associated coordinate system. 2) Equivalently, marginal medians are DIC if and only if they are computed with respect to a basis such that, for any vector in the basis, any linear combination of the other vectors is Birkhoff-James orthogonal to it. 3) We show how semi-inner products and normality provide an analytic method that can be used to find all DIC marginal medians. 4) As an application, we derive all DIC marginal medians for $l_{p}$ spaces of any finite dimension, and show that they do not depend on p (unless $p = 2$ ).
Spatial implementation
2015, Games and Economic Behavior
Citation Excerpt :
He establishes that the only rules which are efficient, anonymous, and strategy-proof select a “generalized” median of the announced peaks of the agents. Multi-dimensional extensions establish characterizations of coordinate-wise medians which hold even in the Euclidean case (see Border and Jordan, 1983; Kim and Roush, 1984; Barberà et al., 1993, 1998; Le Breton and Sen, 1999; Le Breton and Weymark, 1999; Bordes et al., 2012). Coordinate-wise medians are intuitive, but suffer from the drawback that the choice of coordinates determines the rule.
Strategyproof Social Choice
2011, Handbook of Social Choice and Welfare
Arrow's Possibility Theorem for one-dimensional single-peaked preferences
2008, Games and Economic Behavior
In one-dimensional environments with single-peaked preferences we consider social welfare functions satisfying Arrow's requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of $2^{N}$ strictly quasi-concave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter's preference is strictly quasi-concave.
Anonymous voting and minimal manipulability
2007, Journal of Economic Theory

View all citing articles on Scopus

View full text

Nonmanipulability in two dimensions

Abstract

J. Econom. Theory

Straightforward elections, unanimity, and phantom voters

Rev. Econom. Studies

Properties of consistent voting systems

Int. J. Game Theory

Competitive Economics: Equilibrium and Arbitration