Elsevier

Journal of Economic Theory

Volume 189, September 2020, 105103
Journal of Economic Theory

Monotonic norms and orthogonal issues in multidimensional voting

https://doi.org/10.1016/j.jet.2020.105103Get rights and content

Abstract

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 lp spaces of any finite dimension, and show that they do not depend on p (unless p=2).

Introduction

We analyze a canonical social choice/mechanism design problem where several privately informed agents take a multidimensional, collective decision. The main results identify the particular issues that can be put to vote in order to obtain incentive compatible mechanisms when issue-by-issue voting by (possibly qualified) majority is used to determined the outcome of the collective choice. Issue-by-issue voting by majority yields as outcome the issue-by-issue (or marginal, or coordinate-wise) median. Due to the multidimensionality of the decision space, the dimensions on which voting takes place are not uniquely defined. In other words, the issues that are put on the ballot are endogenous, and each feasible set of issues yields a potentially different multidimensional median.

Consider, for example, a legislature that has to decide how much money to allocate to several programs in a given fiscal year. One budgeting procedure, called “bottom-up”, is to vote on each program separately, in which case the total budget will be the sum of the individual budgets. An alternative, called “top-down”, is to vote on the total budget first, and then vote on how to divide the total budget among the individual items.1 Of course, there are potentially many other different budgeting procedures with different welfare properties. In order to find the one that is “optimal” according to some criterion (e.g., budget size, or utilitarian welfare), one first has to characterize the set of voting-based budgeting procedures that have good incentive properties.2

The goal of this paper is to provide such a characterization for an important class of mechanism design problems where there are no monetary transfers among voters, and where the utility of each agent is determined by the distance between a privately known, individual peak (or ideal point) and the taken decision. This distance is derived from a norm on the decision space, assumed here to be a vector space. The norm can vary across agents, may itself be private information, and it need not be generated by an inner product. In particular, it need not be the Euclidean norm.

Our first main result shows that, with preferences induced by norms, marginal medians are dominant-strategy incentive compatible (DIC henceforth) if and only if they are calculated with respect to coordinates determined by an algebraic basis such that the norm is orthant-monotonic in the associated coordinate system.3 Norm monotonicity compares all possible pairs of vectors that are ordered with respect to a lattice structure defined on the underlying space, and requires that the norm of a vector with larger coordinates (in absolute values) is larger. Orthant monotonicity applies the same condition, but requires it to hold only for pairs of comparable vectors in the same orthant. By selecting a coordinate system that aligns the norm structure to the lattice structure, this result allows us to translate to a multidimensional space the one-dimensional insight that a deviation from truthful reporting should move the median peak farther away from one own's true peak.

For any two-dimensional normed space, we prove the existence of at least two distinct DIC marginal medians. This is done by invoking an elegant result that goes back to Hermann Auerbach4: any convex body that is point-symmetric around a center has at least two pairs of conjugate diameters.5

Our next main result shows that, for normed spaces of any dimension, 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 (BJ)-orthogonal to it. BJ-orthogonality translates to any convex set that is point-symmetric around a center (and hence can serve as a unit ball of some norm) the insight that a circle's radius is orthogonal to the tangent through the point where the radius hits the circle's boundary. BJ-orthogonality can be defined for any normed vector space, and it reduces to the usual orthogonality relation in terms of an inner product. Its main “defects” are a lack of symmetry and a lack of additivity: as a consequence, with more than two dimensions, mutual BJ-orthogonality of the vectors in the selected coordinate system is not sufficient in order to induce a DIC marginal median.

To understand the intuition behind the above result, consider a three-dimensional normed space, and choose a set of three mutually orthogonal issues (this always exists by Auerbach's construction, which can be performed for any dimension). Assume that a decision has already been taken on two issues (by consecutive majority votes, say): the obtained decision can be any arbitrary vector in the respective two-dimensional subspace. DIC constraints require the already taken decision to be BJ-orthogonal to the remaining third issue. While this property is automatically satisfied by an initial orthogonal basis in spaces endowed with an inner product, it needs to be additionally imposed in general normed spaces.

A purely geometric approach cannot yield all BJ-mutually orthogonal vectors (and hence, a priori, not all bases with the above described additivity property). Thus such an approach is not very helpful in mechanism design exercises where a comprehensive class of incentive-compatible mechanisms needs to be identified before maximizing some goal over it.

Our third main result shows how an alternative, analytic approach based on semi-inner products (SIP) can be used to overcome this difficulty. An SIP is a special bivariate form that can be defined for any pair of vectors in any normed space. This analytic approach can, in principle, be used to obtain all DIC marginal median mechanisms as the set of solutions to a system of non-linear equations.

As a main application of the SIP approach, we characterize for any finite dimension d2 and for any p1 the set of coordinate systems yielding DIC marginal medians in the standard lp(d) space.6 We also show that, surprisingly, these systems do not depend at all on p (unless p=2, the only inner-product space in this class). Thus, the characterized marginal medians remain DIC even for situations where the norm is allowed to vary across agents within the lp class, and is their private information.

For example, for d=2 and any p1, p2, there are exactly two distinct DIC marginal median mechanisms for a lp(d) space7: they correspond to medians taken with respect to the standard Cartesian coordinates, or with respect to a 45-degree rotation of these coordinates.8 Combining this observation with a result of Peters et al. (1993) implies that these are the only DIC, anonymous and Pareto optimal mechanisms in those settings.

As a further illustration, we consider a setting where agents use individually weighted Euclidean norms that are their private information. All these norms are generated by inner-products, and they do not allow for cross-interactions among issues. Then, under a genericity condition on the set of possible weights, there is exactly one DIC marginal median mechanism on this class: the standard Cartesian coordinates are the unique jointly orthogonal ones for all the norms in this class. Introducing even the slightest degree of interaction among the issues – by allowing utility functions derived from other, more general, inner-product norms – yields an impossibility result.

The issue-by-issue median is the prime example for a “structure-induced equilibrium” in the spirit of Shepsle (1979).9 Since the median is not a linear function of its inputs, the issue-by-issue (or coordinate-wise) median varies with the underlying system of coordinates (see Haldane, 1948).

Besides its ubiquity in practice, this type of voting mechanism (together with its generalization to the so-called “generalized medians” that allow for the presence of additional “phantom” voters with fixed, known peaks) exhausts the set of DIC mechanisms in various settings where the preference domain is sufficiently rich. The first, fundamental result in this vein was obtained by Moulin (1980) for the one-dimensional case.10 Common examples of generalized medians are obtained by issue-by-issue voting with a qualified majority, and the voting thresholds may differ across dimensions (e.g., a bill where one aspect requires a constitutional amendment and hence a higher majority). Our analysis easily generalizes to such mechanisms as well.

With a few notable exceptions, the literature on incentive compatible multidimensional voting and its applications to Political Science and Economics has focused on quadratic loss functions. When applied to normed vector spaces, this assumption yields utilities derived from variations on the Euclidean norm (see, for example, the textbook by Austen-Smith and Banks, 2005).11 The reason behind this choice is technical: it allows the use of familiar mathematical methods from Euclidean geometry and/or familiar mean-variance statistical methods associated to the quadratic formulation.

But, quadratic loss functions derived from an Euclidean norm are not always suitable for multidimensional applications. Consider, for example, the choice of a budget on two items. Then, under Euclidean distance, the two items are equally weighted, equal deviations from a preferred budget on each item are perceived in the same way, and equal deviations upwards and downwards from the wished spending on one item are also perceived in the same way. Moreover, since utility is separable in the two dimensions, there is no cross-interaction among spending deviations on the two items. While it is possible to extend some of the results based on Euclidean norm to the more general class of quadratic preferences generated by inner-products, and to hereby address some of these concerns,12 there is an obvious need to understand more general models where preferences do not display such a high degree of spatial symmetry. Eguia (2013) offers a critical discussion of these issues, and references papers that empirically test this and alternative distance functions.

Even for actual location problems (of a facility, say) the relevant distance function need not be Euclidean: think about the proverbial cab driver in Manhattan who needs to use the “taxicab” norm, driving along the right angles imposed by the array-like city street map. Indeed, distance in US cities is often colloquially measured in “blocks”. Such a distance function is not generated by an inner-product norm.

Barbera et al. (1993) (BGS henceforth) assumed that the decision set is a product of lines. They fixed a system of directions, but did not focus on norm-based preferences. Instead, they studied a richer class of preferences called multidimensional single-peaked (m.s.p.) and showed that, on the class of m.s.p. preferences, a mechanism is DIC if and only if it is a generalized marginal median.13 BGS also showed that their class is maximal in the sense that, if an agent has a preference outside it, there exists a marginal median that is not DIC. In an earlier paper, Border and Jordan (1983) considered a different rich domain of preferences which they called star-shaped and separable, obtained similar results and generalized Moulin's one-dimensional finding.14

While our analysis strongly focuses on the dependence on the chosen coordinate system this dependence does not play a role neither in the BGS' nor in Border and Jordan's analysis. 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 For two-dimensional spaces equipped with a strictly convex norm, Peters et al. (1993) constructed a BJ-mutually orthogonal pair of vectors, and therefore proved the existence of at least one DIC marginal median mechanism. We show here that their results hold in this form only for two-dimensional normed spaces.

Gershkov et al. (2019) maximize utilitarian welfare over the class of marginal median mechanisms in Hilbert spaces, including the Euclidean norm l2.18 We note here that the analog exercise becomes much easier for any lp norm with p2, because the set of marginal medians, which does not vary with p, is much smaller (a finite set instead of a continuum for p=2).

The remainder of the paper is organized as follows: Section 2 presents the social choice model and marginal median mechanisms. Section 3 introduces monotonicity properties of norms and connects DIC constraints to orthant monotonicity. We also relate our insights to those obtained by BGS and Border and Jordan. Section 4 connects orthant monotonicity (and hence DIC mechanisms) to bases consisting of BJ-mutually orthogonal vectors that satisfy a left-additivity condition. Section 5 shows how to analytically use semi-inner products in order to find all DIC marginal medians. Section 6 illustrates the various concepts and findings for lp norms and for inner-product norms, yielding several sharp characterizations of DIC mechanisms. Section 7 concludes.

Section snippets

The social choice model

An odd number of agents n collectively choose a decision vRd where d is a positive integer. We shall endow this vector space with different norms.19 Throughout of the paper, the bold font is used to denote vectors in Rd. We use i=1,...,n to label voters, and j or k=1,...,d to label coordinates.

We denote by {x1,...,xd} a

Incentive compatibility and monotonic norms

In this section, we first define norm monotonicity and orthant monotonicity with respect to a given algebraic basis in Rd. We then show that a marginal median mechanism with respect to a given basis is DIC if and only if the norm is orthant-monotonic with respect to that same basis. Finally, we discuss how this result is related to the insights in BGS and Border and Jordan (1983).

Incentive compatibility and orthogonality

In mechanism design exercises one usually seeks an optimal mechanism in a certain class. Thus, one first needs to characterize the relevant incentive compatible mechanisms. How can we construct all systems of issues that induce DIC issue-by-issue voting? In other words, given a norm, what are the coordinates that render this norm orthant-monotonic? We first discuss a geometric approach towards answering this question, and in the next Section we complement the answer via an analytic device.

Let

How to find all DIC marginal medians?

Theorem 2 reduces the quest for all DIC marginal medians to the quest for all bases consisting of vectors that satisfy property (⁎). We first recapitulate how this is done for the Euclidean norm. We next explain why this geometric procedure does not work for general normed spaces that are not Hilbert. Finally, we introduce an analytic approach based on semi-inner products, and show how it can be used to find all bases with property (⁎).

Illustrations

In this section we offer several illustrations and applications of the above concepts and insights.

Concluding remarks

We have studied issue-by-issue voting by majority in a multidimensional collective decision situation, and we have identified all special systems of coordinates (the “issues”) that render marginal median mechanisms incentive compatible. Our analysis has combined a variety of methods and concepts from geometry/functional analysis. Many of these are novel to the Economics literature.

For fixed, relatively small classes of norm-induced preferences we were able to construct incentive compatible

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  • We wish to thank Tilman Borgers and two anonymous referees for excellent editorial comments. We also wish to thank Xavier Alonso, Matt Jackson and Hans Martini for bibliographic pointers, and Olivier Compte, Francesc Dilme, Philippe Jehiel, Herbert Koch and Roland Strausz for their helpful remarks. Moldovanu would like to dedicate this work to the memory of Joram Lindenstrauss, an unsurpassed teacher and functional analyst. Gershkov's research is supported by grant 1675/18 from Israel Science Foundation, Moldovanu's research is supported by the German Science Foundation through the Hausdorff Center for Mathematics and CRC TR-224(Project B01), and Shi's research is supported by grant 494888 from the Social Sciences and Humanities Research Council of Canada.

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