An equity-efficiency trade-off in a geometric approach to committee selection Daniel Eckert and Christian Klamler

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1 An equity-efficiency trade-off in a geometric approach to committee selection Daniel Eckert and Christian Klamler

2 An equity-efficiency trade-off in a geometric approach to committee selection Daniel Eckert and Christian Klamler University of Graz daniel.eckert@uni-graz.at 30 October Introduction Recently, Brams et al. [3] have introduced a minimax procedure into the growing literature on committee selection and compared it with the application of majority voting to the election of committee members. Their finding that these two rules can lead to antipodal results is allthemore disturbing as these rules can be considered to incorporate the principles of equity and of efficiency respectively (for the use of antipodality in the comparison of voting rules see Eckert and Klamler [4], Klamler [6] and Ratliff [8]). We extend their approach to a general geometric framework as developed by Saari [9] for preference aggregation. 2 Equity-efficiency trade-off: formal framework and antipodality result For a set of candidates, J, a committee can be represented by a vector of binary evaluations x = (x 1, x 2,..., x J ) X {0, 1} J, where x j = 1 means that candidate j belongs to the committee and X denotes the set of all possible committees. Geometrically, X is a subset of the vertices of a J dimensional hypercube. (Restrictions on the set X, which are ignored here, can guarantee non-emptiness and fixed sizes as well as specific compositions of committees. On the other hand, the absence of restrictions on the set of binary valuations establishes a certain analogy to approval voting. 1 ) 1 On approval voting see Brams and Fishburn [2]. 1

3 It is assumed that the individuals have metric preferences over committees which can be derived from their top preferred committees and the Hamming distance 2. For any two committees x, y X, the Hamming distance d(x, y) is the number of components (candidates) in which they differ. This means for example that the distance between the committees x = (1, 0, 0, 0) and y = (0, 1, 0, 1) is d(x, y) = 3. Hence, an individual with top preferred committee z, considers committee x at least as good as committee y if it does not differ in more candidates from her top preferred committee. A voting profile for this problem of committee selection is a vector p = (p 1, p 2,..., p X ) which associates with every binary evaluation x k X the fraction p k of individuals for which x k is the top preferred committee, where X i=1 p i = 1. A committee selection rule f is a mapping that assigns to every profile p = (p 1, p 2,..., p X ) a committee x = f(p). 3 In our framework, for any profile p = (p 1, p 2,..., p X ) the outcome of committee selection by majority voting on candidates S x = MS(p) is defined as follows: For all candidates j J, S x j = 1 if and only if X i=1 p i x j i > 0.5. In this framework, committee selection by majority voting on candidates is equivalent to the distance minimizing rule of minisum [3]: Obviously, for any profile p = (p 1, p 2,..., p X ), the minisum outcome S x is the committee that, among all y X, minimizes X i=1 p id(x i, y). Under the assumption of interpersonal comparability, minisum implements the utilitarian principle of sum-efficiency (see Moulin [7]). Now, the minimax procedure introduced by Brams et al. [3] selects the committee that minimizes the maximal distance to the individuals most preferred committees. The major significance of this selection procedure lies in guaranteeing equity in the well established Rawlsian sense of making the worst off individual as well off as possible. Hence we define for any profile 2 A preference can be qualified as metric if there exists a distance function d : X X R+ on the set of alternatives X and, for every individual i, a most preferred alternative x i such that for any pair of alternatives y and z, y is strictly preferred to z if and only if d(x i, y) < d(x i, z). Euclidean preferences are the most important class of metric preferences. 3 Possible restrictions on the codomain of the committee selection rule, such as fixed numbers of committee members or the exclusion of certain combinations of members are again ignored here. On the other hand, our framework makes the size of the committee endogeneous. 2

4 p = (p 1, p 2,..., p X ) the minimax outcome M x = MM(p) as the outcome that, among all y X, minimizes max d(x k, y) for all k such that p k > 0. The comparison of the minisum (majority voting) outcome S x and the minimax outcome M x shows a strong form of the antagonism between the corresponding principles, namely the possibility of the antipodality of the respective outcomes, i.e. there exist profiles for which S x j = 0 if and only if M x j = 1. Example 1 The following table states a profile of binary evaluations with J = 4 candidates and N = 19 voters. # voters evaluation # voters evaluation # voters evaluation Table 1: Profile with 19 voters Looking at candidate 1, we see that she is supported by 10 out of the 19 voters. Hence majority voting gives S x 1 = 1. The same applies to candidate 2, whereas candidates 3 and 4 are supported by only 9 voters, therefore S x 3 = S x 4 = 0 and thus S x = (1, 1, 0, 0). Now, if we look for the minimax outcome it turns out that there are individuals with every valuation but (1, 1, 0, 0). So for every valuation other than (0, 0, 1, 1) there exists an individual that has exactly the opposite valuation and therefore the minimax distance would be 4. As there is no individual with binary evaluation (1, 1, 0, 0) however, an outcome of (0, 0, 1, 1) must minimize the maximal distance to the individual evaluations as such a maximum is 3. Hence M x = (0, 0, 1, 1). Obviously M x and S x are antipodal, in the sense that the Hamming distance between these evaluations is maximal because for all candidates j {1, 2, 3, 4} M x j = 0 if and only if S x j = 1. The above example can be extended to any number of candidates. For the proof of our results we use the geometric framework developed by Saari [9] for the analysis of preference aggregation. Proposition 1 For any number of candidates larger than 2, there exists a profile of binary evaluations such that the minisum and the minimax outcomes are antipodal. 3

5 The proof proceeds with the help of two lemmas. Lemma 1 For any binary evaluation x k there exists a profile p k = (p 1, p 2,..., p k,..., p X ) such that p k = 0 and p i > 0 for all i k, but such that x k = MS(p k ) = S x. Proof. For any binary evaluation x k let {x a } k a A {1,2,..., X } denote the set of the J adjacent vertices of x k and let p A k = (p 1, p 2,..., p k,..., p X ) denote the profile where p k = 0 and p a = ρ > 0 for all a A, and p l = ɛ > 0 is a small positive share for all other evaluations x l. As x a is a neighbor of x k if and only if d(x a, x k ) = 1, for each component j J, there are J 1 evaluations having x j a = x j k {0, 1}. Hence, for each component j J, X i=1 p ix j i = (n 2)ρx j k nɛ xj k 1. For ɛ being sufficiently small and J 3 this implies that M x j = 1 if and only if x j k = 1 and hence the lemma is true. Lemma 2 For any k {1, 2,..., X } and for any profile p k = (p 1, p 2,..., p k,..., p X ) such that p k = 0 and p l > 0 for all l k, x k = MM(p k ), where x k denotes the antipodal evaluation of x k, i.e. x j k = 1 if and only if xj k = 0. (The easy proof of this lemma being left to the reader.) Proof of Proposition 1. Consider any binary evaluation x k X. By lemma 1 it can be obtained as the minisum outcome of some profile p k = (p 1, p 2,..., p k,..., p X ). By lemma 2 the minimax outcome of p k is x k, the antipodal valuation of x k. An immediate corollary of Proposition 1 is that in the case of antipodality of minisum and minimax outcomes, the minimax outcome is identical to the maxisum outcome, i.e. the worst outcome w.r.t. sum-efficiency. On the other hand, for any minisum outcome, if its antipodal evaluation is held by at least some individual, its minimax value is maximal, i.e. it is worst w.r.t. equity. Thus the trade-off betwenn equity and efficiency might be maximal 4. 3 Discussion and future research The antipodality of the outcomes illustrates the antagonism of the two different principles. However, insofar as these principles are incorporated by objective functions, a compromise can always be realized by their convex 4 It remains however to investigate how large it is on average if all possible or relevant profiles are considered. 4

6 combination. In location theory a similar antagonism has motivated the development of bicriteria models which are based on a convex combination of the minimax and the minisum objective function (see Halpern [5]). For committee selection this means finding the committee y X that minimizes the convex combination for some α [0, 1]. X α p i d(x i, y) + (1 α) max d(x k, y) p k >0 i=1 The following example of a profile which exhibits the strong antagonism of the two criteria and the possibility of a reasonable compromise based on their convex combination motivates this approach. share of voters evaluation 8/ / / / Table 2: Profile Example 2 Obviously S x = (0, 0, 0, 0) with a minisum value of 23, but the 20 maximal minimax value of 4. On the other hand the minimax outcome is M x = (1, 1, 1, 0) with a minimax value of 2 but a minisum value of 39. However, for α ( 10, ) the outcome y X that minimizes the objective function is y = (1, 0, 0, 0) with a minisum value of 25 and a minimax value of References [1] Arrow, K.J. (1963). Social Choice and Individual Values, Wiley, New York, second edition. [2] Brams, S.J. and P.C. Fishburn (1983). Approval Voting, Boston, Birkhauser. [3] Brams,S.J., Kilgour, D.M. and M.R. Sanver (2004). A minimax procedure for negotiating multilateral treaties. Preprint, Department of Politics, New York University. 5

7 [4] Eckert, D. and C. Klamler (2008). Antipodality in committee selection. Economics Bulletin 4 : 1 5. [5] Halpern, J. (1976). Finding minimal center-median convex combination (cent-dian) of a graph. Management Science 16 : [6] Klamler, C. (2004). The Dodgson Ranking and the Borda Count: A Binary Comparison. Mathematical Social Sciences 48, [7] Moulin, H. (2003). Fair Division and Collective Welfare. MIT Press. [8] Ratliff, T.C. (2002). A comparison of Dodgson s method and the Borda count. Economic Theory 20 : [9] Saari, D.G. (1995). Basic Geometry of Voting, Springer, Berlin. 6