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Voting with rubber bands, weights, and strings

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Abstract

We introduce some new voting rules based on a spatial version of the median known as the mediancentre, or Fermat-Weber point. Voting rules based on the mean include many that are familiar: the Borda Count, Kemeny rule, approval voting, etc. (see Zwicker (2008a,b)). These mean rules can be implemented by “voting machines” (interactive simulations of physical mechanisms) that use ideal rubber bands to achieve an equilibrium among the competing preferences of the voters. One consequence is that in any such rule, a voter who is further from consensus exerts a stronger tug on the election outcome, because her rubber band is more stretched.While the R1 median has been studied in the context of voting, mediancentre-based rules are new. Voting machines for these rules require that the tug exerted by a voter be independent of his distance from consensus; replacing rubber bands with weights suspended from strings provides exactly this effect. We discuss some novel properties exhibited by these rules, as well as a broader question suggested by our investigations—What are the critical relationships among resistance to manipulation, decisiveness, and responsiveness for a voting rule? We argue that a distorted view may arise from an exclusive focus on the first, without due attention to the other two.

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... Barthelemé and Monjardet [2] pioneered the application to voting theory of the two-step method we use for the characterization via ellipsoids -first convert Hamming distance to squared Euclidean distance in the hypercube, and then apply Huyghens' theorem on the mean. It has also been exploited in [30], [31], and [4]. This approach has the potential to transform any result that entails minimizing a sum of Hamming distances, and deserves to be better known. ...
... 3 However, a precise definition seems necessary if we wish to address two related issues: QUESTION 1 "Is there always some choice of voting weights that perfectly reflects influence?" QUESTION 2 "How can we choose voting weights for legislators in a representative assembly so that they appropriately reflect population differences among the districts represented?" 4 So, how should we measure the influence of a voter in a simple game? In [1], Banzhaf argues that we should count instances in which a voter is critical or decisive, swinging the outcome of the collective decision. ...
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... Until now, these two approaches have not been systematically connected. Specific distancebased rules have indeed been studied in the simplex or permutahedron, notably by Zwicker and coauthors [24,23,3]. However, a more general approach is lacking. ...
... When using profiles as input, the simplex geometry is hard enough to visualize that some authors have used a fixed projection to the permutahedron and essentially used S as a consensus. The cases p = 2 (mean proximity rules) [24,9] and p = 1 (mediancenter rules) [3] have received attention. These can be interpreted in our framework by changing the distance -detailed formulae might be interesting. ...
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... 21 We denote with π i the number of individuals with preference p i , and with π the corresponding distribution. See Cervone et al. (2012) for a study on preference networks. 22 20 Common feature of networks (g , π ) and (g ,π ) is a type of "structural regularity." ...
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... Baranchuk and Dybvig (2009) assumed Euclidean preferences and used the term 'consensus,' and they applied the concept to analyze decision making by a board of directors. Cervone et al. (2012) used the terminology of 'mediancentre' and 'Fermat-Weber point,' and they discussed computational issues and cited earlier work on the topic. Brady and Chambers (2015) used the term 'geometric median,' and assuming Euclidean preferences and a variable population, they showed that when individual preferences are Euclidean, the geometric median is the smallest rule that is Maskin monotonic and satisfies a number of background axioms; and Brady and Chambers (2016) assumed three individuals with Euclidean preferences, and they showed that the geometric median is the unique rule satisfying Maskin monotonicity, anonymity, and neutrality. ...
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... Other works using the geometric median in economics or political science research includeCervone et al. (2012),Baranchuk and Dybvig (2009) andChung and Duggan (2014). In particular, the latter work describes an interesting generalization of the concept to general convex preferences. ...
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... Other works using the geometric median in economics or political science research include Cervone, Dai, Gnoutcheff, Lanterman, Mackenzie, Morse, Srivastava, and Zwicker (2012), Baranchuk and Dybvig (2009), and Chung and Duggan (2014). In particular, the latter work describes an interesting generalization of the concept to general convex preferences. ...
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Smith [J.H. Smith, Aggregation of preferences with variable electorate, Econometrica 41 (1973) 1027–1041] and Young [H.P. Young, A note on preference aggregation, Econometrica 42 (1974) 1129–1131; H.P. Young, Social choice scoring functions, SIAM J. Appl. Math. 28 (1975) 824–838] characterized scoring rules via four axioms: consistency, continuity, anonymity, and neutrality. In their context a ballot consists of a strict ranking of alternatives, and an election outcome is either a set of (winning) alternatives (Young) or a weak ordering of alternatives (Smith). Many rules fail to fit this context, yet intuitively satisfy one’s notion of a generalized scoring rule; this very broad class GSR includes the Kemeny rule, approval voting, and certain grading systems. We show that GSR is identical with the class MPR of mean proximity rules loosely, rules in MPR are those for which the “average voter” determines the outcome. The techniques in the proof allow us to make some surprisingly direct comparisons between rules (for example, between Kemeny and Borda) that might initially seem to be of completely different sorts. The abstract anonymous voting rules provide the context for GSR, which is of necessity too general to admit a neutrality axiom. A natural question arises: “What happens to the Smith and Young characterizations in the absence of neutrality?” We discuss one answer in the form of a characterization of the rational mean neat voting rules (a class closely related to GSR) as those that are consistent and connected. Connectedness is a strong form of continuity that implies a discrete analogue to the Intermediate Value Theorem.
Article
A mean proximity rule is a voting rule having a mean proximity representation in Euclidean space. Legal ballots are represented as vectors that form the representing polytope. An output plot function determines a location for each possible election output in the same space, and these locations decompose the polytope into proximity regions according to which output is closest. The election outcome is then determined by which region(s) contain the mean position of all ballots cast. Mean neat rules are obtained by relaxing the requirement that the regions be determined by proximity, insisting only that they be neatly separable by a hyperplane. If each of these hyperplanes contains a dense set of rational points (vectors with all rational components), the mean neat voting rule is said to be rational. The aim of this article is to prove that consistency and connectedness are necessary and sufficient conditions for mean neat rationality of any voting rule that is anonymous. Connectedness can be viewed as a strong form of continuity, with an intuitive content related to the Intermediate Value Theorem (or to a discrete analogue of this theorem). The proof relies on a recent result in convexity theory [D. Cervone, W.S. Zwicker, Convex decompositions, J. Convex Anal. 2008 (in press)] and suggests a conjecture: if we relax connectedness to continuity, the class so characterized is that of the mean neat voting rules. This latter class properly contains all intuitive scoring rules.
Article
A procedure is developed to obtain representations for the probability of election outcomes with the Impartial Anonymous Culture Condition and the Maximal Culture Condition. The procedure is based upon a process of performing arithmetic with integers, while maintaining absolute precision with very large integer numbers. The procedure is then used to develop probability representations for a number of different voting outcomes, which have to date been considered to be intractable to obtain with the use of standard algebraic techniques.
Article
This paper extends the work of Gehrlein and Fishburn (1976) and Gehrlein (1982) by providing a general theorem relating to the analytical representation of the probability of an event in a given space of profiles. It applies to any event characterized by a set of linear inequalities regardless of whether the coefficients defining the inequalities are integer or fractional coefficients. An algorithm for the probability calculation is also suggested. This suggested methodology is used to provide a complete characterization of the vulnerability properties of the four scoring rules studied in Lepelley and Mbih (1994) to manipulation by coalitions in a 3-alternative n-agent society.
Article
All social choice functions are manipulable when more than two alternatives are available. I evaluate the manipulability of the Borda count, plurality rule, minimax set, and uncovered set. Four measures of manipulability are defined and computed stochastically for small numbers of agents and alternatives. Social choice rules derived from the minimax and uncovered sets are found to be relatively immune to manipulation whether a sole manipulating agent has complete knowledge or absolutely no knowledge of the preferences of the others. The Borda rule is especially manipulable if the manipulating agent has complete knowledge of the others.
Article
The voting situations at which the Borda rule or the Copeland method can be manipulated by a single voter or a coalition of voters in three-alternative elections are characterized. From these characterizations, we derive (when possible) some analytical representations measuring the vulnerability of these rules to strategic misrepresentation of preferences. Our results suggest that the Borda rule is significantly more vulnerable to strategic manipulation than the Copeland method.
Article
Consider an election in which each of the n voters casts a vote consisting of a strict preference ranking of the three candidates A, B, and C. In the limit as n→∞, which scoring rule maximizes, under the assumption of Impartial Anonymous Culture (uniform probability distribution over profiles), the probability that the Condorcet candidate wins the election, given that a Condorcet candidate exists? We produce an analytic solution, which is not the Borda Count. Our result agrees with recent numerical results from two independent studies, and contradicts a published result of Van Newenhizen (Economic Theory 2, 69–83. (1992)). Copyright Springer 2005
Article
Median Voting Rule (MVR) has been proposed as a voting rule, based on the argument that MVR will be less manipulable than Borda Rule. We find that plurality rule has only a slightly greater probability of manipulability than MVR, and that Copeland Rule has a smaller probability of manipulability than MVR. In addition Borda Rule, plurality rule and Copeland Rule all have both a greater probability of producing a decisive result and a greater strict Condorcet efficiency than MVR. Based on all characteristics, MVR does not seem to be viable replacement for either plurality rule or for Copeland Rule. Copyright 2003 by Kluwer Academic Publishers
Article
Variations of IAC are introduced and simulated. A uniformly distributed point P=(X1, X2,…, Xn+1) in a simplex S is generated by a map (ε1, ε2,…, εn)→P from the unit cube to S (surjective with bijective restriction to interiors) with the εi's rectangular and i.i.d on [0,1]. The fraction xyz of the electorate with preference x>y>z is a sum of Xi's. The variations allow different correlations (e.g. ρ(xyz, xzy)≠ρ(xyz, zyx) while they all are −0.2 under IAC. Simulation of two such variations give smaller Condorcet paradox propability than IAC. This is explained heuristically with a graphic “pictogram” representation of the profile.
Article
Electrical engineers employ some methods of linear algebra, derived from Homology Theory, to decompose the flow of current in a complex circuit into two components. The same decomposition can be applied to a ‘circuit’ containing nodes representing the candidates in a multicandidate election, connected by ‘wires’ carrying flows of net voter preference.In this case, the cyclic component measures the tendency towards a voters' paradox, while the cocyclic component measures the spreads in the Borda counts. When the cocyclic component is stronger, it masks the cycles in the cyclic component, and a voters' paradox is avoided; we call this ‘Borda Dominance’.Methods based on this decomposition provide a host of necessary and sufficient conditions for various degrees of transitivity of majority preference. Sen's well-known sufficiency theorem, together with some stronger theorems, are shown to depend upon a strong ‘double’ form of the masking phenomenon. This mathematically natural generalization of Sen's key hypothesis is revealed to be equivalent to a new, quantitative form of transitivity.Because the approach provides fresh insight into the underlying source of the voters' paradox, it appears to represent a promising new tool in social choice theory, with applications beyond those in the current paper.
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