Christian Klamler

The Dodgson ranking and its relation to Kemeny’s method and Slater’s rule

Abstract.In this paper we provide a binary extension of Dodgson’s non-binary preference aggregation rule. This new aggregation rule is then compared to two other rules which, as Dodgson’s rule, are also explicitly based on distance functions, namely Kemeny’s and Slater’s rule. It is shown that the alternative which is top ranked by any of those rules can occur at any position in the Dodgson ranking.

Proportional pie-cutting

David Gale (Math Intell 15:48–52, 1993) was perhaps the first to suggest that there is a difference between cake and pie cutting. A cake can be viewed as a rectangle valued along its horizontal axis, and a pie as a disk valued along its circumference. We will use vertical, parallel cuts to divide a cake into pieces, and radial cuts from the center to divide a pie into wedge-shaped pieces. We restrict our attention to allocations that use the minimal number of cuts necessary to divide cakes or pies. In extending the definition of envy-freeness to unequal entitlements, we provide a counterexample to show that a cake cannot necessarily be divided into a proportional allocation of ratio p:1−p between two players where one player receives p of the cake according to her measure and the other receives 1−p of the cake according to his measure. In constrast, for pie, we prove that an efficient, envy-free, proportional allocation exists for two players. The former can be explained in terms of the Universal Chord Theorem, whereas the latter is proved by another result on chords. We provide procedures that induce two risk-averse players to reveal their preferences truthfully to achieve proportional allocations. We demonstrate that, in general, proportional, envy-free, and efficient allocations that use a minimal number of cuts may fail to exist for more than two players.

A distance measure for choice functions

This paper discusses and characterizes a distance function on the set of quasi choice functions. The derived distance function is in the spirit of the widely used Kemeny metric on binary relations but extends Kemeny’s use of the symmetric difference distance to set functions and hence to a more general model of choice.

The Dodgson ranking and the Borda count: a binary comparison

Abstract This paper provides a binary comparison of two preference aggregation rules, the Borda rule and Dodgsons rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences and therefore avoid the problem of voting cycles. It will be shown that for certain lists of individual preferences the rankings derived from the Borda rule and Dodgsons rule are antagonistic.

Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm

Many procedures have been suggested for the venerable problem of dividing a set of indivisible items between two players. We propose a new algorithm (AL), related to one proposed by Brams and Taylor (BT), which requires only that the players strictly rank items from best to worst. Unlike BT, in which any item named by both players in the same round goes into a “contested pile,” AL may reduce, or even eliminate, the contested pile, allocating additional or more preferred items to the players. The allocation(s) that AL yields are Pareto-optimal, envy-free, and maximal; as the number of items (assumed even) increases, the probability that AL allocates all the items appears to approach infinity if all possible rankings are equiprobable. Although AL is potentially manipulable, strategizing under it would be difficult in practice.

A distance-based comparison of basic voting rules

In this paper we provide a comparison of different voting rules in a distance-based framework with the help of computer simulations. Taking into account the informational requirements to operate such voting rules and the outcomes of two well-known reference rules, we identify the Copeland rule as a good compromise between these two reference rules. It will be shown that the outcome of the Copeland rule is “close” to the outcomes of the reference rules, but it requires less informational input and has lower computational complexity.

N-Person Cake-Cutting: There May Be No Perfect Division

Abstract A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.

Distance-Based Aggregation Theory

The problem of aggregating several objects into an object that represents them is a central problem in disciplines as diverse as economics, sociology, political science, statistics and biology (for a survey on aggregation theory in various fields see Day and McMorris [17]). It has been extensively dealt with in the theory of social choice (see Arrow et al. [5]), which analyses the aggregation of individual preferences into a collective preference. In this context, the idea of a consensus is normatively particularly appealing. A natural way to operationalize the consensus among a group of individuals is by means of a distance function that measures the disagreement between them. Thus, in particular, the construction of aggregation rules based on the minimization of distance functions inherits the normative appeal of consensus.

Maximizing the Minimum Voter Satisfaction on Spanning Trees

This paper analyzes the computational complexity involved in solving fairness issues on graphs, e.g., in the installation of networks such as water networks or oil pipelines. Based on individual rankings of the edges of a graph, we will show under which conditions solutions, i.e., spanning trees, can be determined efficiently given the goal of maximin voter satisfaction. In particular, we show that computing spanning trees for maximin voter satisfaction under voting rules such as approval voting or the Borda count is -complete for a variable number of voters whereas it remains polynomially solvable for a constant number of voters.

Committee Selection under Weight Constraints

In this paper we investigate the problem of selecting a committee consisting of k members from a list of m candidates. Each candidate has a certain cost or weight. The choice of the k-committee has to satisfy some budget or weight constraint: the sum of the weights of all committee members must not exceed a given value W. While the former part of the problem is a typical question in Social Choice Theory, the latter stems from Operations Research. The purpose of this paper is to link these two research fields: we first characterize reasonable ways of ranking sets of objects, i.e., candidates, and then develop efficient algorithms for the actual computation of optimal committees.