Kemeny's rule and Slater''s rule: A binary comparison

The purpose of this paper is to provide a binary comparison of two distance-based preference aggregation rules, Slater's rule and Kemeny''s rule. It will be shown that for certain lists of individual preferences the outcomes will be antagonistic in the sense that what is considered best according to one rule is considered worst according to the other rule.Distance Functions

A comparison of the Dodgson method and the Copeland rule

This paper compares binary versions of two well-known preference aggregation methods designed to overcome problems occurring from voting cycles, Copeland's (1951) and Dodgson''s (1876) method. In particular it will first be shown that the Copeland winner can occur at any position in the Dodgson ranking. Second, it will be proved that for some list of individual preferences over the set of alternatives, the Dodgson ranking and the Copeland ranking will be exactly the opposite, i.e. maximally different.Copeland Rule

A simple ultrafilter proof for an impossibility theorem in judgment aggregation

We show how ultrafilters can be used to prove a central impossibility result in judgement aggregation introduced by Nehring and Puppe (2005), namely that for a logically strongly interconnected agenda, an independent and monotonic judgement aggregation rule which satisfies universal domain, collective rationality and sovereignty is necessarily dictatorial.judgment aggregation

Antipodality in committee selection

In this paper we compare a minisum and a minimax procedure as suggested by Brams et al. for selecting committees from a set of candidates. Using a general geometric framework as developed by Don Saari for preference aggregation, we show that antipodality of a unique maximin and a unique minisum winner can occur for any number of candidates larger than two.

N-Person cake-cutting: there may be no perfect division

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 in which it is impossible to divide it among three players such that these three properties are satisfied, however many 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 an extension of the example for three or more players.Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good

Divide-and-conquer: A proportional, minimal-envy cake-cutting algorithm

We analyze a class of proportional cake-cutting algorithms that use a minimal number of cuts (n-1 if there are n players) to divide a cake that the players value along one dimension. While these algorithms may not produce an envy-free or efficient allocation--as these terms are used in the fair-division literature--one, divide-and-conquer (D&C), minimizes the maximum number of players that any single player can envy. It works by asking n ≥ 2 players successively to place marks on a cake--valued along a line--that divide it into equal halves (when n is even) or nearly equal halves (when n is odd), then halves of these halves, and so on. Among other properties, D&C ensures players of at least 1/n shares, as they each value the cake, if and only if they are truthful. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed.mechanism design; fair division; divisible good; cake-cutting; divide-and-choose

Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Procedure

Properties of discrete cake-cutting procedures that use a minimal number of cuts (n-1 if there are n players) are analyzed. None is always envy-free or efficient, but divide-and-conquer (D&C) minimizes the maximum number of players that any single player may envy. It works by asking n ? 2 players successively to place marks on a cake that divide it into equal or approximately equal halves, then halves of these halves, and so on. Among other properties, D&C (i) ensures players of more than 1/n shares if their marks are different and (ii) is strategyproof for risk-averse players. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed

Better Ways to Cut a Cake - Revisited

Procedures to divide a cake among n people with n-1 cuts (the minimum number) are analyzed and compared. For 2 persons, cut-and-choose, while envy-free and efficient, limits the cutter to exactly 50% if he or she is ignorant of the chooser\u27s preferences, whereas the chooser can generally obtain more. By comparison, a new 2-person surplus procedure (SP\u27), which induces the players to be truthful in order to maximize their minimum allocations, leads to a proportionally equitable division of the surplus - the part that remains after each player receives 50% - by giving each person a certain proportion of the surplus as he or she values it. For n geq 3 persons, a new equitable procedure (EP) yields a maximally equitable division of a cake. This division gives all players the highest common value that they can achieve and induces truthfulness, but it may not be envy-free. The applicability of SP\u27 and EP to the fair division of a heterogeneous, divisible good, like land, is briefly discussed

Maximin Envy-Free Division of Indivisible Items

Assume that two players have strict rankings over an even number of indivisible items. We propose algorithms to find allocations of these items that are maximin—maximize the minimum rank of the items that the players receive—and are envy-free and Pareto-optimal if such allocations exist. We show that neither maximin nor envy-free allocations may satisfy other criteria of fairness, such as Borda maximinality. Although not strategy-proof, the algorithms would be difficult to manipulate unless a player has complete information about its opponent’s ranking. We assess the applicability of the algorithms to real-world problems, such as allocating marital property in a divorce or assigning people to committees or projects