Heights of representative systems

AbstractRepresentative systems are hierarchical aggregation schemes that are applicable in social choice theory, multiattribute decision making, and in the study of three-valued logics. For example, many procedures for voting on issues—including simple majority voting and weighted voting—can be characterized as representative system. Such systems also include procedures in which vote outcomes of constituencies are treated as votes in a higher level of an election system. The general form of a representative system consists of a “supreme council” which aggregates vote outcomes of secondary councils, which in turn aggregate vote outcomes of tertiary councils, and so forth.An n-variable representative system maps n-tuples of 1's, 0's and −1's into {1,0,−1} through a nested hierarchy of sign functions. The height of a representative system is the fewest number of hierarchical levels that are needed to characterize the system. The height μ(n) of all n-variable representative systems is the largest height of such systems. It was shown previously that μ(n) ⩽ n − 1 for all positive integers n and that μ(n) = n − 1 for n from 1 to 4 inclusive. The present paper proves that μ(5) = μ(6) = 4 and that μ(n) ⩽ −2 for all n ⩾ 6. The height function μ is known to be unbounded

Interval graphs and interval orders

AbstractThis paper explores the intimate connection between finite interval graphs and interval orders. Special attention is given to the family of interval orders that agree with, or provide representations of, an interval graph. Two characterizations (one by P. Hanlon) of interval graphs with essentially unique agreeing interval orders are noted, and relationships between interval graphs and interval orders that concern the number of lengths required for interval representations and bounds on lengths of representing intervals are discussed.Two invariants of the family of interval orders that agree with an interval graph are established, namely magnitude, which affects end-point placements, and the property of having the lengths of all representing intervals between specified bounds. Extremization problems for interval graphs and interval orders are also considered

Paradoxes of two-length interval orders

AbstractA two-length interval order is a partially ordered set whose points can be mapped into closed real intervals such that (i) the interval for x lies wholly to the right of the interval for y if and only if x is ranked above y in the partial ordering, and (ii) only two different lengths are involved in the mapping. With the shorter length fixed at 1, let L denote the set of admissible longer lengths for which (i) and (ii) hold for a given interval order.The paper demonstrates that there are two-length interval orders on finite point sets with the following L sets for each integer m⩾2: L = (1,m); L = (2−1m, 2)∪(m,∞); L = (m,2m− 1)∪(2m−1,∞). The second case shows that L can have an arbitrarily big gap between admissible longer lengths, and the third case leads to the corollary that there can be arbitrarily many gaps or holes in L

On a contribution of Freiman to additive number theory

AbstractAn elementary proof is provided for a claim of G. A. Freiman that if 2 ≤ λ < 4 then there is a positive constant c such that, for every finite set X of points in the plane, if every line in the plane contains fewer than c |X| points of X, then the sum set X + X contains more than λ |X| points

Paradoxes of Fair Division

Two or more players are required to divide up a set of indivisible items that they can rank from best to worst. They may, as well, be able to indicate preferences over subsets, or packages, of items. The main criteria used to assess the fairness of a division are efficiency (Pareto-optimality) and envy-freeness. Other criteria are also suggested, including a Rawlsian criterion that the worst-off player be made as well off as possible and a scoring procedure, based on the Borda count, that helps to render allocations as equal as possible. Eight paradoxes, all of which involve unexpected conflicts among the criteria, are described and classified into three categories, reflecting (1) incompatibilities between efficiency and envy-freeness, (2) the failure of a unique efficient and envy-free division to satisfy other criteria, and (3) the desirability, on occasion, of dividing up items unequally. While troublesome, the paradoxes also indicate opportunities for achieving fair division, which will depend on the fairness criteria one deems important and the trade-offs one considers acceptable.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; ENVY-FREENESS; PARETO- OPTIMALITY; RAWLSIAN JUSTICE; BORDA COUNT.

Fair Division of Indivisible Items

This paper analyzes criteria of fair division of a set of indivisible items among people whose revealed preferences are limited to rankings of the items and for whom no side payments are allowed. The criteria include refinements of Pareto optimality and envy-freeness as well as dominance-freeness, evenness of shares, and two criteria based on equally-spaced surrogate utilities, referred to as maxsum and equimax. Maxsum maximizes a measure of aggregate utility or welfare, whereas equimax lexicographically maximizes persons' utilities from smallest to largest. The paper analyzes conflicts among the criteria along possibilities and pitfalls of achieving fair division in a variety of circumstances.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; PARETO OPTIMALITY; ENVY-FREENESS; LEXICOGRAPHIC MAXIMUM

Multiplicities of interpoint distances in finite planar sets

AbstractWhat is the maximum number of unit distances between the vertices of a convex n-gon in the plane? We review known partial results for this and other open questions on multiple occurrences of the same interpoint distance in finite planar subsets. Some new results are proved for small n. Challenging conjectures, both old and new, are highlighted

Representations of Binary Decision Rules by Generalized Decisiveness Structures

This paper is motivated by two apparently dissimilar deficiencies in the theory of social choice and the theory of cooperative games. Both deficiencies stem from what we regard as an inadequate conception of decisiveness or coalitional power. Our main purpose will be to present a more general concept of decisiveness and to show that this notion allows us to characterize broad classes of games and social choice procedures