The theory of conjoint measurement concerns quantitative measurement of a qualitative aspect of objects that are characterized by a number of attributes. The qualitative aspect is often a subjective notion of value or preference, and its quantitative counterpart is a sum of numerical values or utilities over the attributes. If object x is characterized by levels x1, x2, and x3 of three attributes, its numerical valuation is u1(x1) + u2(x2) + u3(x3), where ui is a utility function for attribute i. A particular model of conjoint measurement for this case is based on axioms or assumptions about preferences between objects which imply the existence of u1, u2, and u3 such that, for any two objects x and y, x is preferred to y if and only if u1(x1) + u2(x2) + u3(x3) > u1(y1) + u2(y2) + u3(y3). The axioms make assumptions about ordering properties for preferences and tradeoffs among attributes that lead to the additive utility representation. A variety of axiomatic schemes arise from different attribute structures and different problem settings.

Risk Measurement, Foundations of† Peter C. Fishburn, Peter C. FishburnSearch for more papers by this author Peter C. Fishburn, Peter C. FishburnSearch for more papers by this author First published: 18 November 2014 https://doi.org/10.1002/9781118445112.stat00015 †This article was originally published online in 2011 in Encyclopedia of Statistical Sciences, © John Wiley & Sons, Inc. and republished in Wiley StatsRef: Statistics Reference Online, 2014. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Wiley StatsRef: Statistics Reference OnlineBrowse other articles of this reference work:BROWSE BY TOPICBROWSE A-Z RelatedInformation

This paper argues that social choice from among more than two feasible alternatives should not be based on social choice from two‐alternative subsets. It considers in some detail the case where one alternative ties or beats every other alternative on the basis of simple majorities, and raises the question of whether such an alternative should be chosen. A condition of ‘stochastic unanimity’, introduced in this context, is shown to be incompatible with the simple majority rule when it can apply. This new condition plus a consideration of ties leads into a brief discussion of the use of individual expected utility in social choice theory.

Interdependent preferences arise in economic theory in the study of both individual decisions and group decisions. We imagine that a decision is required among alternatives in a set X and that the decision will depend on preferences between the elements in X. If the preferences represent different points of view about the relative desirability of the alternatives, of if they are based on multiple criteria that impinge on the decision, then we encounter the possibility of interdependent preferences.

Find the secret to improve the quality of life by reading this approval voting 2nd edition. This is a kind of book that you need now. Besides, it can be your favorite book to read after having this book. Do you ask why? Well, this is a book that has different characteristic with others. You may not need to know who the author is, how well-known the work is. As wise word, never judge the words from who speaks, but make the words as your good value to your life.

Approval voting (AV) is a voting system in which voters can vote for, or approve of, as many candidates as they like in multicandidate elections. In 1987 and 1988, four scientific and engineering societies, collectively comprising several hundred thousand members, used AV for the first time. Since then, about half a dozen other societies have adopted AV. Usually its adoption was seriously debated, but other times pragmatic or political considerations proved decisive in its selection. While AV has an ancient pedigree, its recent history is the focus of this paper. Ballot data from some of the societies that adopted AV are used to compare theoretical results with experience, including the nature of voting under AV and the kinds of candidates that are elected. Although the use of AV is generally considered to have been successful in the societies—living up to the rhetoric of its proponents—AV has been a controversial reform. AV is not currently used in any public elections, despite efforts to institute it, so its success should be judged as mixed. The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key “insider” support.

A set T of linear orders of [n] = {1,2, . . . ,n} is subcyclic if every 3-set in [n] has one order that appears in no order in T. With g(n) the maximum cardinality of a subcyclic set for n, Raz [8] proved that g(n)0, thus resolving a longstanding conjecture of a similar upper bound for maximum acyclic sets. The present paper proves that g(4)=14 and g(5)=42 with both maxima attained only by subcyclic sets with a particular structure for all quadruples in [n]. We conjecture a similar result for larger n, in which case g(n) is the nth Catalan number Open image in new window and min(c)=4 for Raz’s bound.

When electing a committee from the pool of individual candidates, it is not sufficient to elicit voters preferences among individual candidates and one should also take into account voters’ opinions about synergetic effects of candidate interactions if they are jointly elected into a committee. We propose an approval voting method in which each voter selects a set of candidates indicating his or her approval of any committee that has sufficiently many candidates from the selected set. The committee approved by most voters is elected.

We examine the effects of a weak version of expected utility's independence axiom on the probability weighting function in rank-dependent utility. Our weak independence axiom says that a 50–50 lottery between a two-outcome gamble and its certainty equivalent is indifferent to the certainty equivalent. A variety of nonlinear probability weighting functions satisfy this axiom, but most weighting functions proposed by others do not. Nevertheless, the axiom accommodates weighting functions that are quite similar to the inverse S-shaped concave–convex functions of others that overvalue small probabilities and undervalue large probabilities.

An L(2, 1)-coloring of a graph G is a coloring of G's vertices with integers in {0, 1,...k} so that adjacent vertices' colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2, 1)-coloring as a coloring. The span lambda(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is lambda(G), and the hole index rho(G) of G is the minimum number of colors in {0,1,...,lambda(G)} not used in a span coloring. We say that G is full-colorable if rho(G) = 0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span mu(G) of G as 00 if G has no no-hole coloring, otherwise mu(G) is the minimum k for which G has a no-hole coloring using colors in {0, 1,...k}.Let n denote the number of vertices of G, and let Delta be the maximum degree of vertices of G. Prior work shows that all non-star trees with Delta greater than or equal to 3 are full-colorable, all graphs G with n=lambda(G) + 1 are full-colorable, mu(G) less than or equal to lambda(G) + rho(G) if G is not full-colorable and n greater than or equal to lambda(G) + 2, and G has a no-hole coloring if and only if n greater than or equal to lambda(G) + 1. We prove two extremal results for colorings. First, for every m greater than or equal to 1 there is a G with rho(G) = m and mu(G) = lambda(G) + m. Second, for every m greater than or equal to 2 there is a connected G with lambda(G) = 2m, n = lambda(G) + 2 and rho(G) = m. (C) 2003 Elsevier B.V. All rights reserved.

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 with possibilities and pitfalls of achieving fair division in a variety of circumstances.

A cyclic order is a ternary relation that satisfies ternary transitivity and asymmetry conditions. Such a ternary relation is extendable if it is included in a complete cyclic order on the same ground set. Unlike the case of linear extensions of partial orders, a cyclic order need not be extendable. The extension problem for cyclic orders is to determine if a cyclic order is extendable. This problem is known to be NP-complete. We introduce a class of cyclic orders in which the extension problem can be solved in polynomial time. The class provides many explicit examples of nonextendable cyclic orders that were not previously known, including a nonextendable cyclic order on seven points. Let μ be the maximum cardinality of a ground set on which all cyclic orders are extendable. It has been shown that μ≤9. We prove that μ=6. This answers a question of Novák. In addition, we characterize the nonextendable cyclic orders on seven and eight points. Our results are intimately related to irreducible partially ordered set of order dimension three, and to fractional vertices of generalized transitive tournament polytopes. As by-products, we obtain a characterization of cyclically ordered sets of dimension two, and a new proof of a theorem of Dridi on small linear ordering polytopes.

Let f(n) be the maximum cardinality of an acyclic set of linear orders on {1, 2, … , n}. It is known that f(3)=4, f(4)=9, f(5)=20, and that all maximum-cardinality acyclic sets for n≤ 5 are constructed by an “alternating scheme”. We outline a proof that this scheme is optimal for n=6, where f (6)=45. It is known for large n that f (n) >(2.17)n and that no maximum-cardinality acyclic set conforms to the alternating scheme. Ran Raz recently proved that f (n)0 and all n. We conjecture that f (n + m)≤f (n + 1) f (m + 1) for n , m≥ 1, which would imply f (n)<(2.591)n − 2 for all large n.

Voting procedures focus on the aggregation of individuals' preferences to produce collective decisions. In practice, a voting procedure is characterized by ballot responses and the way ballots are tallied to determine winners. Voters are assumed to have clear preferences over candidates and attempt to maximize satisfaction with the election outcome by their ballot responses. Such responses can include strategic misrepresentation of preferences. Voting procedures are formalized by social choice functions, which map ballot response profiles into election outcomes. We discuss broad classes of social choice functions as well as special cases such as plurality rule, approval voting, and Borda's point-count method. The simplest class is voting procedures for two-candidate elections. Conditions for social choice functions are presented for simple majority rule, the class of weighted majority rules, and for what are referred to as hierarchical representative systems. The second main class, which predominates in the literature, embraces all procedures for electing one candidate from three or more contenders. The multicandidate elect-one social choice functions in this broad class are divided into nonranked one-stage procedures, nonranked multistage procedures, ranked voting methods, and positional scoring rules. Nonranked methods include plurality check-one voting and approval voting, where each voter casts either no vote or a full vote for each candidate. On ballots for positional scoring methods, voters rank candidates from most preferred to least preferred. Topics for multicandidate methods include axiomatic characterizations, susceptibility to strategic manipulation, and voting paradoxes that expose questionable aspects of particular procedures. Other social choice functions are designed to elect two or more candidates for committee memberships from a slate of contenders. Proportional representation methods, including systems that elect members sequentially from a single ranked ballot with vote transfers in successive counting stages, are primary examples of this class.

There is a unique eight-point planar configuration H8 in which each point has exactly three distinct distances to the other seven, and it is the only eight-point configuration that minimizes the sum of the points’ distance counts. The points in H8 are the vertices of two same-centered squares at a rotation of 45° with side-lengths ratio 2cos15°. I first heard about H8 from Paul Erdős, who heard about it from Heiko Harborth.