text: We develop new voting mechanisms for the case when voters and candidates are located in an arbitrary unknown metric space, and the goal is to choose a candidate minimizing social cost: the total distance from the voters to this candidate. Previous work has often assumed that only ordinal preferences of the voters are known (instead of their true costs), and focused on minimizing distortion: the quality of the chosen candidate as compared with the best possible candidate. In this paper, we instead assume that a (very small) amount of information is known about the voter preference strengths, not just about their ordinal preferences. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful. Finally, we conclude by quantifying the ideal candidate distortion, which compares the quality of the chosen outcome with the best possible candidate that could ever exist, instead of only the best candidate that is actually in the running.
  @cite_26: Much of the dissatisfaction with Professor Arrow's formulation (1963) of the problem of social choice rests on Arrow's imposition of the controversial "independence of irrelevant alternatives" condition (A3). Arrow himself has ceased to insist upon this condition in its strong original form (Arrow 1967). The chief objection to A3 is that it prevents the collectivechoice rule from responding to intensity of individual preference, making the choice rule unacceptable on a priori grounds (Kemp and Asimakopulos 1952; Hildreth 1953; Coleman 1966; Quirk and Saposnik 1968; Sen 1970). This view is not correct if intensity of preference is measured by an ability to make side payments in order to influence collective decision making. I show that a modified version of A3 which takes explicit account of intensity of preference actually implies the original independence condition for realistic environment sets. This will be true no matter how side payments of private goods and services are evaluated. A modest amount of notation is required before the independence condition can be given formal expression. The set X, the nonnegative orthant of m-dimensional Euclidean space, is the space of hypothetical states of the economy. There are n individuals whose preferences are to be considered by any collective decision-making process, and to each individual i there corresponds a binary relation Ri which is complete on X. The term
  @cite_10: Dinner is over. Mr. and Mrs. Jones and Mr. and Mrs. Smith are having coffee. The question arises: What shall we do this evening? Play bridge? Go to the movies? Listen to some chamber music from the local FM station? Sit and chat? Each, in due course, expresses a “preference” among these four alternatives but with this difference: Mr. and Mrs. Jones and Mrs. Smith, though each has a preference, “don't much care.” Their preferences are “mild” or “marginal.” Not so Mr. Smith. His preference is “strong.” He is tired, couldn't possibly get his mind on bridge, or muster the energies for going out to a movie. He has listened to chamber music all afternoon while working on an architectural problem, and couldn't bear any more. If the group does anything other than sit and chat, he at least will do it grudgingly. He “cares enormously” which alternative is chosen. Now: which is the “correct” choice among the four alternatives? Which, “distributive justice” to one side, is the choice most likely to preserve good relations among the members of the group? Some theorists, it would seem, find these two questions easy to answer. Mr. Smith ought to have his way, and good relations are likely to be endangered if he does not; and these answers are equally valid whether the other three all prefer the same thing or prefer different things. Since, for the latter, the choice is a matter of indifference, it is both “more fair” and “more expedient” (less likely to lead to a quarrel) for the group to do what Mr. Smith prefers to do.
  @cite_21: The concept of preference intensity has been criticized over the past sixty years for having no substantive meaning. Much of the controversy stems from the inadequacy of measurement procedures. In reviewing the shortcomings of existing procedures, we identify three objectives for developing a satisfactory procedure: (1) the capability of validating expressed preference differences by actual choices among naturally occurring options, (2) compatibility with the existing problem structure, and (3) no confounding of extraneous factors in the measurement of preference intensity. Several recently developed measurement procedures are criticized for failing one or more of these objectives. We then examine three different approaches for measuring preference intensity based on multiple perspectives. Thereplication approach emerges as a promising way of satisfying the three objectives above. This methodology applies to problems where an attribute can be replicated by “parallel components” that are independent, identical copies of the attribute. We illustrate the approach with two applications reported in the decision analysis literature. We also offer guidance on how to construct parallel components satisfying the requisite properties.
  @cite_12: This article revisits the problem of preference intensity modelling by proposing and analysing the novel concept of preference intensity functions. These retain what are argued to be the most essential properties expected from the consistent numerical representation of preference intensity orderings and the ordinary preferences induced by them. Their existence is characterized by simple and clearly interpretable axioms, while they are also shown to be genuinely ordinally unique and more general than utility functions associated with utility-difference representations. The empirical content of this model is then pinned down by means of a testable, intuitive and comparatively weak necessary and sufficient condition on observable behavioural data. In line with recent developments in the literature, these data are assumed to include choices and an additional variable foregone resource with intensity-revealing potential such as response times, willingness to pay or desirability ratings.
summary: Attempts to exploit preference strength information have led to various approaches for modeling, eliciting, measuring, and aggregating people's preference intensities in a variety of fields, including Likert scales, semantic differential scales, sliders, constant sum paired comparisons, graded pair comparisons, response times, willingness to pay, vote buying, and many others (see @cite_26 @cite_21 @cite_12 for summaries). In our work we specifically consider only a small amount of coarse information about preference strengths, since obtaining detailed information is extremely difficult. Intuitively, any rule used to aggregate preference strengths must ask under what circumstances an apathetic majority' should win over a more passionate minority @cite_10 , and we provide a partial answer to this question when the objective is to minimize distortion.