Egalitarian-equivalence and the Pareto principle for social preferences

The Pareto principle is often in conflict with the equity criteria in construction of social preferences: there exist two allocations x and y such that x Pareto dominates y, but y is an equitable allocation whereas x is not. The efficiency-first principle requires to rank an allocation x higher than y if (i) x Pareto dominates y or (ii) x and y are Pareto-noncomparable and x is equitable whereas y is not. The equity-first principle reverses the order of application of the two criteria. Adopting egalitarian-equivalence as the notion of equity, we examine rationality of the social preference functions based on the efficiency-first or the equity-first principle. The degrees of rationality vary widely depending on which principle is adopted, and depending on the range of egalitarian-reference bundles. We show several impossibility and possibility results as well as a characterization of the social preference function introduced by Pazner and Schmeidler (1978). We also identify the sets of maximal elements of the social preference relations in the set of feasible allocations. The results are contrasted with those in the case where no-envy is the notion of equity.This paper is a much extended version of a part of Tadenuma (1998). The author is grateful to Marc Fleurbaey, Serge-Chiristophe Kolm, Kotaro Suzumura, and William Thomson for helpful comments on the earlier version of the paper. ‘‘Financial support from the Ministry of Education, Culture, Sports, Science and Technology of Japan for the 21st Century Center of Excellence Project on the Normative Evaluation and Social Choice of Contemporary Economic Systems is gratefully acknowledged’’.     

Comparison functions and choice correspondences

In this paper, we introduce the concept of a comparison function, which is a mapping g that assigns numbers to ordered pairs of alternatives (x,y) with the property that g(x,y)=−g(y,x). The paper discusses how some well-known choice correspondences on tournaments such as the uncovered set, the minimal covering set and the bipartisan set can be extended to this general framework. Axiomatic characterizations and properties are studied for these correspondences. Received: 4 November 1996/Accepted: 13 April 1998     

Does the Borda rule provide more than a ranking?

Consider the following problem: a set of candidates {x, y, z} has to be ranked from best to worse by a committee. Each member of the committee provides his own ranking of the three candidates and you decide to use the Borda method to aggregate the rankings. The resulting scores are as follows: 107 for x, 106 for y and 51 for z. Would you conclude that x is better than y? Probably not, because the difference between the scores of x and y is small. The only conclusion you would draw is that z definitely is the worst candidate. But, is it meaningful to consider differences of Borda scores? We characterize the Borda method in this new framework and find conditions that are very close to those characterizing the classical Borda method. Throughout our paper, we consider a generalization of the Borda method designed to aggregate fuzzy relations. Received: 2 March 1998/Accepted: 5 May 1999     

A quantitative discursive dilemma

The typical judgment aggregation problem in economics and other fields is the following: a group of people has to judge/estimate the value of an uncertain variable y, which is a function of k other variables, i.e., y = D(x ₁, . . . , x _k ). We analyze when it is possible for the group to arrive at collective judgements on the variables that respect D. We consider aggregators that fulfill Arrow’s IIA-condition and neutrality. We show how possibility and impossibility depend on the functional form of D, and generalize Pettit’s (2001) binary discursive dilemma to quantitative judgements.     

Positional information and preference aggregation

A new version of independence (I⁺) is proposed for social welfare functions based on the following notion of agreement. Two weak orders R and R’ on a finite set S agree on a pair {x,y}, denoted byif R|_{x,y} = R’|_{x,y} and [z R*x and z R*y for some z∈S] if and only if [z’ (R’)* x and z’(R’)*y for some z’∈S]. The last part says that x and y are strictly under z with respect to R exactly when x and y are strictly under z’ with respect to R’. Some examples and results on social welfare functions that satisfy (I⁺), Pareto, and nondictatorship are given.I am grateful for the comments and suggestions made by an anonymous referee on an earlier version of this paper.     

An extension of a theorem on the aggregation of separable preferences

An Excess-Voting Function relative to a profile π assigns to each pair of alternatives (x,y), the number of voters who prefer x to y minus the number of voters who prefer y to x. It is shown that any non-binary separable Excess-Voting Function can be achieved from a preferences profile when individuals are endowed with separable preferences. This result is an extension of Hollard and Le Breton (1996). Received: 16 December 1996 / Accepted: 8 October 1997     

Choosing from a tournament

A tournament is any complete asymmetric relation over a finite set A of outcomes describing pairwise comparisons. A choice correspondence assigns to every tournament on A a subset of winners. Miller's uncovered set is an example for which we propose an axiomatic characterization. The set of Copeland winners (outcomes with maximal scores) is another example; it is a subset of the uncovered set: we note that it can be a dominated subset. A third example is derived from the sophisticated agenda algorithm; we argue that it is a better choice correspondence than the Copeland set.     

MEASURING EFFORT INCENTIVES IN A TOURNAMENT WITH MANY PARTICIPANTS: THEORY AND APPLICATION

This article provides an empirical measure to compare incentive effects of different tournament structures. Although there have been many theoretical attempts to analyze incentive effects of a tournament competition, empirical applications are limited as the provided conditions cannot easily be empirically observed. We show that the empirical density of the performance distribution can be used to measure effort incentives in a tournament with many participants. We apply this method to evaluate the grading policy change in the 2008 College Scholastic Ability Test in South Korea. (JEL C70, C51)     

Choosing from a large tournament

A tournament can be viewed as a majority preference relation without ties on a set of alternatives. In this way, voting rules based on majority comparisons are equivalent to methods of choosing from a tournament. We consider the size of several of these tournament solutions in tournaments with a large but finite number of alternatives. Our main result is that with probability approaching one, the top cycle set, the uncovered set, and the Banks set are equal to the entire set of alternatives in a randomly chosen large tournament. That is to say, each of these tournament solutions almost never rules out any of the alternatives under consideration. We also discuss some implications and limitations of this result.     

Banks winners in tournaments are difficult to recognize

Given a tournament T, a Banks winner of T is the top vertex of any maximal (with respect to inclusion) transitive subtournament of T. In this technical note, we show that the problem of deciding whether some fixed vertex v is a Banks winner for T is NP-complete. Received: 22 February 2002/Accepted: 20 June 2002 Supported by the START program Y43-MAT of the Austrian Ministry of Science. I would like to thank two thank the referees for a careful reading of the paper, for helpful remarks, and for many suggestions how to improve the presentation.     

Smallest tournaments not realizable by $${\frac{2}{3}}$$-majority voting

Define the predictability number α(T) of a tournament T to be the largest supermajority threshold for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability ; Gilboa has found a tournament on 54 alternatives (i.e. vertices) that has predictability less than , and has asked whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability , and prove that it is the smallest tournament with predictability <  . Our methodology is to formulate the problem as a finite set of two-person zero-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic. D. Shepardson was supported by a NSF Graduate Research Fellowship during the course of this work.     

A smallest tournament for which the Banks set and the Copeland set are disjoint

Given a tournament T, a Banks winner of T is the first vertex of any maximal (with respect to inclusion) transitive subtournament of T; a Copeland winner of T is a vertex with a maximum out-degree. In this paper, we show that 13 is the minimum number of vertices that a tournament must have so that none of its Copeland winners is a Banks winner: for any tournament with less than 13 vertices, there is always at least one vertex which is a Copeland winner and a Banks winner simultaneously. Received: 2 May 1997 / Accepted: 30 September 1997     

A computational analysis of the tournament equilibrium set

A recurring theme in the mathematical social sciences is how to select the “most desirable” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard, and thus does not admit a polynomial-time algorithm unless P equals NP. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ.     

Manipulability, unanimity, anonymity and merging functions

A merging (or assessment aggregation) function (see [7]) is a rule that synthesizes several individual assessments, or numerical judgements, by creating a unique “collective” assessment. Individual i can manipulate assessment aggregation if the change in the merged assessment due to a change in i's assessment from x to y depends solely on x and y. Some assumptions on the functional form capturing this dependence are put forward and their effects on aggregation investigated under unanimity and anonymity conditions using a functional equation approach. Attention is restricted to three types of results: the merging function does not exist; it is the arithmetic mean; it is dictatorial. Received: 21 November 1997/Accepted: 31 May 1999     

THE OPTIMAL ALLOCATION OF PRIZES IN TOURNAMENTS OF HETEROGENEOUS AGENTS

Tournaments are widely used in organizations, explicitly or implicitly, to reward the best‐performing employees, for example, by promotion or bonuses, and/or to penalize the worst‐performing employees, for example, by demotion, withholding bonuses, or unfavorable job assignments. These incentive schemes can be interpreted as various prize allocations based on the employees' relative performance. While the optimal prize allocation in tournaments of symmetric agents is relatively well understood, little is known about the impact of the allocation of prizes on the effectiveness of tournament incentive schemes for heterogeneous agents. We show that while multiple prize allocation rules are equivalent when agents are symmetric in their ability, the equivalence is broken in the presence of heterogeneity. Under a wide range of conditions, loser‐prize tournaments, that is, tournaments that award a low prize to relatively few bottom performers, are optimal for the firm. The reason is that low‐ability agents are discouraged less in such tournaments, as compared to winner‐prize tournaments awarding a high prize to few top performers, and hence can be compensated less to meet their participation constraints. (JEL M52, J33, J24)     

A counterexample to a conjecture of Schwartz

In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice versa. But in fact there is such a tournament, as we show in this article, and so Schwartz’ conjecture is false. Our proof is non-constructive and uses the probabilistic method.     

A DESIGNER'S CHOICE BETWEEN SINGLE‐PRIZE AND PARALLEL TOURNAMENTS

In this article, I theoretically and experimentally compare a designer's profits from two tournament designs. The first design is a standard winner‐take‐all tournament with a single prize. The second design features two winner‐take‐all (parallel) tournaments with different prizes where individuals choose which tournament to enter before competing. I develop a simple model that illustrates how the relative performances of these designs change as contestants' abilities differ. The theoretical model shows that the designer's profit is higher (lower) in the parallel tournament when contestants' abilities differ greatly (are similar). I complement these findings with experimental evidence. The experiments show that the parallel tournament is more profitable under high heterogeneity, whereas under low heterogeneity, the designer is better off with the single‐prize tournament. Furthermore, high‐ability agents under‐participate and low‐ability agents over‐participate in the high‐prize tournament relative to the theoretical prediction. (JEL C72, D82, J33, M51, M52)     

Affirmative Action Can Increase Effort

Previous theoretical work examining labor tournaments concluded that an affirmative action program will always reduce the effort supplied by agents, thereby reducing output and profit for the tournament administrator; however, experimental results sometime contradict this conclusion. In the context of a labor tournament I demonstrate that there exists an affirmative action program that induces both types of agents to provide greater effort. In some instances the effort maximizing affirmative action program will also give both types of agents an equal chance of winning the tournament.

Improvement indices of well-being

In recent years, various classes of indices measuring the improvement of well-being have been suggested. Formally, an improvement index is a function Q* (x ₁, x ₂), where x _t is the level of some attribute of well-being, e.g., life expectancy, in period t. This paper extends previous works on improvement indices by introducing a class of multidimensional indices which aggregates the improvement of a group of attributes of well-being. We propose a set of axioms which leads inexorably to a multidimensional generalization of Kakwani's class of improvement indices of well-being (Kakwani 1993).Financial support from the Social Science and Education Panel of the Chinese University of Hong Kong (research grant account no: 220201620) is gratefully acknowledged     

THE IMPACT OF MANAGERIAL CHANGE ON PERFORMANCE: THE ROLE OF TEAM HETEROGENEITY

When a key responsibility of a manager is to allocate more or less attractive tasks, subordinates have an incentive to work hard and demonstrate their talents. As a new manager is less well informed, management dismissals reinvigorate this tournament competition—but only in sufficiently homogeneous teams. We investigate this hypothesis using a large dataset on dismissals of soccer coaches, whose main task is indeed the selection of players. We find that dismissals enhance performance (only) in homogeneous teams. Moreover, we show that there is typically a negative selection bias when evaluating succession effects, which reconciles previous contradictory findings. (JEL D22, J44, J63)