Uncovered set choice rules

I study necessary and sufficient conditions for a choice function to be rationalized in the following sense: there exists a total asymmetric relation T (a tournament) such that, for each feasible (finite) set, the choice set coincides with the uncovered set of T restricted to that feasible set. This notion of ‘maximization’ offers testable restrictions on observable choice behavior.     

Partially monotonic bargaining solutions

The purpose of the paper is to study partially monotonic solutions for two-person bargaining problems. Partial monotonicity relates to the uncertainty a player has about the solution before bargaining. If the minimum utility a player can expect is greater in game T than in game S, and if T contains more alternatives than S, this may bring him to expect that his utility at the solution is greater in T than in S. Partially monotonic solutions reflect these expectations.One partially monotonic solution is axiomatized. The axioms of symmetry and independence of linear transformations are not explicitly assumed, although the solution has also these properties. The Kalai-Smorodinsky solution is shown to be the only continuous partially monotonic solution.This study was financed by the Yrjö Jahnsson Foundation, which is gratefully acknowledged. I like to thank an associate editor and a referee for their valuable suggestions, and the Yrjö Jahnsson Foundation Study Group on Public Economics for useful discussions     

Distribution of coalitional power in randomized multi-valued social choice

This paper considers the distribution of coalitional influence under probabilistic social choice functions which are randomized social choice rules that allow social indifference by mapping each combination of a preference profile and a feasible set to a social choice lottery over all possible choice sets from the feasible set. When there are at least four alternatives in the universal set and ex-post Pareto optimality, independence of irrelevant alternatives and regularity are imposed, we show that: (i) there is a system of additive coalitional weights such that the weight of each coalition is its power to be decisive in every two-alternative feasble set; and (ii) for each combination of a feasible proper subset of the universal set and a preference profile, the society can be partioned in such a way that for each coalition in this partition, the probability of society's choice set being contained in the union of the best sets of its members is equal to the coalition's power or weight. It is further shown that, for feasible proper subsets of the universal set, the probability of society's choice set containing a pair of alternatives that are not jointly present in anyone's best set is zero. Our results remain valid even when the universal set itself becomes feasible provided some additional conditions hold. Received: 10 May 1999/Accepted: 18 June 2000 I would like to thank Professor Prasanta Pattanaik for suggesting to me the line of investigation carried out in this paper. I am solely responsible for any remaining errors and omissions.     

The structure of coalitional power under probabilistic voting procedures

We consider probabilistic voting procedures which map each feasible set of alternatives and each utility profile to a social choice lottery over the feasible set. It is shown that if we impose: (i) a probabilistic collective rationality condition known as regularity; (ii) probabilistic counterpart of Arrow's independence of irrelevant alternatives and citizens' sovereignty; (iii) a probabilistic positive association condition called monotonicity; then the coalitional power structure under a probabilistic voting procedure is characterized by weak random dictatorship. Received: 1 March 1999/Accepted: 21 May 2001     

An axiomatization of the Kalai-Smorodinsky solution when the feasible sets can be finite

We axiomatize the Kalai-Smorodinsky solution (1975) in the Nash bargaining problems if the feasible sets can be finite. We show that the Kalai-Smorodinsky solution is the unique solution satisfying Continuity (in the Hausdorff topology endowed with payoffs space), Independence (which is weaker than Nash's one and essentially equivalent to Roth (1977)'s one), Symmetry, Invariance (both of which are the same as in Kalai and Smorodinsky), and Monotonicity (which reduces to a little bit weaker version of the original if the feasible sets are convex). Received: 4 November 1999/Accepted: 6 June 2001     

A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger

Given a tournament T, a Banks winner of T is the first vertex of any maximal (with respect to inclusion) transitive subtournament of T. While Woeginger shows that recognizing whether a given vertex of T is a Banks winner is NP-complete, the computation of a Banks winner of T is polynomial, and more precisely linear with respect to the size of T.The article of G.J. Woeginger appeared in Soc Choice Welfare 20: 523–528 (2003)     

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.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Subcyclic sets of linear orders

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)<cⁿ for some c>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 n^th Catalan number and min(c)=4 for Raz’s bound.     

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     

On probabilistic rationalizability

This paper examines the problem of rationalizability of a choice system without imposing restrictions on the universal set of alternatives and the family of feasible sets. A choice system is the probabilistic counterpart of a choice function which is basic to much of the theory of individual choice behaviour in economics. We provide a necessary and sufficient condition for a choice system to be induced by a probability measure on a set of utility functions. I would like to thank the editor and two anonymous referees for helpful comments.     

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.     

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.     

THE CHALLENGE OF REVENUE SHARING WITH BUNDLED PRICING: AN APPLICATION TO MUSIC

Although bundling can substantially increase profits relative to standalone pricing, particularly for zero‐marginal‐cost information products, it has one major problem: bundling produces revenue that is not readily attributable to particular pieces of intellectual property, creating a revenue division problem. We evaluate several possible solutions using unique song valuation survey data. We find the Shapley value, a well‐motivated theoretical solution, is universally incentive compatible (all bundle elements fare better inside the bundle than under standalone pricing), but revenue‐sharing schemes feasible with readily available consumption data are not. Among feasible schemes, Ginsburgh and Zang's modified Shapley value performs best. (JEL C71, D79, L14)     

Majority rule in multi-dimensional spatial models

A sufficient condition is derived for a policy y to be a Condorcet winner, when the set of feasible policies is some subsety ofR ^N with non-empty interior. Voters are assumed to differ in some scalar characteristic w. The sufficient condition refers only to voters' preferences over the set of preferred policies (of the various other voters). This set is a one-dimensional curve inR ^N> . The condition is that the indifference curves of each typew through the preferred policyy ^* (v) of any typev all be collinear. If the condition holds, then the preferred policyy ^* (m) of the median type of voter will be a Condorcet winner. If there are only three types of voter, then the above condition is also necessary for the existence of a Condorcet winner.This paper is a revision of the first part of my earlier Majority Rule with Multidimensional Economic Choices. I thank Ted Bergstrom, David Donaldson, and Greg Dow for very valuable comments on versions of that earlier paper, without implicating them in this current incarnation.     

Identical preferences lower bound solution and consistency in economies with indivisible goods

We consider the problem of allocating a finite set of indivisible goods and a single infinitely divisible good among a group of agents, and we study a solution, called the Identical Preferences Lower Bound solution, in the presence of consistency properties. This solution is not consistent. We prove that its maximal consistent subsolution is the No-envy solution. Our main result is that the minimal consistent extension of the intersection of the Identical Preferences Lower Bound solution with the Pareto solution is the Pareto solution. This result remains true in the restricted domain when all the indivisible goods are identical, but not when there is a unique indivisible good.This paper was developed during my stay at Rochester University in the summer of 1992. I would like to express my special thanks to Professor William Thomson for all his help and advice. Iam also grateful to my supervisor Luis Corchón, to Koichi Tadenuma and to the anonymous referees for their helpful comments. The remaining errors are my exclusive responsibility. Financial support from the DGCYT under project PB 91-0756 and the Instituto Valenciano de Investigaciones Económicas are gratefully acknowledged.     

Resource-monotonic solutions to the problem of fair division when preferences are single-peaked

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We search for solutions satisfying resource-monotonicity, the requirement that all agents be affected in the same direction when the amount to divide changes. Although there are resource-monotonic selections from the Pareto solution, there are none satisfying the distributional requirements of no-envy or individual rationality from equal division. We then consider the weakening of resource-monotonicity obtained by allowing only changes in the amount to divide that do not reverse the direction of the inequality between the amount to divide and the sum of the preferred amounts. We show that there is essentially a unique selection from the solution that associates with each economy its set of envy-free and efficient allocations satisfying this property of one-sided resource-monotonicity: it is the uniform rule, a solution that has played a central role in previous analyses of the problem.     

Core implementation via dynamic game forms

Implementation of a social choice correspondence (SCC) involves finding a decentralized choice mechanism for which the solution is a manifestation of the SCC. We examine whether dynamic game forms with corelike solutions are feasible candidates for implementation. When the solution is the -, -, or intermediate-cores respectively, such feasibility is found to require a choice structure which embeds a sequential dictatorship, in turn implying the SCC has a partial dictator. Implementation via the strong equilibrium set is found feasible iff society consists of a single citizen. Tapering the range of permissible utility profiles to include only von Neumann-Morgenstern utilities does little to rectify these impossibility results. Effectivity functions are used as analytical tools and several new effectivity function concepts are introduced.The author is grateful for comments by Tatsuro Ichiishi and B. Kelly Eakin. Errors or shortcomings are the sole responsibility of the author.     

On randomized rationality

In contrast to the traditional notion of rationalizability of stochastic choice which requires the preference relations to be strict, we propose a notion of rationalizability without requiring the preference relations to be strict. Our definition is based on the simple hypothesis of a two-stage choice process: stage (i) a preference relation R is drawn according to a probability assignment; stage (ii) an alternative is picked from each feasible set according to a uniform lottery over the R-greatest set in it. We provide a necessary and sufficient condition for rationalizability of stochastic choice. Since our framework is general enough to subsume the traditional case, our result also provides an alternative characterization of the traditional notion of rationalizability. We also show the equivalence between the two notions of rationalizability in a specific case.     

A theory of policy differentiation in single issue electoral politics

Voter preferences are characterized by a parameter s (say, income) distributed on a set S according to a probability measure F. There is a single issue (say, a tax rate) whose level, b, is to be politically decided. There are two parties, each of which is a perfect agent of some constituency of voters, voters with a given value of s. An equilibrium of the electoral game is a pair of policies, b ₁ and b ₂, proposed by the two parties, such that b _i maximizes the expected utility of the voters whom party i represents, given the policy proposed by the opposition. Under reasonable assumptions, the unique electoral equilibrium consists in both parties proposing the favorite policy of the median voter. What theory can explain why, historically, we observe electoral equilibria where the ‘right’ and ‘left’ parties propose different policies? Uncertainty concerning the distribution of voters is introduced. Let {F(t)} _{t ε T} be a class of probability measures on S; all voters and parties share a common prior that the distribution of t is described by a probability measure H on T. If H has finite support, there is in general no electoral equilibrium. However, if H is continuous, then electoral equilibrium generally exists, and in equilibrium the parties propose different policies. Convergence of equilibrium to median voter politics is proved as uncertainty about the distribution of voter traits becomes small.