Koichi Tadenuma

NO-ENVY AND CONSISTENCY IN ECONOMIES WITH INDIVISIBLE GOODS

The authors study fair allocation in economies with indivisible goods. They look for subsolutions of the no-envy solution satisfying the property of consistency which says that the desirability of an allocation for some economy is unaffected by the departure of some of the agents with their allotted bundles. The authors show that essentially there is no proper subsolution of the no-envy solution satisfying consistency. However, many subsolutions satisfy a bilateral version of the condition and many satisfy its converse. But again, there is essentially no proper subsolution satisfying bilateral consistency and converse consistency together. Copyright 1991 by The Econometric Society.

The fair allocation of an indivisible good when monetary compensations are possible

Abstract We consider the problem of allocating a single indivisible good to one of several agents when monetary compensations to the losers are possible. We look for solutions to this problem satisfying the property of consistency , which says that the recommendation made for any problem is never contradicted by the recommendation made for any reduced problem obtained by imagining the departure of some of the agents with their allotted bundles. We show that any subsolution of the no-envy solution satisfying consistency and a minor neutrality condition contains the solution that systematically picks the allocation in the envy-free set the most unfavorable to the winner. This solution is also well-behaved when the amount of money available varies, and it can also be characterized on the basis of a condition of population monotonicity , which says that the arrival of additional agents affects all agents initially present in the same direction.

Efficiency First or Equity First? Two Principles and Rationality of Social Choice

The Pareto efficiency criterion is often in conflict with the equity criteria as no-envy or as egalitarian-equivalence: An allocation x that is Pareto superior to another allocation y can be inferior to y in consideration of equity. This paper formalizes two differnet principles of social choice under possible conflict of efficiency and equity. The efficiency-first principle requires that we should always select from efficient allocations, and when the efficiency criterion is not at all effective as a guide for selection, i.e., when all the available allocations are efficient or there is no efficient allocation, we should apply an equity criterion to choose desirable allocations. The equity-first principle reverses the lexicographic order of application of the two criteria. We examine rationality of the social choice rules satisfying these two principles. It is shown that the degree of rationality varies widely depending on which principle the social choice rules represent. Several impossibility and possibility results as well as a characterization theorem are obtained.

Reduced games, consistency, and the core

This paper establishes an axiomatization of the core by means of an internal consistency property with respect to a new reduced game introduced by Moulin (1985). Given a payoff vector chosen by a solution for some game, and given a subgroup of agents, we define thereduced game as that in which each coalition in the subgroup could attain payoffs to its members only if they are compatible with the initial payoffs toall the members outside of the subgroup. The solution isconsistent if it selects the same payoff distribution for the reduced game as initially. We show that consistency together with individual rationality characterizes the core of both transferable and non-transferable utility games.

Refinements of the no-envy solution in economies with indivisible goods

We consider the problem of fair allocation in economies with indivisible goods. Our primary concept is that of an envy-free allocation, that is, an allocation such that no agent would prefer anyone elses bundle to his own. Since there typically is a large set (a continuum) of such allocations, the need arises to identify well-behaved selections from the no-envy solution. First we establish the non-existence of ‘population monotonic’ selections. Then we propose a variety of selections motivated by intuitive considerations of fairness.

Lexicographic Compositions of Multiple Criteria for Decision Making

This paper considers two distinct procedures to lexicographically compose multiple criteria for social or individual decision making. The first procedure composes M binary relations into one, and then selects its maximal elements. The second procedure first selects the set of maximal elements of the first binary relation, and then within that set, chooses the maximal elements of the second binary relation, and iterates the procedure until the Mth binary relation. We show several distinct sets of conditions for the choice functions representing these two procedures to satisfy non-emptiness and choice-consistency conditions such as contraction consistency and path independence. We also examine the relationships between the outcomes of the two procedures. Finally we investigate under what conditions the outcomes of each procedure are independent of the order of lexicographic application of the criteria.

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.

Implementable stable solutions to pure matching problems

Abstract We consider “pure” matching problems, where being unmatched (“being single”) is not a feasible choice or it is always the last choice for every agent. We show that there exists a proper subsolution of the stable solution that is implementable in Nash equilibria. Moreover, if the number of men | M | and the number of women | W | are less than or equal to 2, then any subsolution of the stable solution is implementable. However, if | M |=| W |≥3, there exists no implementable single-valued subsolution of the stable solution. All these results should be contrasted with the results in the recent literature on the matching problems with a single status.

Trade-off between equity and efficiency in a general economy with indivisible goods

In most of the recent literature on fair allocation in economies with indivisible goods and a single infinitely divisible good, it is assumed that each agent can consume at most one indivisible good. In this paper, we show that if this assumption is dropped, there do not necessarily exist envy-free and Pareto efficient allocations. However, envy-free allocations still exist and so do Pareto efficient allocations. Hence, a trade-off between equity and efficiency arises.

Choice-Consistent Resolutions of the Efficiency-Equity Trade-Off

In a standard framework of choice theory, we formulate two contrasting principles for social choice under the efficiency-equity trade-off. The equity-first principle states that we should select from equitable allocations if any, but if the equity criterion is not at all effective for selection either because all the available allocations are equitable or because no allocation is equitable, we should select from Pareto efficient allocations. The efficiency-first principle switches the roles of the equity criterion and the efficiency criterion above. We examine the choice-consistency properties, known as Path Independence (Arrow, 1963) and Contraction Consistency (Chernoff, 1954), of the social choice correspondences satisfying the equity-first or the efficiency-first principle. Several possibility and impossibility theorems are obtained, which indicate that possibility of consistent social decisions depends crucially on which principle we take as well as what is the precise notion of equity.