Ahmet Alkan

FAIR ALLOCATION OF INDIVISIBLE GOODS AND CRITERIA OF JUSTICE

A set of n objects and an amount M of money is to be distributed among m people. Example: the objects are tasks and the money is compensation from a fixed budget. An elementary argument via constrained optimization shows that for M sufficiently large the set of efficient, envy free allocations is nonempty and has a nice structure. In particular, various criteria of justice lead to unique best fair allocations that are well behaved with respect to changes of M. This is in sharp contrast to the usual fair division theory with divisible goods. Copyright 1991 by The Econometric Society.

Stable schedule matching under revealed preference

In a recent study Baiou and Balinski [3] generalized the notion of two-sided matching to that of schedule matching which determines not only what partnerships will form but also how much time the partners will spend together. In particular, it is assumed that each agent has a ranking of the agents on the other side of the market. In this paper we treat the scheduling problem using the more general preference structure introduced by Blair [5] and recently refined by Alkan [1, 2], which allows among other things for diversity to be a motivating factor in the choice of partners. The set of stable matchings for this model turns out to be a lattice with other interesting structural properties.

The core of the matching game

In the matching game there are two sets of players P and Q and for each pair (p, q) there is a set of payoffs (up, v9) which can be obtained if the players p and q decide to collaborate. We give a constructive proof that the core of this game is nonempty and show that it has a strong connectedness property which may be thought of as a nonlinear generalization of convexity.

NONEXISTENCE OF STABLE THREESOME MATCHINGS

A counterexample shows stable matchings need not exist in societies where threesomes are to form. The stability theorem breaks down, in fact. for K-some formations for all E;z 3, even when preferences are restricted to be separable. Gale and Shapley (1962) proved stability is the rule for ‘marriage’, i.e., matchings of twosomes. This rule says given any society with two kinds of agents to be matched in pairs, each agent having a preference ordering on agents of the opposite kind, there exists at least one stable matching. In this note we show by example that stable matchings may fail to exist in societies with three kinds of agents to be matched in threesomes. The stability theorem breaks down, in fact, for K-some formations for all Kz 3, in quite an unexceptional way and even when preferences are restricted

On preferences over subsets and the lattice structure of stable matchings

Abstract. This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties.

Monotonicity and Envyfree Assignments

SummaryGiven any problem involving assignment of indivisible objects and a sum of money among individuals, there is an efficient envyfree allocation (namely the minmax money allocation) which can be extended monotonically to a new efficient envyfree allocation for any object added or individual removed, and another (the maximin value allocation) extendable similarly for any object removed or person added. Still, the efficient envyfree solution is largely incompatible with the resource and population monotonicity axioms: The minmax money and maxmin value allocations are unique in being extendable.

Existence and computation of matching equilibria

Abstract We give a constructive proof of existence of equilibrium in two-sided matching markets and also show the set of equilibrium prices is pathwise connected. For piecewise linear preferences, an (optimal assignment type) algorithm based on these results can compute an equilibrium in a finite number of steps and likewise reach the buyer-optimal minimum equilibrium prices.

On the properties of stable many-to-many matchings under responsive preferences

We show that the set of stable many-to-many matchings under responsive preferences is a complete distributive lattice but does not possess some of the other nice properties — monotonicity and strategyproofness — associated with the set of stable one-to-one matchings.

Equilibrium in a Matching Market with General Preferences

One of the earliest matching market models to appear was in David Gale’s The Theory of Linear Economic Models (1960) as an interpretation of the optimal assignment problem and its dual. This covered, as it is sometimes called, the ‘linear homogeneous case’ where a buyer’s utility for money is constant and independent of the object he may be assigned. The model has since been substantially generalised and shown to have various remarkable properties, some of which we will refer to below. Our purpose in this study is to give a proof of existence of competitive equilibrium in a matching market allowing rather general preferences, and then to construct a procedure whereby an equilibrium can be reached in an auction. The latter in fact generalises a ‘multi-item auction’ for the linear homogeneous case by Demange, Gale and Sotomayor (1986).

Pairing Games and Markets

Pairing Games or Markets studied here are the non-two-sided NTU generalization of assignment games. We show that the Equilibrium Set is nonempty, that it is the set of stable allocations or the set of semistable allocations, and that it has several notable structural properties. We also introduce the solution concept of pseudostable allocations and show that they are in the Demand Bargaining Set. We give a dynamic Market Procedure that reaches the Equilibrium Set in a bounded number of steps. We use elementary tools of graph theory and a representation theorem obtained here.