Law Of Aggregate Demand Research Articles

Cycles to compute the full set of many-to-many stable matchings

In a many-to-many matching model in which agents’ preferences satisfy substitutability and the law of aggregate demand, we present an algorithm to compute the full set of stable matchings. This algorithm relies on the idea of “cycles in preferences” and generalizes the algorithm presented in Roth and Sotomayor (1990) for the one-to-one model.

A cumulative offer process for supply chain networks

We introduce a version of the Cumulative Offer Algorithm of Hatfield and Milgrom (Am Econ Rev 95(4):913–935, 2005) modified for the supply chain networks setting of Ostrovsky (Am Econ Rev 98(3):897–923, 2008). The algorithm provides an alternative proof for existence of a chain stable network. Moreover, we show that when choice functions satisfy same-side substitutes and cross-side complements, and the Law of Aggregate Demand and Supply, then contracts satisfy Irrelevance of Rejected Contracts condition, a condition implicitly assumed in Ostrovsky (2008).

Binary Operations for the Lattice Structure in a Many-to-Many Matching Model

The lattice structure of the set of stable matchings in many-to-many matching model is well known in literature. If preferences of the agents are substitutable, this result can be obtained by fixed-point methods, for that purpose an algorithm for finding a fixed-point matching is defined. Since the fixed-point set equals the set of stable matchings, the latter has a lattice structure too. In this paper, we consider a many-to-many matching market where the preferences of firms satisfy substitutability and the law of aggregate demand, and workers have responsive preferences. In this many-to-many matching market, we explicitly compute for any pair of stable matchings the least upper bound and the greatest lower bound, without using fixed-point methods.

Random path to stability in a decentralized market with contracts

For a many-to-many matching model with contracts in which the preferences of all hospitals satisfy substitutability and the preferences of all doctors satisfy substitutability, the law of aggregate demand and q-congruence, we show the existence of a convergent blocking path. In other words, we start from an arbitrary allocation and build a finite sequence of allocations leading to a stable outcome, with the special feature that each allocation can be obtained from the previous one by satisfying a unilateral or a bilateral blocking contract. As a consequence, we prove that the process of allowing randomly selected blocks to be satisfied eventually leads to a stable outcome. This explains the fact that some markets with contracts reach stable assignments by means of decentralized decisions.

On stable and strategy-proof rules in matching markets with contracts

This paper studies stable and (one-sided) strategy-proof rules in many-to-one matching markets with contracts. Not assuming any kind of substitutes condition or the law of aggregate demand, we obtain the following results. First, the number of stable and strategy-proof rules is at most one. Second, the doctor-optimal stable rule, whenever it exists, is the unique candidate for a stable and strategy-proof rule. Third, a stable and strategy-proof rule, whenever it exists, is second-best optimal for doctor welfare, in that no individually rational and strategy-proof rule can dominate it. This last result is further generalized to non-wasteful and strategy-proof rules. Due to the weak assumptions, our analysis covers a broad range of markets, including cases where a (unique) stable and strategy-proof rule is not equal to the one induced by the cumulative offer process or the deferred acceptance algorithm.

Equivalence of Pairwise Stability, Set-Wise Stability and the Core under Dynamic Reserves Choice Functions

In this paper I show that the pairwise and set-wise stable allocations in many-to-one matching problems with contracts are equivalent under the choice procedures of schools that are constructed by dynamic reserves. I also show that the set of set-wise stable allocations is equal to the core. Hence, three seemingly different solution concepts coincide under the family of dynamic reserves choice functions for schools that fails the unilateral substitutability and the law of aggregate demand. Since many significant choice procedures developed in practical matching markets fall in this family of choice procedures, the equivalence provide us a way to express set-wise stability notion by the simpler pairwise stability for these real-life problems. It also makes it easier to define stability normatively for specific government goals.

The blocking lemma and group incentive compatibility for matching with contracts

This paper considers a general class of two-sided many-to-one matching markets, so-called matching markets with contracts. We study the blocking lemma and group incentive compatibility for this class of matching markets. We first show that the blocking lemma for matching with contracts holds if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. The blocking lemma for one-to-one matching (Gale and Sotomayor, 1985) and that for many-to-one matching (Martínez et al., 2010) are special cases of this result. Then, as an immediate consequence of the blocking lemma, we show that the doctor-optimal stable mechanism is group strategy-proof for doctors if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. Hatfield and Kojima (2009) originally obtain this result by skillfully using the strategy-proofness of the doctor-optimal stable mechanism. In this paper we provide a different proof for the group incentive compatibility by applying the blocking lemma.

On Stable and Strategy-Proof Rules in Matching Markets with Contracts

This paper studies stable and (one-sided) strategy-proof rules in many-to-one matching markets with contracts. Not assuming any kind of substitutes condition or the law of aggregate demand, we obtain the following results. First, the number of stable and strategy-proof rules is at most one. Second, the doctor-optimal stable rule, whenever it exists, is the unique candidate for a stable and strategy-proof rule. Third, a stable and strategy-proof rule, whenever it exists, is second-best optimal for doctor welfare, in that no individually-rational and strategy-proof rule can dominate it. This last result is further generalized to non-wasteful and strategy-proof rules. Due to the weak assumptions, our analysis covers a broad range of markets, including cases where a (unique) stable and strategy-proof rule is not equal to the one induced by the cumulative offer process or the deferred acceptance algorithm.

Quantile Stable Mechanisms

We construct quantile stable mechanisms, show that they are distinct in sufficiently large markets, and analyze how they can be manipulated by market participants. As a step to showing that quantile stable mechanisms are well defined, we show that median and quantile stable matchings exist when contracts are strong substitutes and satisfy the law of aggregate demand. This last result is of independent interest as experiments show that agents who match in a decentralized way tend to coordinate on the median stable matching when it exists.

Quantile Stable Matchings for Markets with Contracts

We study manipulability of stable matching mechanisms and show that manipulability comparisons are equivalent to preference comparisons: for any agent, a mechanism is more manipulable than another if and only if this agent prefers the latter to the former. In particular, this implies that no two stable mechanisms can be ranked in terms of manipulability for all agents. We then construct quantile stable mechanisms, establish conditions under which they are well defined, and observe that they can be ranked in terms of manipulability for one side of the market, but not for both. In particular, the typically used mechanism that selects the stable matching that is best for one side of the market has no incentive advantage over other quantile stable mechanisms when both sides of the market are strategic. As an auxiliary result, we provide sufficient conditions for the existence of median stable matchings, and more generally, show that quantile stable matchings exist when contracts are strong substitutes and satisfy the law of aggregate demand. This last result is of independent interest as experiments show that agents who match in a decentralized way tend to coordinate on the median stable matching, when it exists.

The Stable Roommates Problem with Choice Functions

The stable marriage theorem of Gale and Shapley states that for n men and n women there always exists a stable marriage scheme, that is, a set of marriages such that no man and woman mutually prefer one another to their partners. The stable marriage theorem was generalized in two directions: the stable roommates problem is the “one-sided” version, where any two agents on the market can form a partnership. The generalization by Kelso and Crawford is in the “two-sided” model, but on one side of the market agents have a so-called substitutable choice function, and stability is interpreted in a natural way. It turned out that even if both sides of the market have substitutable choice functions, there still exists a stable assignment. The latter version contains the “many-to-many” model where up to a personal quota, polygamy is allowed for both men and women in the two-sided market. The goal of this work is to solve the stable partnership problem, a generalization of the one-sided model with substitutable choice functions. We do not quite reach that: besides substitutability, we also need for the result the so called increasing property (or, in other terminology, the law of aggregate demand). Luckily, choice functions in well-known stable matching theorems comply with this property. The main result is a generalization of Irving’s algorithm, the first efficient method to solve the stable roommates problem. We deduce from the algorithm a generalization of Tan’s result on characterizing the existence of a stable matching and prove a generalization of the rural hospital theorem and the so-called splitting property of many-to-many stable matchings. We show that our algorithm is linear-time in some sense and indicate that for general (i.e. not necessarily increasing) substitutable choice functions the stable partnership problem is NP-complete.

Group incentive compatibility for matching with contracts

Hatfield and Milgrom [Hatfield, John William, Milgrom, Paul R., 2005. Matching with contracts. Amer. Econ. Rev. 95, 913–935] present a unified model of matching with contracts, which includes the standard two-sided matching and some package auction models as special cases. They show that the doctor-optimal stable mechanism is strategy-proof for doctors if hospitals' preferences satisfy substitutes and the law of aggregate demand. We show that the doctor-optimal stable mechanism is group strategy-proof for doctors under these same conditions. That is, no group of doctors can make each of its members strictly better off by jointly misreporting their preferences. We derive as a corollary of this result that no individually rational allocation is preferred by all the doctors to the doctor-optimal stable allocation.

The law of aggregate demand and welfare in the two-sided matching market

In the college admission problem, we show that the student-optimal stable matching is weakly Pareto optimal for students if colleges' preferences satisfy substitutability and the law of aggregate demand. We also show that both of these properties are important for the result.

Matching with Contracts

We develop a model of matching with contracts which incorporates, as special cases, the college admissions problem, the Kelso-Crawford labor market matching model, and ascending package auctions. We introduce a new “law of aggregate demand” for the case of discrete heterogeneous workers and show that, when workers are substitutes, this law is satisfied by profit-maximizing firms. When workers are substitutes and the law is satisfied, truthful reporting is a dominant strategy for workers in a worker-offering auction/matching algorithm. We also parameterize a large class of preferences satisfying the two conditions.