Revealed Preferences for Matching with Contracts

08/23/2019 ∙ by Daniel Lehmann, et al. ∙ Hebrew University of Jerusalem 0

Many-to-many matching with contracts is studied in the framework of revealed preferences. All preferences are described by choice functions that satisfy natural conditions. Under a no-externality assumption individual preferences can be aggregated into a single choice function expressing a collective preference. In this framework, a two-sided matching problem may be described as an agreement problem between two parties: the two parties must find a stable agreement, i.e., a set of contracts from which no party will want to take away any contract and to which the two parties cannot agree to add any contract. On such stable agreements each party's preference relation is a partial order and the two parties have inverse preferences. An algorithm is presented that generalizes algorithms previously proposed in less general situations. This algorithm provides a stable agreement that is preferred to all stable agreements by one of the parties and therefore less preferred than all stable agreements by the other party. The number of steps of the algorithm is linear in the size of the set of contracts, i.e., polynomial in the size of the problem. The algorithm provides a proof that stable agreements form a lattice under the two inverse preference relations. Under additional assumptions on the role of money in preferences, agreement problems can describe general two-sided markets in which goods are exchanged for money. Stable agreements provide a solution concept, including prices, that is more general than competitive equilibria. They satisfy an almost one price law for identical items. The assignment game can be described in this framework and core elements of an assignment game are the stable agreements.

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1 Introduction

Matching is a topic of intense research activity and has provided many deep and beautiful theoretical results, solutions to practical problems of great importance and original conceptual insights into the description of preferences, equilibrium theory, social choice theory and the conception of auction mechanisms. It is not possible to survey here even a small part of this activity. It is only those results directly related to the present work that will be mentioned below.

Gale and Shapley [11] described one-to-one and a restricted form of one-to-many matching problems. They showed that the existence of a stable solution was guaranteed for all possible preferences of the agents. They also showed that, in the one-to-one situation there is, for each of the two sides, a stable solution that is optimal for it and that, in the one-to-many situation there is such a stable solution that is optimal for the side interested in a single agent (the doctors’ side). In [17] Knuth attributes to J.H. Conway the remark that the stable solutions to the one-to-many matching problems above form a lattice under a suitable partial ordering. Kelso and Crawford [16] considered a more general situation: the job market, matching firms with teams of workers that receive a salary, whereas a worker can work for only one firm. A discussion of a possible lattice structure for stable solutions to the Kelso-Crawford problem can be found in [26, 28]. Blair [7] considered a generalized job market in which workers can contract with many firms and showed that, even in such a many-to-many matching situation the stable solutions present a lattice structure. The study of many-to-many matchings has been actively pursued since, see, for example, [10].

All the studies mentioned above assume a full description of the preferences of the agents, by a total ordering on the set of all subsets of alternatives. The stream of research in economic theory, initiated by Samuelson [30, 31], developed by [34] and brought to matching theory in [26, 28, 3] proposed to assume only preferences revealed by the agents’ actions. The agents’ preferences are represented by choice functions that pick, out of a set of possible choices, the one preferred.

In their seminal article [15], Hatfield and Milgrom proposed a fundamental shift of focus. Instead of focusing on the preferences that agents on one side have for agents on the other side (or for groups of agents on the other side) they suggested that one should focus on the preferences of the agents on subsets of a set of contracts, where each contract links two agents on opposite sides. They achieved two goals: they broke out of the limited framework of the job market in which only a single salary parameter was considered, and they opened the way for considering preferences that were not preferences on the agents of the other side. For them, choice functions that satisfy a substitutes property describe the preferences of hospitals over sets of doctors, not utility functions. Many researchers pursued the paths opened by [15] and many of them will be mentioned below.

2 About this paper

2.1 Goals and summary

This paper’s goal is to generalize the one-to-many matching with contracts situation of [15] to a many-to-many situation and to study the structure of the stable solutions in such a situation. It can be compared to [10], which does not consider contracts.

In this paper, preferences of the agents over sets of contracts are specified by choice functions without any assumption on the structure of the set of contracts. A set of three conditions on choice functions guarantees

  1. that the aggregation of the choice functions of all agents on the same side satisfies the same conditions and represents the collective preferences of the side in question, thus reducing the many-to-many situation to a one-to-one situation,

  2. that a natural partial order can be defined on stable solutions,

  3. that the set of stable solutions is always a non-empty lattice under this partial order, and

  4. that a simple, elegant algorithm, expressed in terms of two choice functions, that generalizes all previously described matching algorithms, always finds an extremal stable solution.

General two-sided markets can be described in this framework by postulating that consumers prefer low prices and producers high prices, and that none of them cares with whom he or she trades. A Law of two prices is proved: identical goods are traded at an essentially uniform price. An example of such a market is the assignment game of [35]. The core elements of the game are exactly the stable solutions of the market, putting in evidence a proximity between two of Shapley’s works, [11] and [35] that had been sensed in the latter work but could not be formalized.

2.2 Plan of this paper

This paper develops a significant mathematical apparatus, but its purpose is mainly conceptual. Therefore it stresses the change of perspective it proposes on matching and delays the mathematical results as much as possible. Section 3 discusses the representation of preferences over subsets of a given set. Section 4 defines the set of coherent choice functions that will be used to represent preferences. It explains the three properties appearing in the definition, surveys the literature and presents examples of coherent choice functions. Section 5, describes many-to-many matching when the agents’ preferences are specified by coherent choice functions. It defines the collective preferences of each side and shows that those are also coherent. Section 6 redefines many-to-many matching with coherent choice functions as an agreement problem between two parties each of them equipped with a coherent choice function. It proposes a solution concept: stable agreements, that generalizes the stable solutions of the one-to-many matching problems. Section 7 presents an algorithm, in terms of choice functions, that finds a stable agreement. It shows that it reduces to the Gale-Shapley differed acceptance algorithm in the marriage problem and that it is a polynomial-time algorithm. Section 8 is the technical part of this paper and is devoted to the proof of the properties of the algorithm defined in Section 7. It includes a proof of the lattice structure of the set of stable agreements. Section 9 generalizes the framework of many-to-many matching with contracts to economies of producers and consumers in which money is present. It shows that stable agreements define prices for items. Section 10 considers the assignment game of [35]: it shows that core elements and stable agreements coincide in generic situations. Section 11 describes open problems and future work. In particular it suggests that many-to-many matching may be the right framework to understand imperfect finite bilateral markets. Section 12 is a conclusion.

3 The representation of preferences

Preferences, or incentives, are at the center of economic theory and a lot of effort has been devoted to understanding and formalizing them. In situations where the preferences are on the elements of an unstructured set, preferences are described in one of two ways:

  1. a quantitative description that associates a number, its value or its utility, to each element, or

  2. a qualitative description by a binary relation between elements that describes which elements are preferred to which.

Any quantitative description induces a natural total and transitive relation, i.e., a total pre-order, on the elements. Qualitative descriptions are almost always taken to be total pre-orders. Any total pre-order can be defined by a utility function, and therefore one can relatively easily translate from one description to the other.

The picture is quite different when preferences are on a structured set. In this work, following Hatfield and Milgrom’s [15], preferences are on the subsets of a base set, and preferences are assumed to satisfy certain conditions in respect to the subset structure. In [11] the authors present the agents’ preferences in the marriage problem as preferences on the items of an unstructured set: boys have preferences over the set of girls and reciprocally, but this is, in fact, not the most natural description. To express the fact that a solution is a complete matching, i.e., that nobody stays unmatched, the authors have to impose this restriction as a hard constraint: partial matchings will not be considered, whereas, had they chosen to describe preferences of, say boys, as a preference over sets of girls, they could have formulated their restriction as a soft constraint by assuming that boys prefer singletons over all other subsets, and could have shown that their solution indeed is a complete matching. As soon as one considers one-to-many matchings one cannot avoid considering preferences over subsets, e.g., firms have preferences over sets of workers.

In this work we shall assume that preferences are over the collection of all subsets of a set . The elements will be called contracts. The set is the set of all possible contracts. One may describe such preferences in a quantitative way by a valuation that associates a real number to any subset of . This is, for example, the way Kelso and Crawford proceed in [16]. They defined a family of such valuations, the gross-substitutes valuations, and proved their results under the assumption that preferences are gross-substitutes. Since there is, in their framework, no place for a gross-net distinction, this family of valuations will be referred to as v-substitutes in this paper. There is a vast literature on this class of functions and the interested reader will find recent surveys and references in [22, 21]. A Walrasian equilibrium is guaranteed if all agents in a market have preferences described by v-substitutes valuations.

To describe preferences for matching in a qualitative way, one could look for a suitable family of binary relations on . But the revealed preferences method suggests another formalization: describe an agent’s preferences by a function that associates any subset of to the subset of that the agent prefers to all subsets of . In Section 8.1.2 we shall define a transitive binary relation associated with such a function. Hatfield and Milgrom [15] use such choice functions satisfying the Substitutes property of Definition 1 below, but their treatment is not complete as noticed in [5], and they assume additional structure on the set . We shall now propose a definition of suitable choice functions on the set of all subsets of .

4 Coherent choice functions

Section 4.1 defines and discusses the family of choice functions that will be considered in the paper: coherent choice functions. Section 4.2 presents examples of coherent choice functions.

4.1 Definition

A finite set is given. The members of will be called contracts, following [15]. Examples of contracts are: a marriage between a specific boy and a specific girl, a work contract between a specific firm and a specific worker for a specific salary and specific work conditions, a sale from a specific producer to a specific consumer of a specific good at a specific price deliverable in a specific place at a specific date.

Following the revealed preferences approach of [3], we want to describe the preferences of an individual over sets of contracts by a choice function that, for every , provides the preferred set of contracts out of all contracts of . We shall show that, in certain circumstances, the collective preferences of a group of individuals can also be defined by such a choice function.

The following definition encapsulates the properties we want to assume about the function . They seem very reasonable if we think in terms of preferences and a detailed discussion will be provided after Definition 1. It is a claim of this paper that, in all previously studied matching problems, the preferences of the players may be described by coherent choice functions.

Definition 1

A choice function is said to be a coherent choice function iff it satisfies the following three properties, for any :

  • (Contraction) ,

  • (Irrelevance of rejected contracts - IRC) if and , then, , and

  • (Substitutes) if and , then .

Note that we allow the set to be empty, even when is not. An agent may prefer no contracts to any other subset of . A completely different notion has been called by the same name in [32].

The three properties above have been extensively studied, among others, in the social choice literature, where the set is understood as the set of all the best elements of , assuming some global order on . As noticed in [15] (footnote 4) the intuition is different when one is interested in matching: the sets of contracts are ordered in some way and is the best set among all subsets of , assuming there is a unique such set. Those three properties are equivalent to the properties 2.5 – 2.7 of Blair [7]. This set of three properties has been studied extensively in [20] in the context of nonmonotonic logics. The present work builds on results obtained there.

Contraction expresses that the best subset of a set of contracts is a subset of . It is so obviously required that most authors do not care to mention it.

Irrelevance of rejected contracts expresses that the absence in the choice set of a rejected contract cannot cause the acceptance of an additional contract. If a hospital presented with a set of doctors that includes doctor would reject doctor , it would not have proposed to additional doctors if the set presented had been . In fact, Lemma 4 will show that the set of contracts proposed stays unchanged if a rejected contract is omitted: under the assumptions we have . This property has been discussed and named in [5]. Irrelevance of rejected contracts is equivalent to the Local Monotonicity of [20]. Local Monotonicity is the model counterpart of the logical property named Cautious Monotonicity introduced in [18] as the proper weakening of the monotonicity property of classical logics when considering a logic in which an additional assumption can cause the retraction of a previous conclusion. In social choice theory, the importance of this property has been put in evidence by Aizerman and Malishevski [2, 1]. Its importance in matching theory has not been recognized so far. Its purpose is to guarantee that the choice of the best subset of a set (or one of the best subsets of ) is done consistently over all subsets of . Suppose for example that, in a four contracts set, two subsets and are both best. A choice function must choose one of them. Suppose we choose . Choosing would be inconsistent even though is best in . Indeed, even though , , contradicting IRC.

Our third property is named after [15]. It states that, if a contract is accepted when in competition with a set of contracts, it will be accepted when in competition with any subset of . Under the converse view it could have been termed Irrelevance of added contracts: the addition to of new contracts (in ) cannot make a contract rejected from acceptable. Indeed, it says that no added contract can be complementary to a rejected contract. It is one of the remarkable intuitions of Hatfield and Milgrom that this property expresses the fact that contracts are substitutes to one another: if no contract can be complementary to and make us have . The Substitutes property is equivalent to the properties (C2) and (C3) of Arrow’s [4] and to property of Sen’s [33] as will be shown in Lemma 3. It is a kind of antimonotonicity: if , then antimonotonicity would require: , whereas Substitutes only requires that this part of that is included in be included in . It expresses the existence of some kind of coherent test by which the preferred elements of a set are picked up: the test corresponding to a superset must be at least as demanding as the one of a subset. The property appears in Chernoff’s [9]. The persistence property of [3] is very similar. In Section 5, it will be shown that under many circumstances the preferences of collectives such as hospitals or colleges may also be expressed by coherent functions.

4.2 Examples

The identity function is a coherent choice function. It describes the preferences of an agent that is satisfied by what it gets. Note that, if is any strictly monotone valuation, the choice function defined by is the subset of that maximizes , is the identity.

If the set of contracts is equipped with some kind of order relation essentially any that defines as the best elements of , in some sense, will prove to be a coherent choice function. For example, if is equipped with a pre-order , i.e., a reflexive and transitive relation and is the set of all elements of that are maximal in with respect to , i.e.,

then is a coherent choice function. The responsive valuations of [27] are also coherent choice functions.

But, in general, preferences are determined by some kind of ordering on the subsets of , not on its elements. In particular, if every subset of is given a numerical utility, one may consider the choice function that, given a set of contracts, chooses the subset with the highest utility. Note that, for this to be possible, there must be a unique subset of highest utility. Appendix A shows that for every v-substitutes valuation there is a coherent choice functions that picks out an optimal subset of any set of contracts. There is a submodular valuation that defines a choice function that is not coherent. Appendix B contains a different characterization of coherent choice functions, meaningful for social choice theory but whose meaning for matching theory is not apparent, due to Aizerman and Malishevski. It will not be used in this paper.

It is remarkable that the substitutes notion that originates in a utility setting can be expressed at all, and in an elegant way at that, in a revealed preferences setting, by properties of choice functions. The twin notion of complements does not seem to have such an expression.

5 Bilateral many-to-many matching and collective preferences

5.1 Contracts, individual and collective preferences

In matching theory we are concerned with situations involving two parties of agents: e.g., men and women, firms and workers, hospitals and doctors, schools and students, producers and consumers. A contract links two specific individuals, one from each party: a man and a woman, a producer and a consumer and so on. We therefore assume two disjoint finite sets and of agents.

Definition 2

For every contract ,

  • denotes the agent mentioned in contract . For any , denotes the set of contracts of which is part, i.e., .

  • denotes the agent mentioned in contract . For any , denotes the set of contracts of which is part, i.e., .

The preferences of agent are represented by a choice function over . The collective preferences of the agents on the same side are represented by the choice functions and on defined by

(1)

Intuitively: every agent cares only about the contracts of he or she is mentioned in: and from this set picks his or hers preferred subset according to his or hers preferences. In other terms rejects a contract iff it is rejected by the agent of it belongs to, and similarly for . Equation (1) expresses a no externalities assumption: agent is indifferent to the fate of all contracts that do not concern him or her. Under such an assumption the choice functions and defined in Equation (1) faithfully represents the collective preferences of the side considered. Even though men are competing among themselves for women, one may define a collective preference for the men. But such a feat cannot be achieved by defining a collective utility for the men since the collective preferences we want to consider must clearly reflect a very partial ordering: we do not want the collective to prefer the good of one man over another, the collective preferences must only reflect what is the joint interest of all agents on the same side. One of the good news in this paper is that, in the framework of revealed preferences and coherent choice functions, this is possible. We shall now show that if each of the choice functions is coherent then the choice functions and defined in Equation (1) are coherent.

Theorem 1

If , and if is a coherent choice function for every , then the choice function defined in Equation (1) is also coherent.

Proof: Let . For Contraction: . For Irrelevance of rejected contracts: if , there is a single such that and

since is coherent. For Substitutes, assume and . There is a single such that and therefore . Since satisfies Substitutes, , and therefore .    

5.2 Preferences in the Gale-Shapley marriage problem

In this paper, in order to explain the algorithm presented in Section 7, we shall now translate, step by step, the classical marriage problem as presented in [11] into the framework proposed in this paper. We consider a set of men and a set of women. For Gale and Shapley the preferences of man are described by a strict total order on the set , and the preferences of a woman by a strict total order on the set . Our formalization of the marriage problem uses a contract set that contains all possible pairs where and , and the preferences of each agent are described by a coherent choice function on the set of contracts in which participates. We shall now describe the function when , i.e., is a man. The description of the choice function of a woman is similar. For : if is empty, is empty and otherwise the set is the singleton where is such that for any such that . This definition expresses that prefers being married to any woman to staying single, that he prefers one woman to more than one woman and that among the available women his preferences are described by .

Let us check that such a choice function is coherent. Remember that . Contraction is obvious. For IRC, if the woman in is not the preferred one from and . For Substitutes, suppose . If then there is a such that and the contract cannot be the one chosen in .

In matching hospitals and interns, couples have to be given special consideration and have been treated by ad hoc modifications: see e.g., [25, 29, 8]. One could wonder whether, in the perspective proposed in this paper, one could model the preferences of a couple of doctors by a collective choice function. The answer is that one can but this choice function is not expected to be coherent since the two positions looked for are complementary, not substitutes. A couple’s choice function does not satisfy the Substitutes property of Definition 1.

In the marriage problem, the collective choice function for men has the effect of removing from a set of contracts all contracts when there is in another contract such that prefers to , and similarly for women. If one considers a typical situation in the matching algorithm of Gale and Shapley in which men are gathered around women and define the set of all contracts as containing exactly those pairs such that man is one of the men gathered around woman and if is the collective choice function for the women’s side, then is exactly the set of pairs such that has not been rejected by in the curent stage of the algorithm.

6 The agreement problem

6.1 Definition

Section 5 proposed a perspective change: don’t view matching problems as matching individual hospitals and individual doctors each of which have individual preferences but as an agreement problem between two collective preferences described by two coherent choice functions. Such a perspective subsumes many-to-many matching and allows to consider rules or constraints, e.g., the monogamy restriction in the marriage problem, as preferences. In our version of the marriage problem, polygamous or polyandrous marriages are considered and it is only the fact that agents prefer a single mate to more than one mate that ensures a one-to-one pairing. In this framework one may easily introduce soft rules by translating them into strong preferences.

In this Section we shall define the problem, the agreement problem, that generalizes the matching problem, define the solution concept for this problem, a stable agreement and show that this solution concept subsumes stable matchings. In Section 7 we shall propose an algorithm to solve all agreement problems and, then, in the remainder of this paper we shall show that the study of this algorithm enables us to prove that stable agreements exist and present a lattice structure.

We now assume a finite set of contracts and two coherent choice functions: , for . The coherent choice function describes the collective preferences of side . They are obtained from the individual choice functions of agents of side by Equation (1) and are coherent by Theorem 1. Note that, contrary to [15], no structure is assumed on the set of contracts: a contract is always between side and side , we can forget the individuals involved since the individuals preferences have been taken into account in the collective preferences expressed by and . Note also that the situation considered here is that of many-to-many matching: both sides may enter multiple contracts. Roth’s [26] and Blair’s [7] already considered such a symmetric situation and the topic has attracted a lot of attention recently, see [12, 13, 14]. Those works assume that preferences are defined by utilities, i.e., a total order on the sets of contracts.

The basic situation is the following. Both sides wish to agree on a set of contracts, a subset of . If a tentative agreement, i.e., a subset of , is on the table, each side may, if it wishes to, and without any need for a permission from the other side, reject any subset of the tentative agreement but the permission of both sides is necessary before a contract or a number of contracts can be added to a tentative agreement. This leads to the following definitions.

To help the reader’s intuition we shall mention basic properties of the notions defined in Propositions without proofs. No proposition will be used in the sequel and all propositions follow from the results in Section 8.

6.2 Agreements

An agreement is a set no contract from which will be taken away by any of the two parties.

Definition 3

A set of contracts is said to be an agreement iff , i.e., it is a fixpoint for both and .

In a sense, an agreement is a small set of contracts. The proposition below follows from Lemma 6.

Proposition 1

The empty set is an agreement. If is an agreement and , then is an agreement.

Note that the union of two agreements is not in general an agreement.

In the marriage problem one easily sees that an agreement is a partial matching: is an agreement iff for each man there is at most one contract of the form in and for each woman there is at most one contract of the form in . There may be unmatched agents. The explanation is that if an agent had two or more contracts in he or she would prefer to throw all of them away except one as implied by the definition of the choice function in Section 5.2.

6.3 Stable sets

A stable set is a set of contracts to which no addition is supported by both sides.

Definition 4

A set of contracts is said to be a stable set iff for any contract , .

In a sense, a stable set is a large set of contracts. The following proposition follows straightforwardly from Lemma 3. Remember and are coherent.

Proposition 2

The set is a stable set. If is a stable set, then any such that is a stable set.

Note that the intersection of two stable sets is not, in general, a stable set.

In the marriage problem a set is stable iff there is no pair that is not a member of such that prefers to all the s such that and prefers to all the ’s such that . This corresponds exactly to the notion of stability in the marriage environment.

6.4 Stable agreements

We can now formalize our notion of a solution.

Definition 5

A set of contracts is said to be a stable agreement iff it is an agreement and it is a stable set.

A stable agreement is a set of contracts from which no agent would like to reject a contract and no two agents would agree to keep an additional contract between them if offered. In the marriage problem a stable agreement is exactly a stable matching. We may now define the agreement problem in general.

Definition 6

The agreement problem is the following: given a finite set and two coherent choice functions on , and , find a stable agreement, i.e., a set such that and there is no such that .

7 An algorithm for the agreement problem

7.1 The algorithm

We shall now offer a general solution to the agreement problem presented in Definition 6. The algorithm is an iterative one that builds a sequence of sets of contracts: . The first element of the sequence: is taken to be equal to , the set of all contracts. The sequence is then defined by:

(2)

The initial set of contracts on the table is . Then the process proceeds in stages in the following manner: side picks its preferred contracts from the pool and side picks from this set, providing the set of contracts picked by side and accepted, i.e., not rejected by side . This set is put back on the table with all contracts not picked up by side and the process goes into another similar stage. Note that, in the process, all contracts rejected by side are left on the table for the next stage, therefore it is only side that rejects contracts. We shall see below in Lemma 16 that indeed side shall like side ’s offers more and more, i.e., side prefers to .

Lemma 1

If and satisfy Contraction, then the sequence of sets of contracts is a decreasing sequence, i.e., for any and there is an index such that .

Proof: For the first claim:

The second claim holds since is finite.    

In the Gale-Shapley deferred acceptance algorithm: the set of matches that have not yet been rejected (by a woman) decreases (weakly).

We shall prove in the upcoming sections that the set is a stable agreement. We shall also show that is, in some sense, preferred to any stable agreement by side 1.

7.2 A special case: Gale-Shapley

In the marriage problem, the iterative process just described parallels the Gale-Shapley algorithm in which side is the proposing side, i.e., the side that makes the first move. is the set of all possible couples. The set in Equation (2) is the set of pairs where is the partner best preferred by : this describes exactly the first step of the Gale-Shapley process: agents of side gathering each around the agent of the other side that they prefer among all others. The set describes exactly the second step of the process: each agent of side around which gathered a non-empty set of agents from side chooses the one it prefers amongst those gathered, i.e., it rejects all but the prefered one. The definition of says that it contains all contracts of except those rejected by side in the last step. This parallels the fact that, in the upcoming stages of the Gale-Shapley process all pairings are still possible except those pairings that were just rejected by the agents of side . We have seen that, when applied to the marriage situation and when Equation (2) describes exactly the Gale-Shapley process.

7.3 Computational complexity

The algorithm described in Section 7.1 is conceptually straightforward and elegant, but is it efficient from the computational point of view? The answer is: it is remarkably efficient. Note that, by Lemma 1 the number of steps, , of the algorithm is at most the size of the set of contracts . The algorithm finds a subset of with certain properties and a dumb algorithm would consider each subset of and check whether it satisfies the properties or not: this requires a number of operations of the order of , i.e., exponential in the size of . Our algorithm requires only a linear number of operations in the size of . How large is ? Considering the discussion in Section 5, the set will be something like the Cartesian product of the set of agents on one side by the set of agents on the other side by the set of items that can be traded. The considerations in Section 9 show that this may be multiplied by a finite set of prices, but, all in all, the size of is polynomial in the size of the data that defines the problem. This stands in contradistinction with algorithms for finding a competitive equilibrium: in [23] the authors showed that finding an efficient allocation requires exponential communication. For finding a stable agreement a polynomial number of applications of the choice functions is enough.

8 Properties of the algorithm

The purpose of this section is to study the properties of the sequence of ’s above to prove that is a stable agreement and study the structure of stable agreements. To this purpose a significant mathematical apparatus is required. Section 8.1 develops the theory of coherent choice functions. To any coherent choice function it associates a partial pre-order expressing the preferences revealed by . Section 8.2 analyzes the iterative process defined in Section 7 in terms of the partial pre-orders and . It shows that the algorithm produces a stable agreement. Section 8.3 studies the properties of the partial pre-order defined in Section 8.1 on stable agreements. Section 8.4 shows that the set above is a stable agreement preferred by side 1 to any stable agreement. Section 8.5 proves that the set of stable agreements has a lattice structure.

8.1 Properties of coherent choice functions

Section 8.1.1 presents the elementary properties of coherent choice functions. None of the results presented there are original. The most important result is C. Plott’s Lemma 7. A rapid overview may be enough for a first reading. Section 8.1.2 defines the partial pre-order associated with a coherent choice function, first considered by C. Blair. This is a fundamental tool in the sequel. Section 8.1.3 is devoted to the proof of an original technical result that will be used only in Section 8.5 and its reading may be postponed.

8.1.1 First properties

First, IRC is equivalent to the Local Monotonicity property studied in [20].

Lemma 2

The IRC property is equivalent to:

Proof: To show that IRC implies LM, reason by induction on the size of . If , LM holds. Let . Assume then , and we have . We conclude by the induction hypothesis.

Now assume LM and let . We have and .    

The next lemma shows that our definition of the Substitutes property is equivalent to the corresponding formulation in [33, 15]

Lemma 3

A choice function satisfies the Substitutes condition iff it satisfies one of the three following, equivalent,properties:

  1. for any and for any , if , then one has ,

  2. if , ,

  3. for any , if then .

Proof:

  1. Let us show that Substitutes implies property 1. If we have and , then and, since , by Substitutes we have .

  2. Assume property 1 and let , . We have and . By property 1, , but .

  3. We prove now that property 2 implies property 3. Assume property 2, and . We have and therefore we have . But and .

  4. Let us, finally, show that property 3 implies Substitutes. Assume property 3, and . By contraposition property 3 implies .

 

The next results use all three properties of Definition 1.

Lemma 4

If is coherent then, for any :

  • (Cumulativity) if , then , and

  • (Idempotence) f(f(A)) = f(A).

Proof: For Cumulativity, the assumptions imply, by Lemma 2, that , and by Lemma 3 . For Idempotence, note that and conclude by Cumulativity.    

Lemma 5

If is coherent, then .

Proof: By Contraction: . By Lemma 3, and .    

The sets of contracts that are fixpoints of , i.e., such that are particularly interesting. Our next result shows that any subset of a fixpoint is a fixpoint.

Lemma 6

If is coherent, and , then .

Proof: By assumption and Lemma 3 implies but , and . We conclude by Contraction.    

The next lemma is a powerful result of C. Plott [24] and expresses what he calls Path Independence. It is presented there in a Social Choice context where the elements of are not contracts but possible social outcomes. A preferred set of social outcomes contains all individually preferred outcomes and not, as in this work, a set preferred to other sets, as already noticed in [15] (see footnote 4). For completeness sake a proof is provided.

Lemma 7 (C. Plott, Path Independence)

A function is coherent iff it satisfies Contraction and one of the two equivalent conditions below, for any :

  1. ,

  2. .

Proof: First, notice that condition 1 implies and therefore implies condition 2. Then, notice that condition 2 implies Idempotence: . Therefore it implies that and therefore implies condition 1.

Suppose now that is coherent. By Lemma 5 and Contraction . By Cumulativity in Lemma 4 we conclude that . We have shown that satisfies condition 1.

Let now satisfy Contraction and condition 1. Assume . We have

and Local Monotonicity is satisfied. By Lemma 2 we conclude that satisfies Irrelevance of rejected contracts. If, now, we have

We see that . By Lemma 3 we have shown that satisfies Substitutes. We have shown that is coherent.    

8.1.2 The partial pre-order induced by a coherent choice function

We shall now show that any coherent choice function induces a partial pre-order on the set .

The choice function naturally defines a preference relation between sets of contracts. A set is preferred by to a set if the best subset of is , i.e., if the addition of to does not make the agent change her mind in any way: she will stick to . This relation has been considered by Blair [7] in his Definition 4.1.

Definition 7

Let be a coherent choice function and . We shall say that, from the point of view of , is preferable to and write iff . We shall say that is indifferent between and and write iff and .

The reader should note that the relation is a very demanding one: requires that the contracts of do not provide any advantage over those of : if is available in addition to the elements of that are not in will not be used at all. This demanding character of implies that a claim such as the claim in Theorem 3 that any stable agreement is less preferred than a specific set is a powerful claim. If is defined by a v-substitutes valuation as in Appendix A the partial pre-order is not the total pre-order defined by the valuation function. True, if one has but the converse does not hold.

From now on, all choice functions considered are assumed to be coherent and the assumption that is coherent will not be mentioned explicitly.

In the marriage problem, a man prefers a set of contracts to a set iff the most preferred woman in is (weakly) preferred to the most preferred woman in , the empty set being less preferred than any set. Note that, in this specific situation, the relation for man is a total order. This is not the case in general.

The following provides equivalent definitions.

Lemma 8

iff iff .

Proof: The only parts are obvious. Suppose now that . We have and by Cumulativity we have .    

We proceed to study the properties of this binary relation.

Lemma 9

For any :

  1. ,

  2. the relation is reflexive,

  3. .

Proof: By Definition 7. By the previous claim. By Lemma 7.    

Lemma 10

iff and therefore the relation is an equivalence relation.

Proof: If we have and . We conclude that .

Assume that . By Lemma 5 we have . Conclude by Lemma 8.    

Note that does not satisfy the antisymmetry property: and does not imply , and, therefore, is not a partial order.

Lemma 11

If and then , i.e., the relation is transitive.

Proof: By Path Independence (Lemma 7) and our assumptions:

 

We see that, if is coherent, the relation is a partial pre-order, also called partial quasi-order.

The following lemma extends the Substitutes property to the case , instead of .

Lemma 12

Let be such that . If then .

Proof: By Path Independence, Definition 7, and Path Independence again:

But, by Substitutes, implies .    

8.1.3 The operation

The following will prove useful in Section 8.5

Definition 8

For any we define

(3)

The properties of the operation are described below. Note, in particular, item 8 that expresses the preference relation in terms of set containment and the operation .

Lemma 13

For any

  1. if then ,

  2. ,

  3. ,

  4. ,

  5. ,

  6. ,

  7. ,

  8. iff .

Proof:

  1. Let . For any , by Lemma 3, implies . Therefore

    and

  2. Let . Since , by Lemma 3, , contradicting . We see that and we conclude by Cumulativity.

  3. By Lemma 5, Contraction and item 2 above we have

    By Lemma 4 we conclude that . Therefore for any , we have and , a contradiction. We have shown that . By definition .

  4. By item 2 and Lemma 10.

  5. We distinguish three cases. First, for any , and . Secondly, for any , . Let us show that . By Path Equivalence, for any , iff . We see that . Thirdly, for any , iff and iff and the two conditions are equivalent, by Path Equivalence.

  6. By Lemma 5, item 2 above and Contraction

    We conclude by Cumulativity.

  7. Let . If, on one hand, then clearly . If, on the other hand, then , therefore, by Lemma 3, and . Whether or , and similarly .

  8. We use Lemma 8. Assume and . We have

    and by Lemma 4 also . If then , but if , we have and therefore . We have shown that . Assume now that . By Lemma 9, then, and by item 4 above we have and we conclude by Lemma 11.

 

8.2 Properties of the iterative process

We may already see that the end product of our iterative process, the set is an agreement.

Lemma 14

The set is an agreement.

Proof: By Lemma 4, is idempotent and . By construction and therefore and, by Contraction we conclude that i.e., .    

We shall now proceed step by step in the analysis of the iterative process described in Equation (2) and gather all the results in Theorem 3 below. We have seen in Lemma 1 that the sequence is decreasing.

An offer of side , i.e., a contract in , that has been accepted by side will always be offered again by side .

Lemma 15

For any , ,

Proof: Since , by Lemma 3, we have . But, by Equation (2).    

In the Gale-Shapley deferred acceptance algorithm: a man chosen by a woman at stage , will still be available for her at stage .

The result below is a central part of our analysis. From the point of view of side 2, the offers of side 1 get better and better.

Lemma 16

For any , .

Proof: By Lemma 8, it is enough to show . By Lemma 5, we have: . We conclude by Lemma 15 and Contraction.    

In the Gale-Shapley algorithm: after stage a woman is either left in the same situation she was after stage or she has a better (for her) mate.

Our next result is central towards showing that is a stable set.

Lemma 17

For any , if , then, for any , one has and in particular .

Proof: Let . We have and therefore . But, by Lemmas 16 and 11, . Lemma 12 then implies that .    

In the Gale-Shapley algorithm: a woman who has rejected a man at some point will never wish to be matched with him at some later point.

We may now state:

Lemma 18

The set is a stable agreement.

Proof: Lemma 14 showed that is an agreement. Let now . We distinguish two cases. On one hand, if , . But, by Lemma 7, . We conclude that . On the other hand, if , there is a such that . By Lemma 17, then, . We have shown that is a stable set.    

8.3 Ordering agreements and stable sets

We need to prove some basic properties of stable sets before we can prove that is the stable agreement preferred by side . By Definition 4 both sides cannot agree to add any contract to a stable set. We shall show now that both sides cannot agree to add any set of contracts to a stable set.

Lemma 19

If is a stable set, then, for any ,

Proof: Assume . By Contraction and by Substitutes we have , contradicting our assumption that is a stable set.    

Our next result compares the two partial pre-orders and . In the sequel addition has to be understood as addition modulo 2: . If side prefers a set to a stable agreement , then the other side () prefers A to .

Lemma 20

Let . If is a stable set and , then .

Proof: Assume . By Lemmas 9 and 11, , and, by Definition 7 and Idempotence, . Since is a stable set, by Lemma 19,

We see that

and therefore

By Contraction, then, we see that

and we conclude that