It seems appropriate to end this volume with a few words about why we have selected and organized the material as we have, and what we think has been learned. To put it another way, now is a good time to explain the title of the book, which includes both "modeling" and "analysis." The fact that the book contains many theorems should suggest what the analysis consists of. And the fact that we began the book with a detailed description of a particular labor market and with a brief description of some auction phenomena, should suggest at least part of what we mean by modeling. But if that were all there was to it, we could have finished our work in many fewer chapters.

Instead, we analyzed a whole family of closely related models, discrete and continuous, with and without complete information, with and without money, with firms employing one worker or many, and with simple or complex preferences. One purpose of these final remarks is to make clear what we think the consideration of all these models together adds to our understanding and interpretation of each of them.

The purpose of this book is two-fold. First, it reviews and integrates the growing literature about a family of models of labor markets, auctions, and other economic environments. Second, we hope it will illustrate the subtle interactions between modeling considerations and mathematical analysis that characterize the use of game theory to explain and predict the behavior of complex “real-world” economic systems.

We will be concentrating on “two-sided matching markets.” The term “two-sided” refers to the fact that agents in such markets belong, from the outset, to one of two disjoint sets - for example, firms or workers. This contrasts with commodity markets, in which the market price may determine whether an agent is a buyer or a seller. Thus whereas the market for gold has both sellers and buyers, any particular agent might be a buyer at one price and a seller at another, so the market is not two-sided in the sense we will speak of. But a labor market often is, since firms and workers are distinct. For example, as the wages of professors fall, some professors may leave the market, but none will become universities. The term “matching” refers to the bilateral nature of exchange in these markets - for example, if I work for some firm, then that firm employs me. This contrasts with markets for goods, in which someone may come to market with a truck full of wheat, and return home with a new tractor, even though the buyer of wheat doesn't sell tractors, and the seller of tractors didn't buy any wheat.

Chapters 7, 8, and 9 present three models of one-to-one matching in which money plays a prominent role. Unlike the model explored in Section 6.2, money will be modeled as a continuous rather than as a discrete variable, and this will let us employ a different set of mathematical tools. An even more important difference between the models of these chapters and those we have dealt with until now is that in these models, we are going to assume that the preferences individuals have for different matchings are basically monetary in nature.

Chapter 7 presents a model of exchange between one seller, with a single object to sell, and many buyers. This model will be simple enough so that the analysis can be conducted without much technical apparatus. We will explore it in enough detail to see that with one or two significant exceptions, the phenomena uncovered in our presentation of the marriage problem will also arise in these models with money. We will also relate this discussion to our discussion in Section 1.2 of strategic behavior in auctions, and particularly to the options available to coalitions of bidders.

Consider a market consisting of a single seller, who owns one unit of an indivisible good, and n buyers, each of whom is interested in purchasing the object if the price is right. Each buyer b places a monetary value $rb on the object, which is the maximum amount he or she is willing to pay, and the seller similarly places a value $rs on the object, which is the price below which he or she will not sell. We call these monetary values the reservation prices of the agents. Each buyer has cash on hand sufficient to pay his or her reservation price.

We may interpret the reservation price as follows, for example. The seller has in hand an offer of $$rs from some outside party (not one of the buyers in the present market), who will buy the object at that price if it is offered by the seller. Each buyer (whom we may think of as a broker, rather than as a final consumer of the object) has in hand an offer of $rb from a client who will purchase the object from the buyer at that price, should he or she obtain it in the market. Since the seller knows that he or she can earn at least $rs, the seller will not sell at a lower price. And since each buyer b knows that he or she can earn $rb (but no more) by reselling the object, the buyer will not buy at a higher price.

We turn now to a different class of questions, which will require different kinds of models and theories. In the previous sections we investigated the marriage market by exploring the kind of matchings we might expect to observe. Now we will investigate how we should expect individual agents to behave. Specifically, we will want to know to what extent it is wise for men and women to be frank about their preferences for possible mates, and how we can expect them to act in the process of courtship and marriage. Lest the analysis seem a little cold-blooded at times, it is good to remember that our primary interest here is in a simple model of labor markets, and that the phenomena we will be studying can perhaps better be thought of as the courtship that takes place between potential employers and employees. But for simplicity we consider these questions first in the context of the marriage problem, and defer to later chapters more realistic models in which firms may employ more than one worker.

To address these questions of individual behavior, we need to model the decisions that individuals may be called upon to make in the course of a marriage market. We have so far considered only the general rules of the game, which state that for a marriage to take place between some man m and woman w, it is both necessary and sufficient that the two of them agree. These rules are reflected in our definition of a stable matching. But these general rules might be implemented by many different particular procedures. These particular procedures, which constitute the detailed rules of the game, will determine what specific decisions each agent faces. They will determine how an agent goes about making his preferences known, and the order in which decisions are taken. They will, in short, be a description of the mechanics of the market.

This chapter discusses the mathematical structure of the set of stable matchings, including some computational algorithms. Some readers may prefer to simply skim this chapter, at least on first reading.

The core of a game

One of the most important "solution concepts" in cooperative game theory is the core of a cooperative game. In this section we will see that the set of stable matchings in a marriage problem is equal to the core of the game. (In subsequent chapters we will see that in some of the more complex two-sided matching markets we will consider, the set of stable matchings may be only a subset of the core, and in the case of many-to-many matching, [pairwise] stable matchings need not even be in the core at all.)

As discussed earlier, when we formally model various kinds of games, we will generally want to specify a set of players, a set of feasible outcomes, preferences of players over outcomes, and rules, which determine how the game is played. In how much detail we will want and need to specify the rules will depend on what kind of phenomena we are describing, and what kind of theory we are trying to construct. It is often sufficient to summarize the rules of the game by specifying which coalitions (i.e., subsets) of players are empowered by the rules of the game to enforce which outcomes. (Thus in our analysis of the marriage market in Chapter 2, we concentrated on the fact that in order for a marriage to take place, it is necessary and sufficient for the man and woman involved to agree.)

This chapter looks at models of many-to-one matching between firms and workers that generalize the college admissions model in two important ways, by allowing firms to have a larger class of preferences over groups of workers, and by explicitly putting money into the model, so salaries are determined as part of the outcome of the game, rather than specified in the model as part of the job description. Of course, when we look at a model that does both these things together, we will also have to specify how the preferences of firms and workers deal with different combinations of job assignments and wages. Section 6.2 considers a version of a model proposed by Kelso and Crawford, with both complex preferences and negotiated wages. (We model wages here as a discrete variable, which is a natural modeling assumption since, for example, contracts cannot specify wages more closely than to the nearest penny. The next chapters model wages as a continuous variable, which also has some advantages.) But first we construct a simpler model in which we can examine complex preferences while continuing to treat salaries as an implicit part of the job description. Throughout this chapter we continue to make the simplifying assumption that workers are indifferent to which other workers are employed by the same firm.

We close with a few open questions and suggestions of possible directions for further research. In keeping with the emphasis of the book on both analysis and modeling, some of the questions and directions are of each type; that is, some call for the statement and proof of theorems, whereas progress on others will (first) involve the construction of new models.

1. Since many entry level labor markets and other two-sided matching situations don't employ centralized matching procedures, and yet aren't observed to experience the kinds of market failure that seem to be associated with unstable matching, we can conjecture that at least some of these markets reach stable outcomes by means of decentralized decision making. So one of the chief modeling problems that will arise in studying such markets will be to develop decentralized models of stable matching. (We noted that a consequence of Theorem 2.33 is that a random process that begins from an arbitrary matching and continues by satisfying a randomly selected blocking pair must eventually converge with probability one to a stable matching, provided each blocking pair has a probability of being selected that is bounded away from zero. Perhaps this kind of result will provide the building blocks for models of decentralized matching.)

2. …

This chapter presents one of the generalizations of the assignment game, in which agents' preferences may be represented by nonlinear utility functions. So in this model agents are allowed to make somewhat more complex tradeoffs than in the assignment model between whom they are matched with and how much money they receive. Nevertheless, each of the principal results we proved for the marriage market has a close parallel in the present model.

This model is a variant of a model introduced by Demange and Gale (1985). The only difference between the model presented here and their model is in the definition of feasible outcomes, which here allow monetary transfers to be made not only among matched pairs of agents, but also among arbitrary coalitions of agents, as in the assignment model and the one-seller model explored in the previous two chapters. Aside from making this model a generalization of these other models, this change allows us not to rule out a priori the kinds of strategic opportunities available to coalitions of bidders, for example, that we discussed in Sections 1.2 and 7.2.1. However most of the results from Demange and Gale's model carry over unchanged to the case when monetary transfers are allowed between unmatched agents. The reason is that as in the assignment game, no such transfers are made at stable outcomes. That is, we will see that the only monetary transfers that occur at stable outcomes are between agents who are matched to each other.

The formal model

There are clear similarities between the hospital intern market and the simple model of a marriage market studied in the previous chapters. There are two kinds of agents, hospitals and medical students, and the function of the market is to match them. (Strictly speaking, we should speak of hospital programs rather than hospitals, because different internship programs within a hospital are separately administered, and students apply to specific programs.) Because interns' salaries are part of the job description of each position, and not negotiated as part of the agreement between each hospital and intern, salaries will not play an explicit role in our model, but will simply be one of the factors that determine the preferences that students have over the hospitals. Similarly, we will assume that hospitals have preferences over students - that is, they are able to rank order the students who have applied to them for positions, as they are asked to do by the National Resident Matching Program. The major difference from the marriage problem is that each hospital program may employ more than one student, although each student can take only one position. (All the positions offered by a given hospital program are identical, since hospitals offering different kinds of positions must divide them into different programs.) The rules of the market are that any student and hospital may sign an employment contract with each other if they both agree, any hospital may choose to keep one or more of its positions unfilled, and any student may remain unmatched if he or she wishes (and seek employment later in a secondary market).

As in any game-theoretic analysis, it will be important in what follows to keep clearly in mind the “rules of the game” by which men and women may become married to one another, as these will influence every aspect of the analysis. (If, for example, our imaginary village were located in a country in which a young woman required the consent of her father before she could marry, then the fathers of eligible women would have a prominent role to play in the model.) We will suppose the general rules governing marriage are these: Any man and woman who both consent to marry one another may proceed to do so, and any man or woman is free to withhold his or her consent and remain single. We will consider more detailed descriptions of possible rules (concerning, e.g., how proposals are made, or whether a marriage broker plays a role) at various points in the discussion.

The formal (cooperative) model

The elements of the formal model are as follows. There are two finite and disjoint sets M and W: M = {mi m2, …,mn} is the set of men, and W= {w1, w2,..., wp} is the set of women. Each man has preferences over the women, and each woman has preferences over the men. These preferences may be such that, say, a man m would prefer to remain single rather than be married to some woman w he doesn't care for.

In these chapters we will examine in detail the two-sided matching market without money that arises when each agent may be matched with (at most) one agent of the opposite set. For obvious reasons this model is, somewhat playfully, often called a "marriage market," with the two sets of agents being referred to as "men" and "women" instead of students and colleges, firms and workers, or physicians and hospitals. We will follow this practice here. In this whimsical vein, it may be helpful to think of the men and women as being the eligible marriage candidates in some small and isolated village.

The marriage market will be simpler to describe and investigate than a labor market in which a firm may employ many workers. And we will see in Part II that many (although not all) of the conclusions reached about this model will also apply to the hospital intern market, in which a hospital, of course, typically employs many interns. The marriage market will therefore be a good model with which to begin the mathematical investigation. In some of the discussion that follows, it will nevertheless be helpful to remember that much of our interest in this problem is motivated by labor markets, rather than by marriage in its full human complexity. (Thus we will sometimes speak about courtship, but never about dependent children or mid-life crises.)