In this chapter we make a few remarks on the relationship between the existing theory and our work, and on possible generalizations of our results. First, in Section 7.1, we observe that all known restrictions on individual preferences guarantee that cores of voting games are nonempty. Thus, we may have a richer theory of representation (of committees) under the (usual) assumptions of restricted preferences. Next, we remark that restriction of preferences does not eliminate manipulability. Then we argue that there is a need to generalize the notion of exact and strong consistency to allow restricted preferences, and we notice that Dutta [1980b] contains such a generalization (see Section 7.2).

In Section 7.3 we discuss the problem of the existence of faithful and neutral representations of committees. We show how that problem can be resolved by the use of even-chance lotteries on alternatives. Section 7.4 is devoted to a systematic generalization of our results to weak orders. Finally, we discuss possible extensions of our results to infinite sets of alternatives (see Section 7.5).

This chapter is devoted to a detailed investigation of the relationship between committees (formally, simple games) and social choice correspondences and functions. We begin our investigation in Section 3.1 by associating three simple games, which are derived from three different notions of effectiveness, with every social choice correspondence. These games serve to describe the power distribution induced by a social choice correspondence among the various coalitions of voters. We proceed with a study of the properties of, and the interrelations between, the three simple games. Then we consider several examples, including the Borda rule. A social choice correspondence is tight if our three notions of effectiveness coincide for it. We conclude Section 3.1 by demonstrating that certain conditions are sufficient for tightness. In Section 3.2 we consider the converse problem, namely, that of finding representations (i.e., choice procedures or, more formally, social choice correspondences) for committees. We find sufficient conditions for tightness of representations, and we prove, by constructing suitable examples, the existence of tight and “nice” representations. We also find that the solution to the representation problem is closely related to the investigation of cores of simple games.

In this chapter we introduce three families of choice rules that play important roles in social choice theory, namely, social welfare functions, social choice correspondences, and social choice functions, and we investigate their basic properties, such as Pareto optimality, anonymity, neutrality, and monotonicity. The study of social welfare functions culminates in Section 2.2 with Arrow's Impossibility Theorem. The study of social choice correspondences and functions focuses on monotonicity properties and leads to the conclusion that every strongly monotonic social choice function whose range contains at least three alternatives is dictatorial (see Theorem 2.4.11). The Gibbard-Satterthwaite Theorem is shown, in Section 2.5, to be a corollary of the preceding theorem. Simple games and their basic properties are defined in Section 2.6. We conclude with a proof of Nakamura's theorem on cores of simple games (see Theorem 2.6.14).

The theory of voting in committees began with the basic contributions of Borda [1781] and Condorcet [1785]. Borda had been concerned about the inadequacy of choice by plurality voting, and he suggested a different method of assigning marks to alternatives, a method now known as Borda's rule (see Black [1958], pp. 156-9). It is interesting that Borda's method is still a subject of active research (see, e.g., Young [1974] and Gardner [1977]). Indeed, Borda's rule also serves as an important example in this book (see Example 3.1.18).

Condorcet developed an extensive formal theory of voting (see Black [1958], pp. 159-80). One of his profound discoveries was the “paradox of voting,” which is known also as Condorcet's paradox. His most important contribution was the formulation of the Condorcet condition (i.e., the alternative that receives a majority, against each of the other alternatives, should be chosen). This condition plays an important role in so many works in modern social choice theory that it is impossible to give a full record of its use. We, also, apply the Condorcet condition to the theory of committees (see, e.g., Theorem 3.2.5).

Nanson [1882] examined several systems of voting and suggested a modification of Borda's rule that is compatible with the Condorcet condition. (Borda's rule itself does not satisfy the Condorcet condition). Further details on Nanson's work may be found in Black [1958].

This chapter is mainly devoted to a survey of Chapters 2-6. Section 1.1 consists of an almost self-contained presentation of our theory. In particular, it contains a detailed formulation of the central problems that we try to solve in this book and the definitions of the main solution concepts. The reader will get a fairly good picture of the qualitative side of our theory by reading the survey. We conclude in Section 1.2 with brief remarks on possible uses of our results.

Survey

This section presents a summary of the most important results of our study. The main problems that we try to solve and the key solution concepts are presented independent of the following chapters. This should enable the reader to get a fairly good idea of the nature of this book. However, it should be mentioned that only the most important theorems are mentioned in this chapter. Moreover, in order to keep this section as readable as possible, we refrain from discussing proofs. Thus, in order to become familiar with our techniques, one has to look at the proofs of the main theorems. Finally, it should be remembered that we have written the survey as an almost self-contained description of our theory. Hence, for the sake of clarity and briefness, the order in which the various topics are discussed in the book has been slightly changed; however, this has not diminished the usefulness of the survey.

The problem of rule selection

In trying to lay the foundations of the new welfare economics, J. R. Hicks brought forward the following problem:

According to Hicks, there are three possible ways of dealing with this problem:

1. Instead of using the preference scales of the individuals in the community, the investigator may use his own “thermometer,” that is, decide for himself what is good or bad for society in judging the relative performances of the alternative economic organizations.

2. The investigator may seek a method of aggregating the (possibly) conflicting reports of the various thermometers so as to construct an “average” or “social” registration.

3. […]

Introduction

I come now to an examination of the dynamics of the system, the way the variables move out of equilibrium. The main assumptions used to secure stability are those of No Favorable Surprise as discussed in Chapter 4. These are the assumptions that ensure that new opportunities neither arise nor are perceived to do so. However, there are other properties of the motion of the system which it seems reasonable to assume. Since the proof of stability is not the only end of dynamic analysis and since the class of adjustment processes which are consistent with No Favorable Surprise needs to be studied, I discuss such properties as well in the hopes that such discussion will prove useful for further work.

To put it another way, given No Favorable Surprise, the class of models for which the stability result holds is quite general. On the other hand, that very generality means that we do not gain a great deal of information from that result about the workings of the model. To the extent that additional assumptions seem reasonably calculated to restrict the behavior of the model in directions that real economies may be supposed to take, it is useful to discuss such assumptions even though this book will not itself go beyond the proof of stability under No Favorable Surprise.

There is another reason for proceeding in this way. Were I to stop with a discussion of No Favorable Surprise, there would remain some question as to whether models with sensible dynamic assumptions could in fact be fitted into the framework used. By discussing such questions as individual price adjustment, orderly markets, and the problems of non-delivery within the context of the model I show that this is not an issue.

Introduction

In this chapter, I return to the Hahn Process models discussed in Chapter 2 and treat them more precisely than was done there. This enables the introduction of the notation used later in the book. More important, it facilitates understanding of some issues which arise again in the context of more complex and satisfactory models. For the most part, these issues were discussed in the previous chapter and nontechnical readers may proceed directly to Chapter 4 with little loss of continuity.

Two models are discussed in the present chapter. First, I treat the case of pure exchange without the introduction of money. As explained in the previous chapter, this model embodies the basic feature of the Hahn Process. It is very easy to see what is going on in a context that lacks the increasingly complex apparatus of later versions.

The second model treated below adds the complications of firms and of money. However, actual production and consumption out of equilibrium are not permitted. The analysis of this model permits an understanding of the role of firms and introduces a number of the problems which later appear. Production and consumption out of equilibrium will not be separately treated. As explained in the preceding chapter, they are easiest to introduce in the context of a relatively rich disequilibrium model where agents understand what is happening and care about the timing of their actions. Hence, disequilibrium production and consumption are introduced only in the full model of Part II.

Introduction

It cannot be pretended that the theory of stability developed historically with an eye to the considerations discussed in the previous chapter. Nevertheless, that discussion provides a useful vantage point from which to view the development of the subject. In turn, such a review allows us naturally to build an understanding of the steps which lead to ever more satisfactory stability models.

In my view, there have been four major developments in the history of modern stability analysis. These are (1) the realization that the subject was one which had to be studied in a context with a formal dynamic structure; (2) the realization that global, rather than simply local, results could be obtained; (3) the introduction of non-tâtonnement processes; and (4) closely related to this, the insight that attention paid to specifying the disequilibrium processes involved could lead to far more satisfactory results than could be obtained by restricting the excess demand functions. In some ways, the analyses resulting from each of these steps made increasing use of the economic underpinnings of the stability problem (largely, but not exclusively, Walras' Law) and led to correspondingly more and more satisfactory results.

Introduction

Almost every human act, individual or collective, involves choice under environmental constraints, which we can analyze via a generalization of the pure theory of consumer's behavior. In this analysis we start from the set of all conceivable states. The external environment delimits the range of available states without necessarily reducing the range to a single alternative. Those states that could actually be chosen compose a subset of the set of available states.