Social aggregation theory deals with the problem of amalgamation of the values assigned by different individuals to alternative social or economic states in a society into values for the entire society. A social state provides a description ofmaterials that are related to thewell-being of a population in different ways. One fundamental question that arises at the outset is how an individual can rank two alternative states in a well-defined manner. The problem of social aggregation theory can then be regarded as one of clubbing individual rankings into a social ranking in a meaningful way.

Chapter 1 presents an introductory outline of the materials analyzed in the remaining chapters. Chapter 2 of the monograph formally defines individual and social rankings of alternative states of affairs. Chapter 3 provides a rigorous discussion on May’s remarkable theorem on “social choice functions,” which represents group or collective decision rules, limited to two alternatives. In Chapter 4 we analyze Arrow’s social welfare function, a mapping from the set of all possible profiles of individual orderings of social states to the set of all possible orderings. We discuss this with some elaboration in view of the fact that Arrow’s model forms the basis of almost the entire social choice theory. Chapter 5 investigates to what extent the “dictatorship” result can be avoided when some of Arrow’s axioms are relaxed. In Arrow’s framework, when individual preferences are represented by utility functions, utilities are of the ordinal and non-comparable types. Arrow’s theorem with utility functions constitutes a part of Chapter 6 of the book. We use geometric technique to prove this fundamental result. In the recent past, the non-comparability assumption has been relaxed to partial and full comparability assumptions. A discussion on alternative notions of measurability and comparability of utility functions is also presented in Chapter 6. Simple numerical examples have been provided to illustrate the ideas. Possibilities of social welfare functions are expanded as a consequence of interpersonal comparability. Proofs of most theorems in the literature are sophisticated mathematically, which may not be easygoing for non-specialist readers. In view of this, we use simple graphical proofs for many such results.

INTRODUCTION

In this chapter, we explore the extent to which we can escape the dictatorship result if we relax some of Arrow’s axioms. If we relax independence of irrelevant alternatives, then we get the Borda Count which is a well-defined social welfare function satisfying unrestricted domain and weak Pareto. This we have already discussed is Example 4.5 of Chapter 4. In Section 5.2, we relax weak Pareto and replace it by a weaker axiom of non-imposition and then we get the result due to Wilson (1972) that stipulates that a social welfare function satisfying unrestricted domain, independence of irrelevant alternatives, and non-imposition axioms must be null or dictatorial or inverse-dictatorial, given that the number of alternatives is not less than three. Both inverse-dictatorial and null social welfare functions are quite uninteresting. Inverse-dictatorship requires that there exists an agent i whose strict preference over every pair of alternatives is reversed for the society under all profiles in the domain. Like inverse-dictatorial social welfare function, the null social welfare function is also uninteresting since the society is always indifferent across all alternatives for all possible profiles in the domain.

In Section 5.3, we relax transitivity of social preferences and then either we end up in oligarchy or we end up violating the no veto power. Specifically, the oligarchy result, due to Weymark (1984), specifies that a social quasi-ordering that satisfies unrestricted domain, weak Pareto, and independence of irrelevant alternatives must be oligarchic, given that the number of alternatives is not less than three and the set of individuals is finite. In this context, we also derive the Liberal Paradox due to Sen (1970b) that shows that there is no social weak quasi-ordering that satisfies unrestricted domain, weak Pareto, and a very weak form of liberalism. Finally, in Section 5.4, we relax the axiom of unrestricted domain and assume single-peaked preferences and then we can escape Arrow’s dictatorship conclusion. A set of individuals is said to have single-peaked preferences over a set of possible alternatives if the alternatives can be ordered along a line such that the following two conditions hold: (i) each agent has a best (or most preferred) alternative in the set, and (ii) for each agent, alternatives that are further from his or her best alternative are preferred less.

MOTIVATIONS

Often income as the sole dimension of the well-being of a population does not give us an appropriate picture of the living condition of the population since there are non-income dimensions that affect human welfare, for example, habitation, longevity, availability of opportunities from the point of view of contentment, and so on. So, by concentrating on income only, it is implicitly assumed that persons enjoying the same income are considered equally well off regardless of their attainments in the non-income dimensions. However, a non-monetary dimension of well-being need not be perfectly correlated with income. For instance, an income-rich person may not be able to improve the level of sub-optimal supply of a local public good, say, the irregularities of a TV signal that can be received within a specified distance of the signaling station. Thus, apart from the distribution of income, a policy maker might be concerned with the distributions of various non-income goods or dimensions. Examples of such non-income dimensions are health, literacy, housing, and environment. Therefore, to get a complete picture of human well-being it is quite sensible to supplement income with non-income dimensions that influence the contentment of the population. In other words the well-being of a population is a multidimensional phenomenon. This is, in fact, a concretization of our assumption made in the earlier chapters that an individual’s (and hence society’s) utility depends on several states of nature.

In the words of Stiglitz, Sen, and Fitoussi (2009, 14):

Intrinsic to the notion of the capability-functioning approach to the evaluation of human well-being is multidimensionality, where functionings refer to the various things such as income, literacy, housing, life expectancy, communing with others, and so on about which a person cares.

INTRODUCTION

In this chapter, we introduce some basic concepts that will be used throughout this book. In Section 2.2, we start by defining the “at least as good as” relation ≿ that describes the preferences of the individuals over the set of alternatives that they face. We specify certain properties associated with these preference relations. In Section 2.3, we introduce the notion of maximal sets (or choice sets) and link it with the properties of ≿. Finally, in Section 2.4, we discuss social orderings, that is, given a set of agents in a society along with their preferences, how do we aggregate them into social preferences. Specifically, in Section 2.4, we discuss some well-known social aggregation rules like plurality rule, Borda count, anti-plurality rule, oligarchy, and pairwise majority rule.

RELATIONS

Let A = ﹛x,y, z,w …﹜ be the set of alternative states of affairs (alternatives, for short). A relation ≿ on A is a subset of A × A. We shall write x ≿ y if (x, y) ∊ ≿. We say that x and y are unordered by ≿ if neither x ≿ y nor y ≿ x. They are ordered by ≿ if they are not unordered, that is, either x ≿ y holds or y ≿ x holds. We will call ≿ as the “at least as good as” relation defined on the set of alternatives A. Given ≿, let ≻ and ∼ be the asymmetric and the symmetric parts of ≿. That is, x ≻ y if and only if x ≿ y and ￢(y ≿x), where ￢ (y ≿ x) means that y ≿ x is not true. Moreover, x ∼ y if and only if x ≿ y and y ≿ x. We will also refer to ∼ as the strict preference part of ≿ and we will also refer to ∽ as the indifference part of ≿. In words, for any person (society) with the preference relation ≿, between any two alternatives x and y, x ≻ y means that the person (society) strictly prefers x to y and x ∼ y means that the person (society) is indifferent between x and y.

INTRODUCTION

Following the Gibbard–Satterthwaite theorem, several relaxations of the unrestricted domain assumption have been investigated. We have already discussed single-peaked preferences in the previous chapter, which is a relaxation of the unrestricted domain assumption. A second relaxation of the unrestricted domain assumption amounts to assuming that side-payments are allowed among agents (implying the cardinality of their utility functions). Specifically, in this second kind of relaxation, it is assumed that agents have quasi-linear preferences that are represented with quasi-linear utility functions. This assumption of quasi-linearity creates several strategyproof mechanisms. The economic implications of these strategyproof mechanisms are numerous and have been systematically investigated in the vast literature about Vickrey–Clarke–Groves (or VCG) mechanisms (see Vickrey 1961; Clarke 1971; and Groves 1973). The VCGmechanisms, in its most general form, was specified by Groves (1973), though its special cases were identified earlier—first by Vickrey (1961), who specified the second price auction in the context of auction theory, and then by Clarke (1971), who provided the pivotal mechanism for the non-excludible pure public goods problem. There is another class of mechanisms, known as Roberts’ mechanisms for affine maximizers, that generalizes the VCG mechanisms (Roberts 1980). In this chapter, we discuss these strategyproof mechanisms for quasi-linear domains. The problem of designing strategyproof mechanisms is also referred to as dominant strategy mechanism design problem under incomplete information.

In Section 11.2, we introduce a non-excludible pure public goods problem and introduce the VCG mechanisms and show that this is the unique class of mechanisms that satisfies outcome efficiency (that maximizes the sum of utilities of all the agents from the public decision) and strategyproofness (also called dominant strategy incentive compatibility). We then argue that for the pure public goods problems, the VCG mechanisms fail to satisfy Pareto optimality. We then discuss some nice properties of the pivotal mechanism, which is a specific mechanism in the class of VCG mechanisms that was first identified by Clarke (1971). In Section 11.3, we introduce a single indivisible private good allocation problem and show that the pivotal mechanism of Clarke (1971) is identical to the second price Vickrey auction (Vickrey 1961).

INTRODUCTION

In the earlier chapters, it has been assumed that social states are characterized by complete certainty in the sense that they are fully observable by the individuals under consideration. Consequently, each state may be regarded as a certain prospect. An individual can therefore order alternative social states using his preferences without any ambiguity. Thus, the decisions taken by the individuals are taken in an environment of certainty. But when states are affected by uncertainty, the decision criterion may be of a different type. To understand this, consider two farmers for whom the extent of rainfall has a very high impact on their crop outputs from their respective lands. Rainfall conditions may be subdivided into the following categories: (i) flood, (ii) optimum, (iii) hardly sufficient, (iv) less than hardly sufficient, and (v) drought. Each of these categories represents a circumstance of nature. In such a situation, each farmermaximizes his expected (von Neumann–Morgenstern) utility function. A natural question that arises in this context is the following: howare the individual utilities aggregated to arrive at a social utility? John C.Harsanyi (1955) made an excellent recommendation along this line. Harsanyi assumed at that outset that individual and social preferences fulfill the expected utility axioms and these preferences are portrayed by von Neumann–Morgenstern utility functions. The set of alternatives on which individual and social preferences are defined is constituted by the lotteries bred from a finite set of well-defined basic prospects. By including a Pareto principle within the framework, Harsanyi demonstrated that social utility function can be expressed as an affine combination of individual utility functions. In other words, given that the origin of the social utility function has been appropriately normalized, social utility comes to be a weighted sum of individual utilities. This relationship is referred to as theHarsanyi social aggregation theorem (Weymark 1991, 1994).

In Harsanyi’s social aggregation theorem, individual and social preferences are defined on the set of lotteries generated from a finite set of basic prospects. These preferences are expected to satisfy expected utility hypothesis and are represented by von Neumann–Morgenstern utility functions. The only link between the individual and social preferences is the requirement that the society should be indifferent between a pair of lotteries when all individuals are indifferent between them.

INTRODUCTION

Social choices, by which we mean group or collective decisions, are made in twoways. In political voting, collective decision is typically used to make “political” decisions, and in the market mechanism, collective decision is typically used to make economic decisions. In this chapter, we concentrate exclusively on the former. Collective action (or social [group] choices) should be based on the preferences of individuals (or agents) in society. Therefore, an important aspect of social choice theory is a description and analysis of the way preferences of individual members of a group are aggregated into a decision of the group as a whole (see Arrow 1950; and Taylor 2005). We are interested in aggregating preferences of the agents into a social decision, which essentially means that we consider social choice functions that, for each possible individual preference profile, picks an alternative for society from the set of alternatives. This approach is different from social welfare function (discussed in Chapters 2–6) that for each possible individual preference profile picks an ordering for society (not just an alternative like the social choice function).

Social choice methods can be subject to strategic manipulation in the sense that an agent, by misrepresenting his preference, may secure an outcome he prefers to the outcome which would have been obtained if he had revealed his preference sincerely. This issue was first addressed by Gibbard (1973) and Satterthwaite (1975). They consider a situation where a collective decision has to be made by a group of individuals regarding the selection of an outcome. The choice of this outcome depends on the preferences that each agent has over the various feasible outcomes. However, these preferences are known only to the agents themselves, that is, each agent knows only his preference. The Gibbard–Satterthwaite theorem states that under verymild assumptions, the only procedureswhich will provide incentives for each individual to report his private information truthfully is one where the responsibility of choosing the outcome is left solely to a single individual (the dictator).

The chapter is organized as follows. In Section 10.2, we consider the framework and we also provide the statement of the Gibbard–Satterthwaite theorem.

INTRODUCTION

Kenneth J. Arrow (see Arrow 1950) provided a striking answer to a basic problem of democracy: how can the preferences of a set of individuals be aggregated into a social ordering? The answer, known as Arrow’s impossibility theorem, was that every conceivable aggregation method has some flaw. That is, a handful of reasonable-looking axioms, which one thinks an aggregation procedure should satisfy, lead to an impossibility. This impossibility theorem created a large literature and the major field called social choice theory, which is one of the main subject matters of this book. Specifically, Arrow’s impossibility theorem or the general possibility theorem or Arrow’s paradox is an impossibility theorem stating that when agents face three or more distinct alternatives (options), every social welfare function that can convert the preference ordering of all the individuals into a social ordering while also meeting unrestricted domain, weak Pareto, and independence of irrelevant alternatives must be dictatorial. Unrestricted domain (also called universality) is a property in which all possible preferences of all individuals are allowed in the domain. Weak Pareto requires that the unanimous preferences of individuals must be respected, that is, if every agent strictly prefers one alternative over the other, then society must also strictly prefer the same alternative over the other. Social preference orderings satisfy the independence-of-irrelevant-alternatives criterion if the relative societal ranking of any two alternatives depends only on the relative ranking of those two alternatives of all the individuals and not on howthe individuals rank other alternatives.Adictatorial social welfare function is one in which there is a single agent whose strict preferences are always adopted by the society. In this chapter, we present a detailed analysis of Arrow’s impossibility theorem and then provide two proofs of the theorem.

THE FRAMEWORK

Consider a society with n agents and we denote this society by N = ﹛1, … , n﹜. Each agent i ∊ N has an ordering ≿i defined on A and assume that |A| ≥ 3. The assumption that ≿i is an order on A has implications for the strict preference relation ≿i and the indifference relation ≿i. These implications are summarized in the next proposition.

Social aggregation theory is concerned with investigating methods of clustering values that individuals in a society attach to different social or economic states into values for the society as a whole. Loosely speaking, a social state, a state of affairs, represents a sketch of the amount of commodities possessed by different individuals, quantities of productive resources invested in different productive activities and different types of collective activities (Arrow 1950). The values that individuals attach to different social states are reflections of respective preferences. Consequently, the problem of social aggregation is to combine individual preferences into a social preference in an unambiguous way. In this monograph, we will use the terms “social aggregation” and “social choice” interchangeably.

Modern social aggregation theory started with the publication of Kenneth J. Arrow’s pioneering contribution Social Choice and Individual Values, his PhD dissertation, in 1951. It can be regarded as the foundation to laying the groundwork of social aggregation theory in view of its innovative facture and revolutionary influence. The idea of aggregating individual preferences into a collective choice rule predates Arrow (1950) by more than 150 years. In 1785, the French mathematician and philosopher Marie-Jean de Condorcet considered the problem of collective decision-making with regard to majority voting. According to majority voting, in a choice between two alternatives x and y, x is declared as the winner if it gets more votes than y. He established that the method of pair-wise majority voting may give rise to cyclicality in social preference. This paradoxical result, popularly known as the Condorcet voting paradox, appears to draw inspiration, to a certain extent, from an earlier contribution by the French mathematician Jean-Charles de Borda (de Borda 1781). In this alternative voting system, known as the Borda count method, voters rank candidates in order of preference.

One of the major goals of this monograph is the analysis of the Arrovian approach to the theory of collective aggregation and later developments on it. We, therefore, focus now on Arrow’s impossibility theorem, which is generally acknowledged as the formative basis of modern social aggregation rules.Arrow’s seminal work examines the possibility of the aggregation of individual preferences into a social preference in order to obtain a social ranking of alternative states of affairs.

INTRODUCTION

In this chapter, we restrict our attention to the decisive voting rule with only two contesting candidates (or alternatives). A decisive voting rule maps for every possible vote (or preference) of the set of agents over the two contesting candidates (say, x and y) to either a winner or two non-losers. May’s theorem, which is the main subject matter of this chapter, specifies the importance of majority voting rule by arguing that it is the unique rule that satisfies four important democratic principles. These four key democratic principles are decisiveness of the voting rule, anonymity, neutrality, and positive responsiveness. Decisiveness of the voting rule requires that the voting rule must specify a unique decision even if the decision is indifference for any set of individual preferences over the two contesting candidates. Anonymity (or symmetry across agents) requires that a voting rule must treat all voters alike, in the sense that if any two voters traded ballots, the outcome of the election would remain the same. Neutrality (or anonymity across alternatives) requires that a voting rule must treat all candidates alike, rather than favor one over the other. Finally, positive responsiveness (a type of monotonicity property) requires that if the group decision is indifference or favorable to some alternative x, and if individual preferences remain the same except that a single individual changes his or her vote in favor of x, then the group decision should be x. Formally,May’s theoremstates that among the class of all decisive voting rules, the majority voting rule is the only one that satisfies anonymity, neutrality, and positive responsiveness.

The chapter is organized as follows: Section 3.2 provides the framework. In Section 3.3 we state and prove May’s theorem. In Section 3.4 we check the robustness of the axioms used in May’s theorem.

THE FRAMEWORK

We consider preferences of a finite set of agents N = ﹛1, … , n﹜ of a society voting over two alternatives x and y.

INTRODUCTION

Arrow’s theorem is based on both non-measurability (utility of an individual cannot be measured) and non-comparability (the welfare of two different individuals cannot be compared). The non-measurability is implicit in Arrow’s use of orderings to represent individual preferences and non-comparability is implied by the controversial irrelevance of individual alternatives axiom. If either of these assumptions is relaxed, then Arrow’s theorem does not hold.

Sen (1970) was the first to propose a framework for exploring the consequences of relaxing non-measurability and non-comparability. Instead of assuming the set of all preference ordering as a primitive (as in Arrow), he assumed it to be the set of all possible utility functions. A profile is a list of utility functions, one for each agent. A social welfare functional associates a utility function to each profile.

In this framework, Arrow’s theoremholds if the utility functions are only ordinally measurable and interpersonally non-comparable. Arrow’s impossibility theoremis robust in the sense that weakening the requirement that social preferences are orderings, while preserving non-measurability and non-comparability, leaves little room for sensible social binary relations. Sen (1974, 1977a, 1986a) has provided a taxonomy of different measurability and comparability assumptions. In this chapter, we study Arrow’s theoremunder these measurability and comparability assumptions.

Why should we care about measurability and comparability assumptions? d’Aspremont and Gevers (2002) argue that there is a distinction between the problem of determining the “relationship that collective decisions or preferences ought to have with individual preferences” and the separate problem where “an ordinary citizen attempts to take the standpoint of an ethical observer in order to formulate social evaluation judgements.” The first problem is appropriate when looking at the design of constitutions, for instance. When designing a constitution, we will be concerned with the abilities of voters to manipulate the voting process because voting decisions typically involve groups. We also want to make minimal assumptions on measurability and comparability because preferences cannot be measured.

However, when an ordinary citizen takes the viewpoint of an ethical observer, she can make assumptions about measurability and comparability in order to determine her own voting strategy, or to recommend a voting process for society. d’Aspremont and Gevers (2002) also observe that much of the discussion of social welfare prior to Arrow was concerned with the second problem.

INTRODUCTION

Often the literature on inequality measurement regards income as the only yardstick of well-being. One plausible reason for this assumption is that an increase in a person’s income is likely to increase his level of living. However, in the recent period the literature has observed a shift of emphasis from a single-dimensional structure to a multidimensional framework for the purpose of inequality evaluation. (We analyze this issue in greater detail in the next chapter.) As a starting point, in this chapter we treat income as the only dimension of well-being. This chapter forms the basis of multidimensional approaches to inequality evaluation to be analyzed in Chapter 9.

After presenting the basics and preliminaries in the next section, in Section 8.3 we analyze common features of the inequality evaluators that rely respectively on the direct and inclusive measure of well-being approaches. While in the former outlook a social welfare function is defined directly on the set of income distributions, in the latter individual utilities are aggregated to generate a welfare metric of society. The next two consecutive sections deal explicitly with inequality assessment from these two perspectives. Section 8.6 stipulates how some standard inequality indices can be interpreted within the Harsanyi (1953, 1955) framework (Dahlby 1987).

The current literature on the measurement of inequality at times considers both achievement and shortfall inequalities in some dimension of human well-being and establishes a relation between them. Section 8.7 presents a condensed essay on the related literature. A person often may desire to have access to one or more opportunities available in society from the perspectives of happiness and success. The subject of Section 8.8 is a brief discussion on opportunity equality. In the next two sections we deal with inequality for an ordinal dimension of human well-being and an ordinal approach to inequality evaluation. In the context of network engineering, a fairness indicator determines the extent to which users are receiving fair shares of resources. The concern of Section 8.11 is a brief review of the axiomatic literature developed along this line. As we will argue, the use of conventional equality metrics may not be suitable for the evaluation of fairness.

Introduction

The measurement of voting power is a very important topic in social sciences. It is concerned with the power of a member of a voting body or a board that makes yes-or-no decisions on a proposed resolution (or bill) by votes according to some unambiguous criterion. Examples of such decision-making bodies are the United Nations Security Council, the International Monetary Fund, the Council of Ministers in the European Union and the governing body of any corporate house etc.

The voting process of any collective decision-making body is governed by its own constitution, which prescribes the decision-making rule for the body. The individual votes are aggregated using the decision rule to determine the decision of the body as whole. Generally, when a proposal is presented before a voting body, its members are asked to vote either for the proposal (‘yes’) or against it (‘no’). (The more general case when abstention is allowed is discussed later in Section 7.6.) The individual votes are then transformed into a collective decision of the body using the laid down rules.

By voting power of an individual voter, we mean his capability to alter the outcome of the voting procedure by changing his position on the proposed bill. It is an indicator of the extent to which a voter has control over the decision of the voting body. It should rely on the voter's importance in casting the deciding vote. To illustrate this, consider a voting situation where there are three voters, namely a, b and c. These three voters are distinguished by the characteristic that the numbers of votes they have are respectively 9, 4 and 2. Also suppose that the decision rule imposes the condition that at least 12 votes are necessary for any resolution to get through. Now, we may be tempted to conclude that since the number of votes of a is more than two times that of b, the power of a must be greater than that of b. Also since c has a positive number of votes, c should have some positive power.

The objectives of game theory are to model and analyze interdependent decision-making circumstances. A distinction is made in the literature between cooperative and non-cooperative games in the sense that while for the former, obligatory contracts between the participants, referred to as players, is possible, such a possibility is ruled out for the latter.

Cooperative game theory has become very influential in social sciences in the recent years. This book discusses some highly important issues in cooperative game theory with examples from economics, business and sometimes from politics. The book is divided into two parts. Part 1 is composed of Chapters 1—9. Foundations of game theory and a description of the Chapters 2—13 are presented in Chapter 1. Cooperative games with transferable utility are discussed in Chapters 2—6. Chapter 2 explains some basic concepts, definitions and preliminaries. Chapter 3 analyzes set-valued solution concepts like the core, the dominance core, stable sets and different core catchers. An extensive discussion on the relations between alternative solution concepts is also made in this chapter. Two additional set-valued solution concepts, the bargaining set and the kernel that rely on a coalition structure, are presented in Chapter 4. This chapter also discusses the nucleolus, a one-point solution concept, which has interesting relations with the bargaining set and the kernel. In Chapter 5, we consider a well-known one-point solution concept, the Shapley value. A particular type of transferable-utility cooperative game with some especially attractive properties is a convex game, which has been examined in Chapter 6. Relations between the Weber set, an alternative set-valued solution concept, the core and the Shapley value for such games are also reviewed in detail in this chapter. Chapter 7 presents a systematic analysis of voting games that often arise in interactive decision-making situations. The subject of Chapter 8 is stable matching. We discuss the Gale—Shapley basic model of matching men to women or vice-versa, the concept of stable matching, matching problems in two-sided markets, matching problems when participants from one side do not have preferences and housing exchange problems. An investigation of nontransferable utility games is carried out in Chapter 9.