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.

Introduction

This chapter is devoted to the analysis of a two-sided matching market that consists of two sets of non-overlapping agents. The major objective here is to discuss the possibility of matching a set of agents with another set of agents. For instance, in a marriage problem, a set of men and a set of women need to be matched in pairs.

Such a market differs from a standard commodity market in which market price determines whether a person is a buyer or a seller. For example, a person may be a buyer of a good at some price and a seller of another good at some other price—the market is not two-sided. Additional examples of matching problems include: firms have to be matched with workers, hospitals have to be matched with interns, colleges have to admit students and football players require matching with clubs. ‘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’ (Roth and Sotomayor 1990, p.1). These markets are definitely different from markets for goods in which a person may be buyer of one good (say, potato) and a seller of another good (say, rice).

The matching theory is a leading area in economic theory because of its importance and also because of the difficulties involved in the allocation of indivisible resources. The appropriate tools for analysis are linear programming and combinatorics. In recent years, it has become quite popular because of applications game theory to study matching problems.

One very important problem in the analysis of matching problems is stability. The problem is to find a stable matching between two sets of agents given a set of preferences for each agent. An allocation where no person will make any gain from a further exchange is called stable. In their pioneering contribution, Gale and Shapley (1962) defined a matching problem and the concept of stable matching. They also showed that stable matchings always exist and suggested an algorithm for computing stable matchings.

The class of weighted majority games is a special case of the class of simple voting games. Both weighted majority games and simple voting games have been discussed in an earlier chapter. One of the key issues for such games is to measure the power of an individual. Several such measures have been introduced and studied in literature. The question that concerns us in this chapter is the following. Given a weighted majority game, is it possible to actually compute the values of the different power indices for this game? We are interested in efficient algorithms and more generally, in the computational complexity of the problem. Before getting into the algorithmic details, we briefly recapitulate some of the basic notions related to weighted majority games.

Recall that a weighted majority game, which we write as v = (w, q) is given by a set of n players N; a list of weights w = (w1, …, wn) ∈ ℜn, one weight for each player in N; and a quota q. Given a subset S of N, its weight is defined to be w(S) = Σi∈Swi. As mentioned earlier, the subset S is said to be a winning coalition if w(S) ≥ q and a losing coalition otherwise. A minimal winning coalition is a winning coalition S such that if any player is dropped from S, it turns into a losing coalition, i.e., w(S) ≥ q and w(Ti) < q for every i ∈ S with Ti = S \ {i}.

Given a game, the total number of winning coalitions in it and the total number of minimal winning coalitions in it are of interest. For a player i, recall that MWi is the set of all minimal winning coalitions S such that i ∈ S. For the Deegan—Packel index, it is required to obtain the distribution of the cardinalities of the sets in MWi.

Let G = (U, V, E) be a bipartite graph with |U| = |V| = n and the edges in E have one end-point in U and the other end-point in V. A matching μ in G is a set of vertex disjoint edges, i.e., μ is a set of edges such that no two edges are co-incident on the same vertex. Since the edges in a matching have to be vertex disjoint, no matching can have more than n edges. A perfect matching is a matching containing n edges. There are well-known algorithms to find a perfect matching (if one exists).

In this section, we will address a different matching problem. An instance still consists of two sets U and V, but, the constraints are different. Let U = {u1, …, un} and V = {v1, …, vn}. Let π1, …, πn be permutations of the set V and let σ1, …, σn be permutations of the set U. The problem instance is given by the following relations. For 1 ≤ i ≤ n and 1 ≤ j ≤ n,

ui ↦ πi; vj ↦ σj.

The permutation πi is a linear ordering of the vertices in V and similarly, the permutation σj is a linear ordering of the vertices in U. Consider the vertices in U to represent n distinct men and the vertices in V to represent n distinct women. The permutation πi represents the ranking of the n women by the man ui; similarly, the permutation σj represents the ranking of the n men by the woman vj.

A matching μ is a pairing of a man and a woman and can be thought of as a marriage. Let the partner of the man ui be denoted by μ(ui) and the partner of woman vj be denoted by μ(vj). Suppose there is a man ui and a woman vj such that πi(vj) precedes πi(μ(ui)) and σj(ui) precedes σj(μ(vj)).

Introduction

This is an innocuous looking question with a deep answer. We will not attempt to explore the question in its full generality (We refer the reader to Aho, Hopcroft and Ullman (1974) for a comprehensive discussion.). Instead, a high level view will be adopted. An intuitive answer is that an algorithm is a finite sequence of elementary operations with the objective of performing some (computational) task. Let us take this as an acceptable answer and consider several aspects of it in more detail.

Elementary operations: A natural question to ask is how elementary is ‘elementary’? The idea of Turing machines formalises an elementary operation as simply reading or writing a symbol and/or moving a tape head one cell to the left or writing on an infinite tape divided into cells where it is possible to write one symbol on any cell. It is possible to start from this simple notion and obtain algorithms for very complex tasks. In fact, it is a hypothesis that Turing machines capture the exact notion of algorithms. We, however, will not work with the Turing machine model. The reason is that the simplicity of the model makes it quite cumbersome to express higher level ideas. Instead, the elementary operations that we will consider will be at a higher level and include arithmetic and logical operations. This is also the usual practice in the study of algorithms. One works at a higher level knowing that, in theory, all algorithms can be reduced to the Turing machine model.

Finite sequence: The finiteness condition of an algorithm implies that it must always halt. Procedures which continue indefinitely will not be considered as algorithms. The notion of Turing machines can be extended to cover such procedures, but, we will not need to consider them here. A word about the sequence of operations in an algorithm will also be in order.

Introduction

As we have seen in the discussion of the profit-sharing game in Chapter 1, if all the players in a game decide to work together, there arises a natural question concerning the division of profit among themselves. We have also observed that if some of the players in a coalition object to a proposed allocation, they can decide to leave the coalition. In order to understand this formally, a rigorous treatment of the worth of different coalitions of players and the marginal contribution of a player to a coalition is necessary. Often, some structural assumptions about a game, for instance, whether the game is additive, super-additive or sub-additive, make the analysis convenient. Moreover, in some situations, study of issues like equivalence between two games becomes relevant. This chapter makes a formal presentation of such preliminary concepts and analyzes their implications.

Preliminaries

In this section, we present and explain some preliminary concepts and look at their implications. We assume that N = {A1, A2, …, An} is a finite set of players, where n ≥ 2 is a positive integer. The players are decision makers in the game and we will call any subset S of N, a coalition. The entire set of players N is called the grand coalition. The collection of all coalitions of N is denoted by 2N; each coalition has certain strategies which it can employ. Each coalition also knows how best to use these strategies in order to maximize the amount of pay-off received by all its members. For any coalition S, the complement of S in N, which is denoted by N \ S, is the set of all players who are in N but not in S. For any coalition S, |S| stands for the number of players in S.

An n-person cooperative game assigns to each coalition S, the pay-off that it can achieve without the help of other players. It is a convention to define the pay-off of the empty coalition Ø as zero.

'Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision makers. Game theory provides general mathematical techniques for analyzing situations in which two or more individuals make decisions that will influence one another's welfare' (Myerson 1997, p.1). The underlying idea here is that the decisions of the concerned individuals, who behave rationally, will influence each other's interests/pay-offs. No single person alone can determine the outcome completely. Each person's success depends on the actions of the other concerned individuals as well his own actions. Thus, loosely speaking, game theory deals with the mathematical formulation of a decision-making problem in which the analysis of a competitive situation is developed to determine an optimal course of action for a set of concerned individuals. Aumann (1987; 2008) suggested the alternative term ‘interactive decision theory’ for this discipline. However, Binmore (1992) argued that a game is played in a situation where rational individuals interact with each other. For instance, price, output, etc. of a firm will be determined by its actions as a decision maker. Game theory here describes how the firm will frame its actions and how these actions will determine the values of the concerned variable. Likewise, when two or more firms collude to gain more power for controlling the market, it is a game.

To understand this more clearly, consider a set of firms in an oligopolistic industry producing a common output. Each firm must not only be concerned with how its own output affects the market price directly; it must also take into consideration how variations in its output will affect the price through its effect on the decisions taken by other firms. Thus, strategic behaviour becomes an essential ingredient of the analysis. A tool that economists employ for modelling this type of situation is non-cooperative game theory.

As a second example, consider a landowner who owns a large piece of land on which some peasants work. The landowner does not work and requires at least one peasant to work on the piece of land to produce some output.

Introduction

As a solution concept to cooperative games, the core consists of a set of imputations without distinguishing one element of the set from another. It is a useful indicator of stability. However, the core may be quite large or even empty. A more comprehensive solution to cooperative games is the stable set or the von Neumann—Morgenstern solution. However, here also no single point solution exists so that we can associate a single point vector to a coalition form game. These solution concepts cannot predict a unique expected pay-off corresponding to a given game. If an arbiter's objective is the assignment of a unique outcome, which may be decided by the arbiter in a fair and impartial manner, then these solution concepts are inappropriate.

In an axiomatic approach, Shapley (1953) characterized a unique solution using a set of intuitively reasonable axioms. Shapley's solution is popularly known as the Shapley value. The central idea underlying the Shapley value is that each player should be given his marginal contribution to a coalition, if we consider all possible permutations for forming the grand coalition. Therefore, in a sense, the player is paid out his fair share of the value from the coalition for having joined the coalition. The Shapley value of a player is the expected value of the marginal contributions of the player over all possible orderings.

In the next section of the chapter, the Shapley value is defined by an axiomatic approach. The characterization theorem is explained in Section 5.3. Section 5.4 presents a discussion on Young's (1985; 1988) alternative characterization of the Shapley value using an axiom involving monotonicity of the marginal contributions. This section also analyzes the Shapley value using the potential function introduced by Hart and Mas-Colell (1989). Finally, some applications of the Shapley value are discussed in Section 5.5.

The Formal Framework: Definitions and Axioms

In different orders of grand coalition formation, a player's marginal contributions are likely to vary. These marginal contributions indicate how important the player in the overall cooperation is. A natural question here is what pay-off can a player reasonably expect from his cooperation. The Shapley value provides an answer to this.