Introduction

The coalition games we have analyzed in the earlier chapters are transferable utility (TU) games. In such games, each coalition is assigned a pay-off (utility) represented by a real number with the interpretation that the members of the coalition can divide this pay-off in an unambiguous manner. In contrast, for non-transferable utility (NTU) games, the pay-offs for each coalition are represented by a set of pay-off (utility) vectors indexed by the members of the coalition. Transferability of utility is a simplifying assumption which makes the analysis quite convenient. However, the transferability assumption may be undesirable in many applications. To illustrate this, consider a bilateral monopoly, a market situation in which a single seller confronts a single buyer. For concreteness, assume that a monopsonistic supplier of a rare metal, which is needed to produce an alloy, faces a monopolistic buyer, the only producer of the alloy. That is, a monopoly supplier of an input faces a monopoly demander of the input. It is known that in such a situation, the market outcome is indeterminate and the outcome must be settled through bargaining. If the producer ceases production, the supplier will not be able to sell the metal. On the other hand, if the supplier refuses to sell the metal, there will be no production of the alloy. In either case, no positive pay-off will be created for each of them. On the other hand, if the two parties decide to cooperate and come to a settlement, some positive pay-off will be created for each of them. However, the settlement does not involve any transfer of pay-off between the two parties. The settlement between the parties is the outcome of an NTU cooperative game. A unique solution to this bargaining problem emerges if the Nash (1950) bargaining model is adopted.

Extensive studies on NTU games were started only in the 1960s and the literature is not very voluminous. A large part of the literature is devoted to the analysis of bargaining games in which only the individual players and the grand coalition play a role.

One of the most important practical optimization problems prevalent in practically all walks of life is the linear programming (LP) problem. It crops up in various engineering, operations research, scheduling and many other different scenarios. Due to its important, the problem has been extensively studied since the middle of the previous century. As a result, a deep literature has developed around the problem and connections have been established to other sciences.

Our reason for considering LP in this book is its connection to several aspects of cooperative game theory. The problem of determining whether the core of a game is empty can be formulated as an LP problem. Similar formulations can be made for the nucleolus. In this chapter, we provide a brief introduction to LP and describe the connections to the core and the nucleolus. The algorithmic complexity of solving LP is discussed in Chapter 11 and in Chapter 13, LP is used to formulate a notion of fairness in the stable matching problem. Our description of LP is minimal and is intended only to familiarise the reader with the basic idea. For a deeper understanding of the area including its algorithmic issues, we refer the reader to Papadimitriou and Steiglitz (1982).

The Diet Problem

LP is best introduced through a practical example. We start by motivating LP with the so-called diet problem. In this problem, a person wishes to obtain a balanced diet at a minimal cost. The basic idea is that there are several types of nutrients (say, proteins, fats, carbohydrates, minerals, etcetera) and for a healthy diet, a person needs to take in a certain minimum amount of each nutrient. The nutrients are not directly available. Instead, what are available are different kinds of foods (say, rice, wheat, meat, etcetera). Each kind of food contains the basic nutrients in varying proportions. We assume that these proportions can be quantified per unit of each kind of food.

Introduction

Given a particular way of forming the grand coalition in a coalition form game, the marginal contribution of a player to the grand coalition is the amount by which the worth of the coalition increases when he joins the coalition of players that precede him. For instance, the marginal contribution of each share holder in a joint profit-making business is the additional amount of profit that he can guarantee to the coalition of players who have joined the business before him. The Weber set is the smallest convex set containing the set of marginal contribution vectors of the players.

In the next section of this chapter, we will introduce the Weber set of a game and show that it contains the core of the game (Weber 1988). In Section 6.3, we will study the properties of convex coalition form games. Convexity of a coalition form game may be interpreted as the condition where there are higher incentives for joining a coalition as the size of the coalition increases (Shapley 1971). This section also shows that for a convex game, the core is non-empty. It is then shown that the Weber set and the core of a game coincide if and only if the game is convex. It is also demonstrated that the bankruptcy game is an example of a convex game. Next, we show that the Shapley value for a convex game is an element of the core, which in turn demonstrates that the core of a convex game is non-empty. In Section 6.4, we will analyze random order values and their relations with the Weber set and the Shapley value. It is explicitly proven that O'Neill's (1982) random arrival rule, a solution to the bankruptcy problem, coincides with the Shapley value of the corresponding bankruptcy game.

The Weber Set and Core

Recall that given a particular arrangement or permutation of players in the grand coalition of a game, the corresponding marginal contribution vector gives each player his marginal contribution to the coalition formed by his entrance according to the specific permutation. Note that the set of all marginal contribution vectors in a game is a closed set.

Introduction

The bargaining set was introduced by Aumann and Maschler (1964). The central idea underlying this solution concept is that a player may abstain from objecting to a proposed allocation because of the apprehension that the objection might lead to a counter-objection by another player. A player definitely tries to enter a firmly established coalition with the objective that his pay-off will be maximized. An allocation in the current context is regarded as firmly established or stable if all the objections against it can be tackled by counter-objections. Thus, it is not sufficient for a coalition of some players to only improve on a particular allocation to raise objections against it. It is also necessary to guarantee that there does not exist any possibility for members of that coalition to be allured by another coalition that can improve on the allocation proposed by the first coalition as an alternative to the originally proposed allocation.

Davis and Maschler (1965) proposed the kernel as a solution concept to cooperative game theory problems. Essential to the definition of the kernel is the coalitional excess, the difference between the pay-off a coalition can achieve on its own and the sum of the pay-offs of the members of the coalition that the proposed allocation assigns to the members. This excess is a measure of the size of the complaint in the sense that it determines the amount by which the coalition as a group falls short of its potential under the allocation. The kernel can be interpreted with respect to effective objections and counter-objections that are stated in terms of excesses and personal minimums. An individually rational pay-off configuration which is in the kernel is also in the bargaining set. Davis and Maschler (1965) also introduced the pre-kernel as a solution concept. The central idea underlying this notion is that any pair of players is in equilibrium in the sense of equality of maximal excesses of coalitions containing one player but not containing the other.

Introduction

As discussed earlier, if all the players in a game decide to work together, there arises a natural question concerning the division of profit among themselves. Moreover, if some of the players in a coalition object to a proposed allocation, they can decide to leave the coalition. The core is one of the most important solution concepts to such problems in cooperative game theory. It combines the property of Pareto efficiency with individual rationality. A core allocation is based on the idea that no set of players will leave the coalition and take a collective action that will make them better off. An allocation in the core assures that each player is better off in the grand coalition, the coalition of all players of the game. According to Myerson (1997), the core is very appealing since it includes Pareto efficient allocations and reflects the power of the players, as represented by the characteristic function—the function that specifies the worth of any coalition.

After defining and illustrating the core, and looking at some implications of the definition in the next section, we will discuss the relationship between the core and the dominance core in Section 3.3. The topic of discussion in Section 3.4 is the Bondareva (1963)—Shapley(1967) theorem, which provides a necessary and sufficient condition for the core to be non-empty. In Section 3.5, we provide a treatment of two core catchers that contain the core as a subset. Some variants of the core are analyzed in Section 3.6. The von Neumann—Morgenstern solution or stable set, a solution concept closely related to the core, is presented in Section 3.7. Some real-life applications of the core are shown in Section 3.8.

Concepts and Definitions

In this section, we will define the core formally and study some of its properties. The core is probably the most prominent solution concept for allocating pay-offs (or costs) in problems of cooperative game theory.