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.

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.

In the century following the founding of Basra, Kufa, and other Arab-Muslim cities, Arabic literary interests were pursued in conjunction with Quranic, legal, theological, and other religious subjects. An Arabic lingua franca emerged for the diverse tribes. Persian and Aramaic speakers contributed to rapid changes in Arabic’s vocabulary, grammar, style, and syntax. As the Arabic language changed, religious scholars feared that they would lose touch with the Arabic of the Quran and thus lose the meaning of God’s revelations. In eighth-century Basra, philosophical, lexicographical, and grammatical studies were undertaken to recapture the pure Arabic of Mecca and of the desert tribes and to clarify the usage of Arabic in the Quran and hadith. The roots of words had to be specified, vocabulary selected and explained, and proper speech given rules of grammar and syntax. These linguistic efforts persisted for more than a century and produced what we now know as classical Arabic. The grammar of Sibawayhi (d. 796) and the early dictionaries of Arabic were the products of this period.

The cultural ramifications of these studies went beyond linguistic analysis. The basis of linguistic studies was the collection of examples of old Arabic. Much in the manner of contemporary linguists or anthropologists, the scholars of Basra and Kufa sought out the bedouins and recorded their poems and sayings. Gradually, a large body of lore was accumulated and transcribed from oral into written form. This lore included bedouin poetry, as well as information about the life of the Prophet, Quranic revelations, the early conquests, and the behavior of the early leaders of the Muslim community. From the eighth to the tenth centuries, the totality of this literary and religious culture was gathered into several encyclopedic collections. Arabic literary culture was not purely the heritage of the desert but was shaped in the early Islamic era out of the religious and historical concerns of the Umayyad and early ʿAbbasid periods.

For millennia, the central theme in the history of the vast and varied regions that lie between the settled parts of the Middle East and of China was the relationship between nomadic-pastoral and sedentary peoples. Whereas the great civilizations of the Middle East and China were primarily imperial and agricultural, the region between them was a zone of steppe lands and scattered oases. The population was predominantly pastoral and lived by raising horses and sheep. It was also organized into clans and tribes, which were sometimes assembled into great confederations. The settled peoples lived primarily in the oasis districts of Transoxania, Khwarizm, Farghana, and Kashgar, and in scattered towns along the trade routes that linked China, the Middle East, and Europe. Settled and pastoral peoples had close relationships with one another, exchanging products and participating in caravan trade. Pastoral peoples also infiltrated the settled areas and became farmers or townsmen. Sometimes they conquered the agricultural oases and became rulers and landlords. Inner Asia was also the reservoir holding a sea of peoples who, organized into great confederations, from time to time conquered the Middle East and China. From the second millennium BCE to the eighteenth century, the history of the region may be told in terms of ever-repeated nomadic conquests, the formation of empires over oasis and settled populations, and the constant tension between pastoral and agricultural peoples.

The development of an Islamic civilization in Inner Asia was closely related to that of Iran. Islam first spread in this region as a result of the Arab conquests of Iran and Transoxania and the movement of Muslim traders and Sufis from the towns to the steppes. The two regions were also linked by the Turkish migrations of the tenth to the fourteenth centuries that brought Inner Asian peoples into Iran, and Iranian monarchical culture and Islamic civilization into Inner Asia. In the tenth and eleventh centuries, Qarluq and Oghuz peoples converted and founded the Qarakhanid and Saljuq empires. Under the Qarakhanids, the Hanafi school of law and Maturidi school of theology were established in Transoxania, and a new Turkish literature inspired by Persian Islamic literature came into being. The Qarakhanids also favored the diffusion of Islam from Transoxania into the Tarim basin and the northern steppes. Sufi preachers, especially Shaykh Ahmad al-Yasavi (d. 1166), helped to spread Islam among nomadic peoples.

The consolidation of a post-imperial Islamic society was accompanied by the consolidation of Islamic religious literatures, beliefs, and values and by the canonization of an Islamic orthodoxy. Although the basic literatures of exegesis, hadith, law, theology, and mysticism had originated in an earlier era, during the tenth to the thirteenth centuries these literatures were merged into the forms that we now identify as “classical Islam.” A Sunni-scripturalist-Sufi orientation became the most commonly accepted version of Islam. Shiʿism, philosophy, theosophy, and popular religion were the alternatives to the Sunni consensus. The post-imperial era constructed both the normative forms of Islamic religious belief and practice and the alternatives, thus defining the issues that would ever after constitute the problématique of Muslim religious discourse.

Normative Islam: Scripture, Sufism, and Theology

Sunni consensus became grounded in scripture during the medieval period. In the post-imperial era, the Quran was understood to require each person to do the good deeds commanded by God; to be moderate, humble, kind, and just; and to be steadfast and tranquil in the face of his own passions. The true Muslim is the slave of God. He accepts his humble place in the world and takes no pride or consolation in human prowess but recognizes the limited worth of all worldly things and the greater importance of pleasing God.

East Africa, including the eastern Sudan, the East African coasts, and the hinterlands of Ethiopia, Somalia, and Kenya, formed another major zone of Muslim population. Here the sources of Islamic influences were primarily Egypt, Arabia, and the Indian Ocean region, rather than North Africa. (See Map 25.)

Sudan

The history of the eastern Sudan (the modern state of Sudan) was separate from that of the central and western Sudan, due to the fact that Islam reached the eastern Sudan from Egypt rather than from North Africa. Over centuries Arab-Muslims moved deeper and deeper into Sudan. The Arab-Muslim conquerors occupied Egypt as far as Aswan in 641. In the ninth century, Egyptians swarmed to the newly discovered Allaqi goldfields between the Nile and the Red Sea. An early pact, renewed in 975, between Egypt and the king of Maqurra, the northern provinces of the Sudan, provided for a tribute of slaves in exchange for wheat and wine for the Christian Eucharist and for free passage of merchants. In the twelfth and thirteenth centuries Arab bedouins migrated south and married into local families, who adopted Arabic and were drawn into the Islamic cultural orbit. Through matrilineal succession, their children inherited local chieftainships. Arab penetration was followed by the Mamluk conquest of Nubia in 1276, the first of a number of campaigns lasting through the fourteenth century. In 1317 the church of Dongola was rededicated as a mosque. Most of the country, however, was in the hands of local Arab tribal chiefs who continued to push south across northern Kordofan and Darfur and into the Chad basin. In 1517 the Ottoman sultan, Selim I, occupied Nubia as far as the third cataract of the Nile. Nubia remained under Ottoman rule for some 300 years.

The conquests

With the death of the Prophet Muhammad, a new era began, an era of vast conquests and the formation of a Middle Eastern–wide Arab-Islamic empire. The Arab-Muslim conquests began the processes that culminated in the formation of a new empire, which included all of the former Sasanian Empire and much of the Byzantine Empire, and in the emergence in that geographic and political framework of Islamic cultures and societies.

When Muhammad died in 632, he left no instructions concerning succession, and, in the absence of an agreement with regard to a successor, the Muslim community – a conglomeration of diverse elements – was on the verge of disintegrating. To prevent this, some of the tribes and factions elected Abu Bakr – one of Muhammad’s closest associates and his father-in-law – as caliph or successor. Abu Bakr was the first of those who were later identified as the Rashidun, the Rightly Guided Caliphs: Abu Bakr (632–34), ʿUmar (634–44), ʿUthman (644–56), and ʿAli (656–61), who ruled by virtue of their personal connections with Muhammad and Arabian ideas of authority. The conquests made the caliphs the rulers of the newly conquered lands as well.

Islamic societies were built on the framework of already established and ancient Middle Eastern civilizations. From the pre-Islamic Middle East, Islamic societies inherited a pattern of institutions that would shape daily life until the modern age. These institutions included small communities based on family, lineage, clientage, and ethnic ties; agricultural and urban societies, market economies, monotheistic religions, and bureaucratic empires. Along with their political and social characteristics, Islamic societies also inherited many of the religious, literary, and artistic practices of the pre-Islamic past. The civilization of Islam, although initiated in Mecca, also had its precursors in Palestine, Babylon, and Persepolis.

Islamic societies developed in an environment that since the earliest history of mankind had exhibited two fundamental and enduring qualities. The first was the organization of human societies into small, often familial groups. The earliest hunting and gathering communities lived and moved in small bands. Since the advent of agriculture and the domestication of animals, the vast majority of Middle Eastern peoples have lived in agricultural villages or in the tent camps of nomadic pastoralists. Even town peoples were bound into small groups by ties of kinship and neighborhood, with all that implies of strong affections and hatreds. These groups raised the young, arranged marriages, arbitrated disputes, and formed a common front vis-à-vis the outside world.