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## Über dieses Buch

Cooperative game theory is a booming research area with many new developments in the last few years. So, our main purpose when prep- ing the second edition was to incorporate as much of these new dev- opments as possible without changing the structure of the book. First, this o?ered us the opportunity to enhance and expand the treatment of traditional cooperative games, called here crisp games, and, especially, that of multi-choice games, in the idea to make the three parts of the monograph more balanced. Second, we have used the opportunity of a secondeditiontoupdateandenlargethelistofreferencesregardingthe threemodels of cooperative games. Finally, we have bene?ted fromthis opportunity by removing typos and a few less important results from the ?rst edition of the book, and by slightly polishing the English style and the punctuation, for the sake of consistency along the monograph. The main changes are: (1) Chapter 3 contains an additional section, Section 3. 3, on the - erage lexicographic value, which is a recent one-point solution concept de?ned on the class of balanced crisp games. (2) Chapter 4 is new. It o?ers a brief overview on solution c- cepts for crisp games from the point of view of egalitarian criteria, and presents in Section 4. 2 a recent set-valued solution concept based on egalitarian considerations, namely the equal split-o? set. (3)Chapter5isbasicallyanenlargedversionofChapter4ofthe?rst edition because Section 5. 4 dealing with the relation between convex games and clan games with crisp coalitions is new.

## Inhaltsverzeichnis

### 1. Preliminaries

Abstract
Let N be a non-empty finite set of agents who consider different cooperation possibilities. Each subset SN is referred to as a crisp coalition. The set N is called the grand coalition and ∅ is called the empty coalition. We denote the collection of coalitions, i.e. the set of all subsets of N by 2N. For each S ∈ 2N we denote by |S| the number of elements of S, and by eS the characteristic vector of S with (eS)i = 1 if iS, and (eS)i = 0 if iN\S. In the following often N = {1, . . . , n}.

### 2. Cores and Related Solution Concepts

Abstract
In this chapter we consider payoff vectors x = (xi)iN ∈ ℝn, with xi being the payoff to be given to player iN, under the condition that cooperation in the grand coalition is reached. Clearly, the actual formation of the grand coalition is based on the agreement of all players upon a proposed payoff in the game. Such an agreement is, or should be, based on all other cooperation possibilities for the players and their corresponding payoffs.

### 3. The Shapley Value, the τ-value, and the Average Lexicographic Value

Abstract
The Shapley value, the τ-value and the average lexicographic value recently introduced in [111] are three interesting one-point solution concepts in cooperative game theory. In this chapter we discuss different formulations of these values, some of their properties and give axiomatic characterizations of the Shapley value.

### 4. Egalitarianism-based Solution Concepts

Abstract
The principle of egalitarianism is related to the notion of equal share. A completely equal division of v(N) among the players in a game v has little chance to be accepted by all players because it ignores the claims over v(N) of all coalitions S ∈ 2N \ {∅, N}. Communities that believe in the egalitarian principle will rather accept an allocation that divides the value of the grand coalition as equally as possible. Such allocations might result using the comparison of coalitions’ worth, equal share within groups with high worth being highly desirable. The average worth a(S, v) of S ∈ 2N \ {∅} with respect to v, defined by $$a(S,v): = \tfrac{{v(S)}} {{\left| S \right|}}$$ and called also the per capita value, has played a key role in defining several egalitarian solution concepts. We mention here the equal division core (cf. Definition 2.14), the equal split-off set (cf. Section 4.2), and the constrained egalitarian solution (cf. Subsection 5.2.3).

### 5. Classes of Cooperative Crisp Games

Abstract
In this chapter we consider three classes of cooperative crisp games: totally balanced games, convex games, and clan games. We introduce basic characterizations of these games and discuss special properties of the set-valued and one-point solution concepts introduced so far. Moreover, we relate the corresponding games with the concept of a population monotonic allocation scheme as introduced in [107]. We present the notion of a bi-monotonic allocation scheme for total clan games and the constrained egalitarian solution (cf. [45] and [46]) for convex games.

### 6. Preliminaries

Abstract
Let N be a non-empty set of players usually of the form {1, . . . , n}. From now on we systematically refer to elements of 2N as crisp coalitions, and to cooperative games in GN as crisp games.

### 7. Solution Concepts for Fuzzy Games

Abstract
In this chapter we introduce several solution concepts for fuzzy games and study their properties and interrelations. Sections 7.1–7.3 are devoted to various core concepts and stable sets. The Aubin core introduced in Section 7.1 plays a key role in the rest of this chapter. Section 7.4 presents the Shapley value and the Weber set for fuzzy games which are based on crisp cooperation and serve as an inspiration source for the path solutions and the path solution cover introduced in Section 7.5. Compromise values for fuzzy games are introduced and studied in Section 7.6.

### 8. Convex Fuzzy Games

Abstract
An interesting class of fuzzy games is generated when the notion of convexity is considered. Convex fuzzy games can be successfully used for solving sharing problems arising from many economic situations where “cooperation” is the main benefit/cost savings generator; all the solution concepts treated in Chapter 7 have nice properties for such games. Moreover, for convex fuzzy games one can use additional sharing rules which are based on more specific solution concepts like participation monotonic allocation schemes and egalitarian solutions.

### 9. Fuzzy Clan Games

Abstract
In this chapter we consider fuzzy games of the form $$v:[0,1]^{N_1 } \times \{ 0,1\} ^{N_2 } \to \mathbb{R}$$, where the players in N1 have participation levels which may vary between 0 and 1, while the players in N2 are crisp players in the sense that they can fully cooperate or not cooperate at all. With this kind of games we can model various economic situations where the group of agents involved is divided into two subgroups with different status: a “clan” consisting of “powerful” agents and a set of available agents willing to cooperate with the clan. This cooperation generates a positive reward only for coalitions where all clan members are present. Such situations are modeled in the classical theory of cooperative games with transferable utility by means of (total) clan games where only the full cooperation and non-cooperation at all of non-clan members with the clan are taken into account (cf. Section 5.3). Here we take over this simplifying assumption and allow non-clan members to cooperate with all clan members and some other non-clan members to a certain extent. As a result the notion of a fuzzy clan game is introduced.

### 10. Preliminaries

Abstract
Let N be a non-empty finite set of players, usually of the form {1, ..., n}. In a multi-choice game each player iN has a finite number of activity levels at which he or she can choose to play. In particular, any two players may have different numbers of activity levels. The reward which a group of players can obtain depends on the effort of the cooperating players. This is formalized by supposing that each player iN has mi + 1 activity levels at which he can play, where mi ∈ ℕ. We set Mi := {0, ..., mi} as the action space of player i, where action 0 means not participating. Elements of $$\mathcal{M}$$N := ПiNMi are called (multi-choice) coalitions. The coalition m = (m1, ..., mn) plays the role of the grand coalition. The empty coalition (0, ..., 0) is also denoted by 0. For further use we introduce the notation M i + := Mi \ {0} and $$\mathcal{M}$$ + N := $$\mathcal{M}$$N \ {(0, ..., 0)}. For s$$\mathcal{M}$$N we denote by (s−i, k) the participation profile where all players except player i play at levels defined by s, while player i plays at level kMi. A particular case is (0i, k), where only player i is active (at level k). For s$$\mathcal{M}$$N we define the carrier of s by car(s) = {iN | si > 0}. Let u$$\mathcal{M}$$ + N . We denote by $$\mathcal{M}$$ u N the subset of $$\mathcal{M}$$N consisting of multi-choice coalitions su. For t$$\mathcal{M}$$ + N we also need the notation M i t = {1, ..., ti} for each iN and $$\mathcal{M}$$ t N = ПiN M i t . A characteristic function v : $$\mathcal{M}$$N → ℝ with v(0, ..., 0) = 0 gives for each coalition s = (s1, ..., sn) ∈ $$\mathcal{M}$$N the worth that the players can obtain when each player i plays at level siMi.

### 11. Solution Concepts for Multi-Choice Games

Abstract
In this chapter we present extensions of solution concepts for cooperative crisp games to multi-choice games. Special attention is paid to imputations, cores and stable sets, and to solution concepts based on the marginal vectors of a multi-choice game. Shapley-like values recently introduced in the game theory literature are briefly presented.

### 12. Classes of Multi-Choice Games

Abstract
In [82] a notion of balancedness for multi-choice games is introduced and a theorem in the spirit of Theorem 2.4 is proved, which we present in the following.

### Backmatter

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