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# coal - GAME THEORY Thomas S Ferguson Part IV Games in...

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GAME THEORY Thomas S. Ferguson Part IV. Games in Coalitional Form 1. Many-Person TU Games. 1.1 Coalitional Form. Characteristic Functions. 1.2 Relation to Strategic Form. 1.3 Constant-Sum Games. 1.4 Example. 1.5 Exercises. 2. Imputations and the Core. 2.1 Imputations. 2.2 Essential Games. 2.3 The Core. 2.4 Examples. 2.5 Exercises. 3. The Shapley Value. 3.1 Value Functions. The Shapley Axioms. 3.2 Computation of the Shapley Value. 3.3 An Alternative Form of the Shapley Value. 3.4 Simple Games. The Shapley-Shubik Power Index. 3.5 Exercises. 4. The Nucleolus. 4.1 Definition of the Nucleolus. 4.2 Properties of the Nucleolus. 4.3 Computation of the Nucleolus. 4.4 Exercises. IV – 1

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PART IV. Games in Coalitional Form 1. Many-Person TU Games We now consider many-person cooperative games . In such games there are no restrictions on the agreements that may be reached among the players. In addition, we assume that all payoffs are measured in the same units and that there is a transferrable utility which allows side payments to be made among the players. Side payments may be used as inducements for some players to use certain mutually beneficial strategies. Thus, there will be a tendency for players, whose objectives in the game are close, to form alliances or coalitions. The structure given to the game by coalition formation is conveniently studied by reducing the game to a form in which coalitions play a central role. After defining the coalitional form of a many-person TU game, we shall learn how to transform games from strategic form to coalitional form and vice versa. 1.1 Coalitional Form. Characteristic Functions. Let n 2 denote the number of players in the game, numbered from 1 to n , and let N denote the set of players, N = { 1 , 2 , . . . , n } . A coalition , S , is defined to be a subset of N , S N , and the set of all coalitions is denoted by 2 N . By convention, we also speak of the empty set, , as a coalition, the empty coalition . The set N is also a coalition, called the grand coalition . If there are just two players, n = 2, then there are four coalition, {∅ , { 1 } , { 2 } , N } . If there are 3 players, there are 8 coalitions, {∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , N } . For n players, the set of coalitions, 2 N , has 2 n elements. Definition. The coalitional form of an n -person game is given by the pair ( N, v ) , where N = { 1 , 2 , . . . , n } is the set of players and v is a real-valued function, called the characteristic function of the game, defined on the set, 2 N , of all coalitions (subsets of N ), and satisfying (i) v ( ) = 0 , and (ii) (superadditivity) if S and T are disjoint coalitions ( S T = ), then v ( S ) + v ( T ) v ( S T ) . Compared to the strategic or extensive forms of n -person games, this is a very simple definition. Naturally, much detail is lost. The quantity v ( S ) is a real number for each coalition S N , which may be considered as the value, or worth, or power, of coalition S when its members act together as a unit. Condition (i) says that the empty set has value zero, and (ii) says that the value of two disjoint coalitions is at least as great when they work together as when they work apart. The assumption of superadditivity is not needed
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