L8 - A Majority Game Three people form a coalition to get a...

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Page 1 of 36 A Majority Game Three people form a coalition to get a treasure. A team of 3 people gets all (1). A team of any 2 people gets 3 5 . A team of any 1 person gets none (0).
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Page 2 of 36 Coalitional Games with Transferable Payoff There are three players . {1,2,3} N . The worth of the teams is: ({1,2,3}) 1 v . 3 5 ({1,2}) v , 3 5 ({2,3}) v , 3 5 ({1,3}) v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v .
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Page 3 of 36 Coalitional Games with Transferable Payoff D EFINITION . A coalitional game with transferable payoff , Nv consists of a finite set N (the set of players ) a function v that associates with every nonempty subset S of N (a coalition ) a real number () vS (the worth of S ).
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Page 4 of 36 Class Discussion There are three players . {1,2,3} N . The worth of the teams is: ({1,2,3}) 1 v . 3 5 ({1,2}) v , 3 5 ({2,3}) v , 3 5 ({1,3}) v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . Q : How much payoff each player should get ?
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Page 5 of 36 Class Discussion Consider another game with 3 players. The worth of the teams is: ({1,2,3}) 1 v . 4 5 ({1,2}) v , 4 5 ({2,3}) v , 4 5 ({1,3}) v . 2 5 ({1}) v , 2 5 ({2}) v , 2 5 ({3}) v . Q : How much payoff each player should get? Q : Will some set of players break away from the ‘grand coalition’ {1,2,3}?
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Page 6 of 36 Cohesiveness of Coalitional Games with Transferable Payoff D EFINITION . A coalitional game , Nv with transferable payoff is cohesive if 1 ( ) ( ) K k k v N v S for every partition 1 { , , } K SS of N .
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Page 7 of 36 Class Discussion Consider a game with 3 players. The worth of the teams is: ({1,2,3}) 1 v . 3 5 ({1,2}) v , 3 5 ({2,3}) v , 3 5 ({1,3}) v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . Q : Is this game cohesive? A : First, ( ) ___ vN . So, it is/is not cohesive.
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Page 8 of 36 In this course we assume that all coalitional games with transferable payoff are cohesive. 1 ( ) ( ) K k k v N v S
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Page 9 of 36 Feasible Payoff Profiles $ x 1 $ x 2 $ x 3 A feasible payoff profile is a profile 1 ( ) ( , , ) i i N n x x x x  of real numbers, such that () i iN v N x . An S-feasible payoff vector is a vector i i S x of real numbers, such that ( ) ( ) i iS x S x v S .
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L8 - A Majority Game Three people form a coalition to get a...

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