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# L9 - A Majority Game There are three people N cfw_1 2 3 The...

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Page 1 of 30 A Majority Game There are three people. {1,2,3} N . The worth of the teams is: ({1,2,3}) 1 v . ({1,2}) 1 v , ({2,3}) 1 v , ({1,3}) 1 v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . Q : What should be the payoff profile?

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Page 2 of 30 ({1,2,3}) 1 v . ({1,2}) 1 v , ({2,3}) 1 v , ({1,3}) 1 v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . How about this payoff profile: 1 1 2 2 ( , ,0) x . 1. It is a feasible ( i.e. , ( ) i i N v N x ). 2. Everyone feels OK ( i.e. , ({ }) i x v i ). A payoff profile that satisfies these two conditions is called an imputation .
Page 3 of 30 Imputations An imputation is a feasible payoff profile x for which ({ }) i x v i for all i N . The set of imputations is denoted X . {( ) : ( ) and ({ }) for all } i i N i i i N X x v N x x v i i N

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Page 4 of 30 ({1,2,3}) 1 v . ({1,2}) 1 v , ({2,3}) 1 v , ({1,3}) 1 v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . How about this payoff profile: 1 1 2 2 ( , ,0) x . 1. It is a feasible ( i.e. , ( ) i i N v N x ). 2. Everyone feels OK ( i.e. , ({ }) i x v i ). Q : Can you find a better imputation for some coalition S ( that is, an imputation y , such that for some coalition S , i i y x for all i S and ( ) ( ) y S v S )?
Page 5 of 30 ({1,2,3}) 1 v . ({1,2}) 1 v , ({2,3}) 1 v , ({1,3}) 1 v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . How about this payoff profile: 1 1 2 2 ( , ,0) x . 1. It is a feasible ( i.e. , ( ) i i N v N x ). 2. Everyone feels OK ( i.e. , ({ }) i x v i ). But there is an imputation 5 1 1 4 8 8 ( , , ) that is better for the coalition {2,3} . (And other imputations as well, of course.)

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Page 6 of 30 Objections ({1,2,3}) 1 v . ({1,2}) 1 v , ({2,3}) 1 v , ({1,3}) 1 v . ({1}) 0 v , ({2}) 0 v , ({3}) 0 v . The coalition {2,3} is unsatisfied with the current imputation 1 1 2 2 ( , ,0) x , and it can object by suggesting 5 1 1 4 8 8 ( , , ) that is better for all the members of {2,3} and is backed up by a threat to implement 5 1 1 4 8 8 ( , , ) on its own by dividing the worth among its members.
Page 7 of 30 Objections An imputation y is an objection of the coalition S to the imputation x if i i y x for all i S and ( ) ( ) y S v S , in which case we write S y x . It is sometimes said that ‘ y dominates x via S .’ (That is, the coalition S can object to the imputation x by proposing the imputation y , because with the imputation y , all members in S will be better off.)

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Page 8 of 30 The imputation 5 1 1 4 8 8 ( , , ) y
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