LA - Bargaining Set Recall the definition of bargaining...

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Page 1 of 44 Bargaining Set Recall the definition of bargaining sets. D EFINITION . The bargaining set of a coalitional game with transferable payoff is the set of all imputations x with the property that for every objection ( , ) y S of any player i against any other player j to x , there is a counterobjection to ( , ) y S by j . I have an objection ( , ) y S against you to x …! But I have a counterobjec- tion ( , ) z T to your objec- tion ( , ) y S against me …!! j i
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Page 2 of 44 Bargaining Sets and the Core D EFINITION . The core of the coalitional game with transfer- able payoff , N v is the set of feasible payoff profiles ( ) i i N x for which there is no coalition S and S -feasible payoff vector ( ) i i S y for which i i y x for all i S . An imputation is in the core if and only if no player has an objection against any other player. Hence the core is a subset of the bargaining set.
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Page 3 of 44 Objections and Counterobjections G IVEN ANY IMPUTATION x A pair ( , ) y S , where S is a coalition and y is an S -feasible payoff vector, is an objection of i against j to x if S includes i but not j and k k y x for all k S . A pair ( , ) z T , where T is a coalition and z is an T -feasible payoff vector, is a counterobjection to the objection ( , ) y S of i against j if T includes j but not i , k k z x for all \ k T S , and k k z y for all k T S .
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Page 4 of 44 Now we consider another type of objec- tions and counterobjections.
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Page 5 of 44 Excesses of Coalitions For any coalition S , and imputation x , ( , ) ( ) ( ) e S x v S x S is called the excess of S .
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Page 6 of 44 Objections and Counterobjections II imputation: x j i
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Page 7 of 44 Objections of i against j j i Objection! Look at this coalition S without you! With the current imputation x , coalition S sacrifices too much because you are getting too much! A coalition S is an objec- tion of i against j to x if S includes i but not j and ({ }) j x v j .
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Page 8 of 44 Objections Let x be an imputation in a coalitional game with transferable payoff , N v . A coalition S is an objection of i against j to x if S includes i but not j and ({ }) j x v j .
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Page 9 of 44 Counterobjections j i Counterobjection!! But then look at this coa- lition T without you!! With the current imputa- tion x , coalition T sacri- fices not less than your S !! A coalition T is a coun- terobjection to the objec- tion S of i against j if T includes j but not i , and ( , ) ( , ) e T x e S x . i j S T : e ( T , x )≥ e ( S , x )
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Page 10 of 44 Counterobjections Let x be an imputation in a coalitional game with transferable payoff , N v . A coalition T is a counterobjection to the ob- jection S of i against j if T includes j but not i , and ( , ) ( , ) e T x e S x .
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Page 11 of 44 E XAMPLE .
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