32Lecture 3. The bargaining setbutjcan benefit. The objection can take the following (informal) form:I get too little in the imputationx, and agentjgets too much! I can form acoalition that excludesjin which some members benefit and all membersare at least as well off as inx.We recall that in a game with coalition structure(N, v,S), the agents do not try tochange the CS, but only obtain a better payoff. The setX(N,v,S)offeasiblepayoffvectors for(N, v,S)is defined asX(N,v,S)={x∈Rn| ∀C ∈ S,∑i∈Cxi≤v(C)}. Weare now ready to formally define an objection.3.1.1.DEFINITION. [Objection] Let(N, v,S)be a game with coalition structure,x∈X(N,v,S),C ∈ Sbe a coalition, andiandjtwo distinct members ofC((i, j)∈ C2,i=j). Anobjection ofiagainstjis a pair(P, y)where•P⊆Nis a coalition such thati∈Pandj /∈P.