1. To see this, note that if M = M ? {i} where i / ? M , then it must be that judge i is exactly indifferent between joining the majority coalition or not; otherwise, i would have a strictly improving unilateral deviation. This indifference is non-generic and requires an exact alignment of the case, the equilibrium policies chosen by the respective coalitions, and the salience parameter ?;Note by Lemma 2 that M d (z) ? M ? M . WLOG, suppose d = 1. Then, by part 1 of Assumption 1, ?(M ) ? ?(M \ {j}) ? ?(M ) for every j ? M \ M , since M ? M \ {j}. Moreover, for all j ? M \ M , ?(M ) ? ?(M ) ? z < x j . Now, since M is a Nash equilibrium coalition, then u P (?(M ), x j ) + ?l(z, x j ) ? u P (?(M \ {j}), x j ) for each j ? M \M , and given the above ordering
2. Work:
3. Since the deviation from the deviation is profitable;Claudio Scott;we have: u P (?(M \ C), x k ) > u P (?(M , x k ) + ?l(z, x k ) ? u P (?(M \ {k}, x k ), where the second inequality follows from the fact that M is an equilibrium coalition. Hence u P (?(M \ C), x k ) > u P (?(M \ {k}), x k ), which cannot be since ?(M \ C) ? ?(M \ {k}) < x k . Hence, the deviation is stable. Proof of Lemma 4. Let (d, M ) be an adjudication (Nash) equilibrium, and suppose M is not connected. WLOG, suppose d = 1, so that, by Lemma 2, M 1 (z) ? M . Since M 1 is a connected coalition and M is disconnected, M must contain members of M 0 (z). Then there References Baker,2012
4. A Bargaining Model of Collective Choice;Jeffrey S Banks;The American Political Science Review,2000