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 Authors:  Bezalel Peleg and Peter Sudholter Abstract: We consider the weighted majority game (N, v0) which has the tuple (3; 1, 1, 1, 1, 1, 0)as a representation (see (3)). The maximum payoff to the dummy (the last player) in thebargaining set of (N, v0) is shown to be 2/7 (see Remark 2). If we now increase v0(N) byδ, 0 < δ < 2/3, then the maximum payoff to the last player in the new game, in whichthis player is no longer a dummy and contributes δ to N, is smaller than 2/7 and strictlydecreasing in δ (see Lemma 1).We recall some definitions and introduce relevant notations. A (cooperative TU) gameis a pair (N, v) such that ∅ = N is finite and v : 2N → R, v(∅) = 0. For any game (N, v) letI(N, v) = {x ∈ RN | x(N) = v(N) and xi ≥ v({i}) for all i ∈ N}denote the set of imputations. (We use x(S) = i∈S xi for every S ⊆ N.) Let (N, v) be agame, x ∈ I(N, v), and k, l ∈ N, k = l. LetTkl = {S ⊆ N \ {l} | k ∈ S}.An objection of k against l at x is a pair (P, y) satisfyingP ∈ Tkl, y(P) = v(P), and yi > xi for all i ∈ P. (1)We say that k can object against l via P, if there exists y such that (P, y) is an objectionof k against l. Hence k can object against l via P, if and only if P ∈ Tkl and e(P, x, v) > 0,where e(S, x, v) = v(S) − x(S) is the excess of S at x for S ⊆ N.A counter objection to an objection (P, y) of k against l is a pair (Q, z) satisfyingQ ∈ Tlk, z(Q) = v(Q), zi ≥ yi for all i ∈ Q ∩ P and zj ≥ xj for all j ∈ Q \ P.  This Item Is Currently Unavailable.   Special Focus Titles   