By Annick Laruelle

Each day hundreds of thousands of selections are made by means of all types of committees, parliaments, councils and forums through a 'yes-no' balloting strategy. occasionally a committee can in basic terms settle for or reject the proposals submitted to it for a call. On different events, committee individuals have the potential of editing the idea and bargaining an contract ahead of the vote. In both case, what rule can be used if every one member acts on behalf of a different-sized team? it sort of feels intuitively transparent that if the teams are of other sizes then a symmetric rule (e.g. the straightforward majority or unanimity) isn't compatible. The query then arises of what balloting rule can be used. balloting and Collective Decision-Making addresses this and different matters via a examine of the speculation of bargaining and balloting energy, displaying the way it applies to actual decision-making contexts.

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N (v)) ∈ RN to be considered as a rational evaluation of the prospect of playing game v. i (v) 1. Efﬁciency. = v(N). i∈N Given a permutation π : N → N and a game v, πv denotes the permuted game deﬁned by πv(π(S)) := v(S), where π(S) = {π(i) : i ∈ S}. That is, πv is the game that results from v by relabelling the players according to π, so that i is in v what π(i) is in πv. 2. Anonymity. For any permutation π, π(i) (πv) = i (v). A player i is a null player in a game v if his/her entering or leaving any coalition never changes its worth, that is, if v(S ∪ i) = v(S), for all S.

We omit the proof, which is an easy exercise. e. functions such that u¯ = u¯ , if and only if there exist α ∈ R++ and β ∈ R such that u = αu + β. Remarks. (i) The assumption that there are both most and least preferred alternatives is not crucial. It has been made only to simplify the proof of Theorem 3, but the results remain valid without this assumption. Only in Theorem 3 do conditions (ii) and (iii) need to be required for any two alternatives a, b, such that b a, condition (ii-1) has to be required for any x, such that b x a, and (ii-2) for any lottery with support between a and b.

In a weighted majority what should be the weight of a seat in order for it to be a dictator’s seat? What should be the weight of a seat in order for it to have a veto? 4. Let W be an N-voting rule. t. 12 ≤ q < 1. Preliminaries 27 5. Consider an eight-seat committee divided into two subgroups: N = A ∪ B with A = {1, 2, 3, 4, 5} and B = {6, 7, 8}, and two possible rules: Rule 1: A proposal is accepted if it has the support of at least 3 votes from A and at least 2 votes from B. Rule 2: A proposal is accepted if it has the support of at least 5 votes, of which 3 votes must be those from B.