Social Choice_Lecture 6_Yes-No Voting

# Moreover the losing coalition weight must be below

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Unformatted text preview: t exceeds the quota. Moreover the losing coalition’ weight must be below the quota. s The di¢ culty in this example is that we must put extra weight on the permanent members to account for the fact that they have veto power. Van Essen (U of A) Y/N 14 / 30 The U.N. Security Council is a Weighted Voting System Since all non-permanent members of the council have the same in‡uence lets assign them all the same weight of 1. The …ve permanent members all have the same in‡uence which is unknown so we’ designate it with the variable x . ll The quota is also unknown so we’ call it q . ll Van Essen (U of A) Y/N 15 / 30 The U.N. Security Council is a Weighted Voting System A coalition of 10 non-permanent members and 4 permanent members is the largest losing coalition so we need 10 + 4x < q In addition, 5 permanent members and 4 non-permanent members is the smallest winning coalition. Therefore we also need that 5x + 4 q Thus, we must …nd values for x and q that satisfy the two above equations. For instance, if x = 7 then q must satisfy 38 < q 39 so q = 39 seems like a good choice. Van Essen (U of A) Y/N 16 / 30 The U.N. Security Council is a Weighted Voting System Theorem The UN security council is a weighted voting system Proof. Assign each non-permanent member a weight of 1, each permanent member a weight of 7, and the quota equal to 39. It is straightforward to verify that these weights and quota work. Van Essen (U of A) Y/N 17 / 30 Weighted Voting Systems Are all yes-no voting systems weighted? No. How does one prove that a weighted system is not voted? Find an easy to verify property that all weighted systems must have. If we can show that a particular voting system does not have this property, then we have shown that the voting system is not weighted. Note: Proof by contrapositive: If A ! B , then ~B ! ~A. Van Essen (U of A) Y/N 18 / 30 Swap Robustness De…nition A yes-no voting system is said to be swap robust if any one-for-one exchange of players (i.e., a swap) between two winning coalitions X and Y leaves at least one of the...
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