Unformatted text preview: t exceeds the quota. Moreover the
losing coalition’ weight must be below the quota.
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 .
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
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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