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Unformatted text preview: two coalitions winning. One of the players in the
swap must belong to X, but not Y, and the other player in the swap must
be in Y, but not X.
To prove swap robustness we must start with two arbitrary winning
coalitions X and Y, an arbitrary player x 2 X and x 2 Y , and an
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arbitrary playery 2 Y and y 2 X .
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Let X 0 = X nfx g [ fy g Let Y 0 = Y nfy g [ fx g We need to show either X 0 or Y 0 is winning.
Van Essen (U of A) Y/N 19 / 30 Swap Robustness
We are interested in this property because every weighted voting
system is swap robust. Theorem
Every weighted voting system is swap robust. Proof.
Assume that we have a weighted voting system and two arbitrary winning
coalitions X and Y with X containing at least one voter who is not in Y
and Y containing at least one voter y not in X . Form X 0 and Y 0 by
exchanging x for y .
If x and y have the same weight, then both X 0 and Y 0 are winning since
X 0 weighs the same as X and Y 0 has the same weight as Y . Van Essen (U of A) Y/N 20 / 30 Swap Robustness Proof.
[Proof cont.] If x is heavier than y , it follows that Y 0 weighs strictly more
than Y and is therefore winning.
If y is heavier than x , it follows that X 0 weighs strictly more than X and is
therefore winning. Van Essen (U of A) Y/N 21 / 30 US Federal System is not Weighted
Theorem
The US federal system is not swap robust. Proof.
Let X consist of the president, the 51 shortest Senators, and the 218
shortest members of the house.
Let Y consist of the president, the 51 tallest Senators, and the 218 tallest
members of the house.
Both X and Y are minimum winning coalitions. Let x be the shortest
Senator and let y be the tallest member of the house. Form X 0 and Y 0 by
swapping x and y. Both coalitions are losing (why?). Thus, US fed. is not
swap robust. Corollary
The US federal system is not swap robust.
Van Essen (U of A) Y/N 22 / 30 Trade Robustness All weighted voting systems are swap robust, but not all voting
systems that are swap robust are weighted.
It turns out that the procedure to amend the Canadian Constitution is
swap robust, but it is not weighte...
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This note was uploaded on 04/08/2014 for the course ECON 497 taught by Professor Vanessen during the Spring '12 term at Alabama.
 Spring '12
 Vanessen
 The Land

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