Unformatted text preview: d.
We desire a stronger property that characterizes when a system is
weighted. In other words, a voting system is weighted if and only if it
has this property. The property we are looking for is called “trade
robust.” Van Essen (U of A) Y/N 23 / 30 Canadian Bacon Theorem
The procedure to amend the Canadian Constitution is swap robust. Proof.
Suppose X and Y are winning coalitions in the system to amend the
Canadian Constitution and that province x 2 X and x 2 Y . Also suppose
/
0 and Y 0 by swapping x and y . We must show
y 2 Y and y 2 X . Form X
/
that either X 0 or Y 0 is still winning.
Both X 0 and Y 0 have at least 7 provinces (why?). If x had more
population than y , then Y 0 is still a winning coalition (why?). If y had
more population that x , then X 0 is still a winning coalition. Van Essen (U of A) Y/N 24 / 30 Trade Robustness De…nition
a yesno voting system is said to be trade robust if an arbitrary exchange
of players (i.e., a series of trades involving groups of players) among
several winning coalitions leaves at least one of the coalitions winning. Van Essen (U of A) Y/N 25 / 30 Trade Robustness 1 In trade robustness, player exchange need not be oneforone as they
are in swap robustness. 2 In trade robustness, the trades may involve more than two coalitions. Van Essen (U of A) Y/N 26 / 30 Trade Robustness
Theorem
Every weighted voting system is trade robust. Proof.
First, a series of trades among several winning coalitions leaves the
number of coalitions to which each voter belongs unchanged. Thus, the
toal weight of all coalitions added together is unchanged. Moreover, since
the number of coalitions is unchanged, the average weight of the coalitions
is unchanged.
If we start with several coalitions whose weight meets the quota, their
average weight of these coalitions must meet the quota as well. After the
trades, the average weight still meets the quota. Hence, there must be at
least one coalition whose weight still meets the quota and is winning. Van Essen (U of A) Y/N 27 / 30 Trade Robustness and Canada
Theorem
The procedure to amend the Canadian Constitution is not trade robust. Proof.
Let X and Y be
X
PE Island (0%)
Newfoundland (2%)
Manitoba (4%)
Saskatchewan (3%)
Alberta (11%)
British Columbia (13%)
Quebec (23%) Van Essen (U of A) Y
New Brunswick (2%)
Nova Scotia (3%)
Manitoba (4%)
Saskatchewan (3%)
Alberta (11%)
British Columbia (13%)
Ontario (39%) Y/N 28 / 30 Trade Robustness and Canada Proof.
[Proof cont.] Let X 0 and Y 0 be formed by trading PE Island and
Newfoundalnd for Ontario. Now X 0 is losing because it has too few
provinces and Y 0 is losing because it does not have enough population. Corollary
The procedure to amend the Canadian Constitution is not a weighted
voting system. Van Essen (U of A) Y/N 29 / 30 Trade Robustness: Characterization Theorem Theorem
A yesno voting system is weighted if and only if it is trade robust. Van Essen (U of A) Y/N 30 / 30...
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 Spring '12
 Vanessen
 The Land, United Nations Security Council, van Essen, yesno voting

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