Social Choice_Lecture 6_Yes-No Voting

We desire a stronger property that characterizes when

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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 yes-no 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 one-for-one 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 yes-no 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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