Somehow, another maillist I'm subscribed to has turned into another
election-methods list. (I wonder who's responsible for that! :-) One the subscribers, Mike Saari, has a litmus test which Condorcet fails. See below. Is the scenario farfetched? --Steve ------- Forwarded Message Follows ------- Date: Sat, 30 Mar 1996 21:28:18 -0500 From: [hidden email] To: [hidden email] Subject: Condorset counter-example >Steve's definition of Condorset voting: > "Each voter ranks the candidates from most preferred to least > preferred. The info provided in the voters' rankings is used to > determine the winners of each of the possible candidate pairings. > The winner is the candidate whose worst pairing defeat is smallest." OK, here's a situation where Condorset voting fails the "Twins" Litmus Test. It's a variation on the Apple-Chocolate ambiguous example in my paper. This example works because there "just happened" to be a "voting cycle" between the 3 versions of Apple. The example is a bit hairy, but I trust that other motivated people will check my math/logic and let us know if I screwed up. Basically, with this example Condorset voting yields winner=Apple when the ballot contains (Apple, Chocolate), but yields winner=Chocolate when the ballot contains (Apple-1, Apple-2, Apple-3, Chocolate). Ready? Here goes! Assume the following ratings (opinions): Apple-1 Apple-2 Apple-3 Chocolate (Ranking) 20% Exc(99) Exc(98) Exc(97) Exc(95) A1>A2>A3>Ch 20% Exc(97) Exc(99) Exc(98) Exc(95) A2>A3>A1>Ch 15% Exc(98) Exc(97) Exc(99) Exc(95) A3>A1>A2>Ch 15% Bad(-30) Bad(-40) Bad(-50) Exc(95) Ch>A1>A2>A3 15% Bad(-50) Bad(-30) Bad(-40) Exc(95) Ch>A2>A3>A1 15% Bad(-40) Bad(-50) Bad(-30) Exc(95) Ch>A3>A1>A2 If the ballot contains only (Apple-x vs.Chocolate) then the Condorset winner is Apple: A > Ch 55%-45% Apple wins easily. But if the ballot contains (Apple-1, Apple-2, Apple-3, Chocolate) then the Condorset winner is Chocolate: A1 > A2 65-35 A2 > A3 70-30 A3 > A1 65-35 A1 > Ch 55-45 A2 > Ch 55-45 A3 > Ch 55-45 (Chocolate is the candidate whose worse pairing defeat is smallest.) Because a situation can be created where adding "Twins" to the ballot alters the outcome, I conclude that Condorset voting fails the "Twins" Litmus Test and therefore is not worthy. Thanks, Mike Saari |
If I understand this correctly, the problem outlined below is a problem
with not picking a candidate from the Smith set, which plain Condorcet indeed may fail at. The proposal Mike has made for those concerned with this is to add a clause to Condorcet's which states that the winner must be picked out of the Smith set, i.e. the smallest set of candidates who beat all candidates outside the set in pairwise elections. After seeing Bruce's example and this example cropping up, I think the best preemptive strategy might be to go with Mike's old proposal and add the Smith set clause on. At this point, it might not be fair to call it Condorcet's method anymore, but certainly, Condorcet-Smith would be fair. Rob Lanphier [hidden email] http://www.eskimo.com/~robla On Sun, 31 Mar 1996, Steve Eppley wrote: > Date: Sun, 31 Mar 1996 02:53:41 -800 > From: Steve Eppley <[hidden email]> > Reply-To: [hidden email] > To: [hidden email] > Subject: [EM] (Fwd) Problem with Condorcet? > > Somehow, another maillist I'm subscribed to has turned into another > election-methods list. (I wonder who's responsible for that! :-) > > One the subscribers, Mike Saari, has a litmus test which Condorcet > fails. See below. Is the scenario farfetched? > > --Steve > > ------- Forwarded Message Follows ------- > Date: Sat, 30 Mar 1996 21:28:18 -0500 > From: [hidden email] > To: [hidden email] > Subject: Condorset counter-example > > >Steve's definition of Condorset voting: > > "Each voter ranks the candidates from most preferred to least > > preferred. The info provided in the voters' rankings is used to > > determine the winners of each of the possible candidate pairings. > > The winner is the candidate whose worst pairing defeat is smallest." > > OK, here's a situation where Condorset voting fails the "Twins" > Litmus Test. It's a variation on the Apple-Chocolate ambiguous > example in my paper. This example works because there "just happened" > to be a "voting cycle" between the 3 versions of Apple. > > The example is a bit hairy, but I trust that other motivated people > will check my math/logic and let us know if I screwed up. > > Basically, with this example Condorset voting yields winner=Apple > when the ballot contains (Apple, Chocolate), but yields > winner=Chocolate when the ballot contains (Apple-1, Apple-2, Apple-3, > Chocolate). Ready? Here goes! > > Assume the following ratings (opinions): > > Apple-1 Apple-2 Apple-3 Chocolate (Ranking) > 20% Exc(99) Exc(98) Exc(97) Exc(95) A1>A2>A3>Ch > 20% Exc(97) Exc(99) Exc(98) Exc(95) A2>A3>A1>Ch > 15% Exc(98) Exc(97) Exc(99) Exc(95) A3>A1>A2>Ch > > 15% Bad(-30) Bad(-40) Bad(-50) Exc(95) Ch>A1>A2>A3 > 15% Bad(-50) Bad(-30) Bad(-40) Exc(95) Ch>A2>A3>A1 > 15% Bad(-40) Bad(-50) Bad(-30) Exc(95) Ch>A3>A1>A2 > > If the ballot contains only (Apple-x vs.Chocolate) then the > Condorset winner is Apple: > A > Ch 55%-45% > Apple wins easily. > > But if the ballot contains (Apple-1, Apple-2, Apple-3, Chocolate) > then the Condorset winner is Chocolate: > A1 > A2 65-35 > A2 > A3 70-30 > A3 > A1 65-35 > A1 > Ch 55-45 > A2 > Ch 55-45 > A3 > Ch 55-45 > (Chocolate is the candidate whose worse pairing defeat is smallest.) > > Because a situation can be created where adding "Twins" to the > ballot alters the outcome, I conclude that Condorset voting fails the > "Twins" Litmus Test and therefore is not worthy. > > Thanks, > Mike Saari > |
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