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Subject: Election-methods Digest, Vol 8, Issue 4
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Today's Topics:
1. Re: simulating an Approval campaign/election (Rob LeGrand) 2. Comparative Effectiveness of Approval and Condorcet in the case of a three candidate cycle. (Forest Simmons) 3. apology for "no subject" posting (Forest Simmons)
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Message: 1 Date: Tue, 1 Feb 2005 16:06:44 -0800 (PST) From: Rob LeGrand <[EMAIL PROTECTED]> Subject: [EM] Re: simulating an Approval campaign/election To: Election Methods Mailing List <[EMAIL PROTECTED]> Message-ID: <[EMAIL PROTECTED]> Content-Type: text/plain; charset=us-ascii
Russ wrote:Interesting. Do you mind if I ask why you are interested in Declared-Strategy Voting as opposed to Undeclared-Strategy Voting?
DSV is the invention of Lorrie Cranor and the subject of her dissertation (http://lorrie.cranor.org/dsv.html). "Declared" just means that a voter declares a strategy for the DSV system to use to vote for him in the simulated election(s).
Does another strategy converge even if no Condorcet winner exists?
In the example
A B C 9: 100 0 90 (9 voters have utility 100 for A, etc.) 8: 90 100 0 6: 0 10 100
the only equilibrium when all voters use strategy T is
9:A 8:B 6:CB
The same is true for strategies B (Poll Assumption (Approval) from 7.4 of Brams and Fishburn's Approval Voting) and I (change the approvals from your last ballot just enough so that you approve of one of the two frontrunners and not the other). Note that the equilibrium disappears if the 9 A-favorite voters realize that they can improve the outcome from their perspective by approving C in addition to A, as strategy A would recommend.
I assume you mean that plurality can be manipulated by throwing in spoilers (e.g., Nader or Perot).
No, I mean that when everyone votes strategically (giving "spoiler" candidates no votes), those that are voting insincerely are manipulating the election, usually into equilibria that allow little chance for candidates that might turn out to be Condorcet winners. Not that there's anything wrong with this manipulation-- it's the smart way to vote if you want to affect the outcome.
And as for multiple equilibria, it seems to me that all but one of those equilibria is practically inaccessable if it requires a third party to switch places with one of the two dominant parties.
I agree that plurality leads to a static two-party structure. Other systems that lead to equilibria more often than Approval also lead to party systems that limit entrance of new candidates and parties. Approval is the most stable voting system I know that still allows a dynamic (and party-less?) political system; at the extremes, plurality gives you stability at the expense of openness and Borda gives you dynamism (too much!) at the expense of ultra- instability. Also, plurality encourages each ideology to run at most one candidate (and often none), Borda encourages each ideology to run many candidates, and Approval encourages running those candidates that have a chance to win.
You seem to have confirmed my hypothesis that, in the idealized case (DSV batch mode), Approval voting almost always converges on the Cordorcet winner if one exists, but rarely (never?) converges if one does not exist.
Yes, that's true, if all voters use strategy A or something very much like it, which according to my investigations is in their best interest.
If that is true, then it seems to me that Approval may be roughly equivalent to Condorcet with random selection of the winner from the Smith set. Do you agree with that?
Only very roughly. The selection from the Smith set is random (assuming that the number of rounds is large and at least pseudorandom, e.g. based on the number of voters in a large election in a way that makes it very difficult to predict), but some members of the Smith set may have no chance of winning.
If so, has anyone shown that the Condorcet winner based on a "good" Condorcet resolution method would at least be favored in the random selection process?
That depends on the random selection process. If you use regular batch DSV and stop after a predetermined number of rounds, you'll get different winning probabilities than if you use cumulative batch mode or ballot-by-ballot mode. According to my simulations, the above example election run in ballot-by-ballot mode for a large random number of rounds would give A, B and C win probabilities of approximately 45.8%, 32.0% and 22.2%. Most of us would agree that A has the best claim to victory, followed by B. Batch mode gives A 50%, B 25% and C 25% (tight loop); cumulative batch mode gives roughly A 30.4%, B 47.8% and C 21.7%. (The win probabilities for cumulative batch mode are relatively difficult to measure because the intervals between leader changes become longer and longer as the number of rounds increases.) To me, these results confirm my intuition that ballot-by-ballot (with a random voter order in each round) is a fairer way to find a winner than the batch modes given a large number of rounds.
===== Rob LeGrand, psephologist [EMAIL PROTECTED] Citizens for Approval Voting http://www.approvalvoting.org/
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Message: 2 Date: Tue, 1 Feb 2005 17:19:48 -0800 (PST) From: Forest Simmons <[EMAIL PROTECTED]> Subject: [EM] Comparative Effectiveness of Approval and Condorcet in the case of a three candidate cycle. To: [email protected] Message-ID: <[EMAIL PROTECTED]> Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed
Russ brought up the issue of effectiveness of Approval.
I think that we are mostly in agreement now that Approval locks on to the CW fairly quickly when there is a CW. "Quickly" can even mean during the first election if DSV is used, or if partial results are made available to the voters before most of them cast their approval ballots.
Suppose that we have a three candidate cycle. How effective is Approval compared to Condorcet in this setting?
In this setting, Approval voters may have a hard time applying Strategy A, especially if all of the candidates appear to have nearly equal support at all ranks.
In this case Approval voters should ask themselves if their middle candidate is better or worse than half way between the other two candidates. If better, then approve, otherwise, not.
In the borderline case, go with the decision of a friend, or wait for partial results to come out (if possible).
If none of these possibilities are available, flip a coin. If the coin flip result gives you a bad feeling, go the other way. Your subconscious is wiser than you think.
But let's consider the worst possible case: you have absolutely nothing to help you decide. Then just approve your favorite only. As we showed in a recent posting this is exactly as likely (in this zero info three candidate case) to work in your favor as approving both favorite and middle.
In fact, we showed that as long as you approved your favorite and did not approve the candidate you considered worst, then given that your ballot is pivotal, there is a two thirds probability that your approval ballot will tip the election outcomein a direction that you consider favorable. [Satisfaction of the Participation criterion guarantees that it cannot make the outcome worse.]
[If you make your decision on the basis of any information at all, this 2/3 probability is improved drastically.]
So, by way of comparison, let's see if Condorcet can match this:
Suppose that your sincere preference ballot is A>B>C, and that there is a cycle among these candidates. There are two possible directions for the cycle:
Case I. A beats B beats C beats A. Case II. (the reverse of case I): A beats C beats B beats A.
What is the setup that would put two of these candidates in a Condorcet near tie?
The two weakest defeat strengths would have to be within one of each other.
Case I.i The strong defeat is A>B.
Subcase I.i.a The B>C defeat is equal to the C>A defeat. In this subcase Condorcet gives the win to A. Your ballot neither helps nor harms.
Subcase I.i.b The B>C defeat is stronger than the C>A defeat by one. (Same result as previous case)
Subcase I.i.c The B>C defeat is one weaker than the C>A defeat. In this subcase your ballot changes the winner from candidate C to A, definitely in your favor.
Case I.ii The strong defeat is B>C.
Subcase I.ii.a The A>B defeat is equal to the C>A defeat. In this subcase your ballot changes the winner from candidate B to A, in your favor.
Subcase I.ii.b The A>B defeat is one less than the C>A defeat. After your ballot is taken into account B is still the winner: no help, no harm.
Subcase I.ii.c The A>B defeat is one greater than the C>A defeat. A is the winner before and after your ballot is counted. No help, no harm.
Case I.iii The strong defeat is C>A. In all three subcases of this case the two weak defeats are both increased by the same amount (one) so the winner C is not changed (no help, no harm).
Case II.i The cycle is A>C>B>A and A>C is the strong defeat.
Your ballot does not affect the result in any of the three subcases of Case II.i, because it does not change either of the two weak defeats ( C>B and B>A ) since they are both contrary to your ballot (still A>B>C).
Case II.ii Cycle A>C>B>A and C>B is the strong defeat.
Subcase II.ii.a The A>C and B>A defeats are equal in strength. Your ballot changes the winner from C to A.
Subcase II.ii.b The A>C defeat is one less than the B>A defeat. The winner remains C.
Subcase II.ii.c The A>C defeat is one greater than the B>A defeat. The winner remains A.
Case II.iii Cycle A>C>B>A and B>A is the strong defeat.
Of the three subcases, the only one that your ballot improves is the one in which A>C is one weaker than C>B. Your ballot improves the winner from C to B.
Of the eighteen cases, your ballot only improves the result in four cases. Of course, your favorite was already the winner in six of those cases, so no improvement was possible. So taking that into account, we can say that your ballot improved the result in four of the twelve possible cases, about half as effective as Approval.
Of course we didn't consider the use of truncation in Condorcet. But that's only fair, since the advantage of Condorcet over Approval is supposed to be that you can vote your sincere preferences without loss of voting power.
This little case by case study seems to show that this supposition is not true, at least in three candidate cycle case that we are considering here; use of fully ranked ballots is less powerful than ballots that rank two of the three candidates equally (i.e. approval ballots).
Forest
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Message: 3 Date: Tue, 1 Feb 2005 17:28:02 -0800 (PST) From: Forest Simmons <[EMAIL PROTECTED]> Subject: [EM] apology for "no subject" posting To: [email protected] Message-ID: <[EMAIL PROTECTED]> Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed
That no subject posting was just a slip of the Return key while scrolling down the EM digest. Sorry for the bother.
Forest
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