From: Brian Olson <[EMAIL PROTECTED]>
Date: Thu, 20 May 2004 20:50:28 -0700
...I think it's simply the case that with 1 issue, all voters' CR profiles are precisely correlated (i.e., any two profiles differ only by a multiplicative scale factor), so all these methods become equivalent.
This doesn't make sense to me. If a voter has a random preferred stand on an issue, and a candidate has a random position on an issue, then every voter should have a different opinions of the candidates.
Brian,
Here's a (hopefully) clearer explanation of how I modeled candidate issues:
A particular candidate, say candidate 1, has a set of "candidate position" indices CP1[1] for issue 1, CP1[2] for issue 2, etc. Each position index is in the range -1 to +1, with positive meaning the candidate is a proponent of the issue, and negative meaning an opponent. Similarly, candidate 2 has position indices CP2[1], CP2[2], etc. (The simulations are based on randomly generated CP's.)
A particular voter, say voter 1, assigns "weights" to the issues, W1[1] for issue 1, W1[2] for issue 2, etc. These are signed numbers (no range limit), with positive meaning the voter is a proponent and negative meaning an opponent. Similarly, voter 2 has corresponding issue weights W2[1], W2[2], etc. (The W's are also randomly generated.)
Voter 1 determines a sincere cardinal rating CR[1,1] for candidate 1 by taking a weighted average of the candidate's position indices,
CR[1,1] = (W1[1]*CP1[1] + W1[2]*CP1[2] + ...)/(|W1[1]| + |W1[2]| + ...)
(The denominator implicitly scales the weights so that their absolute values add up to 1. This guarantees that CR[1,1] is in the range -1 to 1.) Similarly, voter 1 determines a cardinal rating CR[1,2] for candidate 2; voter 2 determines a cardinal rating CR[2,1] for candidate 1, etc.
Now, if there's only one issue the CR's for voter 1 are
CR[1,1] = (W1[1]/|W1[1]|)*CP1[1],
CR[1,2] = (W1[1]/|W1[1]|)*CP2[1],
etc.
Thus, the voter's CR profile matches the candidate positions, except for the factor of (W1[1]/|W1[1]|), which is either +1 or -1. Similarly, the CR profile of voter 2 matches the candidate positions, except for a factor of (W2[1]/|W2[1]|). All the voters' CR profiles are identical, except for the sign difference. Their candidate rankings will hence also be identical, except for reversal.
One way to look at this is that with one issue, you basically have two types of candidates: liberal (positive CP) and conservative (negative CP). Candidates can be "strong" or "weak" liberals (or conservatives) based on the magnitude of their CP's. Similarly, voters are either liberal (normalized W equal to +1) or conservative (normalized W equal to -1). In the simplest no-strategy case, all liberal voters vote for liberal canditates and conservative voters vote for conservative candidates. If there are 5 liberal candidates, 5 conservative candidates, 51 liberal voters, and 49 conservative voters, then under Approval all 5 liberal candidates will get 51 votes and all 5 conservative voters will get 49 votes. Thus the majority candidates will all be tied, and whatever method is used to break the tie will not likely result in the most liberal candidate (the CR winner) being selected.
Ken Johnson
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