[EM] Efforts to improve on CR's strategy

[Ken Johnson]

>From: Brian Olson <bql at bolson.org>
>Date: Thu, 20 May 2004 20:50:28 -0700
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>>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.
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>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.
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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