<div dir="ltr">One little technical detail in step (4) of the method (see correction in line below)<br><div class="gmail_extra"><br><br><div class="gmail_quote">On Wed, May 7, 2014 at 3:02 PM, Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div>First we need the concept of a covering of a set S of candidates by a subset K of S: a subset K of S covers S iff every member of S that is beaten (pairwise) by another member of S is also beaten by a member of K. A minimal covering is a covering with as few members as possible.<br>
<br></div>Suppose that your favorite election method FEM is clone dependent. <br><br>Let us designate the winner of the clone fixed version of FEM applied to a set S of candidates by the notation CloneFix(FEM, S). To compute this winner ...<br>
<div><div><div><div><div class="gmail_extra"><br>(1) Among minimal coverings of the set S let K be the one with maximum total approval.<br><br></div><div class="gmail_extra">(2) If K is a singleton, elect its only member. Otherwise ... <br>
</div><div class="gmail_extra"><br>(3) Let k be the member of K that is elected by (unadorned) FEM restricted to K.<br><br></div><div class="gmail_extra">(4) Let S' be the set of candidates that have the same pairwise relation to the other members of K that k has, i.e. candidate s' is a member of S' whenever (for each candidate x in K) candidate s' is beaten by x if and only if candidate k is beaten by x.<br>
</div></div></div></div></div></div></blockquote><div> </div><div>Step (4) should start "Let S' be the subset of S whose members have the same pairwise relation ..."<br></div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr"><div><div><div><div><div class="gmail_extra"></div><div class="gmail_extra">Elect CloneFix(FEM, S') by recursion.<br><br></div><div class="gmail_extra">Example: Suppose that FEM is Random Candidate, and that the pairwise defeats are given by <br>
<br>A>B>C>A for every member A of the clone set {A1, A2, A3}, <br>
<br>and that within the clone set the defeats are A1>A2>A3>A1.<br><br></div><div class="gmail_extra">In this case all three minimal covering sets are of the form {A, B, C} where A is a member of the clone set {A1, A2, A3}. <br>
<br>Suppose that A is the clone with greatest approval. Then K = {A, B, C}, and each member of this set has equal likelihood of being candidate k.<br><br></div><div class="gmail_extra">If k is B or C, respectively, then B or C is elected.<br>
<br></div><div class="gmail_extra">If k is A, then S' is the clone set {A1, A2, A3}.<br><br></div><div class="gmail_extra">In that case, we elect CloneFix(FEM, S') which is A1, A2, or A3 with equal likelihood.<br>
<br></div><div class="gmail_extra">In sum the probabilities are 1/9 for each member of {A1, A2. A3}, <br>and 1/3 for each member of {B,C}.<br><br></div><div class="gmail_extra">This distribution of probability respects the clone structure, unlike the raw FEM (i.e. random candidate) distribution which would assign a probability of 1/5 to each of the five candidates.<br>
</div><div class="gmail_extra"><br></div><div class="gmail_extra">Note that in our example the approval scores tuned out to be irrelevant.<br><br></div><div class="gmail_extra">Some other way (besides approval scores) of selecting the best minimal covering K from among minimal coverings of S could be used. For example, the covering with the least Sum (over the candidates in the covering) of the Max (over the candidates in S) pairwise opposition.<br>
<br></div><div class="gmail_extra">If approval scores are used, I suggest singling out the best minimal covering as the one with the highest Proportional Approval Voting score (PAV score).<br><br></div><div class="gmail_extra">
If approval scores are not used, and randomization is acceptable, I suggest going with the minimal covering that ranks the highest according to some random ballot order. (There are various ways of doing this.)<br><br></div>
<div class="gmail_extra">Also note that FEM = Copeland gives the same result as FEM = Random Candidate when Copeland ties are broken by random candidate, at least in examples where the minimal coverings have three or fewer members, which is probably true for almost all public elections (even when the Smith set is much larger).<br>
<br></div><div class="gmail_extra">For those that remember it, note the similarity with the "Condorcet Lottery." Only this clone fixed version of random candidate is fully monotonic if I am not mistaken.<br></div>
<div class="gmail_extra"><br></div><div class="gmail_extra">Does anybody else like this idea? Does anybody have a FEM method and ballot set that they would like to see it applied to?<span class="HOEnZb"><font color="#888888"><br>
<br></font></span></div><span class="HOEnZb"><font color="#888888"><div class="gmail_extra">
Forest<br>
</div><div class="gmail_extra"><br></div><div class="gmail_extra"><br></div></font></span></div></div></div></div></div>
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