[EM] When Voters Strategize, Approval Voting Elects Condorcet

[Forest W Simmons]

Mr. Schudy's article reinforces the rationale behind DYN: that with 
reliable partial information, Approval does as well or better than 
Condorcet.

Mr. Schudy treats the case in which there is a clear frontrunner and a 
clear runnerup.  In that case he shows that (what we usually call) 
"approval strategy A" is rational, and that it gets the Condorcet 
Winner elected, a result well known on this listserv.

Of course, that requires reliable polling information.  I think that a 
version of DYN suggested by Juho is the simplest method to meet this 
requirement without requiring voters to return to the polls.

DYN works well whether or not there are two leading candidates.

Juho's version of DYN requires each candidate to publish their rankings 
of the other candidates before the election, and allows only one proxy 
per voter.

Voters approve (with Y for yes) some candidates and disapprove (with N 
for no) others.  If there are any left over, each voter designates (D) 
one candidate as proxy for making the remaining Y/N decisions.  After 
the statistics of the partial results are in, the candidates (as 
proxies) use their strategies to make the remaining Y/N decisions, 
which have to be consistent with their pre-election rankings.

Consistency means that if the proxy ranked candidate A ahead of 
candidate B, and she gives a Y (for yes) to candidate B, then she must 
also give a Y to candidate A.

It was Juho's suggestion that to simplify things we should allow only 
one proxy per voter. Also, Juho's suggestion of not giving too much 
leeway to the proxies inspired the idea of making their Y/N proxy votes 
be consistent with their pre-published rankings.  That's why I call 
this "Juho's version of DYN."

On another related topic.  How best to use sincere range ballots?

I think maximizing the Gini score is the best (except in situations in 
which the spoils of the election are freely shared by the voters).

For candidate X the Gini score is obtained by 

1. (first) sorting the ballots in order of how they rate X from best 
rating to worst.

2. Then computing a weighted average of the ratings, where the rating 
on the j_th ballot in the sorted order is given a weight of (2*j-1), 
i.e. the consecutive odd numbers are the weights.

For example,

50 A>B>C
50 C>B>A

with B at midrange (50%) on all ballots, hence with an average of 50%, 
no matter the weights.

It turns out that  A and C are tied for last with a common weighted 
average rating of

   (1+3+...+99)/(1+3+...+ 199), 

which is exactly 25%.

Of course, Gini optimal strategy is the same as ordinary Range optimal 
strategy, which is just Approval strategy, and in strategic voting the 
exact symmetry would have to be broken to determine a winner for this 
example.  

If exactly half of the voters approved B and the other half disapproved 
B, then it would be a three way tie, whether measured by Approval, 
Gini, or Range.

Forest