<html><body><div style="color:#000; background-color:#fff; font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif;font-size:13px"><div>A few months ago, Forest Simmons made a series of interesting posts on this subject. I was thinking about it again recently, and on further reflection have a few further questions. Forest's quotes here come from more than one post.</div><div><br></div><div>From: Forest Simmons <fsimmons@pcc.edu></div><div><br></div><div>>Suppose that there are two extreme groups A and B supported by two individual ethnicities, as well as a more moderate group M >with preferences like</div><div><br></div><div>>10 A(100)<br>>30 A(100)>M(80)<br>>45 B(100)>M(80)<br>>15 B(100)</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color:
transparent;"><br></div><div>>(The numbers in parentheses represent voter expectations of relative benefits.)</div><div><br></div><div>>In ordinary party list PR methods the parliament would be formed by 40 representatives from A and 60 representatives from B. >The moderate party would be shut out entirely.</div><div><br></div><div>>Here are my questions:</div><div><br></div><div>>1. What method(s) would take this information and elect a parliament with respective party strengths of 10, 15, and 75 for A, B, >and M?</div><div><br></div><div>>2. What election method could possibly get the two middle factions to honestly convey this information via their ballots? In other >words, how to keep the two middle factions from defecting from their common interest?</div><div><br></div><div>[snip]</div><div><br></div><div>>The simplest answer to question one that I know of is based on an idea that Martin
Harper came up with 12 years ago as a way of >showing that ordinary Approval satisfies "one voter one vote" in the same strict sense that IRV does (through vote transfer):</div><div><br>>First list the candidates in order of most approval to least approval. Then on each ballot transfer the entire support of the voter to >the highest candidate on the list that is approved on the ballot. In other words, the voter's one and only vote is for the candidate >she approves that is most approved by other voters. As Martin pointed out, this assignment of votes still elects the ordinary >Approval winner in the single winner case. (Half a dozen years later Jobst pointed out that this same idea can be used to assign >probabilities in a single winner lottery method.)</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style:
normal; background-color: transparent;"><br></div><div>Is this definitely the best system? If you have these approval ballots:</div><div><br></div><div>2: A<br>10: A, B<br>1: B, C<br>9: C</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><br></div><div>Then the approvals are A=12, B=11, C=10. If I've understood what you said correctly, this would transfer to A=12, B=1, C=9. Would it be better to do it sequentially? So A has the most approvals and takes all its voters, and this leaves us with:</div><div><br></div><div>1: B, C<br>9: C</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><br></div><div>Applying the same procedure again gives us A=12, C=10 leaving B
with nothing. Leaving B with one seems wasteful since B is unlikely to get a seat like this.</div><div><br></div><div>[snip]</div><div><br></div><div>>In our last post we showed that direct approval voting for parties does not yield a satisfactory answer to question two, i.e. there are >strategic incentives against directly approving the moderate party, even when many of the voters consider it almost as good as >their favorite extreme party.</div><div><br>>The solution to this problem comes from Jobst Heitzig, in an idea which he came up with in the context of lottery methods, but >which applies equally well in this PR context. Let's call the idea "Conditional Approval."</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><br></div><div>>The idea is to give the voters a way of saying
"I'll cooperate if they will cooperate, and I'll defect if they defect." In other words, "Put >me down for approving M (the moderate compromise party) as long as there are 74 other voters who are willing to do the same, >otherwise no."</div><div><br></div><div>>Now suppose that all 75 voters in the middle two factions indicate this conditional approval of M on their conditional approval >ballots (along with unconditional approval of their favorites). I claim that this position is a strategic equilibrium, because this >position makes impossible the pesky unilateral defections we had to deal with in our direct approval approach. If any of the 75 >voters from the two middle factions defect, then the deal is off, and the payoffs jump from the (cooperate, cooperate) corner of the >(old) payoff matrix to the (defect, defect) corner, resulting in a loss for both of the middle factions.</div><div><br></div><div>This may
work in this case, but I'm unsure as to how the voters would have any clue in advance how many other voters their approval of M should be conditional upon. Where does the number 74 come from other than a calculation with knowledge of the final result?</div><div><br></div><div>Could you take a score ballot and instead of using the scores as scores in a normal way, take them as approvals that are conditional upon the right number of other people also approving/conditionally approving the right candidates? So the calculation process works out the conditional number for them. Would there be a simple way of calculating how it should be done that would avoid or ameliorate against strategic incentives against giving the moderate party an honest score?</div><div><br></div><div>Toby</div></div></body></html>