<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>Normal Proportional Approval Voting would give it to ABC, ABD or ABE based on satisfaction. Everyone has voted for one of the elected candidates and some get two. Whereas with CDE, it's purely one each, but as you say this is envy-free. While I can see the merits of the envy-free argument, I would probably go for the ABC, ABD or ABE option. Perhaps a slightly weird analogy - you have an apple to give away on Thursday. John is the only person around, so you give it to him. Then on Friday you have two apples to give away. Fred is the only person to come along. You can give him one apple which would be envy-free, but the other one would get thrown away and wasted. Or you could give him both. But he might tell John who would then experience "envy". John doesn't materially lose
anything by Fred getting two apples, so really I think Fred should get them both.</DIV>
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<DIV>It's slightly different with voting, because the people who get the extra candidate could get more of their preferred legistlation through at the expense of the others - so it's not purely envy, but possible material loss. But where do you draw the line? Maybe I'll think of an example later but perhaps you could have a case where the only envy-free solution is for everyone to end up with much less representation than they would in the envy result. So would our ideal system be the one that flattens representation out as much as possible?<BR></DIV>
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<B><SPAN style="FONT-WEIGHT: bold">From:</SPAN></B> Andy Jennings <elections@jenningsstory.com><BR><B><SPAN style="FONT-WEIGHT: bold">To:</SPAN></B> EM <election-methods@lists.electorama.com><BR><B><SPAN style="FONT-WEIGHT: bold">Sent:</SPAN></B> Fri, 22 July, 2011 19:42:06<BR><B><SPAN style="FONT-WEIGHT: bold">Subject:</SPAN></B> [EM] General PR question<BR></FONT><BR>Forest and I were discussing PR last week and the following situation came up. Suppose there are five candidates, A, B, C, D, E. A and B evenly divide the electorate and, in a completely orthogonal way, C, D, and E evenly divide the electorate. That is:
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<DIV>One-sixth of the electorate approves A and C.</DIV></DIV>
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<DIV>One-sixth of the electorate approves A and D.</DIV></DIV>
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<DIV>One-sixth of the electorate approves A and E.</DIV></DIV></DIV></DIV>
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<DIV>One-sixth of the electorate approves B and C.</DIV></DIV>
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<DIV>One-sixth of the electorate approves B and D.</DIV></DIV>
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<DIV>One-sixth of the electorate approves B and E.</DIV></DIV></DIV>
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<DIV>It is obvious that the best two-winner representative body is A and B. What is the best three-winner representative body?</DIV>
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<DIV>CDE seems to be the fairest. As Forest said, it is "envy-free".</DIV>
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<DIV>Some methods would choose ABC, ABD, or ABE, which seem to give more total satisfaction.</DIV>
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<DIV>Is one unequivocally better than the other?</DIV>
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<DIV>I tend to feel that each representative should represent one-third of the voters, so CDE is a much better outcome. Certain methods, like STV, Monroe, and AT-TV (I think) can even output a list of which voters are represented by each candidate, which I really like.</DIV>
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<DIV>I also note that if there was another candidate, F, approved by everybody, it is probably true that ABF would be an even better committee than CDE. Is there a method that can choose CDE in the first case and ABF in the second case?</DIV>
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<DIV>Andy</DIV></DIV></DIV></div></body></html>