<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">On 18.9.2012, at 18.03, Kristofer Munsterhjelm wrote:<br><br><blockquote type="cite">On 09/16/2012 02:35 PM, Juho Laatu wrote:<br></blockquote><blockquote type="cite"><blockquote type="cite">On 16.9.2012, at 9.57, Kristofer Munsterhjelm wrote:<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">(More precisely, the relative scores (number of plumpers required)<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">become terms of type score_x - score_(x+1), which, along with SUM<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">x=1..n score_x (just the number of voters), can be used to solve<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">for the unknowns score_1...score_n. These scores are then<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">normalized on 0..1.)<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">It seems to work, but I'm not using it outside of the fitness<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">function because I have no assurance that, say, even for a monotone<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">method, raising A won't decrease A's score relative to the others.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">It might be the case that A's score will decrease even if A's rank<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">doesn't change. Obviously, it won't work for methods that fail<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">mono-add-plump.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">What should "candidate's score" indicate in single-winner methods? In<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">single-winner methods the ranking of other candidates than the winner<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">is voluntary. You could in principle pick any measure that you want<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">("distance to victory" or "quality of the candidate" or something<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">else). But of course most methods do provide also a ranking as a<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">byproduct (in addition to naming the winner). That ranking tends to<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">follow the same philosophy as the philosophy in selecting the winner.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">As already noted, the mono-add-plump philosophy is close to the<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">minmax(margins) philosophy, also with respect to ranking the other<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">candidates.<br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">What should "candidate's score" indicate? Inasfar as the method's winner is the one the method considers best according to the input given, and the social ordering is a list of alternatives in order of suitability (according to the logic of the method), a score should be a finer graduation of the social ordering. That is, the winner tells you what candidate is the best choice, the social ordering tells you which candidates are closer to being winners, and the rating or score tells you by how much.<br></blockquote><br>Here "suitability" is close to what I called "quality of the candidate".<br><br>I note that we must have a suitable interpreation for "social ordering" because of the well known paradoxes of social ordering. Maybe we talk about "scoring" the candidates (transitively, numerically). That would make it "social scoring" or someting like that.<br><br>I also note that if we talk about "list of alternatives" in the sense that we want to know who should be elected in case the first winner can not be elected, then there may be different interpretations. We may want to know e.g. who should be elected if the winner would not have participated in the election, or in the case that the winner participates but can not be elected (=> the question is, do we measure losses to the winner).<br><br>One more note. Term "closer to being winners" does refer to being close in quality / scores. It does not refer to being close to winning e.g. in number of voters that could change the results. (I used earlier term "distance to victory".)<br><br><blockquote type="cite"><br></blockquote><blockquote type="cite">If the method aims to satisfy certain criteria while finding good winners, it should do so with respect to finding the winner, and also with respect to the ranking and the score. A method that is monotone should have scores that respond monotonically to the raising of candidates, too.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite">I note that some methods like Kemeny seem to produce the winner as a<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">byproduct of finding the optimal ranking. Also expression "breaking a<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">loop" refers to an interest to make the potentially cyclic socielty<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">preferences linear by force. In principle that is of course<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">unnecessary. The opinions are cyclic, and could be left as they are.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">That does not however rule out the option of giving the candidates<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">scores that indicate some order of preference (that may not be the<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">preference order of the society).<br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">I think most methods can be made to produce a social ranking. Some methods do this on its own, like Kemeny. For others, you just extend the logic by which the method in question determines the winner. For instance, disregarding ties, in Schulze, the winner is the candidate whom nobody indirectly beats. The second place finisher would then be the candidate only indirectly beaten by the winner, and so on.<br></blockquote><br>I guess in Schulze we can have the two options that I mentioned above. We can consider the ballots/matrix with or without the first winner, when determining the second winner.<br><br>In real life these cases could be compared e.g. to the situation where the winnig candidate has died or has just decided to be in opposition instead of becoming elected. The ideal winner may be different depending on what kind of opposition he/she will have. This difference is obvious e.g. in the case of a loop of three vs. majority decision between two candidates.<br><br><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Turning rankings into ratings the "proper" way highly depends on<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the method in question, and can get very complex. Just look at this<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">variant of Schulze: <a href="http://arxiv.org/abs/0912.2190">http://arxiv.org/abs/0912.2190</a> .<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">They seem to aim at respecting multiple criteria. Many such criteria<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">could maybe be used as a basis for scoring the canidates. Already<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">their first key criterion, the Condorcet-Smith principle is in<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">conflict with the mono-add-plump score (there can be candidates with<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">low mono-add-plump score outside the Smith set).<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">My favourite approach to scoring and picking the winner is not to<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">have a discrete set of criteria (that we try to respect, and violate<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">some other criteria when doing so) but to pick one philosophy that<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">determines who is the best winner, and also how good winners the<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">other candidates would be. The chosen philosophy determines also the<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">primary scoring approach, but does not exclude having also other<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">scorings for other purposes (e.g. if the "ease of winning" differs<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">from the "quality of the candidate").<br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">If you do that, you get into a problem when comparing methods, however. Every method can be connected to an optimality measure that it optimizes. That measure might be simple or it might be very complex, but still, there's a relation between the method and something that it attempts to optimize. Discussing methods could then easily end up on cross purposes where one person says: "but I think minmax is the obvious natural thing to optimize", and another says "but I think mean score is the obvious natural thing to optimize", and nobody gets anywhere.<br></blockquote><br>I think that might get us somewhere. For example it makes sense to me to discuss what kind of a winner we should have. Should the winner be one that is not hated by anyone? Or, should the winner be one that is so strong that he/she can beat any other candidate that would try to oppose him/her (alone without teaming with other candidates)? Or maybe we should count the number of voters who can say "the winner would have lost to someone better".<br><br>After deciding who the best winner would be (this function can well be different in different elections), one could study if there are strategic problems with the method that implements the ideal "score function". (I note that many of the well known criteria talk about strategic vulnerabilities, not so often about which candidate is good.) If there are too many problems, we may be forced to use some other method than the one that gives best outcome with sincere votes. So, in this second phase I see already a compromise between "who should be elected" and "how to fix the strategic problems without making the method too bad".<br><br><blockquote type="cite"><br></blockquote><blockquote type="cite">At least with criteria, we have some way of comparing methods. We can say that this method behaves weirdly in that if some people increase the ranking of a candidate, the candidate might lose, whereas that method does not; or that this method can deprive candidates of a win if completely alike candidates appear, whereas that method does not.<br></blockquote><br>For me the "ideal winner" / "ideal score function" and "small enough strategy problems" are targets. Different criteria are technical tools that we can use for exact discusson. Different methods may meet different criteria more or less well. I prefer not to talk about on/off (strategy related) criteria but rather about meeting them as well as needed.<br><br>For example FBC is an important criterion, but I can accept methods that do not meet it, but that are good enough in the sense that they allow voters to rank their favourite always first, as a safe enough rule of thumb. I don't lke methods that fail FBC in the sense that voters often have to betray their favourite, or if voters have to decide whether to betray or not based on some complex analysis. In the same way many other criteria can be met "well enough".<br><br><blockquote type="cite"><br></blockquote><blockquote type="cite">Or perhaps it's more appropriate to say that if we want to compare methods by some optimization function or philosophy, we should have some way of anchoring that in reality. One may say "I think Borda count is the obvious natural thing to optimize", but if we could somehow find out how good candidates optimizing for Borda would elect, that would let us compare the philosophies. Yet to do so, we'd either have to have lots of counterfactual elections or a very good idea of what kind of candidates exist and how they'd be ranked/rated so that it may be simulated, because we can't easily determine how good society would be "if" we picked candidate X instead of Y.<br></blockquote><br>It is hard for me to find a choice/election where a society should follow the Borda philosophy. Certainly there are such choices, but I think Borda is not a "general purpose optimization criterion for typical decisions". On the other hand it it is easy to imagine situations where e.g. Range would yield ideal winners. Maybe in olympics range is a good philosophy for many events. Many election method experts think that Condorcet criterion is a good approach to making decisions in the typically very competitive political environment. Certainly we can also fine-tune that discussion and discuss which candidate is the ideal winner in the case that there is no Condorcet winner.<br><br>And if we take also strategies into account, the answer could be at some balance point where we don't always pick the best winner, but we do almost that, and add some resistance to strategies.<br><br><blockquote type="cite">(Well, we might in very limited situations: for example, one could take the pairwise matrix for a chess tournament once it's halfway through and use that to find the winner, then determine how often each election method gets it right. However, it's not obvious if "accuracy at predicting chess champions" is related to "being able to pick good presidents", say.)<br></blockquote><br>In elections we in some sense have complete information. I have noted many discussions where people also make guesses on what those voters that didn't vote might think. Often this appears in discussions about truncated votes. Did the voter truncate because he thought that all others are tied last, or was he just lazy. Or maybe he thought that not ranking some candidates means that they are unacceptable. I guess the main rule is that we can assume that the information we have is complete (or that we can ignore those who didn't give us their opinion).<br><br>Juho<br><br><blockquote type="cite"><br></blockquote><br></body></html>