<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div>The ordering need not be random. One can use also a tree structure as given by the candidates. Clones and near clones will form branches. The simplest approach is to consider defeats within a branch to be weaker than defeats between branches.</div><div><br></div><div>Juho</div><div><br></div><div><br></div><br><div><div>On Jun 26, 2010, at 4:01 AM, Jameson Quinn wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><br><br><div class="gmail_quote">2010/6/25 <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</a>></span><br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> >1a. Probabilistic dilemma. If truncation causes a cycle, then there is some<br> probabilistic tiebreaker which always includes some chance of C winning.<br> This can act as a goad to A and B voters to cooperate. I suspect that some<br> system like this might be the theoretical optimum response if voters were<br> pure rational agents; however, real people tend not to like probabilistic<br> election systems.<br> <br> Sports fans don't object to the use of a certain amount of randomization in deciding the order of contests<br> in a tournament.<br></blockquote><div><br>True, and good point.<br><br>Still, sports aren't elections. Sports are intended to be exciting, and so an element of chance can be a positive advantage. Also, since only a limited number of sequential two-way contests are possible, there is really no alternative to a seed order, unlike with election systems which have (too) many alternatives.<br> <br></div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> <br> A single elimination tournament, for example, needs a "seed" order. If a random order is not used, then<br> (if I remember correctly) the contestants with the better records are seeded near the end of the<br> tournament, to avoid anticlimatic contests at the end of the tournament.<br> <br> I suggest the following way of picking a seed order SO for an election: use the order of a random ballot<br> refined by the orders of additional ballots until the order is complete.<br> <br> One way to use the seed order SO is by single elimination, starting at the bottom of the list and working<br> up.<br> <br> Another (distinct!) way is to elect the lowest alternative on the list that pairwise beats every alternative<br> listed above it.<br> <br> Here's my favorite: initialize X as the highest alternative in the SO. While X is covered, replace X with the<br> highest alternative on the SO that covers X. Then elect the final value of X.<br> <br> When the seed order SO is the refined random ballot order as given above or any other social order that<br> is monotone and clone free, these methods will pick from the Smith set, while preserving the clone<br> independence and monotonicity. Furthermore, the last of these (my favorite), satisfies Independence<br> from both Smith and Pareto Dominated Alternatives, and will elect from the uncovered set.<br> <br> Do these methods solve the truncation dilemma?<br></blockquote><div><br>Mostly. With a given seed ordering, if you are a B>A>C voter, a B>A=C vote cannot change the winner from A to B, unless it causes C to cover B. This is only possible if, with honest ballots, C beats B and B beats A. Neither of these are consistent with a truncation dilemma scenario. <br> <br>There's still a possibility that your ballot is one of the random ballots that helps define the seed, and so your strategic vote causes C to come first in the seed order, AND causes A not to cover C, so that B is then elected. So, your "refined by additional random ballots until the order is complete" could break the truncation resistance. I suspect - but am not sure - that a "single random ballot refined by random choices" seed order would not have a truncation-dilemma.<br> <br>... On a separate note, perhaps the "covering" concept is too hard to explain. How much better is that than simply a single bubble sort pass up from the bottom of the seed order? That would also guarantee Smith set.<br> </div></div><br>JQ<br> ----<br>Election-Methods mailing list - see <a href="http://electorama.com/em">http://electorama.com/em</a> for list info<br></blockquote></div><br></body></html>