2010/6/25 <[email protected]> > >1a. Probabilistic dilemma. If truncation causes a cycle, then there is > some > probabilistic tiebreaker which always includes some chance of C winning. > This can act as a goad to A and B voters to cooperate. I suspect that some > system like this might be the theoretical optimum response if voters were > pure rational agents; however, real people tend not to like probabilistic > election systems. > > Sports fans don't object to the use of a certain amount of randomization in > deciding the order of contests > in a tournament. >
True, and good point. 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. > A single elimination tournament, for example, needs a "seed" order. If a > random order is not used, then > (if I remember correctly) the contestants with the better records are > seeded near the end of the > tournament, to avoid anticlimatic contests at the end of the tournament. > > I suggest the following way of picking a seed order SO for an election: > use the order of a random ballot > refined by the orders of additional ballots until the order is complete. > > One way to use the seed order SO is by single elimination, starting at the > bottom of the list and working > up. > > Another (distinct!) way is to elect the lowest alternative on the list that > pairwise beats every alternative > listed above it. > > Here's my favorite: initialize X as the highest alternative in the SO. > While X is covered, replace X with the > highest alternative on the SO that covers X. Then elect the final value of > X. > > When the seed order SO is the refined random ballot order as given above or > any other social order that > is monotone and clone free, these methods will pick from the Smith set, > while preserving the clone > independence and monotonicity. Furthermore, the last of these (my > favorite), satisfies Independence > from both Smith and Pareto Dominated Alternatives, and will elect from the > uncovered set. > > Do these methods solve the truncation dilemma? > 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. 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. ... 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. JQ
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