Since the topic of "Later no harm" came up again, I would like to point out that some randomization can make a Condorcet methods fulfil that criterion in a certain sense.
As we know, no deterministic method can fulfil "Later no harm". For the sake of completeness, here's again a simple counter-example to prove that: Preferences: sincere cast 1 A>B>C A>B>C 1 B>C>A B>C>A 1 C>A>B C>A>B 2 C>B>A C 2 B>A>C B 2 A>C>B A Assume that A wins with certainty with these cast ballots. Then the last two voters cannot cast a later preference by voting A>C without making the less preferred C the Condorcet Winner. When B or C is the original winner, the same is true for another pair of voters, hence the method is either not Condorcet or fails "Later no harm" or must involve randomness. Now let us look what happens when we use a simple randomized method such as "Elect the CW when s/he exists, otherwise use Random Candidate". Then each of A,B,C will get probability 1/3 with the above cast ballots. Now when the last two voters switch to voting A>C, then C is the CW and gets probability 1. Although this is "harm" to their favourite A since A's probability decreases from 1/3 to 0, it is also of some positive use for the two voters because it also decreases their LAST choice's probability from 1/3 to 0 ! So, randomized methods can be both Condorcet-efficient and fulfil the following version of "Later no harm" for randomized methods: Definition: "Later no probability moves down only" (LNPMDO) ----------------------------------------------------------- Assume that the candidates are A1,...,An, and that some voter who has sincere preferences A1>A2>...>An switches from voting A1>...>Ak to voting A1>...>Ak>A(k+1) for some k. Assume further that for some p, the probability that one of A1,...,Ap wins is decreased by this change. Then there must be some q>p such that the probability that one of A1,...Ap,...,Aq wins is increased instead! In other words: If some amount of probability moves down in the voter's ranking, then below that position some amount of probability must also move up in the voter's ranking. In particular, the least preferred possible outcome can be no worse than before, so that voting later preferences helps avoiding the worst choices! For a deterministic method, this is equivalent to "Later no harm", since then it just demands that the winner (=probability 1) cannot move down the voter's ranking, am I right? So, here we have a version of "Later no harm" which is consistent with Condorcet, but perhaps there are still better versions of that criterion? Perhaps we can even require that the amount of probability moving up in the voter's ranking must be at least as much as the amount of probability moving down? Also, the method which I used to show the consistency of the two criteria above is of course bad in most other respects, so we should now look whether good randomized methods such as Condorcet Lottery, RBCC, or RBACC also fulfil LNPMDO... Jobst ---- Election-methods mailing list - see http://electorama.com/em for list info