[EM] rainbow lotteries

[Simmons, Forest]

In my last message I claimed that the rainbow method based on MMPO was clone proof like the one based on Approval.   However, I now doubt that the one based on MMPO is clone proof because MMPO itself fails clone winner.  I don't think that Rainbow can convert a clone independence failure into a clone independence complier, but it will keep a complier compliant:-)
 
In general if a method can be used to create a social order, a rainbow can be created by arranging the candidate color bands in the social order with the band widths proportional to the number of first place votes (counted in such a way that the total is equal to the number of ballots).  Slicing the rainbow in half (or some other fixed, pre-specified fraction) and using random ballot to choose a candidate represented by a remaining color is the Rainbow version of the method that created the social order.
 
If the method used to create the social order is monotone and clone free (like  Approval) then the Rainbow version will be, too.  If  the method satisfies the majority criterion, then so will the Rainbow version based on it.
 
Forest
 
 
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