I like Raphael's idea of giving each voter three copies of the current lottery in exchange for one lottery's worth of papers. Let's call his idea the DLE(1/3) enhancement in contrast to my original suggestion of DLE(1/2). In general, suppose that n copies of the current lottery are exchanged for m<n lottery's worth of papers. In that case if each candidate has more than the fraction m/n of first place support, then the random ballot lottery will be an equilibrium lottery of DLE(m/n). So for example, if there are three candidates with first place support, and each of them has more than 30 percent first place support, then the random ballot lottery will be unchanged by the enhancement that exchanges ten copies for three copies worth. Suppose a country has three major distinct ethnic groups with more than 20 percent of the population each. Why not use a DLE(1/5) equilibrium lottery? If there is more than one equilibrium, use the one among those with greatest support (i.e. greatest number of candidates with positive probability) that maximizes the minimum probability. I also suggest that an additional rule be put into effect that each voter can declare at the outset a subset of the candidates that she does not have to support, not withstanding the m/n rule. In other words, if the m/n probability extends down into the voter's truncated candidates, the voter does not have to return a full m/n fraction of the papers (but all of the papers with non-truncated names must still be returned). In the case of the normal distribution of candidates, the DLE(1/6) lottery enhancement would (roughly) amount to requiring each candidate to return the papers corresponding to the the candidates that they percieve to be more than one standard deviation above the expected winner, since in a normal distribution about 16 percent of the probability lies more than one standard deviation to the right of the mean. A DLE(1/4) lottery enhancement would ask the voters to return the papers of the upper quartile candidates. The DLE(1/3) lottery doesn't seem so far out when considered in comparison with these others. Which would work best in the case of 49 C 24 B 27 A>B given the truncation rule? Are there any other interesting cases that could help calibrate the DLE(m/n) choice? Personally, another reason that I like m/n < 1/2 is that DLE(1/2) is too likely to make all of the lottery support go to the Condorcet Winner when there is one. Such behavior tends to encourage burial strategy in cases like 40 A>C>B 30 B>C>A 30 C>A>B , since going from a pure C win to a non-trivial lottery is likely to benefit the 40 A>C>B faction if they bury C below B. Thanks for reading. Forest
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