Jobst! Here's a connection to approval. A lottery L is undefeated in mean iff every candidate would end up with less than 50 percent approval if voters were to use above mean approval strategy (based on prior winning probabilities borrowed from L) . A lottery L is undefeated in median iff every candidate would end up with less than 50 percent approval if voters were to use Joe Weinstein's approval strategy (based on prior winning probabilities borrowed from L). Joe Weinstein's approval strategy is (for each candidate X) to approve X iff it is more likely that the winner will be someone you rank below X than someone you rank above X. This brings up something I mentioned several months (8 or 9 months) ago: The hardest thing about approval voting is getting reliable, meaningful estimates of prior winning probabilities. If voters had a definite lottery as a standard of comparison, then this problem would be solved: let the lottery choose the winner if no candidate gets more than fifty percent approval in comparison, otherwise the candidate with the most approval beyond 50% wins. The problem is how to come up with the lottery for use as a standard of comparison. What about the lottery that minimizes the maximum approval when Joe Weinstein's strategy is applied? If your conjecture is true, then this lottery would be the winner. If this lottery is not unique, then the one that equalizes the probabilities as much as possible while maintaining this minmax property? (Just thinking out loud.) Forest
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