Dear EM List participants, When last I wrote on the topic "Fair and Democratic versus Majority Rules" my purpose was to set forth some of the advantages of using chance to advance the principles of fairness and democracy as a remedy for the Tyranny of the Majority problem in single winner elections.
Immediately the thread got off into Proportional Representation as the solution to Tyranny of the Majority, so the context of single winner methods was forgotten. When I tried to get back on track most readers were doubtful about the advantages of "lotteries" for use in serious elections. Although Jobst Heitzig elocuently answered all objections, and gave some examples in which the almost sure lottery solution was clearly preferable to the majority favorite, the thread dwindled into oblivion. I would like to try to resurrect the thread by showing how single winner lottery techniques can lead to better deterministic multi-winner PR methods. I alluded to the analogy between deterministic multi-winner PR methods and single winner lottery methods in my original post on the thread, but nobody except Kristofer really picked up on it, and he was doubtful of the value of the analogy. I just realized that the problem was psychological. Psychologically it is better to show how the analogy can be used to improve deterministic PR methods (which most list participants already believe in) than to use the analogy to convince participants of the value of single winner lottery methods (for which there is a mental barrier). To see the precise nature of the analogy consider two possible interpretations of Jobst's challenge scenario 60 A 100, C 80, B 0 40 B 100, C 70, A 0 In the single winner lottery interpretation, A, B, and C represent the alternatives. The 80 next to C in the majority faction row of the preference schedule means that those voters would prefer C to the lottery 79%A+21%B, but would prefer the lottery 81%A+19%B to the sure election of alternative C. Expressed as a compund inequality this information looks like this: 79%A+21%B < 100%C < 81%A+19%B This compound inequality is the content of the assertion that 100%C ~ 80%A+20%B. Similarly, the 70 next to C in the minority faction row entails the following approximate equality: 100%C ~ 70%B+30%A In both cases we have 60%A+40%B < 100%C, because in the first faction 60%A+40%B < 80%A+20%B ~ 100%C , and in the second faction we have 40%B+60%A < 70%B+30%A ~ 100%C . Now for the second interpretation: This time the context is a multi-winner Proportional Representation (PR) election. Now the letters A, B, and C represent parties, and the numbers next to them represent the confidence had by the voters in the respective factions that the indicated parties will represent their interests. This time the inequality 80%A+20%B~ 100%C means that the majority faction voters would rather have all representatives come from party C than for 79% of them from party A and 21% of them from party B, but would rather have 81% of the representatives from party A and 19% from party B than having 100% of them from party C. In this interpretation, the inequality 60%A+40%B < 100%C represents the fact that both factions would prefer all of the representatives to come from C over the alternative that 60% come from A and 40% from B. In this PR context, note that any extant party list system will almost surely result in 60% of the representatives coming from party A, and 40% of the representatives from party B and none from party C, even though every voter would much rather have all of the representatives come from party C. All naïve attempts at overcoming this problem fail. For example, getting everybody together and saying, "Since 100%C is preferred by all of us to the standard party list system result, let's just all promise each other that we will vote for party C." If ballots are secret, and the voters of the minority faction are honest, but the voters of the majority faction are low on scruples, the result will be 60%A+40%C, which rewards the defecting majority faction while penalizing the honest and loyal minority faction. In fact, under list PR rules, the game theoretic optimal strategy for both factions is to defect. Is there a way around this? The answer is "yes," and we can thank Jobst Heitzig for most of the work and inspiration behind the methods that best solve the problem, because he is the one who has consistently held our feet to the fire on the analogous issue in the single winner lottery case. To see this correspondence let's return to the single winner lottery interpretation. There the analogous problem is that if voters are allowed to directly assign their shares of the lottery probabilities according to their desires, the resulting lottery will be the "random favorite" lottery 60%A+40%B in which alternatives A and B have the respective winning probabilities of 60% and 40%, with C getting none of the probability, even though every voter prefers the 100% C lottery over the random favorite lottery. As in the PR analogue, all naïve attempts at overcoming this problem fail. Non-binding agreements fail in the same way because defection is the optimal strategy when voters (by secret ballot) get to decide which alternatives get their share of the probability. At this stage many EM list participants would say, "The obvious solution is to forget lotteries and use some form of Range, since C is the obvious Range winner." There are two problems with this approach: (1) Because of the Tyranny of the Majority problem, rational voters who are well informed about each others' preferences will not elect alternative C under the rules of Range. More importantly: (2) Even if there were a deterministic method that reliably elected C, that would not help us in the PR analogue. Fortunately, Jobst has had the vision to see the value of lottery solutions to the single winner fair-compromise problem, since these solutions do transfer directly across to the deterministic party list PR context. Ironically (in the single winner context), any of Jobst's proportional probability lottery methods would almost surely elect C under the same assumptions about rationality and information that forced the failure of alternative C in the deterministic Range setting. I say "ironically," because only by making "chance" an essential part of the method can we make "sure" that C is elected in the single winner setting. Before continuing I want to emphasize that I am not proposing electing PR assemblies by use of chance. I am proposing that we use the analogy between stochastic single winner methods and deterministic multi-winner list PR to convert (mutatis mutandis) Jobst's single winner lottery methods into the best deterministic list PR methods the world has ever seen! To be continued: ---- Election-Methods mailing list - see http://electorama.com/em for list info
