A possible criterion for cardinal methods that take lottery information is this: "If there exists a single lottery that, no matter what affine scaling we apply to a voter, every voter prefers to every other lottery, then that lottery should win." For a deterministic voting method, only consider deterministic lotteries (i.e. 100% election probability for some candidate, 0% for the rest). (There's also a Smith set analog: it should elect from the smallest group for which some lottery inside is preferred to every such outside.) But this gives rise to a utilitarian/OMOV tradeoff problem that I've mentioned earlier: it might be the case that one of the voters feels *very* strongly about the outcome, so that by the intensity of his preference, he would have dictatorial powers over the outcome, if we were aiming to maximize utility. So even if we assume complete honesty, there might exist an inherent dictator by the logic of utilitarianism itself. Thus there's a tension between OMOV (which limits the relative power of one voter over another) and utilitarian maximization. A cardinal voting method based on utilitarian reasoning has to set that limit somewhere. One reason that it's so hard to construct cardinal methods based on lottery information might be that we haven't decided just where that limit should be, or even thought about how it factors into the design of the method itself. To set the limit, there seem to be three alternatives: - The method can directly set the tradeoff as a tunable parameter, or - The method can be designed not to be directly utilitarian, just be better at distinguishing "weak centrist" scenarios from "true consensus" scenarios, or - The method can allocate some maximum power to each ballot and then let the voters voluntarily claim only a portion of this power if they feel that their opinions are not sufficiently strong to warrant full intensity. I *think* MJ is in the second category and Range in the third. Applying DSV to the third type of method would result in something in the second, because the voters who deliberately choose not to use the full range of their ballots could strategize depending on who's in the running. But this normalization doesn't have to look like Range - for instance, a type three method could be cumulative voting with a maximum on the Euclidean norm of the ballot; and then the corresponding strategic method chooses a ballot with Euclidean norm at that exact maximum. Methods of type one (I don't know of any) would try to resolve the problem by saying "suppose each voter's utilities are within some interval; then with honest voting with lottery information, if there exists a dominating lottery in the sense above, then that's sure to be the one that maximizes social utility". Another thought: suppose that the method normalizes lottery information to get the maximum power out of any comparison. This scale has to involve more than two candidates - otherwise the normalization is just "100% power to whichever I prefer", which turns into Condorcet. Would it be possible to make a three-candidate variant with its own analogs of a Condorcet winner and Smith set? Such a method might end up majoritarian, but it's possible to be majoritarian and cardinal -- at least more cardinal than ranked -- as e.g. shown by MJ.

PR lottery methods avoid a lot of these utilitarian philosophical conundra by conceding at the outset that a faction that bullet votes for X will guarantee at least a P percent probability of X being elected, where P is the percentage of voters in that stubborn faction. Determinism puts up a high hurdle for proportionalty that can be surmounted in multi-winner contexts, but not really in single winner contexts except in rotation/time/resource sharing that replaces probability with infinitely divisible goods and/or "bads." Other than that, as Jobst has pointed out, all of the popular "consensus building" methods in the literature rely on subtle psychological bullying on the one hand, and acceptance of the same by weaker more abnegated personalities on the receiving end. They will shame you and wear you down until you finally concede with a strained smile, a handshake, and one final chorus of Zumbaya! [my words, not Jobst's] FWS El mar., 12 de oct. de 2021 6:44 a. m., Kristofer Munsterhjelm < km_elmet@t-online.de> escribió: > A possible criterion for cardinal methods that take lottery information > is this: > > "If there exists a single lottery that, no matter what affine scaling we > apply to a voter, every voter prefers to every other lottery, then that > lottery should win." > > For a deterministic voting method, only consider deterministic lotteries > (i.e. 100% election probability for some candidate, 0% for the rest). > (There's also a Smith set analog: it should elect from the smallest > group for which some lottery inside is preferred to every such outside.) > > But this gives rise to a utilitarian/OMOV tradeoff problem that I've > mentioned earlier: it might be the case that one of the voters feels > *very* strongly about the outcome, so that by the intensity of his > preference, he would have dictatorial powers over the outcome, if we > were aiming to maximize utility. > > So even if we assume complete honesty, there might exist an inherent > dictator by the logic of utilitarianism itself. Thus there's a tension > between OMOV (which limits the relative power of one voter over another) > and utilitarian maximization. > > A cardinal voting method based on utilitarian reasoning has to set that > limit somewhere. One reason that it's so hard to construct cardinal > methods based on lottery information might be that we haven't decided > just where that limit should be, or even thought about how it factors > into the design of the method itself. > > To set the limit, there seem to be three alternatives: > > - The method can directly set the tradeoff as a tunable parameter, or > - The method can be designed not to be directly utilitarian, just be > better at distinguishing "weak centrist" scenarios from "true consensus" > scenarios, or > - The method can allocate some maximum power to each ballot and then let > the voters voluntarily claim only a portion of this power if they feel > that their opinions are not sufficiently strong to warrant full intensity. > > I *think* MJ is in the second category and Range in the third. Applying > DSV to the third type of method would result in something in the second, > because the voters who deliberately choose not to use the full range of > their ballots could strategize depending on who's in the running. But > this normalization doesn't have to look like Range - for instance, a > type three method could be cumulative voting with a maximum on the > Euclidean norm of the ballot; and then the corresponding strategic > method chooses a ballot with Euclidean norm at that exact maximum. > > Methods of type one (I don't know of any) would try to resolve the > problem by saying "suppose each voter's utilities are within some > interval; then with honest voting with lottery information, if there > exists a dominating lottery in the sense above, then that's sure to be > the one that maximizes social utility". > > Another thought: suppose that the method normalizes lottery information > to get the maximum power out of any comparison. This scale has to > involve more than two candidates - otherwise the normalization is just > "100% power to whichever I prefer", which turns into Condorcet. Would it > be possible to make a three-candidate variant with its own analogs of a > Condorcet winner and Smith set? Such a method might end up majoritarian, > but it's possible to be majoritarian and cardinal -- at least more > cardinal than ranked -- as e.g. shown by MJ. > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >