Maximal Lotteries
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I just learned about this now. It’s a Smith compliant method, and is participation compliant up until necessary randomization.
https://en.wikipedia.org/wiki/Maximal_lotteries
Basically, you form the majority margin matrix Mij over candidates. Then consider the two player game where each player chooses one candidate: if player 1 chooses i, and player 2 chooses j, then player 1 gets Mij and player 2 gets -Mij.
This game has at least one mixed strategy Nash equilibrium, which is a distribution over candidates. A maximal lottery selects a candidate at random according to one such mixed strategy Nash equilibrium. In this sense, a maximal lottery is a mixed “candidate” that cannot be challenged by a majority.
Occasionally maximal lotteries are non-unique, but they often are. A trivial case is when a Condorcet winner exists, in which case the maximal lottery selects the Condorcet winner as a pure strategy. Orthogonally, if there is an odd number of voters, and if preferences are strict, the maximal lottery is always unique. Otherwise, symmetries like multiple dominance components in the Smith set can induce degeneracies.
Still, any maximal lottery satisfies probabilistic notions of participation, where participation cannot reduce the chances of a preferred outcome for a voter.
I think this is nearly the “right” notion of compliance. There is only slight room in choosing which maximal lottery to implement. For example, in case of degeneracy, one could choose a maximal lottery that optimizes expected score. Or, maybe compute Jeffrey’s prior over maximal lotteries, and then sample a maximal lottery accordingly.
What thoughts do others have?
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@cfrank I think lottery methods in general are interesting and worth looking into. Looking at the Wikipedia page, it seems interesting that this would satisfy participation but not monotonicity, which is the opposite of most Condorcet methods. So while some might criticise it for failing monotonicity (seen as easy to get in a Condorcet method), the prize is arguably better, since elections are really about voters rather than candidates.
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@toby-pereira yes monotonicity fails, but I think pairwise probability margins are apparently monotonic.
I think maximal lotteries are interesting too, and I wonder if it makes sense to compensate the dissatisfied majorities with electoral credit to spend in subsequent election(s).
For example, say the maximal lottery (or maybe any other Condorcet method) elects A when there is no Condorcet winner. Let beat(X) be the set of candidates that beat X in a head-to-head majority.
Then for each B in beat(A), it seems like the majority preferring B to A should be compensated in some way. My thinking is that in aggregate, that majority should be compensated by something like the smallest extra weight they would have needed to secure B as the (Condorcet) winner against each C in beat(B). This might get complicated, and there may already be literature on this kind of thing.