Lottery-condercet seems like a nicer condition than condercet and this seems like a very compelling argument against condercet. Lottery independence seems like a hard property to reject when randomization is available (and "we didn't ask the players how they felt about lotteries" is not a very compelling defense). Overall seems like a very interesting perspective, though it would certainly be nicer if maximal lottery-lotteries existed.

Maximal lottery-lotteries also satisfies consistency and participation, since they are just maximal lotteries over a larger set of candidates.

Actually, I think there is a decent chance I am wrong here. It might be merging two electorates that produce the same lottery-lottery over candidates also produces that lottery-lottery, but we can't say the same thing if we collapse that to lotteries, which is what we would need for consistency.

In its suggested form Maximal Lottery-Lotteries is still a majoritarian system in the sense that a mere majority of 51% of the voters can make sure that candidate A wins regardless how the other 49% vote. For this, they only need to give A a rating of 1 and all other candidates a rating of 0.

One can also turn the system into a non-majoritarian system in which power is distributed proportionally in the sense that any group of x% of the voters can make sure that candidate A gets at least x% winning probability, similar to what is true of the MaxParC voting system used in vodle

The only modification needed to achieve this is to replace $\Delta \left(C\right)$ (the set of all lotteries on C) in your formula by the set of those lotteries on C which every single ballot rates at least as good as the benchmark lottery. In this, the benchmark lottery is the lottery of drawing one ballot uniformly at random and electing the highest-rated candidate (as in the "random ballot" or "random dictator" method).

A voting system satisfies lottery independence if introducing lottery candidates does not change the probability that any candidate is elected.

So far I'm not getting the point of this. Suppose my Lottery-Lottery has one element, the Maximal Lottery (TM). I'm no mathematician, so prove me wrong at-will, but I don't think there are any other lotteries you can add that do not change the probability that any candidate is elected, that aren't identical to the Maximal Lottery (TM). 

Why would I ever want a Lottery-Lottery that has elements that aren't the Maximal Lottery (TM)? Surely my voting system is better if I just replace whatever random lotteries are in the Lottery-Lottery with my single Maximal Lottery (TM).

How to elicit the necessary information in the voting booth? Surely we can't have a ballot with all possible gambles on it.

We could use score voting (and then process the votes differently, much like the zoo of Borda count alternatives process the same ballot differently). But score voting has strategy problems where you're incentivized to turn it into approval voting, right? Are there similar strategies in MLL counting?

Not expecting this to be a solved problem, more just reacting in the form of a question.

I'm having trouble with the solution to the dictatorship vs anarchy example. Is a $\frac{1}{3}$ split actually maximal? Is "A dominates B" transitive?

Let's say $ϵ=0.1$. Consider an unfair 50/50/0 split in which the first two potential dictators throw the third under the bus. As I understand, this dominates the equal split, since 2 out of the 3 voters prefer this new lottery.

Then say voter #3 counterproposes a 0/60/40 split. This dominates again. But then 50/0/50 dominates that... so domination is non-transitive. And I don't see a maximal randomized outcome.

Am I missing something here?

Can someone explain to me how the giry monad factors in? For some $A\in \Delta \left(\Delta \right(C\left)\right)$, executing $A\text{>>=}(\lambda x.{\delta }_x)$ to get a $\Delta \left(C\right)$ would destroy information: what information, and why not destroy it? (Am I being too hasty comparing probability monad to haskell monad?)

There are three voters trying to elect a leader. If any one of them is elected leader, that person gets utility 1, and the other two get utility 0. However, there is also a third anarchy option where they have no leader. Anarchy is bad. Each person gets $\epsilon$ utility where $0<\epsilon <\frac{1}{3}$. Anarchy is the Condorcet winner, and thus maximal lotteries will choose anarchy with probability 1. This is dumb. Everyone would prefer to just randomly pick between the three candidates, which is what maximal lottery-lotteries does.

Whoa whoa whoa. This isn't guaranteed. 0 utility is really bad! Like, if the winner gets to just torture me and my loved ones to death, I don't want that to happen! I'd prefer anarchy than submitting to that with a 2/3rd probability! Part of the point of the election is that it should mean, 'this kinda sucks, but I'll get another chance to have my candidate win next time'. Any voting outcome which is worse than the alternative of 'must fight an underdog rebellion' should be expected to turn into an underdog rebellion. Underdog rebellion is worse for the group than anarchy (equal footing, no established leader), but still better than no utility at all. Or did you mean 'gets 0 utility' to mean a perfectly neutral state where things are neither positive or negative? That's a very different and not to-my-mind default meaning.

It seems like an important feature of democracy that winners are in some sense partial, the condition is temporary and intended to have largely reversible effects where possible. This isn't always the case, like Candidate A committing the country to a war that the other candidates wouldn't have, but in many cases it is. If the laws passed by one regime are, in practice, found to be intolerable by a large minority, then reversing those laws will become a party ticket in the subsequent campaign.

Here's another thought. Maybe voting method is less important than campaigning rules by a large margin. Maybe if campaigns were structured in such a way that important information about the relative competence and biases of the proposed candidates were exposed, the voting method would become mostly irrelevant. Forcing all the candidates to undergo public trials of their ability to do legislative decision making in realistic simulations, and then simply summarizing and publicizing their scores seems to me that it could do more to get better candidates elected. If a particular group thought they wanted candidate A, but then saw that actually candidate B's decisions regularly resulted in much better outcomes for them in the simulations as well as for B's base (because the world isn't zero sum, and better decisions can be the tide which floats all boats)... Seems like the shift in favor of B would likely outweigh subtleties of voting methods around edge cases. If I had to pick between candidate-simulation-trials plus a random choice of voting method between <theoretically perfect method> / ranked choice / approval voting, or not have candidate-simulation-trials and be sure to get to use <theoretically perfect voting method>, I'd strongly prefer the option with simulation trials.