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).
It is not the case that voters on average necessarily prefer the maximal lottery to any other lottery in expected utility. It is only the case that voters on average prefer a candidate sampled at random from the maximal lottery to a candidate sampled at random from any other lottery. Doing the first thing would be impossible, because there can be cycles.
This is how anarchy wins in maximal lotteries in the example in the next section. If you compare anarchy to choosing a leader at random, in expected utility, choosing a leader at random is a Pareto improvement, but if you sample a candidate at random, they will lose to Anarchy by 1 vote.
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).
It is not the case that voters on average necessarily prefer the maximal lottery to any other lottery in expected utility. It is only the case that voters on average prefer a candidate sampled at random from the maximal lottery to a candidate sampled at random from any other lottery. Doing the first thing would be impossible, because there can be cycles.
This is how anarchy wins in maximal lotteries in the example in the next section. If you compare anarchy to choosing a leader at random, in expected utility, choosing a leader at random is a Pareto improvement, but if you sample a candidate at random, they will lose to Anarchy by 1 vote.
Answer: The next section gives a situation where a maximal lottery gives a worse result than a certain loterry-lottery.