In the last post we saw how by introducing lotteries, we could resolve one of the major conflicts in voting theory, between Condorcet and consistency (while also satisfying participation and clone independence). Now, we are going to make it better. How? By adding even more lotteries!

As far as I know, unlike the previous posts, the content of this post is a new proposal by me. Everything in this post should be treated as conjecture, rather than proven.

The Proposal

Fix a finite set $C$ of candidates, and an electorate $V \in Δ (C \to [0, 1])$ .

First, recall the following definition of maximal lotteries. Given lotteries $A, B \in Δ (C)$ , we say that $A$ dominates $B$ if $P_{a \sim A, b \sim B, v \sim V} (v (a) > v (b)) \geq P_{a \sim A, b \sim B, v \sim V} (v (b) > v (a))$ . A maximal lottery is an $M \in Δ (C)$ such that for all other $L \in Δ (C)$ , $M$ dominates $L$ .

Now for maximal lottery-lotteries. Given lottery-lotteries $A, B \in Δ (Δ (C))$ , we say that $A$ dominates $B$ if $P_{A \sim A, B \sim B, v \sim V} (v (A) > v (B)) \geq P_{A \sim A, B \sim B, v \sim V} (v (B) > v (A))$ . A maximal lottery-lottery is an $M \in Δ (Δ (C))$ such that for all other $L \in Δ (Δ (C))$ , $M$ dominates $L$ .

Note that when we say $v (A)$ , since $A$ is not a candidate, but a distribution of candidates, we mean the expected value of $v (a)$ when a is sampled from $A$ , and similarly for $v (B)$ . Unlike in maximal lotteries, to determine whether $v (A) > v (B)$ , we need the full information of $v$ as a utility function (up to affine transformation), not just the preference ordering over candidates.

$M$ is a distribution on distributions on candidates, but we need to output a distribution on candidates. Thus, we imagine sampling a distribution at random from $M$ , and then sampling a candidate at random from that distribution. The output of the maximal lottery-lotteries voting system is the distribution that assigns each candidate the probability that they would be sampled from the above procedure.

Lottery Independence

To argue for maximal lottery-lotteries, I introduce a new criterion: lottery independence.

Given an initial set of candidates, $C$ , we could imagine fixing any distribution over candidates $P \in Δ (C)$ , and introducing a new ...

Maximal Lottery-Lotteries