I invented a new STV rule that appears to be independent of covered alternatives for a single-winner case (IRV). I am not sure what basic approach to use to formally prove this is independent of covered alternatives (a new criterion as far as I know); I may not have the proper background to do so.
If anyone could give me an idea of what direction to take for formally proving this is ICA that would be great.
The Rule as IRV
Specialized to IRV, it looks as below:
- Canonicalize all ballots: All rankings progress from 1 (highest preference) down (1, 2, 3…). Renumber ballots for any gaps.
- Determine any winner: If any candidate has more than 50% of the votes, elect that candidate and end.
- Begin at the lowest rank $R$. Initialize $R$ to the number of non-eliminated candidates (the lowest possible rank).
- Count the Bucklin score at rank $R$. Count the number of times each candidate $C$ appears on a ballot up to and including at rank $R$, but not further. This is $B_C$. (So if a candidate appears ranked anywhere from 1-4 on a ballot, and $R=4$, don't count ballots where that candidate appears ranked at 5.)
- Decide if eliminations happen: If any candidate's score $B_C$ falls below 50% of valid ballots, eliminate the candidate with the lowest $B_C$ and return to 1.
- Move back one rank: Set $R=R-1$ and go to 4.
This turns out to eliminate all non-Smith candidates before eliminating any Smith candidates, making it independent of Smith-dominated alternatives (ISDA). More than that, if $C_A$ has a majority over $C_B$, $C_A$ has a majority over $C_C$, $C_B$ has a majority over $C_C$, and $C_B$ has no majority over any other candidate, then $C_B$ must be eliminated before $C_A$. That is to say: this process must necessarily eliminate all covered alternatives, after which all candidates lose a majority $B_C$ simultaneously in each round, and so it is independent of covered alternatives (ICA).
I haven't taken the time to figure out if it eliminates all non-mutual-majority candidates, then all non-Smith candidates, then all covered candidates, in that order. For that matter, I don't know what constitutes a formal proof that it is in fact ICA, or how to begin that. Any thoughts?
Aside, the multi-winner rule generalized to STV, on which this IRV rule is based, is…even weirder. I've found two manipulations. The first is equivalent to just not ranking a candidate on your ballot: put a loser between your favorite and second-favorite so the process eliminates your second-favorite. The second manipulation is outright dangerous: rank another coalition's candidate above yours—particularly the one you dislike the least—and somehow maintain the delusion that this is going to do anything beyond lose you (and possibly your entire coalition) your vote (instead of electing Warren instead of Bernie, you will likely end up electing Warren and Bernie, and not electing anyone you actually wanted to elect). The second one is only reasonable to even consider if you have information about all the ballots, and even then it's NP-Complete.
It also has an inversion issue where e.g. with a quota of 25, 24 Giorgi>Ingrid plus 1 Ingrid>Hans>Giorgi causes Giorgi to be eliminated (if Hans isn't eliminated first) and Ingrid to be elected. (In any case, if Hans is a winner, Giorgi actually doesn't have a quota of votes and can't win; but Ingrid does, so it's…weird…since a lot of strange coincidences have to occur for this to matter, but I can't dismiss it as a significant pathology.)