Set Approval Voting

[Brian Langstraat]

Set Approval Voting (SAV) may be a simple solution for optimal proportional representation.

I was inspired by the article "Optimal proportional representation" multiwinner voting systems, which mentions a problem and a possible solution:

One way to design a multiwinner voting system is to define a "quality function" which examines the votes, voters, candidates, and winner-set, and outputs a number. Then the "optimum voting system" would simply be to choose the winner-set yielding maximum quality.  But finding that best subset might be computationally infeasible if the number of candidates and winners is too large.

 

Forest Simmons suggests the following brilliantly simple (?) trick for dodging the computational roadblock: ask the candidates, voters, and/or other interested entities (anybody who wants) to suggest winner sets. (An automated web site could accept submissions.) If anybody succeeds in finding a new-record higher quality set, we switch to it. If not, then we just tell the disgruntled candidates "if you lost but think you should have won, it is your own damn fault for not suggesting a better winner set than you did."

In Set Approval Voting (SAV), voters would suggest winner sets.

With large numbers of candidates and winners, the suggested number of winner sets would have a maximum of the number of voters and (typically around a hundred winner sets) instead of a bzillion (approaching infinity).

Overview of SAV:

Voters approve or disapprove each of the N candidates.

Voters may approve any number of candidates, but only ballots with N approvals will be used as suggested N-winner sets.

For each N-winner set, the total number of ballots that approve at least one candidate in the set is calculated.

The N-winner set with the most total ballots is the winning set.

[If there are no valid suggestions for N-winner sets, then the N most approved candidates win.]

I am not sure how optimal or proportional this voting system would be.

Likely, the winning set would include the most approved candidates from a variety of political factions that at least one voter suggested.

If every voter bullet voted, then it would devolve into Single Non-Transferable Voting (SNTV).

[parker friedland]

I don't think this voting system is too proportional. Like in SNTV, it is only proportional if voters are strategic. Consider the following example:

                                            Democrat 1, Democrat 2, Democrat 3, Republican 1, Republican 2, Republican 3,

1000 Liberal voters:                    X                   X                  X
2500 Liberal voters:                    X                   X
2500 Liberal voters:                    X                                        X
2000 Conservative voters:                                                                                                                        X
2000 Conservative voters:                                                                                                  X
2000 Conservative voters:                                                                          X

1 voter:                                         X                   X                  X                   X
1 voter:                                         X                   X                  X                                            X
1 voter:                                         X                   X                                        X                      X
1 voter:                                         X                                       X                    X                      X
1 voter:                                                              X                  X                    X                      X
1 voter:                                         X                   X                  X                                                                  X
1 voter:                                         X                   X                                        X                                            X
1 voter:                                         X                                       X                    X                                            X
1 voter:                                                               X                 X                    X                                            X
1 voter:                                         X                    X                                                               X                    X
1 voter:                                         X                                       X                                            X                    X
1 voter:                                                                X                X                                            X                    X
1 voter:                                         X                                                              X                     X                    X
1 voter:                                                                X                                       X                     X                    X
1 voter:                                                                                   X                    X                     X                    X

If there are 4 winners, the winning outcome should be: Democrat 1, Republican 1, Republican 2, Republican 3, even though there were just as many liberal voters as conservative voters.

A voting system that is proportional when voters are honest though is PSI voting: http://scorevoting.net/QualityMulti.html

And in this conversation, I mention the approval voting version of PSI voting: https://groups.google.com/forum/#!searchin/electionscience/parker$27s$20system|sort:relevance/electionscience/yQkesuINOHs/68Z8xxuvCAAJ. I called it Parker's system because I didn't know that PSI voting existed. You can see that is system isn't very different from that.

[Jeremy Macaluso]

The two methods are the same method - just with different deltas. The method Brian mentioned has a delta of 0, whereas psi voting has a delta of 1. A delta of 1 is more proportional, but both of those methods have the same calculatability requirements, which only using councils from ballots attempts to solve. However it does not do a good job of solving the problem. For small elections where there are fewer than a dozen winners, the problem does not exist at all. The problem only begins to appear in elections with more than a dozen winners, which often will have millions of voters. Because the necessary winner sets come from other ballots, the results will not be precinct summable - each precinct will need to report all possible winner sets to a central system or each other before the counting of each possible winner set can start.

[parker friedland]

It almost resembles PSI when you use delta of 0 (which is not recommended. In PSI, it is recommended that delta is greater then or equal to 0.5 and less then or equal to 1 if you want the voting system to be proportional) but not exactly, because in Brian's set approval, he said that the only N-winner sets that are considered are the ones that correspond to a ballot that approves all of the candidates in that set and no more. That is why at the end of my example, I included a voter for every possible set of N candidates that a voter could possibly approve of. Without that weird rule, then yes, Set approval voting is PSI when delta = 0.

[Brian Langstraat]

Parker,

Excellent analysis!

After I wrote the original post for Set Approval Voting (SAV), I realized that a subset of "bullet-ed" candidates could have an advantage when placed into a set with the most popular candidate.
Your analysis shows that what I said was too true:

Likely, the winning set would include the most approved candidates from a variety of political factions that at least one voter suggested.

The 3 Republican candidates could be considered "political factions", so SAV would tend to result in parties splitting into smaller factions and eventually behave like fairly proportional Single Non-Transferable Voting (SNTV).

SAV and PSI seem too complex for precincts.

If a little precinct complexity is okay, then I would prefer Approval Remainder Voting (ARV).