Algorithmic Asset Voting

A flowchart explaining how Algorithmic Asset works.

Asset Voting can, depending on which relevant assumptions are made about how negotiators act, be turned into a Smith-efficient Condorcet method in the single-winner case, and a Condorcet PR method in the multiwinner case (akin to CPO-STV and Schulze STV). Its variants, Sequential Asset Voting and Bloc Asset Voting, can also be algorithmized.

Assumptions

Some of the assumptions are:

• Voters submit ranked or rated ballots.
• The negotiators strictly follow the preferences of those ballots and try to maximize those ballots' satisfaction with the outcome i.e. if a negotiator is asked to negotiate on behalf of a voter whose ballot was A>B>C, and the negotiations are at such a stage that the negotiator can use their assets to decide which of A, B, or C will win, then the negotiator must help elect A.
• The candidates with the most votes at the end of the negotiations are sequentially elected until all seats are filled.
• The negotiators have as much time as necessary to reach a final outcome or set of outcomes.
• The negotiators move one negotiating step at a time (i.e. if some negotiators agree to support a candidate, they must first all give their votes to that candidate before any further negotiating actions occur)

Some optional assumptions are:

• When a negotiator is indifferent between certain outcomes (i.e. because their voters equally ranked those outcomes), they use their assets to help pick the socially best of those outcomes.
• In the multiwinner case, when a voter submits a rated ballot, and their negotiator can choose between electing, say, the voter's first choice, or both of their second and third choices, the negotiator somehow uses the rated information to decide which outcome is preferable (i.e. they might add up the utilities for the voter in either outcome and pursue the higher-utility outcome.)
• A resolution method is applied when there are multiple outcomes in the Smith Set, and the candidates' preferences can change in order to change the Smith Set in favor of maximizing their voters' satisfaction (though this might break the algorithm or make it fail to be Condorcet-efficient in certain scenarios). As a further possibility, the candidates might also be allowed to try to induce Condorcet cycles or otherwise grow the Smith Set in ways that allow them to then resolve the election in favor of their voters' satisfaction (though this also might break the algorithm).

Procedure

Do pairwise matchups between all possible outcomes. (To speed up computation, some matchups can be ignored if the outcome is certain. Also, some outcomes can be checked first if they're more likely than others).

When doing a matchup, transfer votes in such a way as to ensure a voter gets as many of their favorite candidates as possible. (If the voter submitted a rated ballot, optionally attempt to instead maximize their overall utility i.e. if electing their 1st choice is worth 9 utility points, and electing their 2nd and 3rd choice is 8+5= 13, prioritize electing the latter two).

If there isn't a stable winner in a pairwise matchup (the negotiators are never able to make one outcome beat the other without some negotiators making strategically rational moves that help the other outcome win), declare the matchup a tie.

Use a Smith-efficient Condorcet method to elect from the Smith Set of outcomes (or use any Condorcet cycle resolution method).

Example

3-winner election, Hagenbach-Bischoff quota 25:

26 A
34 C>B>D
6  B>D
8  D>B
5  D>S
21 S>D

It can immediately be observed that A and C have over a quota of 1st choices, so they will win. (D, S) form a Droop solid coalition of 26 votes, so one of them must win. Therefore, the only outcomes to compare are (A, C, D), and (A, C, S).

(A, C, D) vs. (A, C, S):

26   A
26.5 C
26.5 D
21   S

Here, 7.5 CBD votes and the 19 other votes that prefer D to S give their votes to D. A, C, and D are then the 3 candidates with the most votes, and since the voters who prefer S>D don't have enough votes to change the outcome, (A, C, D) win this matchup. Since we already know (A, C, D) can win all other matchups, (A, C, D) is the Condorcet winner and wins.

Another example:

If there are 51 voters who prefer Party A and 49 for Party B, then in a 5-winner election, the 49 B voters can always ensure two of their candidates are in the final winner set.

Explanation of how Asset Voting is, under certain assumptions, a Condorcet method (and how this enables it to be done as an algorithm)

Algorithmic Asset to elect a Condorcet winner.

Algorithmic Asset identifying the Smith Set.

In the single-winner case, if the negotiators are honest, strictly follow voter preferences, and have enough time to negotiate, then Asset becomes a Smith-efficient Condorcet method, and in the multiwinner case, resembles Condorcet PR methods such as CPO-STV and Schulze STV (these transformations can be observed by turning Asset Voting into an algorithm using various assumptions, as mentioned below). The reasoning for this can in part be linked to the fact that Asset is an iterative voting method (it is almost like an iterative version of FPTP; iterative voting methods are generally more Condorcet efficient than their non-iterative equivalents) where the voters/negotiators are constantly updated on who is about to win if no change in votes occur (i.e. which set of candidates of a size equal to the number of seats to be filled have more votes committed to them than all other candidates so far), and they can, therefore, plan to defeat such candidates. Pairwise comparison is implicitly involved in this planning, as the negotiators must see if the candidates they prefer over those about to win can obtain more votes from all negotiators than those who are about to win.

Asset Voting can be done algorithmically on ranked or rated ballots when certain assumptions are applied, such as the ones mentioned above (here is a visualization of the algorithm). One main assumption is that every negotiator attempts to maximize their assigned voters' satisfaction with the outcome.

Lewis Carroll's own likely observations that Asset is intended to be Condorcet-efficient

Lewis Carroll is the first known inventor of Asset Voting. In a passage in an article describing his thought process in developing Asset, Carroll appears to have noted that the ideal PR scheme would involve voters forming into sets of coalitions, and then, when enough voters prefer a different set of coalitions than the one that is currently formed, a different set of coalitions is formed, and this repeats until no more improvement is possible. He also notes that it is possible to, by looking at ranked ballots, figure out what sets of coalitions might occur if the voters were acting "rationally" (most likely meaning maximally strategically), and he even solves a few examples where such an approach yields different results from STV, though it is noted that at the time, the full computation for this process when looking at the ballots for a large-scale election would've been "indeterminate". This all appears to establish that Carroll was interested in finding a Condorcet equilibrium for the final set of winners, and saw Asset as the easiest way to do it, because he believed the candidates' preferences would be close enough to the voters' preferences to make the candidates' Condorcet equilibrium more Condorcet-efficient by voter preference standards than STV.

Notes

It could be possible for a voter to submit a ballot indicating partial preferences in certain pairwise matchups i.e. that they'd want to only give 0.2 votes to help Candidate A be in the winner set rather than candidate B, but 1 full vote to help Candidate A and/or B be in the winner set over Candidate C. So for example, if they voted with a rated ballot A10>B8, A10>C0 and B10>C0. See the "connections to cardinal methods" section of the Condorcet methods page for more information. Note that this may be possible even in the multiwinner case.

The use of the KP transform on rated ballots and then converting those Approval ballots to ranked ballots allows voters to submit, for example, a rated ballot A5 B4 C3, and have it treated either as 0.2 votes A>B, 0.2 votes B>C, and 0.4 votes A>C (score for preferred candidate minus score for less preferred candidate divided by max score yields the number of votes in each matchup) or 1 vote A>B, 1 vote B>C, 1 vote A>C.

Note that in a pairwise match-up between two winner sets, a third winner set can actually emerge as the winner. This is because some voters may prefer some candidates from one set and some from the other. For example, in a matchup between a set with only the majority's preferred candidates and another set with mostly candidates with almost no support, but one candidate whom a quota prefer, the final winner will actually be a new winner set where most of the candidates are the majority's preferred candidates and the final candidate is the quota's preferred candidate.