Asset is Smith-efficient for negotiators’ preferences, if they are given enough time to do all the relevant pairwise comparisons.
We can define an “algorithmic Asset Voting” procedure as one where the voters themselves negotiate, are honest, altruistic where it doesn’t hurt them, strictly follow the preferences submitted on their ranked (or rated) ballots, and where the candidate(s) with the most votes at the end of the negotiations are elected until all seats are filled. The use of ratings in such a procedure might be if, say, a voter has to choose between a guarantee of getting their favorite and third choice, or can create a tie which would give them a 1/3rd probability of getting their favorite and second choice (an improvement), a 1/3rd probability of (Favorite, 3rd choice) (no change) and a 1/3rd probability of (2nd Choice, 3rd Choice) (a worsening).
In algorithmic Asset, if each voter is given as many votes as there are winners, then I believe the final winner set will be a “Smith Set committee”, where the chosen winner set is majority-preferred over all other winner sets except those in the Smith Set of winner sets. This would be a majoritarian multiwinner voting procedure. (Edit: I got it wrong; it doesn’t matter how many votes each voter has, so long as they each get an equal amount, the procedure will be proportional. The way to make algorithmic Asset more majoritarian is if each seat is elected sequentially: we find the Smith Set outcomes for the first seat, pick one, and spend votes. If less than a Droop Quota is spent, the final winner set will be biased away from PR to a majoritarian outcome.)
If each voter is given only one vote, then algorithmic Asset should return a “Smith Set PR committee” which would be a winner set that receives more votes in Asset negotiations than all other winner sets except those in the Smith Set of winner sets. This is a fully proportional procedure.
I believe “Smith Set PR commitees” are the same or almost exactly the same outcomes that CPO-STV and Schulze STV produce.
Considering that with cardinal methods, we can go from consensus (Bloc Score) to fully proportional (SMV, or I suppose for purposes of Vote Unitarity, maybe SSS but select the winner based on most-supporting Hare Quota?), and in between, we get Thiele (not exactly one vote per winner per voter, but also not exactly one vote total per voter), does this mean we can do something similar with algorithmic Asset to come up with a “Condorcet-Thiele” PR method?
And also, are there other standards of proportionality that could apply to algorithmic Asset and/or Condorcet?
Edit: Most of this can be found at https://civs.cs.cornell.edu/proportional.html for Condorcet PR; I think the connections to “algorithmic Asset” are pretty strong.
Since, in RRV, a ballot giving a candidate maximal support loses half of its weight, does that mean that in a sequential algorithmic Asset election (where multiple rounds of negotiations occur and each round, the candidate with the most votes is elected; this would be “Bloc Asset Voting” where a majority wins every seat if no reweighting), Thiele can be applied by reducing the weight of the votes supporting the winner in each round by half?