Algorithmic Asset Voting

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.

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).

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

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. When there is a Condorcet cycle of negotiating outcomes in this algorithm that would give a voter incentive for Favorite Betrayal in most Condorcet methods, it is sometimes possible to prevent that in Algorithmic Asset if a cycle resolution method is applied and the algorithmic negotiators are then allowed to optimize their preferences for candidates in the cycle in response to the cycle resolution method's chosen winner. As an example:

|2|A>B>C| |3|C>A>B| |4|C=B>A| |2|A>B>C|

All 3 candidates are in a Condorcet cycle. Schulze picks C, so that would be the default outcome if no negotiation occurs. Based off of this, the algorithm can flip the 4 A>B>C voters to B>A>C to help resolve the cycle and elect B (since B would then be the only member of the Smith Set, B can't be overtaken by anyone else), because this change in expressed preference benefits these voters' actual preferences. Then, not enough C voters would have an incentive to negotiate to elect someone other than B. It may be possible for some cycles to only be resolvable when certain cycle resolution methods are used as default methods in Algorithmic Asset and not others.     

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 (starting at the sentence "Suppose we have a multimember constituency..." on Page 4 and ending at the sentence "In general, however, an operational answer to the problem is again not feasible." on Page 5), 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.