Sunday, September 16, 2012

extending the model: salutary log-rolling

A few weeks ago I wrote about voting for a single winner; I'd like to extend this to more general situations, in which we can amplify the "intensity" property that was obtained in the special case in which there was no Condorcet winner. This is related somewhat to the idea of storable votes, which has worked well in a lab with human subject experimentation.

Let's start with an example, in which we have three misanthropes.  The three misanthropes are going to vote on three binary issues: do we kill misanthrope 1, do we kill misanthrope 2, and do we kill misanthrope 3?  (The killing is not done until all three decisions have been made; misanthrope 1 gets to vote on all three questions, regardless of the answer to the first question.)  Each would prefer to kill each of the others, but would much more strongly prefer not to be killed himself.  By way of usual, issue-by-issue voting, each would be killed; intensity of preference can't be accounted for.  However, by considering this a choice of 8 possible outcomes, we can make it a (no longer binary) single-issue vote; it has no Condorcet winner, as the result "don't kill anyone" beats any result that kills at least two people, each of which beat two of the results in which one person is killed, each of which beats the "don't kill anyone" optimum.

Now, this is actually a bit simpler on some level to analyze in the many-voter limit; let's suppose there are three factions of approximately equal sizes, each voting whether or not to annihilate each faction.  Now, with approval voting, in equilibrium everyone votes against every outcome in which they would die and votes for every outcome in which they survive and at least one of the other factions is annihilated.  If it looks like there is a good chance that they will be annihilated, they will vote for the outcome in which nobody is killed as well; if they believe themselves to be practically safe, they might vote against this outcome.  In equilibrium, the outcome in which nobody is killed will usually win, with "usually" getting more and more probable as the importance to each person of surviving becomes more important to them relative to their own bloodlust.

In situations with many votes, some of which may not be binary, this can get much more complicated.  The ability of the system to take account of intensity improves as the number of votes aggregated increases; we might, for example, want Congress to vote at the end of the session on every bill brought to the floor in the previous two years.  This, however, would be entirely impractical, even supposing none of the votes was on something of particular urgency.  (Say, a declaration of war.)

What might make it less impractical is the fact that the equilibria will be unaffected by the addition of irrelevant alternatives.  In the case of the misanthropes, the equilibrium consists of a large probability of nobody being killed and small, equal probabilities of one person being killed; if we remove some of the other four options from the set of allowed possibilities, the equilibrium stays the same.  In the case of millions or billions of combinations of possible vote outcomes, if there is a Condorcet winner, then if that Condorcet winner could be found, then if that were offered along with any smallish set of other combinations of outcomes to voters to then vote, based on approval voting, for the whole combination, that result would win the vote, regardless of what the alternatives were.  On the other hand, if the Condorcet winner were not included in the set of options, adding it would upset the equilibrium that was obtained from the set that excluded it.  Thus we might hope that allowing some means of offering competing slates of results to "challenge" the current winner of the vote would allow the Condorcet winner to be found; as long as someone is clever enough, at some point along the way, to get that option nominated, it will win, regardless of what other nominees are in place.

It seems then that a good first guess would be to do the vote in the traditional way — have each voter vote issue-by-issue, but use approval voting for natural ternary options. (If there are two competing jobs bills, and nobody thinks they should both happen, combine into a single vote "do jobs bill A, do jobs bill B, or don't do either".) Take the resulting combination of results, and allow the nomination of challenges to it.  If the bill on which I feel the most strongly went against me, and I think that a lot of people would be willing to change that result if we also changed some other set of results on which my opponents feel more strongly and I feel less strongly, it might make sense to nominate that.  (For example, after we all vote to kill everyone, I can propose letting everyone live, which would get everyone's vote against the first equilibrium; I prefer letting myself live very strongly, and others oppose it weakly, so I propose that we change that in the direction of my preference while changing the other results against my preference.)  If more than about 10 or 15 combinations have been proposed, you can drop those that are getting the fewest votes, allowing for new nominations of new potential combinations while keeping the number of combinations under consideration manageable.

In the spirit of this blog, there is probably a lot that I haven't worked out; in particular, for certain common classes or distributions of preferences, it seems likely that there is some way of further tamping down on the informational complexity I'm asking voters to manage.  In particular, perhaps there's a way to solicit from voters their preference intensities on different issues so as to compute likely alternative combinations of results that can be placed into nomination for that revision phase.

No comments: