Russ Paielli wrote:
Here's what I modeled. I have three candidates only. I randomly generate votes, with equal probabilities for all six possible preference orders. The only control variable for each vote is where the voter "draws the line." In this case, that amounts to whether or not the voter approves the middle candidate of his preference list. I initialized the middle-candidate state of each vote randomly, with an expected mean of half approved and half not.
Then I started an iterative simulation of polling cycles and voter re-evaluation of his vote. I simply assumed that complete and perfect polling data is available to every voter. Then I have each voter re-evaluate his approval/disapproval of his middle candidate based on Forest Simmons elegant strategy rule (special case for three candidates only): if the voters first choice has more votes than his third (last) choice, the middle candidate does not get approved, but if the third choice has more votes than the first choice, the middle candidate gets approved (if they are equal I leave it unchanged).
You're simulating a DSV (Declared-Strategy Voting) election with Approval. My current research is on just that topic, though I'm also interested in using DSV with other point-count systems such as plurality, Borda and several others. That Approval strategy is identical to strategy A in the 3-candidate case.
Interesting. Do you mind if I ask why you are interested in Declared-Strategy Voting as opposed to Undeclared-Strategy Voting?
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The first few runs I tried showed rapid convergence within a cycle or two. Then I wrapped the whole thing in another loop to simulate many separate elections. I found that most of them converged within 2 or 3 iterations. However, roughly 1 in 10 fails to converge either to a stable vote count or a stable winner.
1 in 10 agrees with Merrill's figure: 91.6% of random elections with 3 candidates and 25 voters have a Condorcet winner. You used more voters, but that would decrease the percentage only very slightly. Actually, Approval DSV in batch mode using strategy A doesn't always converge even when there's a Condorcet winner, but the examples are quite contrived and require more than 3 candidates. Ballot-by-ballot mode, when the voter order is weakly fair (no voter is shut out for more than 2n steps, say), always finds an equilibrium eventually in my simulations. There's always a path of changes that leads to an equilibrium, anyway. When no Condorcet winner exists, strategy A can't lead to an equilibrium because any poll leader can and will be toppled.
Does another strategy converge even if no Condorcet winner exists?
So the bottom line is that, even in the simplest, most idealized case, Approval Voting can be unstable. In such cases, the ultimate winner would essentially be a random function of when the election happened to be held. A sort of random lottery. And many voters would regret their decision.
Any voting system for which you can't say the same (like plurality) is easily manipulated and leads to multiple equilibria, some of which may not elect an existing Condorcet winner. If you find
I assume you mean that plurality can be manipulated by throwing in spoilers (e.g., Nader or Perot).
And as for multiple equilibria, it seems to me that all but one of those equilibria is practically inaccessable if it requires a third party to switch places with one of the two dominant parties.
convergence more important than competitive elections and sincere voting, you may prefer plurality to Approval. But I see plurality's many equilibria as false ones that hide much about the electorate's wishes. Approval only fails to converge when the electorate's wishes are collectively irrational, in a sense, and in that case Approval will eventually cycle only among the sincere Schwartz set.
Note that all Condorcet-compliant ranked-ballot voting systems are sometimes manipulable and nonconvergent when there's no Condorcet winner. Some prefer Condorcet methods to Approval because they see them as harder to manipulate and thus more stable, but I'd rather voters know the rules of the game they're playing. Alex Small wrote on the ApprovalVoting list:
Legitimacy should come from a transparent connection between the decisions people make in the voting booth and the final outcome. If it takes a game theorist to sketch out a flow chart and explain why voting for A allowed B to win, how much respect will the system command?
That's actually one reason why I like Approval Voting: Although there are sometimes risky decisions to be made (do I approve my second choice or only my first? Do I risk my least favorite winning or risk hurting my favorite?), at least the cause and effect is clear. We won't need a game theorist with a flow chart to explain things to us the next morning.
I second that. Besides, Approval can make a sincerity guarantee that no ranked-ballot system can: You should always vote the maximum for your favorite candidate and the minimum for your least favorite. If all you're given is poll information, you should never vote for B and not for A when you prefer A to B; it never pays to express a false pairwise preference. I still haven't found another system that has that property of weak sincerity.
Anyway, the point is that I think Approval has the best combination of manipulation-resistance, convergence and quality of winners, not to mention simplicity. A little divergence is worth the better equilibria.
That all seems reasonable to me, but let me outline my evolving view, and you can let me know if you think I am on the right track.
You seem to have confirmed my hypothesis that, in the idealized case (DSV batch mode), Approval voting almost always converges on the Cordorcet winner if one exists, but rarely (never?) converges if one does not exist.
If that is true, then it seems to me that Approval may be roughly equivalent to Condorcet with random selection of the winner from the Smith set. Do you agree with that? If so, has anyone shown that the Condorcet winner based on a "good" Condorcet resolution method would at least be favored in the random selection process?
That all applies to the idealized case, of course. Once you start adding uncertainty and other "real-world" effects, things could change dramatically.
--Russ
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