Richard said:
However, there is a difference between extrapolating Approval voting Nash equilibria when all voted preferences are known (and assuming voters decide in blocks) I reply: Sometimes a number of same-voting voters change their strategy in the same way, or in a few same ways. Writing definitions that way, then, doesn't seem unrealistic or unreasonable. After all, when only one voter changes his strategy, there's rarely any result. As for all voted preferences being known, the equilibria that we've been discussing are useful no matter what we assume voters know. If a strategy-configuration & outcome can be improved on by some voters, or if it can't be, that's relevant in itself. Of course often voters would notice that they could improve on the most recent outcome, by a different strategy, and would likely use the different strategy next time. As you said, a group of same-voting, same-utilities voters can sometimes benefit by dividing their vote between 2 or more strategies. But that could be considered just one strategy--one probabilistic strategy that they all use. "Flip a coin; if it's heads, use strategy1, and if it's tails use strategy2." Of course it's been pointed out that it won't be so easy to get people to vote based on flipping a coin. Richard continued: , and counting actual Approval votes cast by individual voters with their assorted fine-grained utilities and probability estimates. I reply: It would be interesting to consider a game in which the voters have probability-info about eachother's preferences, or maybe about eachother's votes, or ties & near-ties. Maybe someone here is familiar with game theory enough to study Approval, IRV, wv, Margins, etc. in that way. Richard continues: So real-world Approval voting likely won't always converge to a stable Nash equilibrium. I reply: No doubt it's premature to say for sure what would happen. But the fact that Approval & wv always have equilibria in which the CW wins, and no one order-reverses, and the fact that Margins often has only equilibria in which defensive order-reversal is used-- that says something about the methods. One thing for sure is, a strategy-configuration and outcome that someone can improve on is unstable, compared to an equilibrium. On the next topic, if voters know what they're doing, the'll elect the candidate at the voter-median, if there is a candidate there. Richard continues; Now on to voting strategy. The following Approval strategies have some valid mathematical reasoning behind them. Let u(i) be your utility for candidate i. 1. delta-p strategy: If delta-p(i,j) is the increase (positive or negative) in the probability of candidate i winning caused by your vote approving candidate j, then the strategic value of approving candidate j is s(j) = sum_over_i( delta-p(i,j)*u(i) ) Approve candidate j if s(j) > 0; disapprove if s(j) < 0; if s(j) = 0 then it doesn't matter. This is the only exact optimum strategy formula I know of. The others are approximations. Of course, the delta-p values are hard to know, which makes an approximate strategy desirable. I reply: Yes, the delta-p are difficult to know, and if they're difficult enough to know, that would make that method the less useful one. You say, as you said when this was discussed before, that yours is the only exact optimum strategy, because the others are approximations. But when I asked you how you'd determine the delta-p values, you either derived them from the Pij, or assumed that all ties are 2-way ties. Your way of determining your delta-p values relied on the same approximations as the methods that you say are more approximate. This became evident when we began discussing the strategy difficulties when there are few voters. I thought that then you knew that your method suffers from the same problems (unless you ask an oracle what the delts-p values are--while you're at it, ask him what the best candidate is :-) ). This is illustrated by the fact that you yourself agreed that none of us have a few-voters strategy even for the relatively simple 0-info case. You described several rough approximations for 0-info strategy. 0-info allows us to regard many terms as equal, and that's a simplification. If you don't have a precise 0-info strategy, you don't have a precise probability-info strategy either. And you said that you don't have a precise 0-info strategy when voters are few. But if you're sure that more useful results can be gotten by your strategy than by the Pij strategy, then I hope that you'll notify professors Weber & Merrill. Richard continues: 2. p(i,j) strategy: If p(i,j) is the probability, given a two-way first-place tie, that it will be between candidates i and j, then the utility of approving j is s(j) = sum_over_i( p(i,j)*(u(i)-u(j)) ) In http://groups.yahoo.com/group/election-methods-list/message/6706, I showed this to be almost equivalent to the delta-p strategy. The source of error (I think) is that this strategy only considers two-way ties. For a large electorate, that error term will be exceedingly small. Again, we have some values -- p(i,j) in this case -- that are hard to know. I reply Again, saying that the delta-p approach is more error-free depends on telling us how you can get your delta-p values without approximations, or in ways that are less approximate than the Pij method's assumptions and its Pij estimates. Richard continues: 3. geometric mean strategy: This is the approximation discussed by Mike at http://www.barnsdle.demon.co.uk/vote/strat.html, wherein the p(i,j) values are each replaced with sqrt( p(i)*p(j) ). So this method is an approximation of an approximation. I reply: In public elections the assumption that ties are 2-way is much more reliable than any of our probability assumptions, or even our candidate-ratings guesses. In few-voter elections, your way of determining your delta-p values used the same approximating assumptions that the Pij approach uses. By the way, would you re-post your suggestion for estimating the delta-p values? As for Tideman's geometric mean suggestion for estimating the Pij, of course it's an approximation. So is whatever way you'd determine your delte-p values. Approximations will be needed for any of these methods. But geometric mean isn't really a separate strategy method. It's a way of estimating the Pij for the Pij method. Tideman's Pij estimate for Weber's strategy method. Richard continues: 4. above-expected-utility strategy: Forest Simmons, Joe Weinstein, and I have all mentioned this method in the last couple of months. The expected utility of the election is EU = sum_over_i( p(i)*u(i) ) and we vote for candidate j if u(j) > EU. This strategy can easily translate into an intuitive strategy. In just about every election, we all have some expectation going into the election about how much we will like the outcome. I reply: That approach seems convincing, but, unless I made an error (I didn't recheck all the calculation), it gives an Approval cutoff-utility that doesn't look anything like the one that can be derived from the Weber-Tideman approach, using the same winning probabilities and candidate utilities. I'm not claiming that one of those 2 procedures is better, only that they seem to disagree. Does the expected utility approach have the obvious motivating principle that the Pij has? Just a question, not a judgement. Richard continues: Like strategy #3, strategy #4 has underlying assumptions about the distribution of ballots. Those assumptions can sometimes lead to problems. Here is a (very contrived) example: Suppose, in a four-way election, your utilites are A(100), B(60), C(40), D(0). Strategies 3 and 4 would both fail in the following scenario. x1: AB x2: CD x3: A x4: B x5: C x6: D I reply: What do x1, x2, etc. stand for? Which strategy method is that? Richard continues: Richard continues: This of course goes against the conventional wisdom, that voters should never reverse a preference in Approval. I reply: It's known that combinations of Pij and Ui can be written for which the voter's utility-expectation-maximizing strategy involves "skipping", not voting for a candidate though you vote for someone whom you like less. But of course that never means not voting for one's favorite. And I believe it was Merrill who quoted another author as pointing out that situations that cause skipping are so implausible and unlikely that the possibility can be ignored for practical purposes. Bart, do you have that quote? Mike Ossipoff _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp.
