Remember that Joe W. once suggested approving as far down your preference order as you can without exceeding 50 percent probability of the winner coming from your approved set.
Whether or not you should approve the next candidate (the one that would tip the scales to more than fifty percent) depends on how far it would tip the scales, how much you like that candidate, etc. I would like to hear some ideas for rules of thumb to help decide whether or not to approve this borderline candidate. Joe's idea is sound from an information theoretic point of view. The expected information from a Bernoulli (i.e. binary) random variable is greatest when the two values are equally likely. For this reason, to get the most information about a person from a True/False test or questionnaire, you should ask questions that (a priori) have a fifty/fifty chance of being answered True. If you ask questions for which you already know the likely answers, then in all likelihood you won't learn much about your respondent. As a math instructor I like to gear my tests so that the average student has a fifty/fifty chance of correctly solving a typical problem. This is the best kind of test for separating the sheep from the goats, but students don't like them, so I usually put in several easy problems near the front for psychological purposes. Back to Joe's idea: It seems to me that the idea of his strategy is to maximize the probability that your ballot will be positively pivotal in determining the winner. In other words, this strategy actualizes your potential voting power. In the case of many viable candidates, no single candidate has a large probability of winning, so each voter can approach the fifty/fifty optimum, and each voter that makes use of this strategy is equally likely to be positively pivotal in the outcome. I say "positively" pivotal, because if one voter is pivotal, then all are pivotal, but not necessarily in the direction that they would desire. Take plurality, for example. Suppose that A beats B by one vote, but you voted for C. Your vote (or lack of vote) was pivotal in preventing a tie between A and B. But it could not be considered positively pivotal if you actually preferred B to A. In Approval suppose that A beats B by one vote and that you approved neither or both, then you missed your chance of positively affecting the outcome. Joe's strategy minimizes the probability that you will end up with this kind of regret. With the above idea of minimizing this kind of regret in mind, I would like to make the following rule of thumb suggestion: Include or exclude the borderline candidate according to whether or not it makes the approved/unapproved winning odds closer to or further from fifty/fifty. Suppose for example, that p1, p2, and p2 are the winning probabilities for candidates C1, C2, and C3, and that your preference order is C1>C2>C3. When should you approve C2 ? If you want to maximize the probability that your vote is positively pivotal, then approve C2 only if (p1+p2) is closer to 50% than p1 is. If both are equally close to 50% then approve C2 only if C2 has above average utility for you. For example, suppose that p1=.3 and p2=.4, then (p1+p2)=.7 which is the same distance (.2) from .5 as p1 is. If candidate C2's utility is more than fifty percent of the max, then approve C2. In practice it is difficult to know the probabilities p1, p2, p3, etc. reliably. But what I have in mind is more along the line of Lorrie Cranor's Declared Strategy Voting (DSV). Remember the DSV CRAB race? CRAB stands for Cumulative Repeated Approval Balloting. You submit your CR ballot, check the box for your favorite CRAB strategy, and the DSV machine implements your CRAB strategy for you. Joe's strategy could be the default strategy, for example. Where would the p1, p2, p3, etc. values come from? Well during the typical CRAB race, various candidates cycle through first place. The percentage of time spent in first place (so far) is an estimate of that candidate's probability of winning. Or (alternatively) the p values (having started equal) adjust (according to Bayesian principles, for example) as the race advances. That's the general idea. There's lots of room for imagination in the details. Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
