Kevin, I misread what you wrote, but now I see that you were indeed measuring which method maximized expected range value for the voter. However, there may be voters that wish to maximize the probability that their ballot will be positively pivotal, i.e. they might wish to maximize their voting power. For these voters the "above median" approval strategy is better than the "above mean" strategy. Given a free chance, would you rather have a fifty percent chance of winning a million dollars or a one percent chance of winning sixty million dollars? Your answer tends to depend on whether or not you are already a millionaire. Even more to the point, it depends on whether this is a once in a lifetime chance or not. A presidential election is usually once per four years, which might not be frequent enough for some voters to bank on straight expectation. In the once in a lifetime scenario, I would take the fifty percent chance, even though its expected payoff is only $500,000 compared to the $600,000 expectation of the riskier choice. If I knew the chance were going to be repeated every year on my birthday, I would take the higher expectation choice as long as I had sufficient for my needs. For readers who weren't around last time we discussed approval strategies, here's Joe Weinstein's generalization of the "above median" strategy from the zero info case to the near perfect info case: Approve candidate X if and only if the winner is more likely to be someone you don't like as much as X, than someone you like more than X. Here's a refinement that is not quite as robust, but slightly increases your voting power if the probabilities are very reliable: Approve candidate X if and only if X is more likely to be tied for first place with someone that you don't like as much as X, than with someone that you like more than X. Both of these strategies reduce to the "above median" approval strategy in the zero information case, but as I said, Joe's strategy, which always yields approval blocks without holes in them, is more robust. [When there are holes in the approval blocks, it can happen (though with small probability) that your ballot is negatively pivotal, i.e. is pivotal in breaking a tie in favor of the tied candidate that you like less.] The "above mean" strategy has two analogous generalizations, one with and one without holes. The "without holes" method is to take a weighted average of the range values as a probability cutoff, where the weights are simply the winning probabilities for the respective candidates. The "with holes" method is based on probabilities of ties multiplied by differences in utility. If the sum of such products is positive for a candidate, then that candidate is approved. This method is the most sensitive to errors in the utility and probability estimates, so it is the least robust of all. These two methods both reduce to the "above mean" strategy in the zero info case. Forest
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