You wrote:
(I) 40% chance: A: 101 B: 102 C: 104
(II) 60% chance: A: 100 B: 104 C: 105
Now, if Voter X's true preference is A > B > C, the most effective ranking in this particular case is:
A: 5 B: 3 C: 1
In case (I), this means his preferred A can win, whereas in case (II) it at least his second-choice B will beat the despised C.
Of course this is a highly unusual level of information, and probably can only happen if the other voters are acting non-strategically,. However, the very fact such a counterexample exists means that the most effective CR voting strategy not always precisely identical to Approval under all circumstances, which is why I think it is better to use less confusing terminology.
I reply:
Actually it probably isn't confusing to give advice that doesn't take into account information that's for all practical purposes impossible, even in a small committee, let alone a public election.
Remember that no one has been claiming that Approval & CR are strategically equivalent in small committees. Only in public elections--many voters.
For one thing, as you yourself said, the information that you specified is not the kind that will be available. It's nearest possiblity of being available would be in a small committee where you knew all of the few voters, and how they were going to randomize their voting. But even then it isn't quite clear how your vote total probabilities would become available, even under those conditions. In a public election, forget it.
Also, in a public election, a 3-way neartie is so vanishingly more unliikely than a 2-way neartie, that 3-way nearties should be ignored, for public elections. In fact, due to the uncertainties in your utility and probabililty estimates, it's reasonable to ignore them in committees too, where 3-way ties & nearties are still significantly less likely than 2-way ties & nearties.
When advising people on Approval & CR strategy for public elections, it isn't confusing to not bring up strategy with information of a kind that they'll never have in a public election, or even in a committee election.
Mike Ossipoff
And I repeat that no one has been claiming that CR & Approval are strategically equivalent in small committees.
-- Ernie P.
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