Take a look at this MinnPost article by Joel Kramer and the comment on it by Paul Birnberg. The subject matter is Minneapolis’s Ranked Choice Voting tabulation process. Kramer offers a proof sketch that a more efficient process would necessarily give the same results. Birnberg offers a proof sketch that the results of Kramer’s process could be different. Clearly there is a fallacy in one argument or the other. Which do you find more convincing? Why? If I were teaching our proofs course, I’d offer extra credit points, but I’m not, so you’ll just have to puzzle through this for the sheer joy of it.
See this video to see how this process happens in a fictional animal universe: http://youtu.be/3Y3jE3B8HsE
The fact that this very effective propaganda video shows losing candidates being eliminated one by one helps explain one aspect of the “faulty logic” debate. Namely, the standard of correctness for a simultaneous-elimination algorithm needs to be equivalence with one-at-a-time elimination. If two simultaneous-elimination algorithms each on their own merits seem to make sense, but can produce different results, then we need to adopt the algorithm that matches the results of the one-at-a-time algorithm, because those are the results that the voters are being told to expect. That seems to be what the Minneapolis City Council did in adopting their ordinance. Kramer seems to be inappropriately swayed by the name that was chosen for the elimination standard. When reading a law, as when reading a mathematical proof, the name of a defined term should be completely ignored. The law means the same thing as if it said “all candidates satisfying the Frobboz Criterion must be defeated simultaneously,” accompanied by a definition of the “Frobboz Criterion” in place of the current definition of “mathematically impossible to be elected.”