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Tyrrell_McAllister comments on Arrow's Theorem is a Lie - Less Wrong
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Your system fails the first two criteria:
Dictatorship: (ETA: My point here was wrong. I missed that the voters could only select numbers from 0 to 10.) Suppose that one of the voters can write numbers much, much larger than any number that that the other voters can write. (Maybe this voter can write faster and knows clever notations for writing very large numbers.) Then this voter can be dictator by writing a number next to its choice that exceeds the sums of the largest numbers that the other voters can write.
Determinism: The same preference orders, but expressed with different numbers, can lead to different outcomes. (Examples available upon request.)
For the first: You can only write numbers up to 10.
For the second: Yes, that's the point, and that's not what determinism means; determinism just means that there is no randomness involved. Relative distances between preferences matter! Suppose the vote was between option A, you get a thousand dollars, option B, you get nothing, and option C, your country gets destroyed in nuclear war. You would want a way to express that you dislike C much, much more than you like A.
I had noticed that and edited my post, but too late for you to see, I guess. Sorry for the hasty conclusion.
No, "determinism" in Arrow's theorem means that the preference orders assigned by the algorithm cannot change unless the preference orders of the voters change. Randomness is one way that an algorithm could be nondeterministic, but it isn't the only way. Your system gives another way to be nondeterministic in the sense of Arrow's Theorem.
If you want to use the word "determinism" in that sense, then a far better definition would be "the voting outcome is not affected by anything other than the votes of the voters", which my system does hold to. As I said above, I haven't claimed to found a flaw in the mathematical proof of Arrow's Theorem, just a mismatch between the content of the theorem and how voting systems work in real life. Certainly, in real life, we should want to distinguish between "a vote between the Democrats, the Greens, and the Republicans", and "a vote between the Democrats, the Greens, and superintelligent UFAI", even if our preference order is the same in both cases.
Fair enough, but it's not a matter of how I want to use the word "determinism". That word in Arrow's Theorem has a certain technical meaning, and your system does not qualify as deterministic under that technical meaning.
You're making the case that determinism, in Arrow's sense, is not such the desideratum that it's usually made out to be. FWIW, I'm coming around to your point there.
How do you break numerical ties without randomness?
You can just choose the first one in alphabetical order, or some equivalent.
I volunteer to choose the language and wording of the preferences.
Okay, then, we all agree before the vote to break ties based on some arbitrary binary digit of pi, which we don't look at until after the vote is done.
Or, equivalently, we can agree to break ties based on the outcome of the rand() function seeded with the time in milliseconds at the time of agreement.
Neither of those fulfills the relevant functional definition of determinism. The same inputs either do or do not lead to the same outputs; if they do, then it is deterministic and subject to Arrow, and if they do not, they they are functionally random.
Exactly. Both do or both don't. As (I assume) you were implying with your earlier question, ties in this system are broken either arbitrarily (limited dictator situation) or functionally randomly. Which of these categories 'alphabetical' and 'PI picking' could vary by assumption but one of the two is implied.
Yes.
The suggested methods for tiebreaking are therefore not helpful, because the voters have to know the voting function beforehand, without even epistemic randomness. I was just noting that this system is extremely sensitive to tactical voting on the margin.