I'm just started lecture myself about arrow's theorem. There are some problems which make me confused.

ARROW'S THEOREM: Any constitution that respects transitivity, independence of irrelevant alternatives, and unanimity is a dictatorship.

This is a kind of statement from john Geanakoplos' paper.

I think unanimity+transitivity and independence of irrelevant alternatives are equivalent. For example:

voter A: a>b>c>d voter B: a>c>b>d voter C: a>d>c>b

So we can conclude that everybody prefers a to b just according to unanimity and transitivity. And I think this conclusion implies independence of irrelevant alternatives. So if I drop the constraint of independence of irrelevant alternatives, Is this still right?

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    $\begingroup$ If you're right, you're the first person to notice an "obvious" mistake in a widely cited sixty-year-old theorem. Seems unlikely. $\endgroup$ – David Richerby Dec 18 '13 at 8:25
  • $\begingroup$ Could you tell me why this is wrong? $\endgroup$ – Alex Dec 18 '13 at 10:11
  • $\begingroup$ Sorry, no. I don't know about this area. $\endgroup$ – David Richerby Dec 18 '13 at 10:14
  • $\begingroup$ david, do you mean the claim is more an unnoticed redundancy & not so much a "mistake"? if it were a mistake then is there some kind of contradiction? in any case it would be interesting to know of the proof that shows of all 3 features, none implies any others. $\endgroup$ – vzn Dec 18 '13 at 22:33
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    $\begingroup$ @vzn - the proof crucially uses all 3 features. Indeed, the Borda-count does not satisfy one of them, and is not a dictatorship. And other examples can be constructed that violate each of the other conditions. $\endgroup$ – Shaull Dec 19 '13 at 4:16

Unanimity and transitivity do no imply IIA.

For example, consider the Borda-count voting mechanism. There is an example of the violation of IIA here.


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