Proofs of Arrow's Theorem I read Philip J. Reny's paper (Arrow’s Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach)
What I cannot understand is step 5 of the proof of arrow's theorem.
I think figure 4 is a special case because the position of a,b,c are fixed in 1,...,N except n. So if everybody randomly chooses a sequence of preference without any fixed alternatives. The social ranking will still the same with Ln ? 
In figure 4, If Ln's ranking is a,b,c....., will the social ranking be a,b,c...? Why?
Another proof in Noam Nisan's book (Algorithmic game theory)
, P234.
I cannot understand the proof of pairwise neutrality. I think it is almost the same with IIA. Is there anything different between them?
 A: Yes, Figure 4 is a special case, which is used to prove the result in full generality in step 5. I believe the answer to your first question is Yes, $n$ is the dictator, so her preference relation coincides with the social welfare function. This is precisely what step 5 proves. For the same reason the answer to your second question is again Yes, but only using the full force of the theorem; I don't think that we could have made that conclusion ("if $L_n$'s ranking is $a \succ b \succ c \succ \cdots$ then the social ranking is $a \succ b \succ c$") at that stage of the proof. Note that $c$ is just an auxiliary alternative, and what we are really interested is the position of $a$ relative to $b$.
The definition of unanimity on Nisam is not the usual one. Normally unanimity (or Pareto efficiency) is defined as: if every agent ranks $a$ above $b$, then so must the social welfare function. Nisam's apparently weaker definition (he requires that the full preferences coincide, not just $a \succ_i b$ for all $i$) is equivalent to this because of IIA (do you see this?).
In any case, yes, pairwise neutrality is almost the same as IIA, the difference being that in IIA we compare $a$ with $b$ in both $\succ$ and $\succ'$, while pairwise neutrality considers a possibly different pair $(c,d)$ for $\succ'$. The claim basically proves that unanimity (as defined by Nisam) plus IIA imply pairwise neutrality. Note that IIA is indeed used, even though only unanimity is mentioned.
