Proof of Arrow's Theorem:

The proof of the theorem (which is the one given by Sen (1970) is described below for three persons X, Y and Z and three alternative choices A, B and C.

1) Let X: A > B > C,
     Y: B > A,
     Z:  B > C.

To fix the comparison in the group choice between (A, B) there is difficulty since for individuals the comparison between (A, B) differs for revealing their preference.

This leads to indecisiveness in the group choice.

Suppose somehow in the group choice A > B appears. Then X is termed as decisive for fixing this pair in the group choice.

∴X is Decisive: A > B

as such B > C  (Paretian condition)

It implies that A >C  (Transitivity relation) 

2) X: B > A > C
    Y: C > A
    Z: B > C
 ∴ X is Decisive A > C
 B > A (Paretian condition)

It implies that B > C (Transitivity)

3) X: B> C >A
    Y: C >B
    Z: C > A
  ∴ X is Decisive B > C
  C > A (Paretian condition)

 It implies that B > A (Transitivity)

4) A: C > B > A
    Y: A > B
    Z: C > B
  ∴ X is Decisive B > A
C > B (Paretian condition)

It implies that C > A (Transitivity)

5)  X: C > A > B
     Y: A > C
     Z: A > B
  ∴ X is Decisive C > A

A > B (Paretian condition)

 It implies that C > B (Transitivity)

6) X: A >C  > B
    Y: B > C
    Z: A > C
  ∴ X is Decisive C > B
 A > C (Paretian condition)

It implies that A > B (Transitivity)

The above formulation proves that if individual X is decisive on any pair it follows that he is also decisive on all other pairs based on  unrestricted domain. In other words X becomes the dictator.  It violates the non-dictatorship axiom. It follows that when one  individual is decisive on any single pair he maintains his decisiveness on all other pairs making him a dictator. At this stage we have to prove that for same pair there exists an almost decisive pair containing just one individual

Suppose X: A > B > C
            Y: B > C > A
            Z: C > A > B

In this example there cannot exist a group choice following the majority rule based on pair wise comparison and transitivity relation.
Majority supports A > B

and B > C  => A > C

but according to majority C > A.  It is a contradiction.

So if A > B > C appears in the group choice X is decisive on A > C.

Similarly if B > C > A appears in the group choice Y is decisive on B > A. For C > A > B appearing as the group choice. Z is decisive on C > B.

It proves that at least an individual is decisive on any single pair and therefore for all pairs and becomes a dictator. This proves that all the axioms of rational collective choice are not fulfilled hence there is impossibility in deriving the rational collective choice, which  is transitive and follows all axioms.

Prof. A.K.Sen has diluted the concept of dictatorship as considered by Arrow and has taken a liberal approach. According to him an  individual is a dictator as far as his personal choice   is concerned. He has published a paper entitled  "Impossibility of Paretian Liberal"  in the Journal of Political Economy in 1970. In this paper he concludes that collective choice can be derived which is rationale provided that individual do not interfere in each other's personal affairs or choices. Otherwise liberal approach to dictatorship contradicts Paretian condition giving rise to Sen's paradox and the derivation of rationale collective choice becomes impossible.