Arrows Impossibility Theorem:

Arrow has made an attempt to translate individual preferences in terms of collective choice. In order to understand the proposition, we require at least three persons and minimum there outcomes.

Further, Arrow has considered the concept of 'Rationale Collective Choice', which need conform to the following axioms necessarily.

1) Unrestricted Domain: (U) Let X, Y and Z are three individuals and A, B and C, are three outcomes (social states).  The preference  patterns of these three individuals are as follows:

X: A > B >C (> means preferred to)
Y: B > C > A
Z: C > B > A

According to Arrow always there will be a pair wise comparison. In this example for each individual there are three possible pair wise comparisons such as between (A, B), (B, C) and (C, A) and there are three possibilities (>, Preferred to), (<, inferior to), or (i,  indifferent to).  No restriction is imposed on any of the individuals to state the comparison of outcomes in any pair in a specific  manner. Therefore, the nine possibilities, applicable to all individuals, are: 

    A > B,        B > C        A > C
or A < B,     or C >  B     or, C > A
or A  i B      or B  i   C      or,   A i  C

2) Independence of Irrelevant alternatives: (1) In the pair wise comparison of  (A, B) C option is treated as irrelevant.  When there is change in the comparison patterns of (B, C) and (A, C) it does not affect (A, B).

3) Paretian Condition: (P) If for X:A > B and     Y:A i B, then A>B. It implies that A > B is the group choice so that the welfare of X is increased without any harm to Y.

4) Non-Dictatorship: (D) If for X: A > B, Y: B > A and Z: B > A, but in the group choice some how X manages to fix A > B, then X is termed as a dictator.  If it does not occur then there is non- dictatorship.

5) Transitivity: (T) For any individual or group if X > Y and Y > Z implies X > Z then transitivity condition gets fulfilled.

Arrow has proved that it is not possible to fulfill all the above axioms simultaneously. At least one of the axioms get violated.

Therefore it is not possible to derive Rational Collective Choice. This is known as Arrow's Impossibility Theorem. According to Arrow no Social Welfare Function (SWF) exists which satisfies the four conditions, U, P, 1 and D, and can produce a transitive preference  ordering over social states.