Arrow's Approach

Arrow assumes that any social ranking must follow certain rules which are seemingly unobjectionable for a social performance order to be acceptable.

Arrow Axioms

1. Completeness: It must completely rank all the social states. i.e., if A and B are two social states then $A{P}^{>}B$ or $B{P}^{>}A$ or $A{I}^{}B$
2. Transitivity: The ranking must b transitive. If $A,B,C$ are three states then If $A{P}^{>}B$ and $B{P}^{>}C$ then $A{P}^{>}C$ If $A{P}^{>}B$ and $B{I}^{}C$ then $A{P}^{>}C$
3. Unanimity: The ranking must be related to individual preferences. i.e. $A{P}^{>}B$ unanimously, the society must also say $A{P}^{>}B$
4. Independence: If new social state becomes feasible then this fact should not afffect the ranking of original state. E.g. if $A{P}^{>}B$, suppose a new state $D$ is added then it must still remain true that $A{P}^{>}B$. In this axiom there appears to be a controversy, e.g. in this context of an election this means that of 2 candidates, X is elected in preference to y. This should remain the same even if some new candidate contested.
5. Non-imposition: The social preference relation shouldn’t be imposed by the way of a custom¹.
6. Non-dictatorial: The relationship should be non-dictatorial. The individual preferences should not dictate the societal preferences by power/force.

According to Arrow, there will be welfare optimization in the society if all the 6 axioms are satisfied but it can be showed that there are some internal contradictions in the axioms. So that, we cannot think of preference order of society which can satisfy all these axioms.

Suppose there are

Footnotes

1. Suppose a tradition is going on in the society ↩