For finite sets of alternatives a social welfare function in

For finite sets of alternatives, a social welfare function in the sense of Arrow always is a social decision function, but the converse does not hold. Why?

Solution

For the finite set of alternatives, a social welfare function in the sense of Arrow is always a social decision function but a social decision function in the sense of Arrow is not always a social welfare function because it is difficult to aggregate the social decisions of the individuals in the society in a social welfare function because aggregation of the preferences of all the individuals in the society in a single welfare function is impossible because of the Impossibility Theorem of Arrow. Thus, the converse of the above statement is not correct.