Some Results for Impossibility Theorems of Social Welfare Function and Social Choice Function

We present some results on social welfare function and social choice function. First, we replace weak Pareto principle of Arrow’s axioms with strong Pareto principle, and then show that assuming unrestricted domain property, independence of irrelevant alternatives and strong Pareto principle on a social welfare function in Arrow’s sense leads to the existence of first to nth commanders whose strict preferences have an influence on social preference along the order of priority of commanders. Second, we define generalized strong positive association for social choice function in Gibbard-Satterthwaite’s sense and show the equivalence of generalized strong association and strategy-proofness. Third, we study the existence and properties of a social welfare function whose individual preference domain is restricted to a subset of the whole set of alternatives. We show that the social welfare function satisfying unrestricted domain property, independence of irrelevant alternatives and weak Pareto principle is always dictatorial when at least one individual is entitled to express his/her preference on the whole set of alternatives. Fourth, we discuss a social choice function whose individual preference domain is restricted in the same way, then show that the social choice function satisfying strategy-proofness is dictatorial whenever there exists an individual whose alternative set contains the range of social choice function. In addition, generalized positive association for restricted individual preference domain, which is equivalent to strategy-proofness, is introduced. Further, we study a social choice function on mutual evaluation as a special case of restricted individual preference domain. Then we show the non-existence of social choice function satisfying strategy-proofness and weaken the condition of strategy-proofness, which leads to the existence of a non-dictatorial social choice function satisfying strategy-proofness.