Weak rationalizability and Arrovian impossibility theorems for responsive social choice

This paper provides representation theorems for choice functions satisfying weak rationality conditions: a choice function satisfies $$\alpha$$ α if and only if it can be expressed as the union of intersections of maximal sets of a fixed collection of acyclic relations, and a choice function satisfies $$\gamma$$ γ if and only if it consists of the maximal elements of a relation that can depend on the feasible set in a particular, well-behaved way. Other rationality conditions are investigated, and these results are applied to deduce impossibility theorems for social choice functions satisfying weak rationality conditions along with positive responsiveness conditions. For example, under standard assumptions on the set of alternatives and domain of preferences, if a social choice function satisfies Pareto optimality, independence of irrelevant alternatives, a positive responsiveness condition for revealed social preferences, and a new rationality condition $$\delta ^{*}$$ δ ∗ (a strengthening of $$\gamma$$ γ ), then some individual must have near dictatorial power.