A topological proof of Eliaz's unified theorem of social choice theory

Recently Eliaz [Social aggregators, Social Choice and Welfare 22 (2004) 317-330] has presented a unified framework to study (Arrovian) social welfare functions and non-binary social choice functions based on the concept of preference reversal. He showed that social choice rules which satisfy the property of preference reversal and a variant of the Pareto principle are dictatorial. This result includes the Arrow impossibility theorem [Arrow, Social Choice and Individual Values, Second ed., Yale University Press, 1963] and the Gibbard-Satterthwaite theorem [Gibbard, Manipulation of voting schemes: a general result, Econometrica 41 (1973) 587-601; Satterthwaite, Strategy-proofness and Arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory 10 (1975) 187-217] as its special cases. We present a concise proof of his theorem using elementary concepts of algebraic topology such as homomorphisms of homology groups of simplicial complexes induced by simplicial mappings.