Equivalence of the HEX game theorem and the Arrow impossibility theorem

摘要

Gale [D. Gale, The game of HEX and the Brouwer fixed-point theorem, American Mathematical Monthly 86 (1979) 818–827] has shown that the so called HEX game theorem that any HEX game has one winner is equivalent to the Brouwer fixed point theorem. In this paper we will show that under some assumptions about marking rules of HEX games, the HEX game theorem is equivalent to the Arrow impossibility theorem of social choice theory that there exists no binary social choice rule which satisfies transitivity, Pareto principle, independence of irrelevant alternatives and has no dictator. We assume that individual preferences over alternatives are strong (or linear) orders, that is, the individuals are not indifferent about any pair of alternatives.