On the equivalence of the HEX game theorem and the Duggan–Schwartz theorem for strategy-proof social choice correspondences

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 for a 6 × 6 HEX game is equivalent to the Duggan–Schwartz theorem for strategy-proof social choice correspondences [J. Duggan, T. Schwartz, Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized, Social Choice and Welfare 17 (2000) 85–93] that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator.