On Smithson's fixed point theorem for order preserving multifunctions

Fixed point theorems are ubiquitous in economic research. Many studies cite Smithson (1971) ``Fixed points of order preserving multifunctions,‘’ yet the original proof contains errors. This note presents a new, concise proof and explains why Smithson’s argument is invalid. It also contains new results on the structure of the set of fixed points and monotone comparative statics.