This paper extends the formulation of complementarity in Milgrom and Shannon (1994) to the case of incomplete but acyclic preferences. It is shown that the problem can be reformulated as one with complete but intransitive preferences. In this case, quasi-supermodularity and single-crossing on their own do not guarantee either monotone comparative statics or equilibrium existence in pure strategies: an additional condition, monotone closure, is required. The results obtained here relax the requirement of convexity in Shafer and Sonnenschein (1975)s existence result with incomplete preferences. In an application, it is shown that pure strategy equilibria exist in incomplete information games with Knightian uncertainty.
Warwick economics research paper, University of Warwick, 2010