Incomplete preferences, complementarity and non-convexity ¤

This paper extends to incomplete preferences the formulation of strategic complementarity in Milgrom and Shannon (1994). It is shown, by example, that quasi-supermodularity and single-crossing no longer su¢ce to guarantee monotone comparative statics with incomplete preferences. However, when incomplete preferences satisfy monotone closure in addition to quasi-supermodularity and single-crossing, both monotone comparative statics and equilibrium obtains in games with incomplete preferences. I show, by example, that in general, the existence result obtained here cannot be deduced the standard existence result for games with complete preferences. The result obtained here weakens the requirement of convexity in the equilibrium existence theorem with incomplete preferences contained in Shafer and Sonnenschein (1975).

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