A Nash equilibrium exists for supermodular games if single crossing property is satisfied. Additionally, if f is submodular then -f is supermodular. Is there any way in which we can link these to find to prove existence of Nash equilibrium in submodular games? In texts that I have read, they basically have dealt with Nash existence in submodular games with two agents. Can someone please help me with references which deal existence of Nash in submodular games with multiple agents?
Your question is for instance addressed by:
Amir, Rabah, 2005, "Supermodularity and Complementarity in Economics: An Elementary Survey," Southern Economic Journal, 71, 636-660.
Cournot competition is an example of a submodular game for which an equilibrium exists (under some rather mild conditions though).