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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?

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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).

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    $\begingroup$ Well, Cournot competition with two firms. $\endgroup$ Sep 29 at 20:45
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    $\begingroup$ @Michael: As shown by Novshek (1985), Cournot equilibrium actually exists for an arbitrary number of firms, with (almost) arbitrary (nondecreasing) cost function provided the game is submodular. See Amir's paper for details. $\endgroup$
    – Bertrand
    Sep 30 at 5:26
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    $\begingroup$ Sorry, what I meant was that standard method based on order theoretic fixed-points work only for the case of two firms, Novshek came up with a proof for a very specific model. It is not what I would call an existence proof for submodular games, as asked for in the question. Amir even explains why there is no such result in general. $\endgroup$ Sep 30 at 6:40

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