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Nice properties of the set of Nash equilibria can be shown for supermodular games(see here). But for a two-player game in which

  1. Strategy spaces for two player, $S_1$ and $S_2$ are finite, and both of them are subsets of $\mathbb R$.
  2. $u_1$ exhibits increasing differences while $u_2$ exhibits decreasing differences.

What can we say about it?

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There is a recent paper which may help you: Games with strategic complements and substitutes (Monaco and Sabarwal (2015))

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