One of the good things about Mathematica is that old code still works. There's a package written in 1991 by John Dickaut that still works in Version 11 to find (all) Nash equilibria in 2 x 2 games. I've attached a copy to this post.
Here's an illustration of the package in use finding all the Nash equilbria of a 5 x 5 2-player game. The game is expressed in strategic (normal) form as a 5x5x2 matrix.
game={{{10, 14}, {0, 12}, {12, 7}, {16, 10}, {11, 11}}, {{1, 7}, {14,
19}, {5, 19}, {13, 15}, {13, 12}}, {{10, 18}, {7, 3}, {17,
16}, {18, 20}, {9, 6}}, {{6, 3}, {20, 19}, {20, 13}, {17, 15}, {13,
4}}, {{18, 12}, {10, 10}, {20, 12}, {10, 11}, {6, 8}}};
Quiet[Nash[game]]
Now, I am confident there have been a few developments in efficiently finding Nash Equilibria, particularly in larger or n-player games since 1991, but if you wanted to get started, you do not have to reinvent the wheel. Also, if your skills extend to RLink, there is an R package GNE that you can access via Mathematica that appears to be a lot more modern.
Here's the link to the Nash package: Microeconomic Analysis
There's a book edited by Hal Varian Economic and Financial Modeling with Mathematica that discusses the package.
Also FWIW, I did some work about 15 years ago in "Foggy Game Theory" in which the assumption that a player can always "see" all of its moves is relaxed and the visible strategies available to a player depend on the current state of the game.
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