Is there any relationship between commutative algebra and game theory? For example, have any tools in commutative algebra been applied to game theory?

A text or reference would be ideal, but I'd be grateful for any insightful comments.

  • 4
    $\begingroup$ I think there is a relation between everything. $\endgroup$
    – ploosu2
    Oct 16 '14 at 11:58
  • 1
    $\begingroup$ Okay, all the rules say we should close this question. It's vague, brief, and hilariously formatted. But I really want to know the answer, so maybe it's possible to coax this into something deeper. $\endgroup$
    – Slade
    Oct 16 '14 at 12:04
  • $\begingroup$ I took the liberty of making it way, way more polite. I don't know if the question is specifically asking for applications of commutative algebra to game theory, game-theoretic interpretations of theorems in commutative algebra, or something else. e-r, if you could elaborate a bit, that would be great. $\endgroup$
    – Slade
    Oct 16 '14 at 12:08
  • $\begingroup$ I've added an example in response to the "too broad" vote to close. I apologize if my interpretation of the question deviates from its intention, but I'm determined to keep this alive long enough to get a decent answer. $\endgroup$
    – Slade
    Oct 16 '14 at 12:49

I think the answer is yes and no. In finite games (finite players, finite strategies), you are mostly working with mixed strategies and your utility functions are polynomial functions in your strategies. But the equilibrium condition is really a stability condition which you can express with a bunch of polynomial inequalities. This and the fact that probabilities are positive makes the set of feasible equilibria a semi-algebraic set. This is explained in more details here:

http://www.jstor.org/stable/2951732 (The Algebraic Geometry of Perfect and Sequential Equilibrium by Lawrence E. Blume and William R. Zame)

So in this sense the answer is no, but in practice what people do is that they relax the inequalities (in a good way) and then reduce the problem to finding solutions to polynomial equations. Then they go back and check if the solutions satisfy the inequalities. This is done here:

http://link.springer.com/article/10.1007/s00199-009-0447-z (Finding all Nash equilibria of a finite game using polynomial algebra by Ruchira S. Datta)

And looking at the suggested links, I learned that you can even make a game over a commutative algebra: The Ring Game on $K[x,y,z]$

there is also a link to this paper, called algebraic games: http://arxiv.org/pdf/1205.2884v2.pdf (Algebraic games by Martin Brandenburg)


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