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Science Forum Index » Math - Symbolic Forum » Nash equilibria and symbolic computation...
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| Robert H. Lewis... |
 Posted: Wed May 14, 2008 2:15 pm |
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Does anyone know how much has been achieved or attempted in computing Nash equilibria symbolically?
I have been reading this paper by Ruchira Datta: Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra.
http://arxiv.org/abs/math.AC/0612462
The point is to solve a system of polynomial equations. In the paper (p. 33) she has the equations for the case of three players with two pure strategies, and solves them with Singular (let's call this the 3-2 case). How about the 4-2 case, or the 3-3 case? Again, I want fully symbolic solutions, not numerical computing.
Thanks.
Robert H. Lewis
Fordham University
New York |
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| pnachtwey... |
| Posted: Wed May 14, 2008 5:48 pm |
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On May 14, 5:15 pm, "Robert H. Lewis" <rle... at (no spam) fordham.edu> wrote:
Quote: Does anyone know how much has been achieved or attempted in computing Nash > > equilibria symbolically?
I am totally clueless and I had to look up Nash Equilibria.
I have a lot of experience at writing chess programs and othello
programs where a minmax or ( alpha - beta ) search can calculate the
principle variation. The Nash Equilibrium does not appear to need a
search but also assumes the evaluations for the combinations of
choices or moves is known.
Quote:
I have been reading this paper by Ruchira Datta: Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra.http://arxiv.org/abs/math.AC/0612462
I am an engineer, not a mathematician. I could only get the general
principles but don't test me on it.
Quote: The point is to solve a system of polynomial equations.
This is interesting to me because I too need to solve systems of
polynomial equations but the Nash Equilibrium part only gets in the
way.
Quote: In the paper (p. 33) she has the equations for the case of three players with two >pure strategies, and solves them with Singular (let's call this the 3-2 case). How >about the 4-2 case, or the 3-3 case? Again, I want fully symbolic solutions, not >numerical computing.
It will be interesting to see what you come up with but as an engineer
I am a skeptic about the evaluation that are put in the grid of
choices. I know that in chess or othello the evaluation routines are
the key to success. There is no point in searching for the wrong
answer. From what I have seen the Nash Equilibrium does not address
the evaluations so I am skeptical.
So how would you apply a symbolic solution to a simple game? Think of
a simple 3D 3x3x3 or 4x4x4 Tic Tac Toe played by three people. One of
the first games I wrote to test my skills had a 10x10 grid with random
numbers between 1 and 10 in each square. The person could pick any
square and add the squares value to his score.
How would a symbolic solution be more practical than a brute force
tree search?
Peter Nachtwey |
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| Posted: Thu May 15, 2008 12:44 am |
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On May 15, 1:15 am, "Robert H. Lewis" <rle... at (no spam) fordham.edu> wrote:
Quote: Does anyone know how much has been achieved or attempted in computing Nash equilibria symbolically?
There is a large literature on algorithmic and complexity
considerations in game theory.
The recent CUP book, Algorithmic Game Theory (q.v.
http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521872829),
seems a introduction to the literature.
The game theory literature in economic theory is largely concerned
with symbolic solutions. Applications of game theory may then
substitute in particular functional forms to obtain numerical results.
Best,
Colin Rowat
Department of Economics, University of Birmingham |
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