# Game Theory - A graphical solution to all 2-person n x n strategy games?

1. Jun 10, 2010

### hyrodi

I was not sure where to post this. It didn't seem to fit in the homework/self-study category, and I believe it is thought-provoking enough to warrant posting in the general forums. I am sorry if I am wrong.

In game theory, 2-person non-zero-sum games with 2n strategies can be graphically solved by by placing the two strategies of player 1 on two endpoints of a line segment that has a length of one, then graphing the expected values of player 2s strategies as lines from a corresponding y-value above one end of the line segment to another. Where these lines intersect and form either an upper or lower envelope, the corresponding x-value denotes the optimal mixed strategy.
My question is, can 2-person 3n strategy games also be represented in this manner by forming a triangular prism in which the vertices of a base are the three strategies for player 1, expected values are graphed as lines between any two of the three player 1 strategies, and various triangular planes are formed, the corresponding xy coordinate value of the intersection of these planes and their respective distances to the vertices denoting the optimal mixed strategies?
Further developing this intuitive line of thought, could any 2-person non-zero-sum game be theoretically represented by an n-polygonal prism, with the intersections of the various planes formed denoting the solution? Though I lack the knowledge in topology to pursue the topic further, I imagine that the shape of any of the polygons past a triangle would assume a "soap bubble" - like behavior once the vertices of the polygon would not form a flat plane?
Help would be appreciated.