# Visualizing Nash Equilibria of a 4 dimensional matrix

Are there any good ways to visualize Nash equilibria of a 4-d matrix?

I have created an game theory model which consists of of four players (P1; P2; P3; P4) who can all choose between a set of 27 different strategies(P1-1..27; P2-1..27; P3-1..27; P4-1..27). By comparing all the strategies to each other I have created a matrix composed of 4 columns and 27^4 rows, in which each row represents a different combination of strategies (e.g. row_297484: P1-16; P2-4; P3-2; P4-26). Depending on the input for my model I find between 30 and 60 Nash equilibria, which can thus be seen as a 4 dimensional coordinate system.

output:

" found 34 Nash equilibria:

Nash coordinate ['G1-26', 'G2-2', 'G3-26', 'G4-26']: Nash coordinate ['G1-26', 'G2-2', 'G3-26', 'G4-27']: Nash coordinate ['G1-26', 'G2-2', 'G3-27', 'G4-26']: Nash coordinate ['G1-26', 'G2-2', 'G3-27', 'G4-27']: " " " " " " " " " etc. "

As I need to present this data in an attractive way in my Master thesis, I am looking for a way to accomplish this.

So does anyone know in what manner a 4d coordinate system can be visualized and presented on a 2d Master thesis? Any other ideas to present this would also be helpful. I promise that the winning tip will be named in the preface section of my thesis :) Anyway thanks for your help.

• I would suggest that visualization for a single Nash equilibrium might be more feasible than visualizing between $30$ and $60$ such points simultaneously. Since your strategy selections seem to be discrete ("a set of $27$ different strategies"), it is not clear that a coordinate system (whether 4D or lower dimension) will convey meaning to your Readers in a natural or conventional way. – hardmath May 10 '16 at 18:37
• I understand your point. The strategies are in fact discrete, however the number indicating a certain strategy, corresponds to an action which I can of course present in a different loop-up table. Now, letting go of the necessity to include all Nash points in one overview, do you perhaps have any recommendations on presenting a single 4dimensional coordinate as discussed above @hardmath ? – Reinier Witteveen May 11 '16 at 14:55
• I would "flatten out" the 4D "matrix" in a different way, namely $27^2\times 27^2$, hoping to see more symmetry in that layout. – hardmath May 11 '16 at 15:18
• @hardmath, thanks for your recommendation although I don't think that I fully understand what you're aiming at. The matrix I am trying to visualize are just four coordinates, one for each player, which directly respond to a specific strategy. In other words, the coordinates are just a way to codify the strategies. As the Nash equilibrium is based on the complete set of strategies, it would be cool to capture this in one overview somehow. Is that also what you are aiming at with the 27^2 x 27^2 matrix? – Reinier Witteveen May 14 '16 at 14:45