# Visual Representation of mixed strategy profile using a simplex in $R^3$

I am looking for a reference to learn how to plot mixed strategy profile in a two player game with three pure strategies.

Looking forward to any advice or suggestion.

(Here is the question regarding simplex I posted in mathematics.stackexchange)

I'm not sure what you're asking is possible.

Start with the intuition from the case with 2 pure strategies. A player's mixed strategy is a 2-vector $\sigma_i(p)=(p,1-p)$, which can be visualized on a 1-dimensional simplex (or simply a line). Therefore, a profile of two mixed strategies can be represented as a point in a 2-dimentional plane, because this profile is controlled by two parameters (say $p$ for player 1 and $q$ for player 2) like the following

With 3 pure strategies, a mixed strategy is a 3-vector $\sigma_i(p_1,p_2)=(p_1,p_2,1-p_1-p_2)$. It is controlled by two parameters, and so we need a 2-dimensional simplex to represent one (mixed) strategy. For a profile of two mixed strategies, we need two 2-dimensional simplices, or planes. Naturally they should be perpendicular to each other just like the two 1-dimensional simplices are in the 1-dimensional case, although this is not a hard requirement. The difficult part is that we now have four controlling parameters, e.g. $p_1,p_2$ for player 1 and $q_1,q_2$ for player 2. So the strategy profile lives in a 4-dimensional space, which is impossible to visualize.

The following is a failed example. Player 1's mixed strategy $\sigma_1(\hat p_1,\hat p_2)$ lives in the blue simplex and Player 2's mixed strategy $\sigma_2(\hat q_1,\hat q_2)$ lives in the orange simplex. The two simplices share the vertical axis, but this is not necessary. As the two red dashed lines show, it's not always possible to find an "intersection" of the two strategies (the two lines have different heights).

• I've seen a graphical method for a two player game where one player has two and the other three strategies in Hans Peter's Game theory a multi-level approach. In that case you get two equilateral triangles connected by lines in a tent-like shape. The three-strategy player chooses a point in the triangle, the two-strategy player chooses along the lines connecting the triangles. If this player would have three strategies as well, that would no longer work though. – Maarten Punt Oct 12 '17 at 16:36
• @MaartenPunt That makes sense because there are only three controlling parameters which can be visualize in a 3D space. In my second example above, it can be achieved by setting q1=0. – Herr K. Oct 12 '17 at 17:17

If what you actually want to do is to plot a strategy, not a strategy profile, then you can use a Kolm triangle. The Kolm triangle is an equilateral triangle that is a representation of the three dimensional unit simplex. Any point in the triangle defines three numbers by taking the distances of the point from each side. In the equilateral triangle the sum of these distances is always constant. If the sides are of length one then the sum of the distances is also one.

Kolm's book wherein the notion is introduced is called Public Value/La Valeur Publique, and you can read a bit about his triangles for example in Economies with Public Goods: An Elementary Geometric Exposition.