oc.optimization control - Five Front Battle

I am reposting notzeb's solution to the 3 front case here, and making some comments on it. In particular, I will point out that the solution is not unique; while notzeb used fractal methods, I will give a piecewise smooth solution using the same ideas.

Idea of solution:

I claim that it is enough to find any probability distribution on ${(p,q,r): p+q+r=1, p,q,r geq 0 }$ whose projection to each coordinate is the uniform measure on $[0,2/3]$.

Proof that such a measure works:

(For simplicity, I disregard ties.) Observe that it is impossible for either general to win on all fronts. Therefore, if I find a strategy that guarantees that I am expected to win at least 1.5 fronts against any opposing strategy, this means that I have probability at least 1/2 of winning 2 fronts against any opposing strategy. (This logic does not extend to the 5 front case.)

Suppose my enemy sends $p$ troops to the first front. I beat him with probability $max(1-(3/2)p, 0)$. By linearity of expectation, if my enemy sends troops $(p,q,r)$, my expected number of victories is

$$max(1-(3/2)p, 0)+max(1-(3/2)q, 0)+max(1-(3/2)r, 0)$$
$$geq 3-(3/2)(p+q+r) = 3/2.$$
If my opponent adopts a mixed strategy, linearity shows that I still have expected number of victories at least $3/2$. QED

notzeb's measure:

Take the triangle of possible solutions and inscribe a hexagon in it, with vertices at the permutations of $(0,1/3,2/3)$. All our solutions will be inside that hexagon.

Now, take that hexagon and place 6 smaller hexagons in it as shown below.

Choose one of those 6 hexagons uniformly at random. Place 6 smaller hexagons inside that one, and choose one of these uniformly at random again. Keep going. The hexagons shrink in size each time; the limiting point is your army distribution.

Notice that the space of possible solutions is a Sierpinski-gasket-like figure, of Hausdorff dimension $log 6/log 3$. It is cute to observe that the white star of David in the center becomes a Koch snowflake of excluded points in the final solution.

My alternate measure

Inscribe a circle in the triangle. On that circle, place the measure $dA/sqrt{1-r^2}$, as described in Harald Hanche-Olsen's answer to a different question.

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