It seems to me that this game is illuminated a little if one considers a huge generalization. Take a probability space $(X,\mu)$ and define on it a measurable directed graph, which we can think of as a measurable subset $A$ of $X\times X.$ The first player chooses a point $x$ and the second player chooses a point $y$. The first player wins if and only if $(x,y)\in A$. (It might also be nice to add the conditions that the measure is absolutely continuous and that if $x\ne y$ then $(x,y)\in A$ if and only if $(y,x)\notin A$, but I'm not sure that affects the discussion too much.)

Now consider a randomized strategy for the first player. This consists in choosing a different probability measure on $X$, which for convenience I'll assume is a density $f$ with respect to $\mu$ (though I may have to drop that assumption later). If the second player knows $f$, then the second player will choose $y$ such that $\int f(x)\mathbb{1}_A(x,y)d\mu(x)$ is minimized.

This produces a problem that's a continuous version of the following problem: given an $n\times n$ matrix $A$, find a non-negative vector $v$ with coordinates summing to 1 such that the smallest coordinate of $Av$ is as large as possible. If the rows of $A$ are $a_1,\dots,a_n$ then this is asking us to maximize the minimum of the inner products $\langle a_i,v\rangle$ subject to the coordinates of $v$ being positive and adding up to 1, which is similar in flavour to a linear programming problem. (Can it be turned into one? I don't see it immediately. The difference is that the objective function is a minimum of linear functions rather than a linear function. So it is a convex programming problem but with the convex function of a relatively simple form.)

I imagine that all this is either incorrect or very standard game theory. Apologies in advance if there's a Wikipedia article that says similar things more clearly and authoritatively.