Following my previous post of yesterday, online here, assume now that I do not know if my son came when I turned my back at time, and missed me… Then the payoff function is the one propose by Vincent, i.e.

In that particular case,

becomes

i.e.

If we draw those functions, on [0,1], the optimal value is solution of 

i.e.  (we focus only on solutions in [0,1]). Because here the game is symmetric, my son should also shoot at time 
Thus, the payoff is then

 

Since we consider here a  zero-sum game, this cannot be a solution of the game. So the game does not have pure strategy solution.

Assume that now I have a mixed strategy, i.e. a distribution of the optimal time to shot. My strategy has distribution , with density  (we assume here that the density exists, or we seek only solution that are differentiable). Assume further that there exists >0 such that the support of my optimal shooting time (the time to shoot is now a random variable) is (,1] (or [,1] since we assume that  is differentiable). There is a discussion at the end of Vincent’s post where he needs that assumption, at the end. Actually, I think we can make it now, since we can rationally assume that
I will not shot at time 0 (even on a neighborhood of 0 since I have zero chance to hit my son).
The expected payoff function, assuming that my son shoots at time y is

Since the zero-sum game is symmetric, again, the expected payoff should be zero. It comes that necessarily,

if . Hence, if we differentiate (with respect to y), we have

and if we differentiate one more time, it comes

i.e. a general solution should be of the form .
Here, we have the same solution as the one considered in Vincent’s blog. His solution is obtained as follows (with slightly different expressions) conditional to , my expected payoff is

i.e.

With a simple integration by parts,

where , i.e.

Thus,

So, if we want to be indifferent to ‘s strategy, , where

with ,

Consider solutions , then =1, i.e. either =1 and then  is constant, or =-1. This means that  is in proportional to .
If we substitute in the equation we had, initially, it comes that

i.e.

If we consider =a and =1, it comes that =1/3 while =1/4 (but we don’t really care about that normalizing constant).
It means that we should not start shooting before 1/3 of the tank is fulled. Actually, it makes sense, since

if <1/3 (while =0 if  >1/3).