# A game theory problem mixed strategy over a continuous set

I have two players $$A$$ and $$B$$, the action of $$A$$ is $$x_A\geq 0$$ and the action of $$B$$ is $$x_B\geq 0$$. Let $$c_0\in(0,1)$$, $$c_3>0$$ and $$c_2>c_1>0$$ be constants. The payoff functions of $$A$$ and $$B$$ are:

$$U_A(x_A;x_B)=\begin{cases} 1-c_3x_A,~&\mbox{if }c_0>\frac{c_1+x_B}{c_2+x_A+x_B},\\ 1/2-c_3x_A,~&\mbox{if }c_0=\frac{c_1+x_A}{c_2+x_A+x_B},\\ 0-c_3x_A,~&\mbox{otherwise}. \end{cases}$$

$$U_B(x_B;x_A)=\begin{cases} 0-c_3x_B,~&\mbox{if }c_0>\frac{c_1+x_B}{c_2+x_A+x_B},\\ 1/2-c_3x_B,~&\mbox{if }c_0=\frac{c_1+x_A}{c_2+x_A+x_B},\\ 1-c_3x_B,~&\mbox{otherwise}. \end{cases}$$

So basically, $$A$$ and $$B$$ are competing on two fractions, $$c_0$$ and $$\frac{c_1+x_A}{c_2+x_A+x_B}$$. If $$c_0>\frac{c_1+x_A}{c_2+x_A+x_B}$$, then $$A$$ gets everything and $$B$$ gets nothing (they also need to pay the cost associated with their action), otherwise, it is the other way round.

I am not sure where to start to analyze this game. I was thinking of taking the first order condition, but it seems not to work for this game. I think that this game will have mixed strategy over the continuous set. But I only the textbook example with discrete action space. Any suggestions are appreciated!