Game Theory (continuous strategies, pure strategy)
I think I figured this out--I forgot to imposed the constraints on Alice and Beatrice, using the fact that they know this about Ashok and Bob and will strategize accordingly to maximize payoff.

Thus if 

$$q_{A} = -\frac{1-2p_{A}+p_{B}}{3}$$

and 

$$q_{B} = -\frac{1-p_{A}-2p_{B}}{3}$$

then Alice maximizes her payoff by 

$$\frac{\partial B_{alice}}{\partial p_{A}} = \frac{\partial }{\partial p_{A}}\left(p_{A}\left[-\frac{1-2p_{A}+p_{B}}{3}\right]\right)$$

$$=-\frac{1}{3}(1-4p_{A}+p_{B})=0$$

With the same analysis given for Beatrice you get two linear equations in $p_{A},p_{B}$.