Game Theory (continuous strategies, pure strategy)

Question

So I have this Game Theory problem, and I have a solution, but at a certain point I assume the symmetry of the problem to finally get my answer. I'd like to be able to avoid using symmetry, though, so that in the future I can solve problems that are not symmetric.

So Alice and Beatrice are suppliers, Ashok and Bob buy from them to then sell again at retail. Ashok only buys from Alice and Bob only buys from Beatrice. First Alice and Beatrice set their prices simultaneously, $p_{A},p_{B}$ respectively. Then Ashok and Bob set their quantities $q_{A},q_{B}$, and their price is determined by

$$P=1-q_{A}-q_{B}$$

The payoffs for Alice, Beatrice, Ashok, and Bob respectively are $p_{A}q_{A}, p_{B},q_{B}, q_{A}(P-p_{A}), q_{B}(P-p_{B})$. I want to find a sub-game perfect equilibrium.

I first look at Ashok and Bob and, for any fixed prices from Alice and Beatrice, find the intersection of their best response curves.

$$\frac{dB_{A}}{dq_{A}} = 1-2q_{A}-q_{B}-p_{A}=0$$

$$\frac{dB_{B}}{dq_{B}} = 1-q_{A}-2q_{B}-p_{B}=0$$

We're solving for $q_{A},q_{B}$ so we get

$$1-3q_{A}-2p_{A}+p_{B}=0 \Rightarrow$$

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

Similarly for $q_{B}$. Once we know those, we can substitute into the first equation for price and solve for $p_{A},p_{B}$. But the solution will not be unique. It becomes unique when I assume $p_{A}=p_{B}$ but if anyone can point out how I can solve this without that equation, I'd appreciate it.

Answer 1

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 $p_{A},p_{B}$.