Consider three agents $A_i$, who engage in a Bertrand game. All agents have perfect knowledge on all parameters and the distribution $F()$.
- $A_1$ moves first and selects price $0\leq p_1\in \mathbb{R}$. $A_2$ moves second and selects price $0\leq p_2\in \mathbb{R}$. $A_3$ moves third and selects $0\leq q_1\in \mathbb{R}$ and $0\leq q_2\in \mathbb{R}$, that is, the quantities that it buys from $A_1$ and $A_2$, which it sells at market clearing price.
- $A_1$ tries to maximize $ q_1 p_1$.
- $A_2$ tries to maximize $ q_2 p_2$.
- $A_3$ tries to maximize $$((A - q_1 - q_2) (q_1+q_2) - q_1 p_1 - q_2 p_2) (1-F(-q_1 p_1))+ ( (A - q_2) q_2- q_2 p_2+B) F(- q_1 p_1) $$, with $F$ being a continuous cumulative distribution function. $A_3$'s profit function captures that buying from $A_2$ comes at no risk, while $A_1$ might be unable to deliver. However, the higher $A_1$'s profits the smaller that risk and the higher the chance to capture the benefits/losses $B$.
Note, all agents have perfect knowledge on all parameters and the distribution $F()$.
Does $A_3$'s profit increase in $B$? How to prove this?
(If $F$ is uniform, $A_3$ will only buy from one of the agents and it is easy to show that the answer is yes. If $p_1$ and $p_2$, then it is easy to show that the answer is yes.)
In the general case, which is outlined above, my numerical studies suggest that the answer is yes, but I struggle to write down the analytical proof.