Two firms compete in a Bertrand duopoly with differentiated products. For any of the two firms $i$, demand for firm $i$ is equal to $q_i(p_i,p_j)=a−p_i−b_ip_j$ where $p_i$ is the price set by firm $i$ and $p_j$ is the price set by the other firm $j$. Costs are zero for both firms. The sensitivity of firm $i$’s demand to the firm $j$’s price is either high or low. That is, $b_i$ is either $bH$ or $bL$ where $bH>bL>0$. For each firm $i$, $b_i=bH$ with probability $\theta$ and $b_i=bL$ with probability $1−\theta$, independent of the realization of $b_j$. Each firm knows its own $b_i$, but not its competitor’s. All of this is common knowledge.
- Define the corresponding Bayesian game.
- Define the associated strategic form game.