The new widget production process that firm 1 is developing is equally likely to have high or low costs. Firm 1 will learn whether the production process has high costs or low costs at the beginning of next quarter.
Then firm 1 can choose whether to build a new plant or not. Firm 2 will not be able to observe the costs of firm 1's new process, but firm 2 will be able to observe whether firm 1 builds a new plant or not. Firm 2 will subsequently decide whether to enter the widget market against firm 1 or not. Firm 2 will make $2 million (in present discounted value of long-run profits) from entering the widget market if firm l's process has high costs, but firm 2 will lose $4 million from entering the widget market if firm l's process has low costs.
Lower costs in the new process will increase firm l's profits by $4 million. Building a new plant would add $2 million more to firm l's profits if the new process has low costs (because conversion to the new process would be much easier in a new plant), but building a new plant would subtract $4 million from firm l's profits if the new process has high costs. In any event (whether the new process has high or low costs, whether firm 1 builds a new plant or not), firm 2's entry into the widget market would lower firm l's profits by $6 million. (Both firms are risk neutral, so we may identify utility with monetary profit.)
a. Describe this game in extensive form.
b. Construct the normal representation of this game in strategic form.
c. Analyze the normal representation by iterative elimination of weakly dominated strategies.
d. Show the multiagent representation of the extensive-form game. Does any agent have a dominated strategy in the multiagent representation?
e. Suppose now that the game starts at the point in time just after firm 1 learns whether its costs are high or low (but before firm 1 decides whether to build the new plant). Write down a Bayesian game with incomplete information to describe this game.
f. Write down the type-agent representation of the Bayesian game from (e). What is the set of players in this strategic-form game?