Principal-Agent game Hillary manages a technology development company. A company customer asks Hillary to implement a particular project. Because it is unclear whether or not the project is feasible, the customer offers to pay Hillary $2 million at the start of work on the project, and an additional $4 million upon its completion (if the project is never completed, the customer pays nothing beyond the initial $2 million payment). Hillary seeks to hire Bill to implement the project.
The success of the project depends on the amount of effort Bill invests in his work: if he fails to invest effort, the project will fail; if he does invest effort, the project will succeed with probability p, and will fail with probability 1 - p. Bill assesses the cost of investing effort in the project (i.e., the amount of time he will need to devote to work at the expense of the time he would otherwise give to his family, friends, and hobbies) as equivalent to $1 million.
Bill has received another job offer that will pay him $1 million without requiring him to invest a great deal of time and effort. In order to incentivize Bill to take the job she is offering, Hillary offers him a bonus, to be paid upon the successful completion of the project, beyond the salary of $1 million. Answer the following questions:
(a) Depict this situation as an extensive-form game, where Hillary first determines the salary and bonus that she will offer Bill, and Bill afterwards decides whether or not to take the job offered by Hillary. If Bill takes the job offered by Hillary, Bill then needs to decide whether or not to invest effort in working on the project. Finally, if Bill decides to invest effort on the project, a chance move determines whether the project is a success or a failure. Note that the salary and bonus that Hillary can offer Bill need not be expressed in integers.
(b) Find all the subgame perfect equilibria of this game, assuming that both Hillary and Bill are risk-neutral, i.e., each of them seeks to maximize the expected payoff he or she receives.
(c) What does Hillary need to persuade Bill of during their job interview, in order to increase her expected payoff at equilibrium?