First, the principal commits to an accept/reject rule.
Second, the agent proposes a project if he wishes.
Third, if an acceptable project was proposed, it is implemented.
Project A: (9,10), probability = 1.
Project B: (7,100), probability = .7.
Rule Alpha: Accept the proposed project if V < 8.
Rule Beta (best) Accept the proposed project for sure if V < 8. Accept with probability .5 if V > 8.
Rule Gamma (worst): Accept any proposed project. There are two possible states of the world. With probability .7, both A and B are feasible; with probability .3 only A is. Under Rule Alpha, the agent will propose either B or nothing. The principal's expected payoff is .7(100) + 0 = 70. Under Rule Beta, the agent will propose B if it is feasible and A otherwise. The principal's expected payoff is .7(100) + .3(.5)(10) = 71.5. Under Rule Gamma, the agent will propose A. The principal's expected payoff is 1. What's happening is that since the principal can't use cash to reward a high-V proposal (or punish a low-V one), he commits to destroying some of the value of a low-V proposal. Could we change the example to have probability .001 of every other project on the convex set containing A and B?
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