A Model of Probabilistic Rules for Project Acceptance. This is inspired by a recent working paper by Vickers and Armstrong. Project i has payoff (Ui,Vi) to agent and principal and is feasible with probability theta_i. Both players must agree to implement a project; otherwise they get (0,0). They can agree to one project at most. Only the agent observes which projects are available. He can keep silent or he can truthfully reveal the (U,V) of a project, but he cannot lie. (Click here to read more.)
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?