There exists a unique x1 , where 0 ? x1 ? 1 , given that Vt(q1, q2) is an increasing function in q1 .
Proof. When the strategic RA (RA1) gets a bad project, it will get pay-off if it gives the project a GR, and if it refuses rating. 1 and is increasing in x1 . Given that Vt(q1, q2) is increasing in q1 , it is easy to see that ?(lie) is decreasing in x1 and that ?(honest) is increasing in x1 . Thus, if we define x1 such that
2 Proof Proposition dos
Proof. Suppose that the strategic RA (RA1) gets a good project and that its strategy is x1 . Let us examine whether RA1 wants to deviate:
•if x1 = 1 , we have ?(lie) ? ?(honest) , or . If the RA1 gives NR to the good project, it will get and otherwise. Since RA1 does not want to deviate.
•if x1 = 0 , , hence reputation becomes irrelevant and the RA does not have an incentive to give NR to the good project.
•if 0 < x1 < 1 , we have ?(lie) = ?(honest) , so , and hence RA1 does not want to deviate.
step 3 Proof Corollary step one
Proof. Suppose that the equilibrium strategy is x1 = 0 . Then and we must have I + ?Vt(q1, q2) ? ?Vt(q1, q2) . This is impossible as long as I > 0 . Continue reading “Observe that and you may , which is, are decreasing inside x”