and the payoffs to the competitor are given by
Vc(0, 0,0 ,7) = δ (θ) [τc(0, 0 )π(70) + Tc(0, 0 )] .
(14)
Note that these payoffs do not include the bribes. When augmented by the bribes, the payoffs are
. _ c ^c . . _ c ^c . _ . _ c ^c . _ . _ c ^c
given by V(θ,θ,θ ,7) + B(θ,θ,θ ,7) to the innovator and Vc(θ,θ,θ ,7) + B (θ,θ,θ ,7). Here, 0
denotes the quality of the idea, 0 denotes the report by the innovator, and 0 denotes the report by
the competitor.
LEMMA 1. The truth-telling equilibrium of any direct mechanism must satisfy the bribe-proofness
condition :
V (θ, θ, θ, δ (0)) + Vc (θ, θ, θ, δ (0)) ≥ V (0, 0, θc, δ (0)) + Vc (0, θ, θc, δ (θ)) . (15)
Proof. The proof is by contradiction. Suppose for some 0, 0, and 0 , truth telling is an equilibrium
and the bribe-proofness condition (15) is not satisfied. Suppose that, at the report 0, 0 , the innovator
is strictly better off if both misreport so that V (0, 0, 0 ,δ (0)) > V (0, 0, 0, δ (0)) and the competitor
is strictly worse off so that Vc (0, 0, 0 , δ (0)) < Vc (0, 0, 0, δ (0)) . Consider a bribe by the innovator
that offers the competitor all the surplus the innovator gains by misreporting, so that the bribe equals
V (0,0,0c ,δ 0)) — V (0, 0, 0, δ (0)) . Since, by assumption, (15) does not hold, this bribe makes the
competitor’s payoffs higher than under truth telling, so that truth telling is not an equilibrium. We
have a contradiction. Q.E.D. ■
Note that this lemma relies upon the assumption that the bribe payments are not observable
to the mechanism designer. Note also that the proof of this lemma fails if the size of bribes is
sufficiently limited. To see that the lemma does not hold if bribes are limited, suppose we restrict
bribes to be less than some upper bound T. If T is sufficiently small, the innovator will not be
able to offer all the surplus gained by misreporting, V (0, 0, 0 , δ (0)) — V (0, 0, 0, δ (0)) .Then the
innovator will not be able to induce the competitor to misreport. A planner can then always set
the transfer to the competitor T2 (0, 0c) sufficiently greater than T if 0 = 0C. With such prizes, the
innovator cannot bribe the competitor and the planner can implement the efficient allocation.
We use this lemma to show that the solution to the social planner’s problem in this environ-
ment with bribes coincides with the solution to that in the environment without market signals.
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