Auction Theory, 2nd Edition

66 Mechanism Design

REVENUE EQUIVALENCE (REDUX)

The payoff equivalence derived here immediately implies a general form of the

revenue equivalence principle.

Proposition 5.2. (Revenue Equivalence) If the direct mechanism

(

Q,M

)

is

incentive compatible, then for all i and x

i

the expected payment is

m

i

(

x

i

)

=m

i

(

0

)

+q

i

(

x

i

)

x

i

−



x

i

0

q

i

(

t

i

)

dt

i

(5.8)

Thus, the expected payments in any two incentive compatible mechanisms with

the same allocation rule are equivalent up to a constant.

Proof. Since U

i

(

x

i

)

=q

i

(

x

i

)

x

i

−m

i

(

x

i

)

and U

i

(

0

)

=−m

i

(

0

)

, the equality in

(5.7) can be rewritten as (5.8).

Proposition 5.2 generalizes the revenue equivalence principle from Chap-

ter 3, Proposition ...