Auction Theory, 2nd Edition

206 Some Revenue Considerations

On the other hand, if <0, then taking the limit as  ↑0 we get the opposite

inequality. Thus, if V

i

is differentiable at t ∈(0, 1),

dV

i

(t)

dt

=q

i



tx

i



·x

i

Now substituting in (14.4) we obtain that for all x

i

∈X :

U

i



x

i



=U

i

(

0

)

+



1

0

q

i



tx

i



·x

i

dt (14.5)

Thus, at any point x

i

, the payoff U

i

is determined by the probabilities q

i

up to

an additive constant.

Since

m

i



x

i



=q

i



x

i



·x

i

−U

i



x

i



the expected payments in any two auctions with the same allocation rule are also

the same. We have thus shown that the revenue equivalence principle holds for

multiunit auctions as well.

Proposition 14.1. The equilibrium payoff (and payment) functions