Auction Theory, 2nd Edition

230 Nonidentical Objects

Thus, “truth-telling” is a weakly dominant strategy in the VCG mechanism.

A fortiori, the VCG mechanism is incentive compatible. Using (16.5) agent

i’s payoff in equilibrium (when z

i

=x

i

) is just W (x) −W (0, x

−i

)—that is, the

difference in social welfare when i reports x

i

versus when he reports 0. Moreover,

(16.5) implies that the payoff to a buyer with value vector 0, is 0 and for any

x

i

the expected payoff in the VCG mechanism is nonnegative. Thus, the VCG

mechanism is also individually rational.

Now suppose that the value vectors x

i

are independently distributed. The

following result is a generalization of Proposition 5.5 on page 76 to the case of

multiple objects. Its proof is identical to that of Proposition 5.5.

Proposition ...