102 S. Dasgupta and R.G. Hansen
2.7. Interpreting the optimal auction: The marginal revenue view
Bulow and Roberts (1989)provide an intuitive interpretation of the “virtual valuations”
Ji(vi)according to which the object is allocated in the optimal selling scheme. Interpret
vias a “price” and 1−Fi(vi)as a demand curve: if a pricepis set as a take-it-or-leave-it
price, 1−Fi(p)gives the probability of a sale, i.e., the “quantity”q(p)sold at pricep.
We can then calculate a marginal revenue curve in the usual way, but using 1−Fi(vi)
as the demand curve:
Total Revenue=viq(vi)(20)
⇒(Marginal Revenue)=d(Total Revenue)
dq=vi+q(vi)dvi
dq=vi+(
1 −Fi(vi)) 1
dq/dvi=vi−1 −Fi(vi)
fi(vi).
Thus, the virtual valuations are marginal revenues, and the optimal mechanism
awards the good to the bidder with the highest marginal revenue.Bulow and Roberts
(1989)in fact provide the following “second marginal revenue” auction interpretation of
the optimal selling scheme. Each bidder is asked to announce her value, and the value is
converted into a marginal revenue. The object is awarded to the bidder with the highest
marginal revenue (M 1 ), and the price she pays is the lowest value that she could have
announced without losing the auction (i.e.,MR 1 −^1 (M 2 )).^18
Why does the “second marginal revenue” auction call for the winner to pay the lowest
value she could announce without losing the auction? This is, in fact, a property of the
optimal selling mechanism discussed in the previous section. To see this, definesi(v−i)
as the smallest value (more precisely, the infimum) ofvifor whichi’s virtual valuation
(marginal revenue) would be no less that the highest virtual valuation from the rest of
the values. Clearly,Qi(zi,v−i)=1ifzi>si(v−i)and 0 otherwise. Thus,Qi(zi,v−i)
is a step function, and this implies that
∫vi
0 Q(zi,v−i)dzi=vi−si(v−i)ifvi>si(v−i)
and 0 otherwise. Sincevi >si(v−i)impliesQi(·,·)=1 and
∫vi
0 Q(zi,v−i)dzi =
vi−si(v−i), from(19)we getPi(vi,v−i)=si(v−i)for the winning bidder. Thus,
the bidder with the highest marginal revenue pays the lowest value that would win
against all other values when the object is allocated according to the marginal revenue
rule.
(^18) If no bidder has positive marginal revenue, the seller keeps the object; if only one bidder has a positive
marginal revenue, then she pays the price at which her marginal revenue is zero. It is easy to check that truthful
reporting is a dominant strategy in this auction.