To translate this question into
mathematics, reconsider the Vickrey
(second-price) auction for selling a
single good. Each bidder i has a private
willingness-to-pay vi and submits to the
auctioneer a bid bi. The auction com-
prises two algorithms: an allocation al-
gorithm, which picks a winner, namely
the highest bidder; and a payment al-
gorithm, which uses the bids to charge
payments, namely 0 for the losers and
the second-highest bid for the winner.
We argued intuitively that this auction
is truthful in the following sense: for ev-
ery bidder i and every set of bids by the
other participants, bidder i maximizes
its “net value” (its value for the good,
if received, minus its payment, if any)
by bidding its true private value: bi = vi.
Moreover, no false bid is as good as the
truthful bid for all possible bids by the
other participants. Assuming all bid-
ders bid truthfully (as they should), the
Vickrey auction solves the social welfare
maximization problem, in the sense that
the good is allocated to the participant
with the highest value for it.