Auction Theory, 2nd Edition

13.3 Efficiency in Multiunit Auctions 189

as a result, the allocations are efficient. While Vickrey auctions are always effi-

cient, in some circumstances the outcome of a Vickrey auction may be deemed

to be unfair. This is seen most easily in the context of a simple example. Suppose

that there are two bidders with values x

1

=

(

10,6

)

and x

2

=

(

9,2

)

. In a Vickrey

auction each bidder bids his value vector so that each wins one unit. But notice

that bidder 1 pays only x

2

2

=2 for the unit he wins, whereas bidder 2 pays x

1

2

=6.

Thus, while bidder 1 attaches higher values to both units than does bidder 2, and

indicates this by bidding truthfully, he ends up paying less than what bidder 2

pays. More generally, suppose there are two bidders i and j such that x

i

≥x