On Genelized Vickrey Auction
Ou Ruiqiu
What is an auction?
An auction is a selling institution in which
(i) buyers make “bids”;
(ii) the bids determine who wins;
(iii) the bids and who wins determine how much each buyer pays to the seller.
An auction is a game:
Players: buyers who make “bids”, denote by i  1, 2,
,n
Buyer i's strategy: his bid, denote by bi
Buyer i's payoff under strategy profile b   b1 ,
vi  piW (b),
ui (b)   L
 pi (b),
, bn  :
i f buyer i wi ns
(1)
i f buyer i l oses
W
L
where vi is his valuation for the good, pi (b) is his payment if he wins, pi (b) if loses.
Some common auction forms:
English auction
Dutch auction
sealed-bid fist-price auction
sealed-bid second-price auction or Vickrey auction (Vickrey 1961).
Vickrey auction
A (single undividable object) Vickrey auction:
(i) buyers submit nonnegative numbers in a sealed envelop as bids
(ii) the good is allocated to the only one highest bidder or randomly allocated to one of the
highest bidders if there are more than two highest bidders.
(iii) the winner (who get the good) pays the second-highest bid, others pay nothing.
A Vickrey auction described as a game
Players are buyers who bid.
For all i, buyer i's strategy set is 0,  .
Player i's payoff is
v  mi ,
ui (b)   i
0,
i f buyer i wi ns
i f buyer i l oses
(2)
where m i  max{b j : j  i} .
Basic assumptions:
(1) Individual private values (a good for consumption or be used alone).
(2) Buyers are all risk-neutral.
(3) Buyers can not collude with others.
Proposition 1 (Vickrey 1961)
In a Vickrey auction, it is (weakly) dominant for each buyer i to bid his valuation vi (i.e.
regardless of other buyers bid, it is optimal for buyer i to bid bi  vi ).
Without considering bi  mi , we can also assume buyers are all risk-averse.
Property of Vickrey auction (compare with first-price auction)
In Vickrey auction, each buyer i has a (weakly) dominant-strategy bi  vi , so
(1) Each buyer i's rational decision-making—bidding truthfully (i.e. bi  vi )—is very simple.
(2) b  (v1 ,
, vn ) is a (weakly) dominant-strategy equilibrium.
(3) (2) ensure Vickrey auction is always efficient (i.e. in equilibrium b  (v1 ,
, vn ) , the good is
allocated to the buyer with the highest valuation).
In contrast, no buyer has any (weakly) dominant-strategy in first-price auction and
(1) Each buyer i's rational decision-making is complex because he must estimates others’ bids.
(2) First-price auction is not always efficient.
How to design an efficient auction
(1) The highest bidder wins;
(2) bi  vi is each player i’s (weakly) dominant-strategy.
Generalized Vickrey auction
Proposition 2 (Jerry Green and Jean-Jacques Laffont 1977; Bengt Holmström 1979; Laffont
and Maskin 1980)
Suppose that, for all i, vi can assume any value in 0,1 . Then an auction is efficient and bidding
truthfully is (weakly) dominant for each buyer i if and only if
(a) the highest bidder wins
(b) for all i, buyer i's payment satisfies
mi  ti (bi ),
pi (b)  
ti (bi ),
where bi   b1 ,
i f buyer i wi ns
i f buyer i l oses
（4）
, bn  。
, bi 1 , bi 1,
The auctions of Proposition 2 are generalized Vickrey auctions. Generalized Vickrey auctions
have the same property with Vickrey auction.
Example
Assume buyers must pay the seller a fixed and nonrefundable fee e  0 prior to a (standard)
Vickrey auction in order to be able to submit. Then the new auction satisfies
(a) the highest bidder wins
(b) for all i, buyer i's payment satisfies
m  e,
pi (b)   i
e,
i f buyer i wi ns
i f buyer i l oses
so it is a generalized Vickrey auction.
Other generalized Vickrey auctions may be not feasible.
In the two Propositions above, no player yet owns the good. Now let’s consider two cases below:
(1) The seller is also a bidder (e.g. in a standard Vickrey auction, the seller set a reserve price r. If
the price is lower than r, the seller has the right to not sell the good.)
(2) Allocate a common good among owners.
In the two cases above, the sum of all bidders’ payments should be equal to 0 which we call
payments balance.
Proposition 3 (Green and Laffont 1977; Laffont and Maskin 1980)
Under the hypotheses of Proposition 2 there exists no generalized Vickrey auction in which the
n
payments balance i.e.
p
i 1
i
0.
Proposition 3 indicate there exists no generalized Vickrey auctions either to allocate a common
good among owners or if the seller plays as a bidder.
Proposition 4 (d’Aspremont and GérardVaret 1979)
Suppose that, for all i, vi is drawn independently from a distribution with c.d.f. Fi and support
0,1 . Then there exists an efficient and payment-balanced auction in which bidding truthfully
constitutes a Bayesian equilibrium.
Proposition 4 shows that the failure of balance in Proposition can be overcome by relaxing the
solution concept from dominant-strategy to Bayesian equilibrium which also ensure the auction is
efficient if only the highest bidder wins.
Example
n  2 and F1  F2  x, x 0,1 .
Considering the auction that
(a) the highest bidder wins
(b12  b22 )
(b22  b12 )
(b) two buyers’ payment functions are p1 (b1 , b2 ) 
 c and p2 (b1 , b2 ) 
c
2
2
(c is a constant).
Shortcomings:
(1) Payment functions are unusual, so the auction may be not feasible.
(2) Unreasonable to allocate a common good among owners.
In Proposition 4, we didn’t consider individual rationality.
Let n  2 , suppose player 1 owns the good.
An auction is efficient: the good would be transferred to player 2 if and only if v2  v1 .
Payment-balanced: p1  p2  0 or p1   p2
Thus, in a payment-balanced and efficient auction, individual rationality for player 1 is

1
0
p2  b1 (v1 ), b2 (v2 )  f 2 (v2 )dv2  
1
v f (v2 )dv2  0 for all v1 .
v2  v1 1 2
individual rationality for player 1 is

v2
v1  0
v2 f1 (v1 )dv1   p2  b1 (v1 ), b2 (v2 )  f1 (v1 )dv2  0 for all v2
1
0
Proposition 5 (Laffont and Maskin 1979; Roger Myerson and Mark Satterthwaite 1983)
Let n  2 . Under the hypotheses of Proposition 4, there exists no efficient and payment-balanced
auction that is individually rational for both players when Bayesian equilibrium is the solution
concept.
Proposition 5 demonstrates there may not exist an auction is always efficient when the seller is
also a bidder. Thus, if the government want to transfer public assets to private hand, the primal
allocation is very important. Once the assets have been distributed, efficiency may no longer be
attainable.
Reference:
Eric Maskin, The Unity of Auction Theory: Milgrom’s Masterclass, 2004
Vijay Krishna, Auction Theory, 2002