Economics Letters 56 (1997) 195–200
Bid shading and risk aversion in multi-unit auctions with many
bidders
a
b,
D. Nautz , E. Wolfstetter *
a
¨
Institut f. Statistik u. Okonometrie
, Boltzmannstr. 20, 14195 Berlin, Germany
b
Institut f. Wirtschaftstheorie I, Spandauer Str. 1, 10178 Berlin, Germany
Received 27 March 1997; accepted 23 May 1997
Abstract
We extend the analysis of optimal price taking bidding in multi-unit auctions to allow for risk aversion and a continuous
random stop-out price. We show that in a discriminatory auction risk averse bidders should bid less aggressively than risk
neutral bidders. However, bid shading is optimal at each price, which implies, with complete certainty, inefficient trade. This
is in sharp contrast to competitive auctions where truthful bidding is optimal even under risk aversion.  1997 Elsevier
Science S.A.
Keywords: Optimal price taking bidding; Multi unit auctions; Risk aversion; Continuous random stop-out price; Bid shading
JEL classification: D44
1. Introduction
Multi-unit auctions, like the U.S. Treasury Bill auction, are typically ‘multi’ in the double sense
that the auctioneer puts up many units for sale and bidders demand many units. Therefore, bids
generally take the form of demand schedules that indicate how many units a bidder wants to buy,
contingent upon how much he has to pay. In practice two formats are prevalent: discriminatory
auctions where bidders pay their bid for each successive unit and competitive auctions where bidders
pay the lowest accepted bid—the so-called stop-out price—for each unit they get.
The computation of equilibrium bidding strategies in multi-unit auctions is extremely difficult.
Therefore, the analysis of bidding behavior requires simplifying assumptions. Several contributors
circumvented the difficulties of strategic bidding by assuming that bidders act as price takers (see, for
example, Scott, Wolf, 1979). This seems particularly plausible if the number of bidders is large as in
many financial auctions. In this tradition Nautz (1995) analyze optimal bidding if the auctioneer sets a
discrete price grid. Assuming risk neutral bidders, he showed that in a discriminatory auction optimal
bidding requires bid shading almost everywhere whereas in a competitive auction truthful bidding is
*Corresponding author: Tel.: 0049 3 20935652; fax: 0049 3 20935619; e-mail: [email protected]
0165-1765 / 97 / $17.00  1997 Elsevier Science S.A. All rights reserved.
PII S0165-1765( 97 )00144-4
D. Nautz, E. Wolfstetter / Economics Letters 56 (1997) 195 – 200
196
optimal. Nautz (1997) provides a case study on German repo auctions that supports the relevance of
bid shading in discriminatory auctions.
The present paper extends the analysis of price taking bidding to allow for risk aversion and a
continuous random stop-out price. We show that in a discriminatory auction a risk averse bidder
should bid less aggressively than a risk neutral bidder, at each price. However, even under risk
aversion, optimal bidding requires bid shading everywhere. Therefore, trade is definitely inefficient.
This is in sharp contrast with competitive auctions where truthful bidding is optimal even under risk
aversion.
2. The model
Consider a multi-unit auction with price taking bidders where bidders are asked to submit a demand
function, B, as a function of the random stop-out price P. When P 5 p is observed, each bidder obtains
B( p) units and makes a payment that depends upon whether the auction is discriminatory or
competitive.
Bidders are characterized by a monotone decreasing true demand function D, a concave utility
function U defined on the pecuniary gain p, a log-concave probability distribution of the stop-out
price F: [p, p¯ ] → [0, 1], with f :5 F9.0, which reflects their expectations concerning P. Without loss
]
of generality we assume D(p¯ )50.
The submitted demand function B may—and typically will—differ from the true demand function
D. ‘Bid shading’ is said to occur at p if B( p),D( p), or equivalently Z(B( p)).p, resp. p.ZB (D( p)),
where Z:5 D 21 is the true and ZB :5B 2 1 the submitted marginal willingness to pay function. Of
course, B must be monotone decreasing, just like D, because otherwise bidders would have to bid
negative quantities at some prices.
3. Discriminatory auction
In a discriminatory auction bidders have to make distinct payments for each of the awarded B( p)
units, according to the submitted marginal willingness to pay function, ZB . Therefore, given P5p, a
bidder’s total payment is e0B( p) ZB (x) dx which leads to the following payoff
SE
B( p)
U(p (B( p))): 5 U
0
D
(Z(x) 2 ZB (x)) dx .
(1)
Bidders solve the optimization problem maxB E[U(B(P))] which we now translate into an optimal
control problem. First, rewrite bidders’ expected utility as follows, using integration by parts, and
p (p¯ ) 5 0, p 9(B( p)) 5 Z(B( p)) 2 p
E U(p(B( p))) dF( p)
p̄
E[U(p (B(P)))]: 5
(2)
p
]
E U 9(p(B( p)))[ p 2 Z(B)( p))]B9( p)F( p) dp .
p̄
5
p
]
(3)
D. Nautz, E. Wolfstetter / Economics Letters 56 (1997) 195 – 200
197
Next, define the control variable u( p) and the state variables x 1 ( p), x 2 ( p): 1 u( p): 5 B9( p), x 1 ( p): 5
p (B( p)), x 2 ( p): 5 B( p). Then, the decision problem can be stated as the optimal control problem 2
E U 9(x ( p))[ p 2 Z(x ( p))]u( p)F( p) dp
p̄
max
hu,x 1 ,x 2 j
p
]
1
2
(4)
s.t. x 91 ( p) 5 [Z(x 2 ( p)) 2 p]u( p) ,
(5)
x 92 ( p) 5 u( p) ,
(6)
x 2 (p) 5 D(p) ,
]
]
(7)
x 1 (p) 5 p (D(p)) .
]
]
(8)
The associated Hamiltonian is
H( p, x 1 , x 2 , u, l1 , l2 ): 5 U 9(x 1 ( p))[ p 2 Z(x 2 ( p))]u( p)F( p)
1 l1 ( p)[Z(x 2 ( p)) 2 p]u( p) 1 l2 ( p)u( p) .
(9)
And the optimality conditions are (if there is no risk of confusion, the arguments of functions are
omitted)3
≠H
0 5 ] 5 ( p 2 Z)(U 9F 2 l1 ) 1 l2 ,
≠u
(10)
≠H
l 19 5 2 ] 5 U 0(Z 2 p)Fu ,
≠x 1
(11)
≠H
l 29 5 2 ] 5 (U 9F 2 l1 )Z9u .
≠x 2
(12)
Totally differentiate (10) with respect to p and one obtains, using (11) and (12),
U 9(p (B( p)))[ f( p)(Z(B( p)) 2 p) 2 F( p)] 5 2 l1 ( p) ,
(13)
or equivalently,
l1 ( p)
F( p)
Z(B( p)) 5 p 1 ]] 2 ]]]]] .
f( p)
f( p)U 9(p (B( p)))
(14)
With these preliminaries, we can now characterize optimal bidding.
1
For ease of access we stick to the notation used in Kamien, Schwartz (1991).
The initial conditions (7), (8) are due to the fact that, at P 5p, it is obviously optimal to bid one’s true demand,
]
x 2 (p) 5 B(p) 5 D(p).
]
]
]
3
See Kamien, Schwartz (1991), Part II where necessary and sufficient conditions are discussed in detail.
2
D. Nautz, E. Wolfstetter / Economics Letters 56 (1997) 195 – 200
198
Proposition 1 (Discriminatory Auction). Consider a discriminatory auction.
1. Under risk neutrality the optimal submitted demand function is
S
D
F( p)
D p 1 ]] .
f( p)
(15)
It exhibits bid shading for all p [ (p, p¯ ).
]
2. Under risk aversion the optimal submitted demand function B( p) also exhibits bid shading but
pointwise less so than under risk neutrality
S
D
F( p)
; p [ (p, p¯ ): D( p) . B( p) . D p 1 ]] .
f( p)
]
(16)
Proof. Regardless of the attitude towards risk, in a discriminatory auction it cannot be optimal to
exaggerate one’s true marginal willingness to pay because otherwise the bidder could obviously
improve his bid. Therefore, Z(B( p))$p or equivalently B( p)#D( p). The initial condition (7) implies
p 5 Z(B(p)). Combined with (13) this entails
]
]
l1 (p) 5 0 .
(17)
]
(1) If U 0;0, one has l 91 ; 0 by (11), and thus also l1 ; 0 by (17), which proves (15) by (14). It
follows that a risk neutral bidder should shade bids everywhere, since 4
S
D
F( p)
; p [ (p, p¯ ) D p 1 ]] , D( p) .
f( p)
]
(2) Next we show that bid shading is optimal even under risk aversion. By (14), bid shading is
equivalent to U 9F . l1 . Since l1 (p) 5 U 9(x 1 (p))F(p) 5 0, it is sufficient to show that (U 9F )9 . l 19 .
]
]
]
This holds because fU 9 . 0 and (U 9F )9 5 fU 9 1 l 9i (by (11)). Hence, Z(B( p)).p (bid shading).
Finally, we show that risk aversion reduces the amount of bid shading, i.e. B( p) . D( p1(F( p)) /
f( p)). If U 0,0, then (11) and (17) imply l 91 $ 0 and thus l1 $ 0. Hence B( p) $ D( p 1 (F( p)) /f( p))
by (14). It only remains to be shown that inequality must be strict everywhere in (p, p¯ ). For this
]
purpose suppose there is a p0 [ (p, p¯ ) with B( p0 ) 5 D( p0 1 (F( p0 )) /f( p0 )) . By (14), this implies
]
l1 ( p0 ) 5 0. Since l1 $ 0 and l 19 $ 0, it follows that
S
D
F( p)
; p [ [p, p0 ]: B( p) 5 p 1 ]] ,
f( p)
]
(18)
which is equivalent to l1 ( p) 5 0, ; p [ [p, p0 ]. Therefore, l 91 ( p) 5 0, ; p [ (p, p0 ). Since Z(B( p))2
]
]
p.0, (11) implies u( p)5B9( p)50, ; p [ (p, p0 ); but this contradicts (18). j
]
It is obvious that in a discriminatory auction the bidder only gains if the demand is understated
somewhere. However, Proposition 1 shows that, regardless of bidders’ attitude towards risk, it is
optimal to understate demand almost everywhere. Therefore, even under risk aversion, bidders are
awarded less units than they would like to acquire at the realized stop-out price; hence inefficiency
occurs with certainty.
4
The assumed log-concavity of F assures that D ( p1(F( p)) /f( p)) is monotonically increasing in p.
D. Nautz, E. Wolfstetter / Economics Letters 56 (1997) 195 – 200
199
We mention that the solution under risk neutrality can be viewed as the continuous counterpart of
the discrete solution obtained by Nautz (1995). However, the continuous modelling is better able to
handle risk aversion and to explore comparative static issues.
Proposition 2 (Comparative Statics). Assume risk neutrality and suppose bid shading is reduced at
each p due to a change in F. Then the random stop-out price must have increased stochastically in the
sense of first-order stochastic dominance.Proof. Consider two probability distribution functions of the
stop-out price F1 and F2 and the associated optimal submitted demand functions BF 1 and BF 2 . By part
1 of Proposition 1, one has for each p
F1 ( p) F2 ( p) f1 ( p)
f2 ( p)
BF 2 ( p) $ BF 1 ( p)⇔]] $ ]]⇔]] # ]] .
f1 ( p)
f2 ( p) F1 ( p) F2 ( p)
If this holds, then
f ( y)
f ( y)
E S]]
2 ]] dy 5 ln(F ( p)) 2 ln(F ( p)) .
F ( y) F ( y) D
p̄
0#
p
2
1
2
1
1
2
Therefore, F2 ( p) # F1 ( p) which proves first-order stochastic dominance. j
Note that the converse of Proposition 2 does not hold true. In fact F1 ( p) /f1 ( p) $ F2 ( p) /f2 ( p) holds
everywhere if and only if F2 ( p) /F1 ( p) is always increasing. But this cannot be assured by first-order
stochastic dominance alone. An example of a first-order stochastic dominance shift where the
implication in Proposition 2 goes in both directions is 5 Fg ( p): 5 F( p)1 / g, g [ (0, 1). There, the
stop-out price gets stochastically larger iff g decreases, and Bg ( p) 5 D( p 1 g F( p) /f( p)).
4. Competitive auction
Now consider a competitive auction where the bidder has to pay stop-out price p for each and every
unit. Therefore, the expected utility associated with a submitted demand function B is
E USE
p̄
E[U(p (B(P)))]: 5
p
]
D
B( p)
(Z(x) 2 p) dx dF( p) .
0
(19)
Proposition 3 (Competitive Auction). In a competitive auction it is optimal to bid truthfully at each p,
i.e. B( p)5D( p), regardless of bidders’ attitude towards risk and their expectations concerning the
stop-out price.
Proof. In a competitive auction, the bidder’s optimization problem is a straightforward variational
problem. The corresponding Euler equations are
f( p)(Z(B( p)) 2 p)U 9
SE
B( p)
0
5
D
(Z(x) 2 p) dx 5 0 .
Similar comparative static issues are explored in Simmons (1996).
(20)
200
D. Nautz, E. Wolfstetter / Economics Letters 56 (1997) 195 – 200
These conditions are necessary and sufficient since U is concave and Z is monotone decreasing. From
(20) it follows Z(B( p))5p; hence, B( p)5D( p), as asserted. j
This indicates that the two advantages of a competitive auction rule already emphasized by Nautz
(1995)—simplicity (no need to compute optimal amount of bid shading) and robustness (independence of expectations)–also hold under risk aversion.
5. Closing remarks
The desirability of competitive and discriminatory auctions has been the subject of recurrent debate.
If one is primarily concerned with efficiency, our results clearly favor the competitive auction where
truthful bidding is optimal and hence efficient trade is assured. This indicates that the competitive
auction can be interpreted as an approximation of the generalized Vickrey auction introduced by
Ausubel, Cramton (1995), if the number of bidders is sufficiently large.
This is in sharp contrast to the discriminatory auction where, due to bid shading, it is not assured
that each unit goes to the bidder who values it most. The introduction of risk aversion reduces bid
shading but cannot ameliorate the inefficiency problem. Moreover, Maskin, Riley (1989) showed that
the discriminatory auction generally fails to maximize the auctioneer’s revenue. Therefore, it remains
a puzzle why discriminatory auctions are predominant in for example financial markets.
Acknowledgements
Research support by the Deutsche Forschungsgemeinschaft (DFG), Sonderforschungsbereich 373,
¨
¨ Berlin, is gratefully
‘Quantifikation und Simulation Okonomischer
Prozesse’, Humboldt-Universitat
acknowledged.
References
Ausubel, L.M., Cramton, P.C., 1995. Demand reduction and inefficiency in multi-unit auctions. Working paper, University of
Maryland.
Kamien, M.I., Schwartz, N.L., 1991. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics
and Management, 2nd ed. North-Holland.
Maskin, E., Riley, J.G., 1989. Optimal multi-unit auctions. In: F. Hahn (Ed.), The Economics of Missing Markets,
Information, and Games. Clarendon Press, pp. 312–335.
Nautz, D., 1995. Optimal bidding in multi-unit auctions with many bidders. Economics Letters 48, 301–306.
Nautz, D., 1997. How auctions reveal information—a case study on German repo rates. Journal of Money, Credit, and
Banking, forthcoming.
Scott, J., Wolf, C., 1979. The efficient diversification of bids in Treasury bill auctions. Review of Economics and Statistics
60, 280–287.
Simmons, P., 1996. Seller surplus in first-price auctions. Economics Letters 50, 1–5.
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