All-Pay Auctions with Weakly Risk-Averse Players
Gadi Fibich
∗
Arieh Gavious
†
Aner Sela
‡
June 2004
Abstract
We use perturbation analysis to study independent private-value all-pay auctions
with weakly risk-averse buyers. We show that under weak risk aversion: 1) Players
with low values bid lower and players with high values bid higher than they would
bid in the risk neutral case. 2) Players with low values bid lower and players with
high values bid higher than they would bid in a first-price auction. 3) Players’
expected utilities in an all-pay auction are lower than in a first-price auction. 4)
The seller’s expected payoff in an all-pay auction may be either higher or lower than
in the risk neutral case. 5) The seller’s expected payoff in an all-pay auction may
be either higher or lower than in a first-price auction.
Keywords and Phrases: : Private-value auctions, Risk aversion, Perturbation
analysis.
JEL Classification Numbers: D44, D72, D82.
∗
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, [email protected]
†
School of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gurion
University, P.O. Box 653, Beer-Sheva 84105, Israel, [email protected]
‡
Department of Economics, Ben-Gurion University, P.O. Box 653, Beer-Sheva 84105, Israel, an-
[email protected]
1
1
Introduction
Consider n players who compete for a single item. Every player submits a bid and the
player with the highest bid receives the item. All players bear a cost of bidding which
is an increasing function of their bids. This setup, which is called an all-pay auction, is
commonly used to model applications such as job-promotion competitions, R&D competitions, political campaigns, political lobbying, sport competitions, etc. The literature on
contests and particularly on all-pay auctions has dealt mostly with risk-neutral players,
since an explicit expression for the equilibrium bidding strategies can be used in the analysis.1 Although it is more reasonable to assume that players are risk-averse, dealing with
risk-averse players is a much harder task, since usually explicit expressions for equilibrium
strategies in contests with risk averse players cannot be obtained. In order to overcome
this difficulty, in this study we consider the case of weakly risk-averse players. The presence of the small risk-aversion parameter allows us to employ perturbation analysis, one
of the most powerful tools in applied mathematics, to calculate explicit approximations
of the equilibrium strategies of risk-averse players. As we shall see, such approximate
solutions can be very insightful, justifying the sacrifice of ‘exactness’. In addition, although formally our results are only proved for weak risk-aversion, as is often the case
in perturbation analysis (see, e.g., Bender and Orszag, 1978), these results remain valid
1
All-pay auctions with linear cost functions and incomplete information about the players’ values
include, among others: Weber (1985), Hillman and Riley (1989), Krishna and Morgan (1997), Kaplan et
al. (2002). All-pay auctions with complete information about the players’ values include, among others:
Tullock (1980), Dasgupta (1986), Dixit (1987), Baye et al. (1993, 1996).
2
even when risk-aversion is not small.
In contrast to all-pay auctions, several studies on the classical auction mechanisms
(first-price and second-price auctions) with risk-averse players appear in the literature. In
independent private-value second-price auctions, risk aversion has no effect on a player’s
optimal strategy which remains to bid her own valuation for the object. In independent
private-value first-price auctions, on the other hand, risk aversion makes players bid more
aggressively (see Maskin and Riley (1984)). Since the (risk-neutral) seller is indifferent
to the first-price and second-price auctions when players are risk neutral,2 she prefers
the first-price auction to the second-price auction when players are risk averse. However,
the seller’s preference relations for auction mechanisms with risk-averse players do not
imply anything about the players’ preference relations for these auctions, since under risk
aversion the combined revenue of the seller and the players is not a constant. Indeed,
Matthews (1987) showed that risk averse players with constant absolute risk aversion are
indifferent to first and second-price auctions, and that players prefer the first-price auction
if they have increasing absolute risk aversion and the second price auction if they have
decreasing absolute risk aversion.3
In this paper we analyze the role of risk aversion in all-pay auctions by comparing the
situation where all players are risk neutral (henceforth referred to as the status quo), with
the case where players are weakly risk-averse. In Section 2 we show that a weakly risk2
This follows from the Revenue Equivalence Theorem (Vickrey (1961), Myerson (1981), and Riley and
Samuelson (1981)).
3
This result was generalized by Monderer and Tennenhltz (2000) to all k-price auctions.
3
averse player with a low valuation bids less aggressively than she bids in the status quo
situation. On the other hand, a weakly risk-averse player with a high valuation bids more
aggressively than she bids in the status quo. This behavior can be explained as follows.
When a player’s value is small, she is most likely to lose. Therefore, as she becomes more
risk averse, she is willing to pay less, that is, she bids less aggressively. On the other hand,
when a player’s value is very high, she is afraid of losing the object, therefore, she bids
more aggressively. Note that this a-posteriori intuition suggests that the above result may
also hold for substantial departures from risk neutrality.
We can learn much about the all-pay auction with risk averse players by comparing it
to the first-price auction. Although the first-price auction is a classical auction whereas
the all-pay auction is a contest, these models are similar since in both the highest player
wins for sure and pays her bid. In order to make this comparison, we first calculate
the equilibrium strategies of weakly risk-averse players in first-price auctions (Section 3).
From the expressions of the equilibrium strategies we can immediately recover the wellknown effect of risk aversion on the equilibrium bids, namely, in first-price auctions weakly
risk-averse players bid more aggressively than they bid in the status quo.
In Section 4 we compare all-pay auctions with first-price auctions. Intuitively, one can
expect that as in the risk-neutral case, the equilibrium bids of risk averse players in all-pay
auctions should be lower than in first-price auctions. We show that, indeed, in all-pay
auctions, low types bid less aggressively than they bid in first-price auctions. However,
high types bid more aggressively in all-pay auctions than they bid in first-price auctions.
According to the above described comparison of the players’ bids in first-price auctions
4
and all-pay auctions, it is not clear in which auction the (ex-ante) player’s expected utility
is larger. Nevertheless, we show that, independent of the distribution of the players’
valuations and the number of players, the expected payoff of every player in the first-price
auction is always larger than her expected payoff in the all-pay auction. Consequently,
a weakly risk-averse player will prefer the first-price auction to the all-pay auction. The
dominance of the first-price auction from the players’ point of view can be generalized
to any auction mechanism in which the player pays part of her bid whether or not she
wins and she pays the rest of the bid only if she wins. In contrast to the players, the
seller’s preference relation among first-price and all-pay auctions is ambiguous. Using
perturbation analysis we calculate the seller’s expected payoff in all-pay auctions and
show that it can be either higher or lower than in a first-price auction.
In Section 5 the results of the perturbation analysis described above are illustrated by
an example with two weakly risk-averse players. We show that even when the risk-aversion
parameter is not small, the agreement between the explicit approximations obtained by
the perturbation analysis and the exact values obtained by numerical analysis is quite
remarkable. This finding supports the claim that our results probably remain valid even
when risk-aversion is not small.
2
All-pay auctions
Consider n players that compete to acquire a single object in an all-pay auction. The
valuation of each player for the object v is independently distributed according to a dis-
5
tribution function F (v) on the interval [v, v].4 Each player submits a bid b and pays her
bid regardless of whether she wins or not, but only the highest player wins the object.
Each player’s utility is given by the function U (v − b), which is twice continuously differentiable, monotonically increasing, normalized such that U (0) = 0, and satisfies U 00 < 0
(i.e., risk-averse players). If we assume that the equilibrium bid function b(v) is monotonically increasing, we can define the equilibrium inverse bid function as v = v(b). The
maximization problem of player i with valuation v is given by
maxVi = F n−1 (v(b))U (v − b) + (1 − F n−1 (v(b)))U (−b).
b
Differentiating with respect to b gives
∂Vi
= (n − 1)F n−2 (v(b))f (v(b))v0 (b)[U (v − b) − U (−b)]
∂b
−F n−1 (v(b))[U 0 (v − b) − U 0 (−b)] − U 0 (−b) = 0.
Therefore, the inverse bid function satisfies the ordinary differential equation
v 0 (b) =
F (v(b))[U 0 (v − b) − U 0 (−b)]
(n − 1)f (v(b))[U (v − b) − U (−b)]
U 0 (−b)
,
+
(n − 1)F n−2 (v(b))f (v(b))[U(v − b) − U (−b)]
(1)
with the initial condition v(0) = v.
Equation (1) is exact in the risk-neutral case U (x) = c · x where c is a constant.5 In
that case its solution is given by
ball
rn (v)
= vF
n−1
(v) −
Z
v
F n−1 (s) ds.
v
4
It is assumed that the seller’s valuation is 0.
5
Without loss of generality, the constant c can be set to be 1.
6
(2)
Since there are no such explicit solutions for a general utility function U , the standard
approach in auction theory for analyzing the role of risk aversion in all-pay auctions would
be to try to analyze eq. (1) directly. However, since a direct analysis of eq. (1) is difficult,
one can still gain a lot of insight by adopting the “applied mathematics approach” of
calculating an explicit approximations of the solution. To do that, we consider the case of
weak risk aversion, i.e., U ≈ x. This is the case, for example, for players with a constant
absolute risk aversion (CARA) utility function U(x) = [1 − exp(−²x)]/², or for players
with constant relative risk aversion (CRRA) utility function U (x) = x1−² , if 0 < ² ¿ 1.
Therefore, in general, the utility function of weakly risk-averse players can be written as
U (x) = x + εu(x) + O(ε2 ),
ε ¿ 1.
(3)
Thus, ε is the risk aversion parameter and ε ¿ 1 implies weak risk aversion. Note that
u(0) = 0 and u00 < 0. On the other hand, u0 can be either positive or negative, since in
either case U = x + ²u is monotonically increasing.
The existence of a small risk aversion parameter enables us to use perturbation methods to calculate explicit approximations to the bidding strategies:
Proposition 1 The symmetric equilibrium bid function in an all-pay auction with weakly
risk-averse players is given by
all
2
ball (v) = ball
rn (v) + εb1 (v) + O(ε ),
7
where ball
rv (v) is the equilibrium bid in the risk-neutral case (2), and
ball
1 (v)
=
u(−ball
rn (v))
−
Proof.
Z
v
v
+F
n−1
·
¸
all
all
(v) u(v − brn (v)) − u(−brn (v))
(4)
F n−1 (s)u0 (s − ball
rn (s)) ds.
See Appendix A.
We found an explicit expression for εball
1 (v), i.e., the leading-order effect of risk-aversion on
the equilibrium strategy. Roughly speaking, for a 10% level risk aversion, we calculated
the corresponding 10% change in the equilibrium strategy with 1% accuracy.
The “payoff” for the lengthy calculations needed to derive the explicit expression of the
equilibrium bids in Proposition 1, is that it enables us to derive some new results on the
role of weak risk aversion in all-pay auctions.6
We now can use the expression for ball (v) to show that risk aversion affects low type players
to bid less aggressively.
Corollary 1 In an all-pay auction the equilibrium bid of a weakly risk-averse player with
low type v (i.e., 0 ≤ v − v ¿ 1) is smaller than the equilibrium bid of a risk-neutral player
with type v.
Proof.
We prove this result by showing that ball
1 (v) < 0 if v is sufficiently close to v.
To see that, we first note that from (2) it follows that
0
n−2
(ball
(v)f (v).
rn ) (v) = (n − 1)vF
6
(5)
Deriving these results directly from the differential equation model (7), even if possible, would prob-
ably require considerably more work.
8
Differentiating ball
1 (v) in (4) and using (5) yields
¡
¢0
(v) = (n − 1)F n−2 (v)f (v)S(v),
ball
1
where
·
S(v) = −v F
n−1
0
(v)u (v −
ball
rn )
+ (1 − F
n−1
0
(v))u
¸
(−ball
rn )
all
+ u(v − ball
rn ) − u(−brn ).
In order to complete the proof it is sufficient to show that S(v) < 0 if v is sufficiently close
to v. Indeed, in this case,
S(v) ≈ −vu0 (−brn ) + u(v − brn ) − u(−brn ) = −v [u0 (−brn ) − u0 (x)] ,
where x ∈ (−brn , v − brn ). Since x > −brn , the concavity of u implies that u0 (−brn ) −
u0 (x) > 0.
¡
The following result shows that risk aversion affects high type players and low type players
quite differently.
Corollary 2 In an all-pay auction, the equilibrium bid of a weakly risk-averse player with
high type v (i.e., 0 ≤ v̄ − v ¿ 1) is higher than the equilibrium bid of a risk-neutral player
with type v.
Proof.
From Corollary 4 we have that b1st
1 (v̄) ≥ 0. In addition, in Proposition 8 we
1st
all
show that ball
1 (v̄) ≥ b1 (v̄). Therefore, b1 (v̄) ≥ 0. ¡
Because of the complex way that risk aversion affects the equilibrium bids, it is not
clear whether, overall, risk aversion leads to an increase or a decrease in the seller’s
expected revenue. In order to address this issue, we use the explicit expression obtained
in Proposition 1 to approximate the corresponding seller’s expected revenue:
9
Proposition 2 In an all-pay auction with weakly risk-averse players, the seller’s expected
revenue is given by
Rall = Rrn +
(Z
εn
v
v̄
"
#
(6)
¡
¢
n−1
(v))
+
1
−
F
(v)
u(−ball
F n−1 (v)u(v − ball
rn
rn (v)) f (v) dv
−
Z
v
v̄
F n−1 (v)(1 − F (v))u0 (v − ball
rn (v)) dv
)
+ O(ε2 ),
where Rrn is the expected revenue in the risk-neutral case.
Proof. See Appendix B.
Example 1 Consider n = 2 risk averse players with distribution functions F (v) = va in
[0, 1], such that U (x) = x − ²x2 . Substituting u(x) = −x2 in (6) and integrating gives
R
all
(2 − a) a2
∆R =
.
(2 + 5 a + 3 a2 ) (a + 2)
2
= Rrn + ²∆R + O(² ),
We thus see that depending on the value of a, ∆R can be either positive or negative.
Hence, we conclude that risk-aversion can lead to an increase, as well as to a decrease, of
the seller’s expected revenue in all-pay auctions.
We now use the explicit expression obtained in Proposition 1 to analyze the effect of weak
risk aversion on the players’ ex-ante utility.
Proposition 3 The weakly risk averse player’s (ex-ante) expected payoff in an all-pay
auction is given by
V
all
= Vrn + ε
Z
v
v
2
F n−1 (v)(1 − F (v))u0 (v − ball
rn (v)) dv + O(ε ),
where Vrn is the (ex-ante) expected payoff in the risk-neutral case.
10
Proof: See Appendix C.
Note that the difference between the expected payoffs of a weakly risk-averse player and
a risk-neutral player does not depends on the value of u, but depends on the value of u0 .
3
First-price auctions
Consider n players who compete to acquire a single object in a first-price auction. The
valuation of each player for the object v is independently distributed according to a
distribution function F (v) on the interval [v, v]. Each player places a bid b and the
highest player wins the object and pays her bid.
Since the equilibrium bid function b(v) is monotonically increasing (Maskin and Riley
(2000)), we can define the equilibrium inverse bid function as v = v(b). The maximization
problem of player i with valuation v is given by
maxVi = F n−1 (v(b))U (v − b).
b
Differentiating with respect to b gives
∂Vi
= (n − 1)F n−2 (v(b))f (v(b))v 0 (b)U (v − b) − U 0 (v − b)F n−1 (v(b)) = 0.
∂b
Therefore
1 F (v(b)) U 0 (v(b) − b)
v (b) =
,
n − 1 f (v(b)) U (v(b) − b)
0
(7)
where f = F 0 is the density. Since the lowest type v has zero utility, the initial condition
for equation (7) is given by
v(b = v) = v.
11
(8)
Equation (7) is exact in the risk-neutral case U(x) = c · x. In that case this equation can
be solved explicitly as follows
b1st
rn (v)
=v−
1
F n−1 (v)
Z
v
F n−1 (s) ds.
(9)
v
There are no such explicit solutions for a general utility function U, except for some very
simple models. Hence, we consider the case of weak risk aversion, i.e., when U is given
by (3). Under the assumption of weak risk aversion we can use perturbation analysis
to obtain explicit expressions of the equilibrium bid functions in first-price auctions as
follows.
Proposition 4 The symmetric equilibrium bid function in a first-price auction with weakly
risk-averse players is given by
1st
2
b1st (v) = b1st
rn (v) + εb1 (v) + O(ε ),
(10)
where b1st
rn (v) is the risk-neutral players’ equilibrium strategy (9),
b1st
1 (v)
−1
= n−1
F
(v)
Z
b1st
rn (v)
F
n−1
1st
(vrn
(b))
v
·
0
u
1st
(vrn
(b)
¸
1st
u(vrn
(b) − b)
− b) − 1st
db,
vrn (b) − b
(11)
1st
(b) is the inverse function of (9).
and vrn
Proof: See Appendix D.
The expression for b1st
1 (v) can be rewritten as follows.
Corollary 3
b1st
1 (v)
= u(v −
b1st
rn (v))
−
1
F n−1 (v)
12
Z
v
v
F n−1 (s)u0 (s − b1st
rn (s)) ds.
(12)
Proof. See Appendix E.
As with all-pay auctions, the “payoff” for the lengthy calculations which are needed to
derive the explicit expression of the equilibrium bids in Proposition 4, is that it enables
us to analyze the role of weak risk aversion in first-price auctions and to derive some
conclusions which, when derived directly from the differential equation model (7), requires
considerably more work.
The first consequence that we can obtain immediately from (11) is7
Corollary 4 In a first-price auction the equilibrium bid of every weakly risk-averse player
with type v is larger than the equilibrium bid of a risk-neutral player with type v.
Proof.
By the mean value theorem
1st
u0 (vrn
(b) − b) −
1st
u(vrn
(b) − b)
= u00 (ζ1 (b))ζ2 (b),
1st
vrn (b) − b
1st
0 ≤ ζ1 (b) ≤ ζ2 (b) ≤ vrn
(b) − b.
Thus, we can rewrite (11) as
b1st
1 (v)
−1
= n−1
F
(v)
Z
b1st
rn (v)
1st
F n−1 (vrn
(b)) [u00 (ζ1 (b))ζ2 (b) ] db.
v
Since u00 ≤ 0, it follows that b1st
1 (v) ≥ 0 for all v. ¡
An immediate consequence of Corollary 4 is that in the case of risk aversion the seller’s
expected revenue is higher than in the risk-neutral case. Indeed, we have the following
result:
7
This result was established by Maskin and Riley (1984) for the general case (i.e., without the as-
sumption of a weak risk-aversion).
13
Proposition 5 The seller’s expected payoff in a first-price auction with weakly risk-averse
players is given by is given by
Z "Z 1st
v̄
brn (v)
R1st (ε) = Rrn + εn
v
v
¶ #
1st
u(v
(b)
−
b)
rn
1st
1st
F n−1 (vrn
(b))
− u0 (vrn
(b) − b) db f (v) dv
1st (b) − b
vrn
µ
+O(ε2 )
= Rrn + εn
(13)
Z
v̄
F
n−1
v
·
¸
1st
0
1st
(v) f (v)u(v − brn (v)) − (1 − F (v))u (v − brn (v)) dv + O(ε2 ),
1st
(b) is the inverse
where Rrn is the expected seller’s revenue in the risk-neutral case and vrn
function of (9).
Let
Proof.
R
1st
=n
Z
v̄
b(v)F n−1 (v) f(v) dv
(14)
v
be the seller’s expected revenue in a first-price auction. Substituting (10) in (14) gives
R
1st
=n
Z
v̄
(brn + εb1 ) F
n−1
2
(v) f (v) dv + O(ε ) = Rrn + εn
v
Z
v̄
b1 F n−1 (v) f (v) dv + O(ε2 ).
v
Substitution of (11) completes the proof of the first equality. Substitution of (12) and
integration by parts leads to the second equality. ¡
We now use the explicit expression obtained in Proposition 4 to analyze the effect of
weak risk aversion on the players’ ex-ante utility.
Proposition 6 The weakly risk-averse player’s (ex-ante) payoff in a first-price auction
is given by
V
1st
(ε) = Vrn + ε
Z
v
v
2
F n−1 (v)(1 − F (v))u0 (v − b1st
rn (v)) dv + O(ε ),
14
where Vrn =
neutral case.
Rv
v
F n−1 (v) (v − b1st
rn (v)) f(v) dv is the (ex-ante) expected payoff in the risk-
Proof: See Appendix F.
4
First-price auctions versus all-pay auctions
One way to gain insight into all-pay auctions is to compare them with first-price auctions.
Since in an all-pay auction a player pays her bid whether or not she wins whereas in a
first-price auction she pays only if she wins, it seems natural that players will bid more
carefully (i.e., have lower bids) in all-pay auctions than in first-price auctions. Indeed, it
is well known that the bid of a risk-neutral player in an all-pay auction is smaller than
her bid in a first-price auction and we can expect this relation to be even stronger for
risk-averse players. However, as we show in Propositions 7 and 8, the relation of bids in
first-price and all-pay auctions with risk-averse players is not so simple.
Proposition 7 The equilibrium bid of a weakly risk-averse player with sufficiently low
type v in an all-pay auction is smaller than her bid in a first-price auction, that is,
ball (v) ≤ b1st (v)
for
0 ≤ v − v ¿ 1.
1st
Proof. It is well known that in the risk-neutral case for every type v, ball
rn (v) ≤ brn (v).
In addition, from Corollary 4 and Corollary 1 we have that for v sufficiently close to v,
all
b1st
1 (v) ≥ 0 and b1 (v) ≤ 0. ¡
On the other hand,
15
Proposition 8 The equilibrium bid of a weakly risk-averse player with sufficiently high
type v in an all-pay auction is larger than in a first-price auction. In particular,
ball (v̄) > b1st (v̄).
1st
all
1st
Proof: Since ball
rn (v̄) = brn (v̄), it is sufficient to show that b1 (v̄) ≥ b1 (v̄). From (12)
and (4),
ball
1 (v̄)
−
b1st
1 (v̄)
=−
Z
v
F
v
n−1
·
0
(v) u (v −
ball
rn (v))
0
− u (v −
¸
b1st
rn (v))
dv.
all
0
In addition, since b1st
rn (v) ≥ brn (v) for all v, by the concavity of u we have that u (v −
0
1st
ball
rn (v)) ≤ u (v − brn (v)). Hence, the result follows. ¡
We thus conclude that there is no dominance relation among the bids in first-price
and all-pay auctions. Nevertheless, first-price auctions dominate all-pay auctions from
the player’s point of view:
Proposition 9 The (ex-ante) expected payoff of every player in the first-price auction is
larger than her (ex-ante) expected payoff in the all-pay auction.
Proof.
By the Revenue Equivalence Theorem, the expected payoff of a risk-neutral
player with valuation v is the same in first-price auctions and all-pay auctions. Thus,
we obtain that the difference between her expected payoffs in these auctions in the case
where all types are weakly risk-averse is
V
1st
−V
all
=²
Z
v
v
0
all
2
F n−1 (v)(1 − F (v))(u0 (v − b1st
rn (v)) − u (v − brn (v)) + O(² ).
16
all
0
1st
Since v − b1st
rn (v) ≤ v − brn (v) for all v and since u is concave, then u (v − brn (v)) ≥
1st
u0 (v − ball
− V all > 0.¡
rn (v)) and therefore V
We now show that there is no simple dominance of the seller’s expected revenue. We have
seen in Example 1 that when a > 2 risk-aversion lowers the seller’s expected revenue in
all-pay auctions. Since risk-aversion always increases the revenue in first-price auctions,
the case a > 2 illustrates that due to weak risk-aversion, R1st > Rall . The following
example shows that the opposite is also possible.
Example 2 Consider n = 2 risk averse players with distribution functions F (v) = v in
[0, 1], such that U (x) = x − ²x3 . Substituting u(x) = −x3 in (13) and integrating gives
R1st = Rrn + ²
1
+ O(²2 ).
40
Substituting u(x) = −x3 in (6) and integrating gives that
Rall = Rrn + ²
2
+ O(²2 ).
35
Therefore, in this case, due to weak risk-aversion, R1st < Rall .
5
Conclusion
The results of the perturbation analysis can be illustrated by the following example.
Consider two players where each player’s valuation is distributed on [0, 1] according to
the distribution function F (v) = v α . Assume that each player’s utility function is U (x) =
x − εx2 .
17
ε = 0.5
0.6
0.6
0.5
0.5
0.4
0.4
bids
bids
ε = 0.25
0.3
first−price
all−pay
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
0
1
first−price
0
0.2
all−pay
0.4
v
0.6
0.8
v
Figure 1: Bids of risk-averse buyers (solid lines) and their explicit approximations (dotted
lines) in first-price and all-pay auctions.
From Proposition 4 the equilibrium bid function in the first-price auction is given by
b1st (v) =
α
α
v+ε
v2 + O(ε2 ).
1+α
(1 + α)2 (2 + α)
(15)
Similarly, from Proposition 1 the equilibrium bid function in the all-pay auction is given
by
µ
¶
α 1+α
α 2+α
α 2+2α
+ε −
+
+ O(ε2 ).
b (v) =
v
v
v
1+α
2+α
1+α
all
(16)
In Figure 1 we compare the approximations (15) and (16) with the exact bid functions
(i.e., the numerical solutions of equations (7) and (1), respectively), for the case α = 1
(i.e., uniform distribution). At ² = 0.25, the approximations are almost indistinguishable
from the exact bids. Although when ² = 0.5 the risk-aversion parameter is not small,8 the
8
In fact, ² = 0.5 is the largest possible value of ² for which U = x − ²x2 is monotonically increasing.
18
1
agreement between the explicit approximations and the exact values is quite remarkable.
Such good agreement was also observed in numerous other comparisons that we made
with different distribution functions and utility functions.
In Figure 2 we compare the (exact) bid functions in the risk-averse and the risk-neutral
cases. As predicted by perturbation analysis, under risk aversion the bids increase for all
types in first-price auctions, whereas in all-pay auctions risk aversion lowers the bids of
the low types but increases the bids of the high-types. In addition, under risk aversion
the low types bid less aggressively in all-pay auctions than in first-price auctions, but the
high types bids more aggressively in all-pay auctions than in first-price auctions.
A natural question that arises is whether in the case of weak-risk aversion one cannot simply approximate the bidding functions using the risk-neutral expressions. In other words,
all
when ² is small, is there an advantage for the approximation ball (v; ²) ≈ ball
rn (v) + ²b1 (v)
over the continuous approximation ball (v; ²) ≈ ball (v; ² = 0) = ball
rn (v)? The answer is that
the accuracy of the first approximation is O(²2 ), whereas that of the second approximation
is only O(²). Therefore, the first approximation is significantly more accurate when ² is
moderately small (but not negligible). Indeed, comparison of Figures 1 and 2 shows that
the (exact) bids in the risk-averse case are well-approximated with the explicit approximation that we derived, but are not well-approximated with the bids in the risk-neutral
case.
19
ε = 0.5
0.6
0.6
0.5
0.5
0.4
0.4
bids
bids
ε = 0.25
0.3
first−price
all−pay
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
0
1
first−price
0
v
0.2
all−pay
0.4
0.6
0.8
v
Figure 2: Bids of risk-averse buyers (solid lines) and of risk-neutral buyers (dashed lines)
in first-price and all-pay auctions.
A
Proof of Proposition 1
We can write the equilibrium bid as v(b) = vrn (b) + εv1 (b) + O(ε2 ), where vrn (b) is the
inverse function of the risk-neural equilibrium strategy in all-pay auctions (2). For clarity,
we drop the superscript all. We first note that when ε ¿ 1,
F (v(b)) = F (vrn ) + εv1 F 0 (vrn ) + O(ε2 ),
f (v(b)) = f(vrn ) + εv1 f 0 (vrn ) + O(ε2 ),
U (v(b) − b) − U (−b) = v(b) + ε[u(v(b) − b) − u(−b)]
= vrn (b) + ε[v1 (b) + u(vrn (b) − b) − u(−b)] + O(ε2 ),
U 0 (v(b) − b) − U 0 (−b) = ε[u0 (v(b) − b) − u0 (−b)] = ε[u0 (vrn (b) − b) − u0 (−b)] + O(ε2 ).
Substitution in (1) and expanding in a power series in ε, the equation for the O(1) term is
20
1
identical to the one in the risk-neutral case and therefore is automatically satisfied. The
equation for the O(ε) terms is
v10 (b) =
F (vrn (b))[u0 (vrn (b) − b) − u0 (−b)]
u0 (−b)
+
(n − 1)f (vrn (b))vrn (b)
(n − 1)F n−2 (vrn (b))f (vrn (b))vrn (b)
v1 f 0 (vrn (b))
(n − 2)v1 (b)
−
−
(n − 1)F n−1 (vrn (b))vrn (b) (n − 1)F n−2 (vrn (b))f 2 (vrn (b))vrn (b)
[v1 (b) + u(vrn (b) − b) − u(−b)]
−
,
2 (b)
(n − 1)F n−2 (vrn (b))f (vrn (b))vrn
subject to v1 (0) = 0. Since, by (1),
1
,
(n − 1)F n−2 (vrn (b))f(vrn (b))vrn (b)
0
vrn
(b) =
(17)
the equation for v10 (b) can be rewritten as
v10 (b) + v1 (b)B(b) = G(b)
(18)
where
·
¸
0
vrn
(b) f 0 (vrn (b)) 0
f (vrn (b)) 0
B(b) =
+
v (b) + (n − 2)
v (b) ,
vrn (b)
f(vrn (b)) rn
F (vrn (b)) rn
and
G(b) =
(
0
(b)
vrn
−
·
¸
0
u(vrn (b) − b) − u(−b) (n − 1)F n−2 (vrn (b))f (vrn (b))vrn
(b) (19)
+ F
n−1
!
)
µ
0
0
0
(vrn (b)) u (vrn (b) − b) − u (−b) + u (−b) .
The solution of (18) is given by
R b̄rn
v1 (b) = e
b
B
Ã
C1 −
Z
b
21
b̄rn
G(x)e−
R b̄rn
x
B
!
dx ,
where b̄rn = brn (v̄). It is easy to verify that (see (17))
R b̄rn
e
R b̄rn
Thus, as b → 0, vrn (b) → v and e
b
B
b
B
0
(b)
vrn
.
0
vrn (b̄rn )
=
→ ∞. Therefore it follows that C1 =
and that
v1 (b) =
0
vrn
(b)
Z
R b̄rn
0
G(x)e−
R b̄rn
x
b
0
G(x)/vrn
(x) dx.
0
0
Since b1 (v) = −v1 /vrn
(b), see (21), we get that
b1 (v) = −
Z
brn (v)
0
G(x)/vrn
(x) dx.
0
Substitution of G from (19) gives
b1 (v) =
Z
0
brn (v)
(
¸
·
u(vrn (b) − b) − u(−b) (F n−1 (vrn (b))0
!
)
µ
−F n−1 (vrn (b)) u0 (vrn (b) − b) − u0 (−b) − u0 (−b) db.
A few more technical calculations completes the proof.
B
Proof of Proposition 2
The seller’s revenue is given by Rall = n
we have
R
all
= n
Z
R v̄
v
b(s)f (s) ds. Substituting b = brn + εb1 + O(ε2 ),
v̄
2
(brn + εb1 ) f (s)ds + O(ε ) = n
v
Z v̄
b1 f(s) ds + O(ε2 ).
= Rrn + εn
v
22
Z
v
v̄
brn f (s)ds + εn
Z
v
v̄
b1 f (s)ds + O(ε2 )
B
dx
Substituting b1 from (4) yields
Z
Z
v̄
Z
v̄
v̄
(1 − F (v))u(−brn (v))f(v) dv +
F n−1 (v)u(v − brn (v))f(v) dv
v
v
¸
Z v̄ ·Z v
−
F n−1 (s)u0 (s − brn (s)) ds f (v) dv.
b1 f (s) ds =
v
n−1
v
v
Integrating by parts the double integral gives
Z
v̄
v
·Z
v
F
n−1
v
¸
Z
(s)u (s − brn (s)) ds f (v) dv =
0
v
v̄
F n−1 (v)(1 − F (v))u0 (v − brn (v)) dv.
Therefore, the result follows.
C
Proof of Proposition 3
The ex-ante expected utility for the players in all-pay auctions in equilibrium is given by
V all =
Z
v
v̄
(
)
F n−1 (v)U(v − b(v)) − [1 − F n−1 (v)]U(−b(v)) f (v) dv.
In the case of weak risk aversion (3),
·
¸)
Z v̄ (
all
n−1
n−1
F
(v)v − b(v) + ε F
(v) (u(v − b(v)) − u(−b(v))) + u(−b(v)) f(v) dv + O(ε2 ).
V =
v
Using the relation b(v) = brn (v) + εb1 (v) + O(ε2 ), we have
V
all
=
all
Vrn
−ε
Z
v
v̄
©
£
¤ª
b1 (v) − F n−1 (v) (u(v − brn (v)) − u(−brn (v))) + u(−brn (v)) f (v) dv + O(ε2 ).
Substituting (4) in the last equation and rearranging yields the result.
23
D
Proof of Proposition 4
We need the following Lemma:9
Lemma 1 Let v(b) be the inverse bid function in a first-price auction that satisfies the
conditions of Proposition 4. Then, v(b) = vrn (b) + εv1 (b) + O(ε2 ), where vrn (b) is the
inverse function of (9) and
v0 (b)
v1 (b) = n−1rn
F
(vrn (b))
Z
b
F
n−1
v
·
¸
u(vrn (b) − b)
(vrn (b)) u (vrn (b) − b) −
db.
vrn (b) − b
0
(20)
In addition, we note that if we differentiate the identity v = v(b(v; ε); ε) with respect to
ε and set ε = 0, we get that
0
v1 (brn (v)) + vrn
(brn (v))b1 (v) = 0.
(21)
0
in (21) completes the proof of Proposition 4.
Substitution of v1 from (20) and vrn
D.1
Proof of Lemma 1
Substituting (3) in (7) yields
F (v(b))
1 + εu0 (v(b) − b)
v (b) =
.
(n − 1)f (v(b)) (v(b) − b + εu(v(b) − b))
0
(22)
We can write the equilibrium bid as v(b) = vrn (b) + εv1 (b) + O(ε2 ), where vrn (b) is the
inverse function of the risk-neural equilibrium strategy in first-price auctions (9). We first
9
For clarity, we drop the superscript 1st.
24
note that when ε ¿ 1,
F (v(b)) = F (vrn (b)) + εv1 (b)F 0 (vrn (b)) + O(ε2 ),
(23)
f(v(b)) = f (vrn (b)) + εv1 (b)f 0 (vrn (b)) + O(ε2 ),
εu(v(b) − b) = εu(vrn (b) − b) + O(ε2 ),
εu0 (v(b) − b) = εu0 (vrn (b) − b) + O(ε2 ).
Substituting v(b) = vrn (b) + εv1 (b) + O(ε2 ) and (23) in (22) and expanding in a power
series in ε gives,
µ
¶
1 F (vrn(b)) + εv1 (b)f(vrn (b))
f 0 (vrn (b))
(vrn ) (b) + ε (v1 ) (b) =
1 − εv1 (b)
×
n−1
f(vrn (b))
f (vrn (b))
µ
¶
1 + εu0 (vrn(b) − b)
v1 (b) + u(vrn (b) − b)
1−ε
+ O(ε2 ).
vrn (b) − b
vrn (b) − b
0
0
By construction, the equation for the O(1) terms is identical to the risk-neutral case and
therefore is automatically satisfied. The equation for the O(ε) terms is
¸
·
1 F (vrn (b)) u0 (vrn (b) − b) v1 (b) + u(vrn (b) − b)
(v1 ) (b) =
−
n − 1 f(vrn (b))
vrn (b) − b
(vrn (b) − b)2
F (vrn (b))f 0 (vrn (b))
1
1
1
1
v1 (b)
+
v1 (b)
,
−
2
n−1
f (vrn (b))
(vrn (b) − b) n − 1
(vrn (b) − b)
0
subject to the initial condition v1 (v) = 0. This equation can be rewritten as
(v1 )0 (b) + v1 (b)A(b) = D(b),
where
µ
¶
1
F (vrn (b))f 0 (vrn (b))
F (vrn (b))
A(b) =
−1+
,
(n − 1) (vrn (b) − b)
f 2 (vrn (b))
f(vrn (b)) (vrn (b) − b)
µ
¶
F (vrn (b))
u(vrn (b) − b) u0 (vrn (b) − b)
D(b) =
−
+
.
(n − 1)f (vrn (b))
vrn (b) − b
(vrn (b) − b)2
25
The solution of this equation is
v1 (b) = e
R b̄rn
b
A
Ã
C−
Z
b̄rn
D(x)e−
R b̄rn
x
A
!
dx ,
b
where C is a constant and b̄rn = brn (v̄) = v̄ −
R v̄
v
(24)
F n−1 (s) ds is the maximal bid in the
risk-neutral case, which is obtained from (9). It can be verified that
R b̄rn
e
R b̄rn
Therefore lim e
b→v
b
A
b
A
R v̄
f (v̄)F (vrn (b)) v F n−1 (s) ds
=
.
R v (b)
f (vrn (b)) v rn F n−1 (s) ds
= ∞. We thus conclude that C =
F (vrn (b))
v1 (b) =
R v (b)
(n − 1)f(vrn (b)) v rn F n−1 (s) ds
Z
b
F
v
n−1
R b̄rn
v
·
D(x)e−
R b̄rn
x
A
dx. Therefore
¸
u(vrn (b) − b)
db.
(vrn (b)) u (vrn (b) − b) −
vrn (b) − b
0
0
The proof is completed since in the risk-neutral case we have from (7) that vrn
(b) =
F (vrn (b))
1
,
(n−1)f (vrn (b)) vrn (b)−b)
which combined with (9) gives that
0
vrn
(b) =
E
F n (vrn (b))
.
R v (b)
(n − 1)f (vrn (b)) v rn F n−1 (s) ds
Proof of Corollary 3
Making the change of variables s = vrn (b) in (11) gives,
b1 (v) =
Z
v
u(s − brn (s)) dbrn (s)
ds −
s − brn (s)
ds
v
Z v
1
dbrn (s)
ds.
F n−1 (s)u0 (s − brn (s))
n−1
F
(v) v
ds
1
F n−1 (v)
F n−1 (s)
26
(25)
Substituting
dbrn (v)
dv
= (n − 1) Ff (v)
(v − brn (v)) ( see (7)) in the first integral and integrating
(v)
by parts gives
(n − 1)
=
F n−1 (v)
Z
v
1
Z
v
(s)f(v)u(s − brn (s)) ds − n−1
F n−1 (s)u0 (s − brn (s))
F
(v) v
v
¶
µ
Z v
1
dbrn(s)
n−1
0
= u(v − brn (v)) − n−1
F
(s)u (s − brn (s)) 1 −
ds
F
(v) v
ds
Z v
1
dbrn (s)
− n−1
F n−1 (s)u0 (s − brn (s))
ds
F
(v) v
ds
Z v
1
= u(v − brn (v)) − n−1
F n−1 (s)u0 (s − brn (s)) ds.
F
(v) v
b1
F
F
n−2
dbrn (s)
ds
ds
Proof of Proposition 6
The ex-ante expected utility for the players in first-price auctions in equilibrium is given
by
V
all
=
Z
v̄
v
F n−1 (v)U (v − b(v))f (v) dv.
In the case of weak risk aversion (3), V 1st =
R v̄
v
F n−1 (v) [v − b(v) + εu(v − b(v))] f(v) dv +
O(ε2 ). If we substitute b(v) = brn (v) + εb1 (v) + O(ε2 ) and expand in a power series in ε
we get that
V
1st
=
1st
Vrn
2
+ εG + O(ε ),
G=
Substituting b1 (v) from (11) yields
ÃZ
Z (
v̄
brn (v)
G=
v
v
Z
v̄
v
F n−1 (v) [−b1 (v) + u(v − brn (v))] f (v) dv.
¸ !
u(v
(b)
−
b)
rn
F n−1 (vrn (b)) u0 (vrn (b) − b) −
db
vrn (b) − b
)
·
+F n−1 (v)u(v − brn (v)) f(v) dv.
27
(26)
In order to simplify the expression for G we first note that
¸
u(vrn (b) − b)
parts
F
(vrn (b)) −
db =
v
(b)
−
b
rn
v
¯brn (v)
Z x
¯
1
n−1
F
(vrn (b))
+
− u(vrn (b) − b)
db¯¯
vrn (b) − b v
v
¶
Z brn (v) µZ b
1
n−1
0
F
(vrn(s))
ds u0 (vrn (b) − b) [vrn
(b) − 1] db.
v
(s)
−
s
rn
v
v
Z
A :=
·
brn (v)
n−1
From (9) we have that in a first-price auction
1
F n−1 (vrn (b))
.
= R vrn (b)
vrn (b) − b
F n−1 (s) ds
v
Using this relation and (25), we have that
Z
b
F
n−1
v
=
Z
1
ds =
(vrn (s))
vrn (s) − s
Z
v
b
b
F n−1 (vrn (s))
ds
F n−1 (vrn (s)) R vrn (s)
n−1 (x)dx
F
v
0
(n − 1)F n−2 (vrn (s))f (vrn (s))vrn
(s) ds = F n−1 (vrn (b)).
v
Substituting this relation in the expression for A gives,
A = −u(v − brn (v))F
n−1
(v) +
Z
v
brn (v)
0
F n−1 (vrn (b))u0 (vrn (b) − b) [vrn
(b) − 1] db.
Substituting this relation in (26), we have
G =
Z
v̄
v
ÃZ
µZ
!
brn (v)
0
F n−1 (vrn (b))u0 (vrn (b) − b)vrn
(b) db f (v) dv
v
¶
=
F
(s)u (s − brn (s)) ds f(v) dv
v
v
¯v̄ Z v̄
µZ v
¶
¯
n−1
0
=
F
(s)u (s − brn (s), ds F (v)¯¯ −
F n (v)u0 (v − brn (v))dv
Z
=
Z
v
v̄
v
n−1
0
v
v
v̄
F n−1 (v)(1 − F (v))u0 (v − brn (v)) dv.
28
v
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