Asymmetric All-Pay Auctions with Incomplete Information, the Two

VERY PRELIMINARY, COMMENTS ARE WELCOME
ASYMMETRIC ALL-PAY AUCTIONS WITH INCOMPLETE INFORMATION,
THE TWO PLAYER CASE REDUX
´
JINGLING LU1 AND SERGIO
O. PARREIRAS2
A BSTRACT. We re-visit the all-pay auction model of Amann and Leininger (1996) allowing for
interdependent values and correlation a` la Lizzeri and Persico (1998) and Siegel (2013). We
study both monotone and non-monotone pure strategy equilibria (MPSE and NPSE). For MPSE
with continuous signals: First, we show the allocation and bidding strategies of MPSE can be
obtained in the same manner as in the independent private values environment. For correlated
private values, the allocation is the same regardless of correlation. For common-values, the
allocation sorts players according to their signals’ percentiles. Second, we present a local
single-crossing condition which is necessary for MPSE, and the standard single crossing, which
is sufficient. Third, we exhibit families of common-value, all-pay auctions that violate the local
single-crossing and thus lack MPSE. Also, we construct a correlated private values example,
where the slightest amount of correlation breaks down MPSE that exists under independence.
Finally, we explicitly obtain NPSE for quadratic interdependent valuations in cases where no
MPSE exists.
1. I NTRODUCTION
In rent-seeking contests, distinct individuals may entertain different estimates of the prize.
Such estimates maybe of varying precision or accuracy; possibly they maybe interdependent
and/or correlated.
Here, we model rent-seeking contests as all-pay auctions. Our aim here is limited to
provide a tractable characterization of pure strategy (monotone or not) equilibria (henceforth
MPSE and NPSE) of the (first-price) all-pay auction with two, possibly asymmetric, players
with interdependent valuations and correlated signals.
Our model can be viewed either as an extension of Amann and Leininger (1996) as we add
correlation and interdependent values and an specialization of Lizzeri and Persico (1998) to
the all-pay auction. It is also closely related to Siegel (2013) who studies a discrete signals
Date: First Version: August, 2013 (This version, March 2014).
We are grateful to suggestions, discussions and/or help from Luciano de Castro, David A. Cox, Nicolas
Figueiroa, Fei Li, Humberto Moreira, David McAdams, Edgardo Cheb-Terrab, Lixin Ye and seminar participants
at Ohio State University. S´ergio Parreiras thanks the hospitality of the NUS, Economics Department (August,
2013) and the CEMS-Kellogg-NWU (Winter-Spring, 2014). All error are ours.
1. National University Singapore. E-mail: [email protected].
2. UNC at Chapel Hill. E-mail: [email protected].
1
2
model. We also study non-monotone equilibria which starkly departs from previous studies
of the all-pay auction, with the exception of Araujo et al. (2008).
As Siegel (2013), we do not restrict attention neither to affiliated signals, as in Lizzeri and
Persico (1998) nor to independent signals as in, Araujo et al. (2008). We allow for positive or
negative correlated signals. Speaking plainly, affiliation is mostly an useless assumption –
in the context of all-pay auctions with interdependent valuations. As we show in Section 4,
there are economically interesting all-pay auctions that lack MPSE when signals are strictly
affiliated but do have MPSE when signals are independent. Moreover, in Lemma 1, we
prove that any equilibrium of the all-pay auction with correlated signals must also be an
equilibrium of some all-pay auction with interdependent valuations and independent signals.
After Lemma 1 is established, the characterization of MPSE is a straightforward application
of the recursive algorithm of Amann and Leininger (1996).1 It first solves for the allocation
rule or tying function and next computes bid functions. Siegel (2013) develops an analogous
version for discrete type spaces.
In any MPSE, the allocation rule (i.e. the assignment of the object given the signal of the
players) only depends on the players’ expected values for the object conditional on their
signals. In particular: For the correlated private values environment, the allocation rule is
the same regardless of the nature of the correlation; it coincides with the allocation derived
by Amann and Leininger (1996) for independent private values. For the common-value
environment, the allocation is dictated by the percentiles of the distribution of the agents’
signals; That is, the type of agent 1 who gets a signal in the p-percentile bids the same amount
of that the type of agent 2 who gets a signal in the p-percentile.2
2. T HE M ODEL
There are two agents, i = 1, 2. Let Vi be the random variable describing the value of the
object for player i. Let X1 and X2 be the agents’ signals. The conditional expected value
is vi ( x, y) = E[Vi | X1 = x, X2 = y]. The cumulative distribution of Xi is Fi and, Fi| j is the
conditional cumulative distribution of Xi given X j . The lower-case f denotes the respective
probability density function. Finally, we also define
def
λi ( x, y) = E[Vi | Xi = x, X j = y] · f Xj |Xi (y| x ) where i, j = 1, 2 and j , i.
1
See also Parreiras (2006).
2In particular, if signals are conditionally independent, the winning probability of both agents are the same.
Einy et al. (2013) and Warneryd (2013) independently obtain this corollary. They study common-values models
where one agent’s signal is a sufficient statistic for the other’s signal and hence, signals are conditionally
independent.
3
We assume:
CONTINUITY:
FXi is absolutely continuous.
FULL SUPPORT:
For all ( x, y) ∈ [0, 1]2 , f X1 ,X2 ( x, y) > 0.
UNIFORM MARGINALS :
Without any loss of generality, Xi ∼ U[0, 1].3
As we are interested in non-monotone equilibria of the all-pay auction, unlike the previous
literature we do not assume λi ( x, y) increasing in x.4
With interdependent valuations, there is no loss of generality in assuming independent
signals in the context of the all-pay auction as the reasoning below shows.
THE FICTITIOUS AUCTION :
Given an all-pay auction, the corresponding fictitious (or auxiliary)
auction is the all-pay auction where signals are independently and uniformly distributed on
def
the unit interval, and expected conditional valuations are v˜i ( x, y) = λi ( x, y).
Lemma 1. The fictitious auction and the original auction are payoff equivalent.
Proof of lemma 1. Pick any strategy profile b = (b1 , b2 ) then
R
R
˜ i (b| x ) =
U
v˜i ( x, y)dy − b =
vi ( x, y) f j|i (y| x )dy − b = Ui (b| x ), i = 1, 2.
{y:b j (y)≤b}
{y:b j (y)≤b}
Best reply correspondences in the fictitious and original auctions coincide, so do equilibria
sets.
3. M ONOTONE E QUILIBRIUM
Let φ1 and φ2 denote the inverse bidding functions of some monotone equilibrium. Define,
as in Amann and Leininger (1996) or Parreiras (2006), the tying function Q (i.e. the equilibrium
allocation rule) as the function that maps the type of player 1 to the type of player 2 that bids
def
the same in equilibrium, that is Q(φ1 (b)) = φ2 (b).
Proposition 1. The tying function solves the differential equation,
Q0 ( x ) =
Also b1 ( x ) =
Rx
0
v2 ( x, Q( x ))
v1 ( x, Q( x ))
and
Q(1) = 1.
v2 (z, Q(z)) f 1|2 (z| Q(z)) dz.
Proof. First-order conditions for an optimal bid are
v1 ( x, φ2 (b)) f 2|1 (φ2 (b)| x )φ20 (b) − 1 = 0
and
v2 (φ1 (b), y) f 1|2 (φ1 (b)|y)φ10 (b) − 1 = 0. (3.1)
3Say S is the original signal, re-parametrize signals by taking as the new signal, X = F (S ) .
Si i
i
i
4See Amann and Leininger (1996),Krishna and Morgan (1997), Lizzeri and Persico (1998), Araujo et al. (2008),
and Siegel (2013).
4
Combine the first-order conditions with the identity Q0 · φ10 = φ20 and remember that, since
wlog. signals are uniformly distributed in the unit interval, the conditional density coincides
with the joint density by Baye’s rule.
Under some assumptions, the equilibrium described by Proposition 1 is unique:
Proposition 2. Assume that v1 (·) is bounded away from zero and, v1 (·) and v2 (·) are continuous
in the signal of player 1 and continuously differentiable in the signal of player 2. If a continuous,
monotone equilibrium exists, it is the unique.
Proof. Since the space of signals is compact and by assumption, the derivative of
v2 ( x,y)
v1 ( x,y)
with
respect to y is continuous, it satisfies the Lipschitz condition: exists K > 0 such that
v2 ( x, y) v2 ( x, yˆ ) 2
v ( x, y) − v ( x, yˆ ) ≤ K · |y − yˆ | for all ( x, y) ∈ [0, 1] .
1
1
the differential equation characterizing the tying function has a unique maximal solution
that satisfies the boundary condition Q(1) = 1. By Proposition 1, the uniqueness of the bid
functions follows from the uniqueness of Q.
Notice that distributions with unbounded support will typically violate the assumptions
of Proposition 2 because valuations are not continuous at the boundary of the signal space.
Proposition 1 says that the interdependence of valuations, as opposed to the correlation
between the signals, is the only factor that matters for determining the tying function. We
illustrate this remark in a couple of interesting environments:
Corollary 1.
CORRELATED PRIVATE VALUES .
In any monotone equilibrium, the tying function is
the identical to the tying function of the independent private values environment.
Corollary 2.
COMMON - VALUES .
In any monotone equilibrium, the tying function is the identity.
Without re-scaling signals, in the common value environment, the tying function is given
byQ( x ) = F2−1 ( F1 ( x )). In the statistical literature, this function is also known as the quantilequantile plot, or simply Q − Q plot.
To establish existence (or not) of a monotone equilibrium we need to define:
( LOCAL SINGLE CROSSING ) At ( x, y) = ( x, Q( x )), λ1 ( x, y) is non-decreasing in x and λ2 ( x, y)
is non-decreasing in y, for all x.
( SINGLE
CROSSING )
˜ λ1 ( x,
ˆ Q( x )) < λ1 ( x, Q( x )) < λ1 ( x,
˜ Q( x )) and
For all xˆ < x < x:
˜ λ2 (y,
ˆ Q−1 (y)) < λ2 (y, Q−1 (y)) < λ2 (y,
˜ Q−1 (y)).
analogously, for all yˆ < y < y,
Proposition 3. The local single-crossing is necessary and the single-crossing is sufficient for b1 ( x ) =
Rx
−1
0 v2 ( z, Q ( z )) f 1|2 ( z | Q ( z ) dz and b2 ( y ) = b1 ( Q ( x )) to be an (monotone) equilibrium.
5
Proof. The function vi (z, φj (b)) f j|i (φj (b)|z)φ0j (b) − 1 satisfies the local single-crossing condition with respect to z if and only if vi ( x, φj (b)) f 1,2 (z, φj (b)) is non-decreasing in z. Again
remember that f 1 = f 2 = 1 in their respective supports. Differentiating the identity,
vi (φi (b), φj (b)) f j|i (φj (b)|φi (b))φ0j (b) − 1 = 0, with respect to b, and assuming φi0 > 0, the local
single-crossing in z at φi (b) is equivalent to the second-order condition for i’s optimal bid.
The local single crossing is thus clearly necessary. The argument to establish single-crossing
is sufficient for a monotone eq. is standard:5 single-crossing implies that, at b = b1 ( x ),
∂Ui
∂Ui
∂Ui
(b| xˆ ) <
(b| x ) = 0 <
(b| x˜ ).
∂b
∂b
∂b
Our model maybe viewed as a continuous version of Siegel (2013)’s discrete signals model.
There, the assumption λi ( x, y) increases in x for all y guarantees the single-crossing condition.
Here, however, as we explicitly characterize the tying function, the conditions for existence
of a monotone equilibrium are weaker as the example below illustrates.
y
Example 1. Consider the symmetric model v( x, y) = x + 2 · (3 − 4x + 2x2 ) with independent
signals x, y ∈ [0, 1]. Since v x ( x, y) = 1 − 2y · (1 − x ), we have v( x, y) is not increasing in x for
all y. Nonetheless, v(z, x ) − v( x, x ), as a function of z, crosses zero only once for all x ∈ (0, 1),
the single-crossing condition holds.
4. N ON -E XISTENCE OF M ONOTONE E QUILIBRIUM
When signals are correlated, the all-pay auction may lack monotone equilibria.
Example 2. (C ORRELATED P RIVATE VALUES ) The signals ( X1 , X2 ) follow a truncated, symmetric, bivariate normal distribution specified by (µ, σ2 , ρ)and truncation points µ − M
and µ + M. The expected value of the object for player i is exp (h( Xi )) where h is a given
increasing function and i = 1, 2.
Proposition 4. Example 2 does not have a monotone pure strategy equilibrium if h0 ( x ) ≥
2ρ
(µ −
σ 2 (1+ ρ )
x ) for some x.
Proof. As players are symmetric, by Proposition 1, if a monotone, pure strategy equilibrium exists then it must be symmetric. However,
using the fact that X j | Xi ∼ N (1 − ρ)µ + ρXi , (1 − ρ2 )σ2
∂
2ρ
we obtain
vi ( x, y) · f Xj |Xi (y| x )
> 0 ⇔ h0 ( x ) > 2
( µ − x ).
∂x
σ (1 + ρ )
y= x
As a result, for a large class of examples, the symmetric monotone equilibrium is not robust
to the introduction of a small degree of correlation:
5See Krishna and Morgan (1997, p. 351), or Lizzeri and Persico (1998, p. 104), or Athey (2001).
6
b
β
b(·)
φ1 ( β)
φ2 ( β)
φ3 ( β)
x
φ4 ( β)
F IG . 1. A piecewise monotone strategy and its local inverse bids.
Corollary 3. Assume k h0 k∞ < K then for any ρ > 0 there is M > 0 such that the private values
model of example 1 has no monotone equilibrium.
Example 3. (C OMMON -VALUE ) In all three families given by the table below: signals and the
value are affiliated; the parameter θ measures the precision of the players’ information; and
there is no monotone equilibrium.
V
Si | V
ln N (µ, τ −1 ) N (V, θ −1 )
Pareto(ω, α) V · B(θ, 1)
Inv − Γ(α, β)
Γ ( θ V −1 )
E[V |Si = x, S j = y]
τµ+θ x +θ y+ 21
exp
τ +2θ
f S j | Si ( y | x )
N
α+2θ
α+2θ −1 max( ω, x, y )
Γ(α+2θ −1)
( x + y + β)
Γ(α+2θ )
τµ+θ x τ +2θ
τ +θ , θ (τ +θ )
( α + θ )1
( α + θ −2)1
[ x <ω ] x
[ x > ω ] y θ −1
(α+θ )θ ω
α
+
2θ
α+2θ
max(ω,x,y)
Γ(α+2θ ) yθ −1 ( x + β)α+θ
Γ(α+θ )Γ(θ ) ( x +y+ β)α+2θ
TABLE 1. Common-value models without MPSE.
5. N ON -M ONOTONE E QUILIBRIA
Consider a pure strategy equilibrium profile in which every bid strategy, bi (·), is piecewise
monotone, that is, bi0 (provided it is well defined) changes sign a finite number of times6. Next
partition i’s type space into finite intervals [0, 1] =
S ni
i
k=1 Ik
such these intervals are maximal
with respect the property each restriction bi | I i (·) is monotone. For exposition purposes, let’s
k
focus on the case where bi (·) is increasing (decreasing) in odd (even) intervals. The cases
where one or both of the bi (·) is increasing in even intervals are analogous.
6This is related to the ’limited complexity strategies’ of Athey’s 1997 working-paper version of Athey (2001)
7
Now define the kth local inverse bid function of player i: φki : b−1 ( Iki ) → Iki . Using inverse
bids, the payoff of a type x of player i who bids b is:

j
 nj
Ui (b| x ) = 
∑
k =1

φ2k−1 (b)
Z
j

λi ( x, y)dy
 − b,
φ2k−2 (b)
j
j
where by convention, φ0 (b) = 0, and for odd n, φ2n j −1 (b) = 1.
j
Later we shall use the ni × n j – matrices Λi (b) with entries given by Λik,l (b) = λi φki (b), φl (b) .
We abuse notation and write ni (b) for the number of types of player i that bid b.
Definition 1. A piecewise-monotone equilibrium b is regular if, ni (b) = n j (b) < +∞ is
constant in a neighborhood7 of b for almost all b and, Λi (b) is full-rank for i = 1, 2.
Any monotone equilibrium is regular. Without regularity there is little hope to pin-down
the equilibrium using the first-order approach. Without regularity, the differential system
corresponding to the first-order conditions is undetermined.
Lemma 2. Consider a regular equilibrium and let b ∈ bi ([0, 1]) with φki (b) for k = 1, ..., ni as
def
the corresponding local inverse bids. Define the tying functions by Qik (φ1i (b)) = φki (b). Let
Qi ( x ) = ( x, Q1i ( x ), . . . , Qini ( x )). The tying functions satisfy the system of differential equations:
j
L
(
x
)
k
∂ i
Qk ( x ) = (−1)k+1 j
∂x
L
(
x
)
1 j
j
j
where ( Lk )r,c ( x ) = λ j ( Q a ( x ), Qb ( x )) if c , k and ( Lk )r,k = 1 for all r.
Proof. Now notice the first-order condition for type x of player i who bids b is:
nj
∑ (−1)
l =1
l +1
λi
j
x, φl (b)
∂φ j
l
∂b
(b) = 1
(FOC)
As the FOC must be satisfied for x = φki (b) for all k, we obtain an ODE system, which in
!
j
∂φ
l
+
1
l
matrix form reads as, Λi (b) · Φ j = 1, where Φ j is the n j –column vector (−1)
(b) .
∂b
Consider the matrices that are Λi but with the kth column replaces by a vector of ones:
i
i
i
Λk (b) = Λ1,...,k−1 (b), 1, Λk+1,...,n (b) . Now, applying Cramer’s rule yields the lemma. i
To make it more concrete, let’s consider the case when two types are pooling at each bid in
a neighborhood of b; players are symmetric and the equilibrium is also symmetric. We have
7In a regular equilibrium, n (b) may vary with b but it can take at most a countable number of values.
i
8
that:
λ( x, x ) − λ( Q( x ), x )
.
λ( Q( x ), Q( x )) − λ( x, Q( x ))
Once we solve for Q we can reduce the first-order conditions to a single ODE (symmetric
Q0 ( x ) = −
equilibrium) or a system with two ODEs (asymmetric equilibrium) as the examples below
illustrate.
Example 4. Let v( x, y) = Ax2 + By2 + Cxy + Dx + Ey + F where x, y ∈ [0, 1], x is the signal
of player i and y is the signal of −i.
We look for a symmetric NPSE. Notice the single-crossing condition holds if and only if
n
o
n
o
D
A+C + D
D
A+C + D
either min − A , − A
> 1 or max − A , − A
< 0.
def −2D
Preliminaries: Define c =
and notice c is the unique solution of v( x, x ) + v( x, c −
2A + C
def
x ) = v(c − x, x ) + v(c − x, c − x ). Also define λˆ ( x, y) = v( x, y) + v( x, c − y).
Remark 1. The function λˆ ( x, y) satisfies λˆ ( x, y) = λˆ (c − x, y) for all x and y.
Remark 2. The derivative λˆ x ( x, y) is linear in x with λˆ x (c/2, y) = 0 for all y, and λˆ xx = 4A.
There are several cases to consider:
Case 1 (C < 2A < 0 and 0 < 2D < −2A − C). In this case the bidding function is bell-shaped
R1
for types in [0, c] and increasing for types in [c, 1]. We have b(0) = c v( x, x )dx and,

Rx

ˆ

b
(
0
)
+
if 0 ≤ x ≤ c/2,

0 λ ( y, y ) dy


R
b( x ) = b(0) + c− x λˆ (y, y)dy if c/2 ≤ x ≤ c, and .
0



R

b(0) − x v(y, y)dy
if c ≤ x ≤ 1.
c
Let’s verify that b(·) is indeed a NPSE by a direct mechanism approach. Let U (z| x ) be the
payoff of a player with type x who bids as if his type were z.

Rz

ˆ ( x, y) − λˆ (y, y) dy if 0 ≤ z ≤ c/2,

U
(
c
|
x
)
+
λ

0


U (z| x ) = U (c − z| x )
if c/2 < z < c, and .




R 1 (v( x, y) − v(y, y)) dy
if c ≤ z ≤ 1.
z
We want to show that x = argmax U (z| x ) for all x. The first-order condition is always
z
satisfied since Uz ( x | x ) = 0.
Clearly, if a type has a profitable deviation to some z ∈ [0, c/2] if and only if it also has a
profitable deviation to some z ∈ [c/2, c].
Pick z in [0, c/2], Uz (z| x ) = λˆ ( x, z) − λˆ (z, z) and Uxz (z| x ) = λˆ x ( x, z) satisfies Uxz (z| xˆ ) >
Uxz (z|c/2) = 0 > Uxz (z| x˜ ) for all xˆ < c/2 < x˜ due to remark 2. For any z ∈ [0, c/2] types
9
x < z want to report a type lower than z and types z < x < c/2 want to report a type higher
than z. So no type in [0, c/2] has a profitable deviation in [0, c/2] or in [0, c]. By remark 1,
U (z| x ) = U (z|c − x ) if x ∈ [0, c/2] and z ∈ [0, c] so it also follows that types in [0, c] do not
have a profitable deviation in [0, c].
For z ∈ (c/2, c), we have Uz (z| x ) = −Uz (c − z| x ) so Uxz (z| x ) = −Uz (c − z| x ) > 0 for all
x > c/2. Thus, also the types in (c, 1] do not have a profitable deviation in z ∈ [0, c]
Pick z in [c, 1], Uz (z| x ) = v(z, z) − v( x, z) and Uxz (z| x ) = −v x ( x, z) = −2Ax − D − Cz.
By the assumptions of our case C < 2A < 0 and D > 0 so Uxz (z| x ) ≥ − D − C · c =
2D
−D + C
= D ( 2A2C+C − 1) > 0. This single-crossing property implies that no types
2A + C
have a profitable deviation to [c, 1].
Case 2 (0 < 2A < C and −2A − C < 2D < 0). In this case the bidding function is U-shaped
R c/2
for types in [0, c] and increasing for types in [c, 1]. We have b(0) = 0 λˆ ( x, x )dx and

Rx

ˆ

b
(
0
)
−

0 λ ( z, z ) dz

R
x ˆ
b( x ) =
c/2 λ ( z, z ) dz




b(0) + R x v(z, z)dz
c
if 0 ≤ x ≤ c/2,
if c/2 ≤ x ≤ c, and .
if c ≤ x ≤ 1.
The proof this is an equilibrium is analogous to the previous example and therefore omitted.
Example 5. Let v( x, y) = x + k · y · (3 − 4 x + 2x2 ) where k ≥ 1. For k = 1, this corresponds
to example 4 of Araujo et al. (2008, p. 421).8 Furthermore, we assume9 signals to be uniformly
q
distributed on[0, x ] where x = 1 + 1 − 1k .
For brevity, we focus on symmetric equilibrium and as Araujo et al. (2008) we look for an
U-shapped equilibrium. We denote the corresponding two local inverse bids as φ (b) and
i
φi (b) where φ (b) ≤ φi (b). To solve for the pooling types, we must find c such that:
i
λ( x, x ) + λ( x, c − x ) − λ(c − x, x ) − λ(c − x, c − x ) = 0,
for all x ∈ [0, 1]. So, we are dealing with a continuum of equations: (4c2 k − 8ck + 4) x − 2c3 k +
q
q
1
2
4c k − 2c = 0 whose solutions are: c = 1 − 1 − k and c = 1 + 1 − 1k . We set c = c so
that inverse bids are related by: φ (b) = c − φi (b). Now applying the reduction, we have10
i
8For k = 1, they show the first-price auction has no monotone eq. The all-pay auction also lacks a monotone
equilibrium for k ≥
9
6
10 .
As to ensure a simple equilibrium and easy the exposition. See definition ?? and subsequent discussion.
10In principle, λ
ˆ may depend on both signals but here it only depends only on the player’s own signal.
10
λˆ ( x ) = ( x )−1 · 2x + c · k · 3 − 4x + 2x2 :
h
i
2
− 2φi (b) + c · k · 3 − 4φi (b) + 2φi (b)
φ0 (b) − x = 0
j
i
h
0
2φi (b) + c · k · 3 − 4φi (b) + 2φi (b)2 φ j (b) − x = 0
Notice that λˆ satisfy a single crossing condition at x =
for xˆ <
x
2
where φ (b) ≤ x/2
i
or equivalently,
where φi (b) ≥ x/2.
x
2
since λˆ x ( xˆ ) < λˆ x ( 2x ) = 0 < λˆ x ( x˜ )
0
< x˜ and φ j (b) = −φ0j (b) > 0.
Now we can recover the equilibrium:

R

b − x λˆ (z)dz
0
b( x ) = R
 x ˆ

x/2 λ ( z ) dz
where
b=
Z x/2
0
if x ≤ x/2,
,
otherwise.
1 + 5k
λˆ (z)dz =
6
r
1−
1
1 5
+ k+ .
k 6
2
6. C ONCLUSIONS
We characterized monotone equilibrium of all-pay auctions in the continuous signals case,
allowing for correlation and interdependent values.
Motivated by the non-robustness of the monotone equilibrium to small degree of correlation, we also study non-monotone pure strategy equilibrium.
The existence of pure strategy equilibria (monotone or not), however, remains an open
question Rentschler and Turocy (2012)’s results suggest that some models may have mixed
strategy equilibrium (where each type mixes). In this paper, we do not discuss mixed strategy
equilibrium.
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