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Economics Faculty Articles and Research Economics
1996
#e All-Pay Auction with Complete Information
Michael R. Baye
Indiana University
Dan Kovenock
Chapman University, kovenock@chapman.edu
Casper G. de Vries
Erasmus University
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Recommended Citation
Baye, Michael R., Dan Kovenock, and Casper G. De Vries. ";e all-pay auction with complete information." Economic eory 8.2
(1996): 291-305.
#e All-Pay Auction with Complete Information
Comments
;is is a pre-copy-editing, author-produced PDF of an article accepted for publication in Economic eory,
volume 8, issue 2, 1996 following peer review. ;is article may not exactly replicate the <nal published
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Springer
;is article is available at Chapman University Digital Commons: h=p://digitalcommons.chapman.edu/economics_articles/121
THE
ALL-PAY
AUCI'IOM WITH
fJ0lIPLE1E
IPIF~ATIOPiw
Michael
R.
Baye
Texas
AS~M
University
Dan
Kovenock
Erasmus
Universiteit
Rotterdam
Purdue
University
Casper
G.
de
Vries
Katholieke
Universíteit
Leuven
Preliminary
draft
:
March
1989
Working
Paper
:
June
1990
AffiI'RACI'
This
paper
provides
an
exhaustive
and
explici[
description
of the set
of
Nash
equilibría
in the
n-player,
first
príce
sealed
bid,
all pay
auction
under
complete
information.
Both
the
cases
of
homogeneous
and
heterogeneous valuations
are
analyzed.
For the
common values
case with
more
than two
players
we
show
there
is
a
unique
symmetric
equilibrium
and a
continuum
of
asymmetric
equilibria.
All of the
equilibria, how-
ever,
are payoff
and
revenue
equivalent.
Wíth
heterogeneous
valua-
tions,
two
new
situations
can
arise. First,
if
the
three
highest
valuations
are
strictly
unequal
then
there ís a
unique
asymmetric
equilíbrlum.
Second,
with
a
single
híghest
valuation
and
more
than
one
player
with
the
second
highest
valuation,
there
is
a
con[inuum
of
asymmetric
equilibria.
In
both
of
these
latter
cases,
the
expected
sum
of
the
bids
is
below
the
second
highest valuatíon,
and
depends
on
the
strategies
of the
agents
with
the
second
híghest
valuation.
Hence,
while
the
equilibria
are
payoff
equivalent,
they
are not re-
venue
equivalent.
The
continua
of
asymmetric equilíbria
were
missed
by
both
the
theoretical
literature,
and
the
applied
literature
on e.g.
rent
seekíng
and
rent
díssipation.
Mailing
address
:
Center
for
Economic
Studies
Katholieke
Universiteit
Leuven
E.
Van
Evenstraat
2B
B-3000 Leuven
BELGIUM
w
We
are
grateful
to
Chuangyin
Dang,
Chaim
Fershtman,
Arthur
Robson,
Heinrich
Ursprung
and
Ton
Vorst
for
helpful conversations.
The
paper
was completed
while
Baye
was
visiting
the
CentER
for
Economic
Re-
search
at
Tílburg
University.
1
1.
II'TI'RODLK.TIOti
Consider
the
public
auction
of
a
dollar,
ín
which
each
of
n-bidders
places
money
in
an
envelope.
The
money
ín
the
envelopes
ís
collected
and
kept
by the seller,
and
the
dollar
ís
awarded
to the
bídder who
placed
the
highest
amount
of
money
in his
envelope
(ties
are
broken
in
an
arbitrary
fashion).
This auction,
whích
is
called
an
all-pay
auction
[cf.
Moulin (1986
a.b)
and
Weber
(1985)],
is
important
because
many economic
problems
under
complete
information have
a
similar
structure.
For
example,
Hillman
and
Samet (1987)
and
Hillman
(1988)
model
lobbying
as an
all-pay
auction,
where
the
lobbying
parties
sweeten
the
decisionmaker
by
making
a
bribe,
and
the
prize
(a
polítical
favor)
is
awarded
to
the
party
having
given
the
highest
bribe.
Similarly,
much
of
the
contest
and principal
agent literature
under
complete
informatíon
is
isomorphic
to
the
all-pay
auction;
cf.
lvalebuff
and
Stiglitz
(1983) and
Baye.
Kovenock,
and
de
Vries
(1989b).
Essentially,
contests
are
an
all-pay auction
in
effort
:
the
person
putting
forth
the
greatest
effort
wins
the
prize,
while
the
effort
of
other
contestants goes
unrewarded.
Thís
paper completely
characterizes
the set
of
IVash
equílíbria
in
the
all-pay
auction
with complete
informatíon.
We
show
that
the
set
of
equilibria
is
much
larger than had
originally
been
thought.
Moreover.
in some
economícally
interesting
cases,
the
equilibria
are
not
revenue
equivalent.
Before
we
present
a
more specific
statement
of
our
results
and
their
proofs,
it
is
useful
to
describe
our
results
vís
à
vis
the
existíng
literature
on
the
all-pay
auction.
Two
cases
have
been
considered
in
the
literature-
(1)
the
case
where
all
players
value
identically
[he
prize,
and
(2)
the
case
where
some
players
value
the
prize
more than
others.
For
the
case
of
homogeneous
valuations,
Hillman
and
Samet
have
shown
that,
in
addition
to
a
symmetric
Wash equilibrium,
there
also
exist
a
finite number
of
asymmetric
equilibria.
We
extend
this
result
by
showing
that there is
actually
a
continuum
of
asymmetríc
2
equilibria.
In
each
equilibrium.
at least two
players
randomize
continuously
over
the
union
of
the
supports
of the
players'
equilibrium
mixed-strategies,
while
up to
n-
2
players
may
have
a
mass
point
at
zero
and
only
randomize
over
a
strlct subset
of
the
union
of
other
players' supports.
The exístence
of
these
a,ddítional
equilibria
has
obvious
empirical
implications.
However,
for the
case
oE
homogeneous
valuations,
all
of
these
equllíbria
are
payoff and
revenue
equívalent-
The
expected
sum
of
the
bids
equals
the
value
of
the prize,
and
the
net
expected
pay-off
to
each
bidder
is
zero
for
all
equílibría.
The
second
case
is
when
several
players
have
heterogeneous valuations
of the prize. For the
case
where
the
second
highest
valuation
of the
prize
ís
strictly
greater
than the third
highest
valuation~,
Hillman
and
Riley
(1989)
have
shown
that there
is
a
unique
equílibrium
and
that
only
the
two
players
with
the
highest
valuations
bid
a
positive
amount
withe
positive
probability.
Furthermore,
they
show
that
if
the
highest
valuation
is
strictly
greater
than
the
second
híghest,
the
expected
sum
of
bids
is
less
than
the
second-híghest
valuatíon.
We
extend
Hillman
and
Riley's
analysis
of the
heterogeneous
valuations
case
by
considering
other
configuratíons
of
individual
valuations.
One
of the
more
important
confígurations
of
valuations
is
where
a
single
player
values
most
the
prize,
while
all
other
players
value
the
prize
at
some
common,
lower value.
This case
is
economícally
ínteresting,
because
in
much
of
the
literature
on
regulation
[cf.
Rogerson
(1982)]
and politícal
contests
[cf.
Snyder
(1989)], one
player
(usually
an
incumbent)
!s
modeled
as
having an
advantage
over
ldentical
challengers.
For this case, we
show
not
only
that
there
ís
a
contlnuum
of
equilibria,
but that
the
equilibria
are
not
revenue
equivalent-
the
expected
sum
of
bids
differs
across equilíbria.
1
Kore
precisely,
i f
the
players
can
be
ordered
i
n
such
a
way
that
vl
Z vy
~
v3
Z...
~
vn,
where
vi
is the
valuatíon
of
player
1.