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ij'C\.'*'t;i^l j;
HB31
.M415
'no.
97'//
working paper
department
of economics
SINGLE CROSSING PROPERTIES AND THE EXISTENCE OF
PURE STRATEGY EQUILIBRIA IN GAMES
OF INCOMPLETE INFORMA TION
Susan Athey
No. 97-11
June, 1997
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
SINGLE CROSSING PROPERTIES AND THE EXISTENCE OF
PURE STRATEGY EQUILIBRIA IN GAMES
OF INCOMPLETE INFORMATION
Susan Athey
No. 97-11
June, 1997
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50
MEMORIAL DRIVE
CAMBRIDGE, MASS. 021329
MACSftCHUSETTS IKSTiTUTE
IMi:.'
'39^
Single Crossing Properties
Games
and the Existence of Pure Strategy Equilibria
in
of Incomplete Information*
Susan Athey
MIT and NBER
FirstDraft:
August 1996
This Draft: June 1997
Abstract: This paper derives sufficient conditions for a class of games of
incomplete information, such as first price auctions, to have pure strategy Nash
equilibria (PSNE). The paper treats games between two or more heterogeneous agents,
each with private information about his own type (for example, a bidder's value for an
object or a firm's marginal cost of production), and the types are drawn from an
atomless joint probability distribution which potentially allows for correlation between
types. Agents' utility may depend directly on the realizations of other agents' types, as
The
in MUgrom and Weber's (1982) formulation of the "mineral rights" auction.
restriction we consider is that each player's expected payoffs satisfy the following
whenever each opponent uses a nondecreasing strategy (that
an opponent who has a higher type chooses a higher action), then a player's best
response strategy is also nondecreasing in her type.
single crossing condition:
is,
The paper has two main
results.
The
first result
shows
that,
when
players are
choose among a finite set of actions (for example, bidding or pricing where
the smallest unit is a penny), games where players' objective functions satisfy this
single crossing condition wUl have PSNE. The second result demonstrates that when
players' utility functions are continuous, as well as in mineral rights auction games and
other games where "winning" creates a discontinuity in payoffs, the existence result can
be extended to the case where players choose from a continuum of actions.
restricted to
The paper then applies the theory to several classes of games, providing
conditions on utility ftanctions and joint distributions over types under which each class
of games satisfies the single crossing condition.
In particular, the single crossing
condition is shown to hold in all first-price, private value auctions with potentially
heterogeneous, risk-averse bidders, with either independent or affiliated values, and
with reserve prices which may differ across bidders; mineral rights auctions with two
heterogeneous bidders and affiliated values; a class of pricing games with incomplete
information about costs; a class of all-pay auction games; and a class of noisy signaling
games. Finally, the formulation of the problem introduced in this paper suggests a
straightforward algorithm for numerically computing equilibrium bidding stiategies in
games such as first price auctions, and we present numerical analyses of several
auctions under alternative assumptions about the joint distribution of types.
Keywords: Games of incomplete information, pure
auctions, pricing games, signaling
single crossing, affiliation.
JEL
Classifications:
games,
strategy
supermodularity,
Nash equilibrium,
log-supermodularity,
C62, C63, D44, D82
*I am grateful to seminar participants at Columbia, Maryland, MIT, UCLA, UC-Berkeley, the 1997 Summer
Meetings of the Econometric Society, as well as Kyle Bagwell, Abhijit Baneijee, Prajit Dutta, Glenn Ellison, Ken
Bob Wilson
Dworak, Jennifer Wu, and Luis Ubeda for excellent research assistance. I
would like to thank the National Science Foundation (Grant No. SBR-9631760) for financial support.
Corresponding address: MIT Department of Economics, E52-251B, Cambridge, MA, 02139; email: athey@mit.edu.
Judd, Jonathan Levin, Eric Maskin, Preston McAfee, Paul Milgrom, John Roberts, Lones Smith, and
for useful conversations,
and
to Jonathan
Introduction
1.
This paper derives sufficient conditions for a class of games of incomplete information, such as
first
price auction
games,
to
The
have pure strategy Nash equilibria (PSNE).
class of
games
is
described as follows: there are / agents with private information about their types, and the types are
drawn from a joint
which allows
distribution
for correlation
a convex subset of 9t, and the joint distribution of types
is
between types. Types are drawn from
across players in the distribution over types as well as the players'
functions
may depend directly on
rights" auction
We
atomless.
allow for heterogeneity
functions, and the
utility
other players' types. Thus, the formulation includes the "mineral
(Milgrom and Weber (1982)), where bidders receive a signal about the underlying
value of the object, and signals and values
may be
The
correlated across players.
existence result
does not make any assumptions about quasi-concavity or differentiability of the underlying
functions of the agents, nor does
The main
utility
it
require that each agent has a unique optimal action for any type.
paper
restriction studied in this
games of incomplete information:
utility
is
what we
for every player
/,
call the
single crossing condition for
whenever each of player
opponents uses a
i's
pure strategy such that higher types choose (weakly) higher actions, player Ts expected payoffs
satisfy
Milgrom and Shannon's (1994)
between a low action and a high
action, then all higher types
single crossing property.
action, if a
of agent
/
low type of agent
wUl weakly
/
In particular,
weakly
(strictly) prefer the
when choosing
(strictly) prefers the
higher action as well.
higher
This
condition implies that in response to nondecreasing strategies by opponents, each player will have
The
a best response strategy where higher types choose higher actions.
contrasts with that studied
by Vives
PSNE
supermodular
is that
the
game
is
(1990),
who shows
single crossing condition
that a sufficient condition for existence
in the strategies, in the following sense: if
strategy increases pointwise (almost everywhere), the best response strategies
must increase pointwise
monotone
(a.e).
However,
The paper has four
satisfies the single
choose from a
parts.
first
part
shows
that
space, a
PSNE
need not be
exists.
Next,
when
utility
this paper.
when a game of incomplete
crossing condition described above, but
finite action
opponents
games where players have supermodular
and log-supermodular pricing games highlighted in
The
one player's
all
in Vives' analysis, the strategies themselves
in types. Vives' condition is applicable
functions, but not in the auctions
of
of
information
the players are restricted to
we show
that
under the further
assumption that players' utUity functions are continuous, or in a class of "winner-take-more"
games such
as first price auctions,
which converges
third part of the
to
an equilibrium
we
in a
can find a sequence of equilibria of finite-action games
game where
actions are chosen
from a continuum.
paper builds on Athey (1995, 1996) to study conditions on
type distributions under which
Games which satisfy
commonly
studied
games
utility
The
functions and
satisfy the single crossing condition.
the single crossing condition include any
first
price auction,
where values
are
(weakly) risk averse, and the types are independent or
private, bidders are
"mineral rights" auctions, the conditions are satisfied
whose types
which
are affiliated. Other applications
when
there are
affiliated. In the class
of
two heterogeneous bidders
satisfy the conditions include all-pay auctions
and multi-unit auctions with heterogeneous bidders and independent private values, noisy signaling
games (such as
limit pricing
with demand shocks),
and a class of supermodular and log-
supermodular quantity and pricing games with incomplete information about costs.
The
of the paper focuses on numerical computation of equilibria, showing that equilibria to
auctions
may be
easily
computed for games with a
finite
number of
final part
first
potential bids.
price
Several
examples of auctions are provided, considering alternative scenarios for heterogeneity, private
versus
common values, and correlated versus
The existence
independent values.
action spaces analyzed in the first part of this paper is
result for finite
straightforward to prove using a reformulation of the problem, which allows us to simplify the
game
to a finite-dimensional
result
proceeds in two steps.
which maps types
problem and then apply standard fixed point theorems. The existence
we
First,
observe that
into actions, the strategy will
problem as determining
restate the player's
to the next highest action, as well as
been reformulated
in this
at
if
a player uses a nondecreasing strategy
be a nondecreasing step function. Thus,
which realizations of his type the
what action
is
taken at the "step point."
we
can
strategy will "step"
Once
the
problem has
way, we can view a nondecreasing strategy as a subset of
finite-
dimensional Euchdean space, and the existence result then relies on Kakutani's fixed point
theorem.
The
strategies
enough such
show
that
it
single crossing condition plays
impUes
that they
that the set
For example, a
roles in this analysis.
of vectors which represent optimal actions
PSNE
argument can be extended
will exist in
shaped strategy by the opponents
is
games where every
However,
U-shaped.
assumption to guarantee that the best response correspondence
are never indifferent
Thus, the
The second
first
between two actions over an open
part of this paper
strategies for
limit of a
every
finite
We
shows
that
show
with
when
that
It is
simplifies the
to
is
convex.
important to
properties such as a U-shape) because
equilibrium strategies for finite
know
when
we
The paper
games with nonmonotonic
player's best response to a
this result requires
is
Second,
convex:
U-
an additional
we assume
that players
finite actions,
PSNE
exist quite generally.
under which these results extend
to
there exists an equilibrium in nondecreasing
game, and when the players' objectives are continuous, there
sequence of equilibrium strategies for
continuous actions.
it
interval of types.
part of the paper proceeds to derive conditions
continuous action spaces.
First,
can be represented with vectors of "step points."
also demonstrates that the logic of the
strategies.
two
finite
games which
is
that the strategies are
the strategies are of
exists a
an equiUbrium in a game with
monotonic (or
bounded
satisfy related
variation, a
sequence of
games has an almost-everywhere convergent subsequence (by
Kelly's selection theorem (Billingsley, 1968)).
However,
in auctions
and
winning gives a discrete change in payoffs, the players' objectives are not
own
their
action,
and thus establishing existence
properties of the equilibria to finite-action
in
games,
games where
related
in general
continuous
such games requires additional work.
we show
in
Using
an auction game satisfies the
that if
single crossing property described above, so that best responses to nondecreasing strategies are
some
nondecreasing, plus
game
action auction
additional regularity conditions, then existence result
from the
finite
will extend to auctions with continuous action spaces.
Analyzing existence in games with a continuum of types and a continuum of actions
because the strategy space
is
not a
finite
Many
subset of Euclidean space.
is difficult
previous results about
existence of pure strategy equilibria are concerned with issues of topology and continuity in the
relevant strategy spaces.
equilibria exist
when
For example, Milgrom and Weber (1985) show
pure strategy
that
type spaces are atomless and players choose from a finite set of actions, types
are independent conditional
on some common
player's utility function depends only
state variable (the utility
on
his
state variable
own
(which
finite-valued),
is
type, the other players' actions,
cannot depend on the other players' types directly).
condition which they call "continuity of information."
show
that players
utility
function depends only on his
choose from a
strategy equilibrium will exist.
finite set
own
Similarly,
and the
They
and each
common
also study a
Radner and Rosenthal (1982)
of actions, types are independent,! ^nd each player's
type, but the type distributions are atomless, then a pure
The authors then provide
faU to have pure strategy equilibria, in particular
several counter-examples of games
games where
which
players' types are correlated.
The
counter-examples of Radner and Rosenthal faU our sufficient conditions for existence in games
with
finite actions
a different reason: the best response of one player
is
always a
litde bit
more
complicated than the strategy of his opponent.
In contrast, our analysis allows arbitrary correlation between types, to the extent that the joint
distributions over types lead to expected
property of incremental returns.
affiliation)
Thus, any required
have economic interpretations.
auction a class of auction
affiliation is
payoff functions which satisfy the single crossing
games where
restrictions
Weber (1994)
studies
on
the distribution (such as
mixed
sti-ategy equilibria in
the affiliation inequality faUs; this paper
not essential except to the extent that
it
shows
that
implies monotonicity properties of the players'
objectives.
Now
consider the special case of
first
price auctions.
The
issue of the existence of pure
strategy equilibria in first price auctions with heterogeneous agents (and continuous actions spaces)
Radner and Rosenthal (1982) also treat the case where each player can observe a
variable which affects the payoffs of all agents.
random
finite- valued "statistic"
about a
many
has challenged economists for
Recently, several authors have
years.
made
substantial
progress, establishing existence and sometimes uniqueness for asymmetric independent private
values auctions (Maskin and Riley (1993, 1996); Lebrun (1995, 1996); Bajari (1996a)), as well as
common value
affiliated private values or
auctions with conditionally independent signals (Maskin
and Riley, 1996)). Lizzeri and Persico (1997) have independently shown
related to the single crossing condition is sufficient for existence
that a condition closely
and uniqueness of equilibrium
in
two-player mineral rights auction games with heterogeneous bidders, but their approach does not
Many
extend to more than two bidders without symmetry assumptions.
auctions with heterogeneous bidders are not treated
where existence
auctions
computing the solution
can be
difficult
Two
is
to a
by
interesting classes
of
the existing analysis, and even for the
known, computation of equilibrium (which involves numerically
system of nonUnear
differential equations
with two boundary points)
due to pathological behavior of the system.^
main approaches
to existence
have been used in the case of
first
price auctions:
(i)
establishing that a solution exists to a set of differential equations (Lebrun (1995), Bajari (1996a),
and Persico (1997)), and
Lizzeri
actions are
drawn from
(ii)
finite sets,
and Riley (1992)). Since there
estabUshing that an equilibrium exists
exist
games (with discontinuous payoffs) which have pure
case (for example, see FuUerton and
different
from
separately
we do
that
where
McAfee
the special structure of the
of the existing
on
limiting arguments,
there is
game
third part
strategy equilibrium in the infinite
at
we
hand.3
The
strategy taken in this paper
treat the issue
it
is relatively
some
classes of auctions
(i.e.
easy to establish that the single crossing condition holds,
can be challenging in other classes (for example, mineral rights auctions with more than
signal and the other players' actions).
existence can also be helpful if it
^ Marshall et al (1994) summarize
is
Knowing
their
own
that the
value from winning based on their
single crossing condition
some of the problems;
Simon and Zame (1990) provide an
impUes
possible to demonstrate that the single crossing condition holds
they provide and implement an algorithm which alleviates
these difficulties for a simple class of distributions, F(Ada)= y^. See Section 5 of this paper for
equiUbria.
to
limit.
two bidders, where players form expectations about
3
However,
of the paper derives conditions under which the single crossing condition holds
private value auctions),
it
of auctions.
and use the special structure of a "winner-take-more" game
for different classes of auctions. This distinction is useful because in
while
is
of existence of equilibrium
strategies in different classes
prove that discontinuities do not in fact arise in the
The
no pure
strategy
(1996)), these limiting arguments generally involve
literature, in that
from the issue of monotonicity of
rely
either types or
and then invoking limiting arguments (Lebrun (1996), Maskin
equilibria for every finite action set, but
more work and use
when
more
discussion.
elegant treatment of limiting arguments in the context of mixed strategy
for a range of parameter values, or if
possible to establish that
is
it
holds in the relevant region
it
using numerical analysis.
The
final part
and computational
difficulties in
assumption
engage
theoretical
analyzing auctions with heterogeneous bidders have confounded
where heterogeneity and
attempts to apply and test auction theory in real- world problems,'*
correlation
The
of this paper analyzes the computation of equilibria to auctions.
between types are the rule rather than the exception, and as well the private values
may be
Further, even if bidders are ex ante symmetric,
tenuous.
in joint bidding,
asymmetries will arise (Marshall
if
they collude or
Since very
et al, 1994).
little is
known
about the theoretical properties of general auction games with heterogeneous bidders, numerical
computation can also play an important role in suggesting avenues for future theorizing.
the growing literature
on
structural empirical analyses of auctions
(i.e.
Further,
Laffont, Ossard, and
Vuong
(1995), Bajari (1996b)) requires that equilibria to auctions be computed in each iteration of an
econometric procedure.
This paper proceeds by observing that the representation of nondecreasing strategies with
actions as a vector of "step points"
very simple algorithms.
Our
imphes
shows
analysis
"close" to equilibria of continuous-action
well-developed numerical techniques
that equilibria to auction
for
games can be computed with
such
that the equilibria to
games
as the
number of
finite-action
actions increases.
games
get
There are
approximating a fixed point in finite-dimensional
We provide a numerical analysis of several
problems (Judd, forthcoming).
finite
examples of
first
price
auctions with alternative assumptions about heterogeneity and the type distribution.
2.
Existence of Equilibrium with Finite Actions
PSNE
This section derives sufficient conditions for the existence of a
information, where the types of the players are
are restricted to choose
between / players,
from a
/=!,..,/,
then takes an action
a.
finite set
of actions.
where each player
Given any
set
first
observes their
/(t_,lf,.).
The
The
^ Only a few studies
distribution
and the players
own
type
f,.e
T=[
utility, M,(^,t),
f ,, f ,]e9t and
may depend on
joint density over player types is /ft), with
objective function for player
of strategies for the opponents,
game of incomplete
Consider a game of incomplete information
from an action space //c9?. Each player's
the actions taken as well as the types directly.
conditional densities
drawn from an atomless
in a
ocf.i tj,
t j]^/^\
/
is
then specified as follows.
j^i,
we
can write player
i's
exist; Hendricks and Porter (1988) analyze the case of informed versus uninformed bidders in
mineral rights auctions, while Pesendorfer (1995) uses a model of an asymmetric private value auction to analyze
asymmetric auctions for school milk contracts. Bajari (1996b) uses a structural econometric approach to study
asymmetric construction auctions.
function
objective
C/,(a,,a,
The following
The
follows
as
0,0=1
We
convention
notational
the
(a.,a_,(t_,))=
M,((a,,a,(t_,)),t)/(t_,|r,Mt_,
basic assumptions are maintained throughout the paper:
types have joint density with respect to Lebesque measure, /ft), which
has no mass points. ^ Further,
all
(using
5 and
all a/.[ tj,
u.({a.,a_^(t_.)),t)f(t_.\t.)dt_^ exists
f
is
bounded and
and
tj\^^\ j^i.
proceed by proving two
for
is finite
(2.1)
results.
The
looks
first
pure
for
strategy
equilibria
in
nondecreasing strategies, while the second potentially allows for more complicated strategies.
However, the second
result requires that players are
an open interval of types
structure, since
two
(this
never indifferent between two strategies over
assumption would not be satisfied for an auction without additional
actions might both
win with
more aggressive
probability zero against a
opponent).
2.1.
Pure Strategy Equilibrium
This section studies games with
The
single crossing will play
two
in
Nondecreasing Strategies
finite action
spaces which satisfy the single crossing condition.
roles in the analysis.
First,
it
Second,
represent each agent's strategy with a vector of finite dimension.
guarantee that each player's best response correspondence
is
convex
will guarantee that
(recall that
it
will
we
we
can
be used
to
have not made
assumptions which could be used to guarantee a unique optimum). These two properties of games
with the single crossing condition allow us to use Kakutani's fixed point theorem to guarantee
existence of a
PSNE.
Before beginning our analysis of equiUbria in nondecreasing strategies,
definition of Milgrom
which
states that
conclusion to hold in problems which
statics
introduce the
and Shannon's (1994) single crossing property of incremental returns (SCP-
IR), as well as the corresponding theorem
comparative
we
SCP-IR
may be
is sufficient
for a
monotone
non-differentiable, non-concave,
or have multiple optima. Consider the following definition:
Definition 2.1 h(x,0) satisfies the (Milgrom-Shannon) single crossing property of
incremental returns (SCP-IR) in (x;d) if, for all Xu>Xij g(6)=h(X[j,0)-h(X[j6) satisfies the
^ In
games with
finite actions, condition (2.1)
distribution, so long as for
throughout the region
[
can be relaxed to allow for mass points
each player, there exists
tj,k).
ak>
tj such that the lowest action
at the
lower end of the
chosen by player 7
is
chosen
following conditions for all Ofj>d^: (a) g{dj>0 implies g(0^)>O, and (b) g{di)>0 implies
The
preferred to x^ for
(strictly)
still
requires
definition
be
(strictly)
increases
from G^
that
/i(%,0)-/j(a;^,0)
is
then x^ must
0^,
preferred
x^
if
x^
to
6
if
In other words,
to Oh-
the incremental returns to
cross zero
jc
at
Figure
most once, from below, as a function of 6
1:
h satisfies
SCP-IR
if,
for
all Xh>Xi^,
the
function h{x„, 6)-h{xij 6) crosses zero at most once, from
(Figure
Note the
1).
relationship
symmetric between x and
Shannon show
set
that
of optimizers
A
Definition 2.2
written A>sB,
function A(t)
Lemma
is
below, as a function of
6.
implies that the
nondecreasing in the Strong Set Order (SSO), defined as follows:
5cSR in the strong set order (SSO),
max(a,6)G A and rmn{a,b)& B. A set-valued
the strong set order (SSO) iffor any T^ > T^ A(Tf,)>sA(Tj.
set Ac,'^ is greater than a set
aeA and any be B,
if, for any
nondecreasing
is
not
Milgrom and
6.
SCP-IR
is
in
(Milgrom and Shannon, 1994) Let h:%^ ^^'^. Then h satisfies SCP-IR
only ifx'{6)= argmaxh{x,0) is nondecreasing in the strong set order for all B.
2.1
Under SCP-IR,
there might be
decreasing on a region; however,
selection of optimizers
is
Using Definition
we
2.1,
ax'ej:*(0J and z.x"ex*{dif) such
can
that
if this is true,
state the sufficient condition for existence
x'>x" so
,
and
if
that
some
then x'gx*(0^) as well.
of a pure strategy Nash
equilibria in nondecreasing strategies.
We
say that a
game
satisfies the
incomplete information
OCj:[tj,
if,
Single Crossing Condition (SCC) for games of
for i=l,..,l, given any set of 7-1 nondecreasing functions
tj\-^/4\j^i, player Vs objective function,
crossing of incremental returns (SCP-IR) in
The
first result is
t/,.
(a,, a_,. (•),?,), satisfies
single
(2.2)
(a.;t).
then stated as follows:
Theorem 2.1 Consider the game of incomplete information described above, where (2.1)
and the Single Crossing Condition (2.2) hold. If /I' is finite for all this game has a PSNE,
i,
where each player's equilibrium
The proof of Theorem
nondecreasing strategy a-
2.
1
strategy, ^ft^, is
relies
(t.) is
on
a nondecreasing function of t^.
the fact that,
when
from a
finite set,
any
simply a nondecreasing step function. Then, the strategy can be
described simply by naming the values of the player's type
action to the next higher action.
players choose
To
at
which the player "jumps" from one
formalize this idea, consider the following representation of
nondecreasing strategies. For simplicity,
action space, but this assumption is
f,.
we treat the case where
made
each player has the same feasible
purely to conserve notation.
Let/^={Ao,A,,..,Aj^^} be
the set of potential actions, in ascending order,
M+1
where
Let
is
number of
the
and
T.'^=X\.ti^t ^,
m=
define
the
of
set
l
nondecreasing vectors in
{x e 7;^^2|xo
Further, let
potential actions.
T.'^
S,^=
as follows:
=t^,x^<x^<-< x^,
x^^,
= r:}.
E^Sf'x-xZf.
Consider a candidate nondecreasing strategy
for
player
we can
function
a,,
xe
according
Z,^
(illustrated in
following
the
to
e
the agent
t
"jumps"
the vector x specify
to a higher action.
there is
xe
that "the vector
some n>m such
Ej*^
on
that a^{t)=A^
otherwise.
^
{x} denote the set
let
we say
a^(t),
= inf|/.|a,.(?,.) > a\ whenever
we say a nondecreasing
Then,
The elements of
2:
algorithm
Given a nondecreasing strategy
an open interval of T^ and x^=
,
Figure
2).
(i)
Sf
such
assign a corresponding vector
represents «,(?,)" if x^
Given xe
any
when
Figure
Definition 2.3
(ii)
For
a,:[r,, f,]->A.
/,
{r,,x,,..,;Cj^^,f,},
"strategy aft^
is
and
m*(t,x)
let
consistent with
x"
= max{m|x„ < t}.
if ocf^t^
=
A^^^,
^^forall
t^
7:\{x}.
Each component of x
function described
by
a,..
is
therefore a point of discontinuity, or a
Simply by naming the x which corresponds
what action the player takes everywhere except
on
actions are specified
To interpret part
(ii),
note that
one nondecreasing strategy.
other players.
on
at the
jump
points
to a^,
we
have specified
and endpoints,
a,(r.) is
consistent with
?,e
x
if
action
that is,
The proof makes use of this
measure zero)
fact,
is
more than
in the distributions
of types, a
will not affect the best responses of
proceeding by finding a fixed point in the space of
filling in
behavior for each player at the jump points. Behavior at the jump points can be assigned
it
won't affect the best responses of the other players; thus,
it
vector
all
consider notation for the objective function faced by a given player.
(x',..,x^),
type fp and
let
which describes a
set
of step functions for each player.
V^(a^;X,ti) denote the expected payoffs to player
1
optimal
arbitrarily
can be assigned to be optimal
without loss of generality, making the resulting strategies best responses for
Now
the
used between x^ and
{x}, a given xeX,. might correspond to
However, because there are no atoms
the set {x} (which has
A^
nondecreasing best responses which can be described by elements of S, and then
since
the step
T^Vfx}.
x^^y Since x does not specify behavior for
player's behavior
"jump point," of
when
types.
Let
X
denote the
Consider player
player
1
1
with
chooses a,e/^
and players
use strategies which are consistent with
2,..,/
Then
(x^,..,x')-
V,(a,;X,?,) can be
written as follows:
,)><xl< •<.,)'
m2=0
mi=0
M
=
where lo(y)
By
is
"2+1
£••£
-
1
J«,(«pA„^,..,A„,,t)-/(t_,|f,Mt_,
an indicator function which takes the value
1
if
ye 5 and
(2.3)
otherwise.
behavior of opponents on sets of measure zero do not affect player
(2.1), the
/,
so player
/
views each opponent as using a nondecreasing strategy whenever they use actions consistent with
x'onr,.\{x'}. Then, by (2.2), V.(a.;X,0 satisfies the SCP-IR in (a,.;^. Let af''(r,.|X)= argmax
V^.(a,.;X,r,.);
)^(f,.)e
this is
nonempty
from the
a^''(t.\X),
members of this
there exists a
for all
t.
by
which
set
is
nondecreasing in
set are nondecreasing; see
ye E. which represents
By Lemma
finiteness of /^.
t.
2. 1, there exists
a selection,
lowest and highest
(in particular, the
Milgrom and Shannon (1994)). As we argued above,
y^t), so that y^t) is consistent
with
Now
one best response vector y which represents an optimal strategy.
y.
Thus, there
is at least
define the set of
all
such
vectors as follows:
r,(X)={y: 3 a,(0 which
consistent with
is
The existence proof proceeds by showing
Once
the
r(X)
is
problem
if
this
way,
it
that
a,(0^ af^ihl^) )
that a fixed point exists for this
is
Observe
that
even
if the
correspondence.
straightforward to verify that the correspondence
nonempty and has a closed graph. However, convexity of
additional work.
even
formulated in
is
y such
player's payoff function
the correspondence requires
is
stricdy quasi-concave,
a change in the player's type changes payoffs everywhere, the player
^4
still
and
might be
^h)
M
A,
w
O. Ui
Aq Aj
*4
A
Figure 3: The
yi
yi
set af^it^ is nondecreasing in the
The
vectors
points" corresponding to optimal strategies.
indicate
A3
A4
y^
x and y represent "jump
The arrows
convex combinations of x and y.
Strong Set Order.
^2
10
Figure
4:
the agent
Even
may be
if
V
is strictly
indifferent
quasi-concave in a„
between two actions for a
range of types.
indifferent
between two actions over a
of types, as shown in Figure
set
Figure
3,
it
implies that the set of best response actions
notice that as
r,
increases, higher actions
action leaves the set of optimizers,
set
is
it
come
important to
is
is
increasing in the strong set order.
into the set of optimizers, but
that the set
first
time.
In
once a lower
When this property is
satisfied,
it
of vectors which represent optimal behavior will be convex.
x and y are both vectors of jump points representing optimal behavior;
In the figure,
the figure
show
it
never reappears. Further, once a given action has entered the
of optimizers, no lower actions enter the set for the
straightforward to
Thus,
The proof makes use of an important consequence of
address the issue of multiple optimal actions.
the SCP-ER.:
4.
show convex combinations of x^ and y^
the arrows in
and any such convex
for m=l,..,4,
combination also represents optimal behavior.
Lemma
2.2: Define Y^as above, i=l,..,I.
correspondence (r,(X),.., r,(X)):Z—>S.
Then there
Proof of Lemma: The proof proceeds by checking
r=(r,(X),..,r/X))
is
With
this result in place, all that
which are consistent with
players
strategies for
that the
nonempty, has a closed graph, and
Kakutani's fixed point theorem will give the
result.
The
is
details are in the
By
r(X)=X.
Thus, for each player
/
we
in the behavior of
correspondence
given
each player
at the
"jump points." Consider an
T was constructed so that each x* represents
X,
call
it j3,(f,).
Then
An
definition,
r(X)
describes
do on
we
Then,
X
such
"jump
like to the
all that
that
remains
is to
Xe T(X).
The
a nondecreasing, optimal strategy for
the vector of nondecreasing strategies (Pi(ti),..,P,(t,))
pure strategy Nash Equilibrium of the original game, since
2.2.
Appendix.
assign strategies to
is to
can assign any behavior
points," {x'}, without affecting the best responses of the opponents.
i
convex-valued. Then
is
consistent with X, and that each player does not care what the other players
a set of measure 0.
player
correspondence
remains in proof of Theorem 2.1
the fixed point of
a fixed point of the
each player which are optimal almost everywhere in response to behavior by the other
players which
fill
exists
j3,(f,)e
af^(r,|X) for i=l,..,I
and
is
a
f.e T..
Existence Theorem for Strategies of Limited Complexity
Now, we
turn to a generalization of
condition (2.2).
The
basic idea
is that
Theorem
2.
1
beyond games with
an equilibrium will exist
if
we
the single crossing
can find bounds on the
"complexity" of each player's strategy such that each player's best response stays within those
bounds when the opponents use
formalization
we
use
for
strategies
representing
which can be represented within those bounds.
strategies
builds
directly
on
our
nondecreasing strategies (there are, of course, altemative representations).
definition, as follows:
11
representation
We
will
The
of
need a
The strategy ajit) has at most
direction changes if there exists a
Definition 2.4
K
nondecreasing vector z e
<
Zf such
that for all
,
Q<k
K the following holds:
z^
t'<t"<
Zt+^j
odd, then for all z^< t'<t"<
z^+i,
If k is even, then
(i)
for
all
a,{t;)<ait;').
Ifk
(ii)
is
We
X2 X^ X^ X^
X^
tj
ait;)>a,{t;').
X^
ATg
Xg t-
say that such a z represents the direction
Figure
changes of cc.(t).
5:
A
representation of a strategy with
two direction changes and four actions (M=3).
This definition, illustrated in Figure
5,
merely formalizes the idea
nondecreasing to nonincreasing, or vice versa,
has
at
most
changes
direction changes.
is that
An
at
most
K times.
that a strategy
changes from
Thus, a nondecreasing strategy
K
important feature of strategies with at most
direction
they are functions of bounded variation; this will imply that our existence results
extend to games with continuous action spaces in Section
3.
We can then generalize condition (2.2) as follows:
A
game of incomplete information
Limited Complexity Condition
satisfies the
there exists a vector of nonnegative integers 'K.={K^,..,K^, such that, for all
player
I's
is finite,
opponents j?^/ use strategies which have
then there exists a best response for player
changes. Further,
open
at
interval
when opponents use such
T|cT^ such
Unlike Theorem 2.1, where
from the SCP-IR, here
we
we
i
/,
if
each of
most Kj direction changes, and/^
which has
at
most K^ direction
strategies, then for all
a^^l, there
that C/,.(a,.,a_,.(),?,.)= C/,.(a;,a_,.(),r,.) for all
/,.e
is
no
(2.4)
7/.
could prove convexity of the best response correspondence
require an additional
assumption which implies that the best response
action
unique for almost
is
all
types.
This in turn
implies that there will be a unique vector of
jump
points representing the best response strategy.
see
why
Figure
6,
this
where a player
actions over
vectors,
assumption
is
required,
is indifferent
two regions of
types.
To
consider
?uc,+0-k)y,
between two
There are two
x and y, both of which represent optimal
However,
the
convex combination of x
and y would assign the player
to use action
A,
12
in
Ax,+(1-A)y,
Figure
6:
actions
Aq and A, over an open
Player
i
is
types, violating (2.4),
behavior.
if
indifferent
and the
representing optimal behavior
set
is
between
intervals
of
of vectors
not convex.
,
the region (Ajc,+(1-X)>',,Xjc2+(1-X)}'2).
which
is
While the Limited Complexity Condition may
not optimal. Condition (2.4) rules this out.
some problems
arise naturally in
of U-
(the case
shaped strategies seems especially promising), another possible motivation for the condition could
be bounded
rationality
Under condition
representation
is
on the
(2.4),
part of the players.
we can extend the logic
of Theorem
2.
in a straight-forward
1
way. The
a natural extension of the one developed for nondecreasing strategies; the vector
which represents our
K
strategy will have
monotonic portion of the
This
strategy.
is
subvectors, each of which describes behavior on a
formalized in the appendix, as part of the proof of the
following theorem.
2.2 Consider a game of incomplete information, where (2.1) and the Limited
Complexity Condition (2.4) hold for some K. Then this game has a PSNE where each
player's strategy, /3.(), has at most K- direction changes.
Theorem
It is
between
interesting to discuss the relationship
literature, especially
we do
general:
Radner and Rosenthal (1982).
not need
it
to hold for all
where a given player's response
On
(2.1)
K, but rather just
Under those conditions, they
and
and the results from the existing
the one hand, condition (2.4)
to a "simple" strategy
and Rosenthal maintain assumption
of types.
this result
for
some K. What
it
seems quite
rules out
is
games
becomes ever more complicated. Radner
finite actions,
and further they require independence
They then
find existence of a pure strategy equilibrium.
present counterexamples, such as an incomplete-information variant of matching pennies, where
correlated information leads to nonexistence of a pure strategy equilibrium.
Since our results do not place ex ante restrictions on the joint distribution over types,
The setup
interesting to revisit their example.
can choose actions Aq or A,.
while
if
When
the players
condition (2.4). Since player
Pr(a^.=Aol?,.)
that
as follows: the
match
whenever player
starting
interval
1
The
triangle 0<f,<?2^1-
types
t^
when
on which player
uses a strategy with
i
is indifferent
However,
2 pays $1
directly affect payoffs,
(i.e.
K+
1
direction changes using Definition 2.4,
in response to
actions.
alternates
wishes to use a strategy with
K+l
direction changes.
induce a response by player 2 represented by
K+ 2
13
K+l
first
issue
A,,
is
A
and A,
K
This can be
change degenerate.
direction changes, player
In turn, such a strategy
direction changes.
The
between Aq and
to avoid a match.
with the
an arbitrary strategy by player 2 with
fails
direction changes, the
with Ag), player 2 potentially has the incentive to switch between
1
game
Pr(a^—Aol?,.)=.5, and since
if
between the two
K direction changes
to player 1,
argue that this
number of
player y uses a finite
times as well, but starting with A, due to the incentive of player
represented as
do not
We now
is
zero-sum, and each player
is
only indifferent between the actions
cannot be constant in
wUl be no open
K times,
/ is
game
their actions, player
they do not match, the players each receive zero.
and are uniformly distributed on the
there
is
it
by player
Notice that
1
this logic
1
will
does
not change
if
from player
we
reorder the action space for player 2; the fact that player 2
I's strategy, but then player
1
tries to
Existence of Equilibrium in
Games with
to construct equilibria of
fails.
a Continuum of Actions
This section shows that the results about existence in games with a
be used
"run away"
"catch" player 2, leads to a situation where
can potentially get progressively more complex, and thus our condition (2.4)
strategies
3.
tries to
games with a continuum of
finite
actions.
number of
actions can
As discussed
introduction, the assumption about finite actions plays a dual role in our analysis.
guarantees existence of an optimal action for every type.
The second
are continuous in actions,
The arguments
actions for the first purpose.
equilibrium in a sequence of
We
actions.
potentially
change
The
finite
games
will
we no
First,
In
games
longer rely on the assumption of
in this section then
it
role is to simplify the
description of strategies so that they can be represented with finite-dimensional vectors.
where payoff functions
in the
show how
finite
the existence of
imply existence in a game with a continuum of
also extend the results to classes of games, such as first price auctions,
which
have discontinuities in payoffs when an increase in a player's action causes a discrete
in the probability
of "winning."
properties of the equilibrium strategies implied
by
the Single Crossing Condition or the
Limited Complexity Condition play a special role in this section.
While
arbitrary sequences
functions need not have convergent subsequences, sequences of nondecreasing functions (or
generally, functions of
bounded
variation,
which can be expressed
as the difference
of
more
between two
nondecreasing functions (BiUingsley, 1986, p. 435)) do have almost-every where convergent
subsequences by Helly's Theorem. Thus, our restrictions play two roles
first,
they guarantees that the strategies in finite
that Kakutani's fixed point
equilibria to finite
show
3.1.
is that
in the existence results:
games can be represented with a
theorem can be applied, and second, they guarantee
finite
that
games have almost-every where convergent subsequences. All
vector so
sequences of
that
remains to
the limits of these sequences are in fact equilibria to the continuous action game.
Games
with Continuous Payoff Functions
This section extends the results of Section 2 to games with a continuum of actions. The
following theorem shows that in a
all finite-action
game
will
games have
game of incomplete
information,
if
payoffs are continuous and
equilibria in ftanctions of bounded variation, then the continuum-action
have an equilibrium as well.
14
Consider a game of incomplete information which satisfies (2.1).
/4= {/4\..y^), where /^=[a^,a.], (ii) for all i, u.(a,t) is continuous in a on
Theorem
that
(i)
(Hi)
for any finite /^(Z/4, a
PSNE
exists in the
where the equilibrium strategies
/f,
Then a
Suppose
3.1
PSNE
game where
P„(t) are functions
game where players choose
exists in the
[a.,a.],
and
the players choose actions from
of bounded variation.
actions from
/4.
Consider a game of incomplete information which satisfies (2.1). Suppose
/4= (/4\..yr^), where /^=[a.,a.], and (ii) for all i, ufsi,i) is continuous in a on [a^,a.].
Corollary 3.1.1
that
(i)
Then:
PSNE,
If the Single Crossing Condition (2.2) holds, then there exists a
(i)
player's strategy
is
nondecreasing
in
her
P'(t),
where each
type.
If the Limited Complexity Condition (2.4) holds for some K, then there exists a
P'(t), where each player i's strategy has at most K- direction changes.
(ii)
While Corollary 3.1.1 does require the regularity assumption
integrable,
it
that u.
is
PSNE,
continuous and
does not require differentiabiUty or quasi-concavity, two assumptions which often
Furthermore,
arise in altemative approaches.
correlation structure
it
between types above what
does not place any additional restrictions on the
is
required for the (economically interpretable)
condition that best responses are nondecreasing.
Despite the generahty of Corollary 3.1.1, the restriction that payoffs be continuous
many
out
interesting
The next
games.
still
rules
games with discontinuous
section extends this result to
payoffs such as auctions.
3.2.
Games
with Discontinuities: Auctions
and Pricing Games
This section studies games whose payoffs exhibit a particular type of discontinuity.
sees a discrete change in her payoffs depending on whether she is a "winner"
in a single-unit auction or the
(first
goal
low price
price or all-pay), pricing
is to allocate
in
a pricing game), or a "loser." Examples include auctions
resources between multiple players. Winners receive payoffs
The
games with only two outcomes.
given a realization of types and actions
= (p,(a)
v,{a,,i)
v,(a,,t),
while
allocation rule (p^ia) specifies the probabihty that the player
receives the winner's payoffs as a function of the actions taken
M,(a,t)
the high bidder
(i.e.,
games, and more general mechanism design problems where the
losers receive payoffs v^{a^,i).
attention to
A player
is
by
all
players.
Under our assumptions,
We
will restrict
then, a player's payoffs
given as follows:
+ (1 - <p.(a))
•
v,(a,,t)
(3.1)
This formulation highlights the second assumption impUcit in this formulation: payoffs depend on
the other players actions only through the allocation, not through payoffs
assumption can be relaxed, but
it
sufficiently
comphcates the analysis
15
that
directly.
we wiU
This
not consider
it
here.
Each player chooses from a
of actions, ^=Qu[q^,a.].
set
each player zero expected payoffs.
The
action 2<min,{ a,
}
guarantees
We will maintain the following assumption about payoffs:
v,(e.t)=v,(e.t)=Oforallt.
For
(i)
a.e [a,, 5^]
all
v,.(a,,t)>0
and
(3.2)
all t,
>
implies v,.(a„t)
v,(a,,t),
There are several classes of examples which have
and
v,.(a,,t)<0
(ii)
this structure. In
implies v,.(a„t)<0.
a first-price auction, the winner
receives the object and pays her bid, while losers get payoffs of v,.(a,,t)=0.
Weber's (1982) formulation of the mineral
to the bidder conditional
(3.3)
In
Milgrom and
rights auction, v,.(a,,t) represents the expected payoffs
on the vector of type
and the vector
realizations,
t is
interpreted as a vector
may be
of signals about each player's true value for the object (where signals and values
correlated). In an all-pay auction, the player pays her bid
some
In
the object.
no matter what, but the winner receives
pricing games, the lowest price (the highest action) implies that the firm
captures a segment of price-sensitive consumers, while having a higher price implies that the firm
only serves a set of local customers.
We will restrict attention to allocation rules which take the following form:
y
<p,(a)=
+ l{KKK|>/-t>K|}/'(^r) nlf
l{K|>/-t}
i-TTlf
1
(3.4)
((Ti.,C7r)<=(l,..,/)\l
s.t
tTiricrr=0
In this expression, m,(-)
probabiUty zero
1
if
if
a
is
strictly
k or more opponents choose actions such
I-k or more opponents choose
"ties," in
which case p:2"'
Player
increasing function.
that
/
receives the object with
mfa^>mf^a^, and with probability
actions such that mj{a^<m.{a^.
""-^[O,!] is the probabiUty that player
/
wins.
The remaining events
We further assume that:
If lG^>0, then P{<5t)<\.
This assumption requires that
player wins with probability
To
(3.5)
if
1.
player
This
/
last
ties for
^=1 and mfa^=-a..
opponents place a higher bid.
That
If ties are
winner with a non-empty subset of players, no
assumption simplifies the proof, but can be relaxed.
interpret expression (3.4), consider the
In this case,
are
is,
example of a
the player
first
price auction for a single object.
wins with probability zero
if
1
or more
broken randomly, the probability of winning given
that
no
opponents place a higher bid and further the player ties with the subset of opponents Oj is given by
p(crj.)
=
i^^-
More
general
mechanism design problems
player's payoffs satisfy the single crossing property,
allocation rule is
also
fall
into this
framework.
When
a
only direct mechanisms in which the
monotonic wiU be incentive compatible; thus, any incentive-compatible rule for
16
winners from / players will take the form of (3.4).
allocating k
mechanism
between
to allocate a single object
risk neutral agents
Further, consider an optimal
whose (independently
distributed)
types represent their valuations for the object. In this case, the actions are reports of types, and the
allocation rule determines the
winner according
who
to the agent
has the highest "virtual type"
(Myerson, 1981).
Thus, the players' expected payoffs can be written as follows:
C/.(a,,a_,(-),r,)
= jM,(a,,a,(t_,),t)/(t_,|0^t_,
= j[(p,(a,,a_,(t_,))
= jv,(«,-,t)
•
+ (1 - <p,(a,,a_,(t_,)))
v,.(a,,t)
/(t_,|0^t_,
+ |[v;.(a,,t) - v,(a,,t)]
•
•
Y,(«„t)] /(t_,|r,)Jt_,
•
(p,(a,,a_,(t_,)) f(tjt,)dt_,
•
We maintain the following additional assumptions:
For
all
/
and a.e
Av,.(a,-,t)=
For
all
a,e
strictly
Since (3.7)
in fact
t,
as
and
is
[a,-,a;],
V ,.(a,.,t)- V .(a,.,!)
[a,-,
a,], all
the
most
in the
Lemma
nondecreasing in
t-
and nondecreasing
restrictive
bounded and continuous
t
and
strictly
supp(/'f,.(t_,.)),
and
(3.6)
(-a,-,?,).
<
£'[Av,.(fl!,,t)|f.,k,
it is
worth pausing
t_,.
<
k2]
is
to note that
it
has
MUgrom and Weber (1982) for the case where Av,. is nondecreasing in
of densities
is
discussed in more detail in Section 4
kj]
is
nondecreasing
log-supermodular
3.2
in
(?,.
,kpk2)/or all
is
g,.
log-supermodular
nondecreasing
(3.1)-(3.7).
in place,
we can
Consider a game which
state
our existence
satisfies (2.1).
result,
existence
is
PSNE,
if and
Then
only
if
P*(t),
proved
Appendix:
in the
Let the action space for each player
Suppose further that the Single Crossing Condition,
Then, there exists a
a.e.
a.e. (equivalently, t is affiliated).
be /^=Q\j[qi,ai], assume that the support of the distribution F(t)
where each player's strategy
is
a product
(2.2),
is
holds for
and assume
set,
a,.e [a^,a^].
nondecreasing
in
her
type.
established for the continuous-action game, standard arguments can be used
to verify the usual regularity properties.
interior
increasing in
(a,,t),
(3.7)
Consider a conditional density f(t_i\t) which
With these assumptions
Once
in
in k^kj.
our these assumptions,
in (3.6) (log-supermodularity
< t_,. <
E\g^(t)\t.,k^
Theorem
v,(a,,t) are
Appendix).
3.2.1
/(t_,lf,) is
is
and
ki<k2 such that kpk^e
increasing in
been characterized by
assumed
v,.(a,.,t)
For example,
strategies are strictly increasing
on
the
of the set of actions played with positive probability, and no player sees a gap in the set of
actions played with positive probability
assumptions,
by opponents. Further, with appropriate
differentiability
we can use a differential equations approach to characterize the equilibrium.
17
The
that
intuition
Theorem
player's
own
behind the proof of Theorem 3.2 can be summarized as follows. The only reason
3.1 cannot
If the
action.
increasing one's
be applied
own
However, observe
payoffs will be continuous in
|r.b3*(fy)<
A:[
are atomless.
a,.,
a
will lead to a discontinuous increase in the
that if each ^,*(r,) is strictly increasing
since the Lebesque measure of the sets
changes continuously in k, and since
If,
potential discontinuity in a
opponents use a particular action a with positive probability, then
action to be just higher than
probability of winning.
game has a
directly is that the
in contrast, P*{t^ is constant at
Z?
we
have assumed
on a
subinterval
on
T-,
expected
Uy|j3J(rp<)t| and
that the type distributions
S of
T-,
then each player yV/
sees a discontinuity in their expected payoffs at aj=b.
Our argument
Let P*(t) denote the equilibrium strategies.
almost every tj^e
possibility of
r,.
sees zero probabiUty of a
tie at
establishes that for each player
her optimal action P*(t^).
Ruling out the
such mass points in the limit involves showing that mass cannot be "adding up" close
to a particular action as the action
Recall that a sequence of nondecreasing
space gets finer.
functions has an almost-everywhere convergent subsequence, and further
converges uniformly except on a set of
strategy involves a
mass point
at
some
arbitrarily small
"jump over"
that player's action
action only slightly higher than
measure.
Thus,
if
b
this
subsequence
a player's limiting
mass of players must
action b, then given a d>0, a positive
be using a strategy on [b-d,b+d] as the action grid gets
incentive to
/,
But then, other players have an
fine.
rather than using an action less than a'—d: with an
b+d, the other players can beat
b+d\. But this will in turn undermine the incentives of the
first
all
of the types playing on [b~d,
player to choose an action on [b—d,
b+d].
This result then generalizes the best available existence results about
first
price auctions.
Previous studies (Maskin and Riley (1993, 1996); Lebrun (1995, 1996), Bajari (1996a)) have
analyzed independent private values auctions, as well as affiliated private values auctions and
common
value auctions with conditionally independent signals about the object's value (Maskin
and Riley (1993, 1996).
The work
closest to ours is Lizzeri
and Persico (1997),
who have
independently established existence and uniqueness of equilibria in a class of games similar to the
one studied above, but with the
differential equations,
approach taken in
this
restriction to
two bidders
while differentiability of
utility
(further, their
functions
paper is different from those of the existing
the issue of monotonicity of strategies and existence,
showing
is
approach
is
based on
not assumed here).
literature, in that
it
The
separates out
that monotonicity implies existence.
Thus, the only role played by assumptions about the joint distribution over types
is to
guarantee
that the single crossing property holds.
The next
section analyzes applications, including
18
first
price auctions.
To preview, however.
.
the weakest general conditions for private value auctions requires
functions,
which amounts
the
form
v.(a,-,t),
for
two bidders we
rights" auctions,
which allow for
for monotonicity in the mineral rights auction with
We are
more than two
(ii)
utihty functions of
and
{a^,t^,
not aware of general conditions
bidders; however, our numerical
results indicate that monotonicity holds in the relevant range (i.e.,
examples with log-normally distributed values and
utility
form V^it-a^, and
require that v,(a,,t) be supermodular in (a,,f,)
increasing in (-a.,t^, and that values are affiliated.
strictly
log-supermodular
to log-concave utility functions if utility takes the
more general "mineral
affiliated types. In the
(i)
near equilibrium) for several
signals.
Characterizing the Single Crossing Condition in Applications
4.
This section characterizes the single crossing condition in several classes of games of
incomplete information.
It
primitives of a game, that
which are
applies results
is,
from Athey (1995, 1996)
of the
utility fianction to satisfy
players use nondecreasing strategies. Thus, applying
The
on the
the utility functions of the players and the joint distribution of types,
sufficient for the expected value
characterize classes of
to describe conditions
Theorems
2.1, 3.1,
SCP-IR when
and
3.2,
we
all
other
are able to
games which have PSNE.
results in this section are
grouped according
to the structure
of the problem: additively
separable problems (such as investment games), multiplicatively separable problems (such as
private value auctions),
in the
and non-separable problems (such as mineral
Appendix summari2:e our analysis from
rights auctions).
Table 7.1
this section, stating the conditions to
check for
games of incomplete information which take a variety of structures. The
theorems about comparative
results
AH
problems from Athey (1995, 1996).
statics in stochastic
of
of these
can be appUed to derive sufficient, and sometimes necessary, conditions on payoff
functions and type distributions so that the
The
will
results are applications
results
make use of
be interested
game
satisfies the Single
Crossing Condition.
the properties supermodularity and log-supermodularity.
in product sets,
we
will state the definitions for that case.
Let
Since
X= x
we
X„ and
n=l,..,N
consider an order for each X„, which wiU be denoted >.
ordered
set.
Suppose
An example of such a product set is SK" with the usual
that
order,
each
X^
where x>y
is
if
Definition 4.1
A
A
sets as
for all x,yeZ,
non-negative function h:X-^3{ is log-supermodular'^
function h-.X-^SH
hixvy)-\-h{xAy)>h(x) + h(y).
for allx,yeX, h(x
a totally
x>.y^ for
We will use the operations "meet" (v) and "join" (a), defmed for product
xvy = (max(x,,>',),..,max(x„,>'j) and x Ay = {Tmn{x^,y^),..,^nnix„,yJ).
n=l,..,N.
follows:
set
is
v y) h(x a y) > h(x)
supermodular
if,
h{y)
^ Karlin and Rinott (1980) called log-supermodularity multivariate total positivity of order 2.
19
if,
Clearly, a non-negative function h(x)
kSi"-^^, and we order
h
differentiable,
is
is
log-supermodular
if ln(h{x)) is
vectors in the usual way, Topkis (1978) proves that
supermodular
if
and only
if
-^^Kx) >
for
additional facts
about these properties are important: (1)
supermodular,
then
h(x,t)
SCP-IR;
satisfies
if
sums
(2)
# j. For
all /
h(x,t)
of
the
if
h
(resp.
log-supermodular),
nondecreasing; (4) a density
(as defined in
then
log-supermodular
is
so
if
supermodular or log-
is
supermodular
functions
General Characterizations
4.1.1.
Additively separable expected payoffs
games where payoffs
First consider
gi(a.,t^)
if
the
random
if
where
/z(a,(x,),..,a„(xj),
and only
twice
are
h(x)
is
a,()
is
variables are ajfiliated
Milgrom and Weber, 1982)7
4.1.
=
is
is
moment, four
supermodular, while products of log-supermodular functions are log-supermodular; (3)
supermodular
When
supermodular.
+ H.{a^,t^.
the expected utiUty
are given
by
M,(a,t)
=
+
gi(a.,t^)
/z,(a,t),
and
A game which fits into this framework is an all-pay auction,
from losing the auction and having
to
pay
a,,
C/,(a,.,a_,(-),?,)
where
gi{a.,t) is
while H.{a.,t) gives the expected
retums to winning as opposed to losing the auction, weighted by the probability of winning.
sum of supermodular
Since the
the
SCP-IR, supermodularity of
g,.
functions
and
//,.
supermodular, and since supermodularity imphes
is
will
imply the SCP-IR.
Milgrom and Shannon (1994), shows function
The following
due
result,
to
that supermodularity is the "right" property for
additive problems.
Lemma
Consider
4.1:
g.,
//,:9t^—>9?. g^{a^,t)+H^(a^,ti) satisfies single crossing
incremental retums (SCP-IR) in (a^;t-)for all H- which are supermodular
is supermodular.
For example,
in
sufficient for the
an aU-pay auction,
SCP-IR
we
in
a.,
might wish to characterize conditions on
to hold without specifying
The next
g,.
which guarantees SCP-IR of payoffs across an unrestricted
step is to characterize
The following Lemma apphes
when expected
results
1981).
is
from Athey (1995, 1996)
also equivalent to the as
is
which
20
are
strategies
the weakest
class of functions
to this
i/,..
problem:
monotone likelihood
ratio
Log-
property in statistics.
property
(MLRP) (Milgrom,
A probability density y(j;0) satisfies the MLEIP if the likelihood ratio y(r;0j)//(r;0J is nondecreasing
all 0,>6li.
if g.
values of payoff functions are supermodular.
^ See Whitt (1982) and Karlin and Rinott (1980) for related discussions of this
supermodularity of a density
g,.
only
any additional structure on opponent
besides monotonicity, and allowing for general type distributions. Supermodularity
condition on
of
if and
in t
for
Lemma
4.2
supermodular
such that
(ii)
k.{a.,i)f.{i_\t.)di_. is
(i)
in (a,, rp, _/=!,..,/, if and
l^Ct,,) is
only
if
i
^(t_,|?,)^t_,. is
nondecreasing
in t.for all
S
nondecreasing.
UA^) " log-supermodular, thenl
l^(t_,.) is
supermodular in(aJ^for allk-whichare
nondecreasing
^(t_,|f,)<it_,. is
nondecreasing
tjor
in
all
S such
that
t_,,
Athey (1995) provides a more thorough characterization of supermodularity
based on altemative assumptions about the payoff functions;
games of incomplete information, and
it
Lemma
in stochastic
4.2
is
problems
most apphcable for
does not rely on higher order derivatives of the payoffs.
Together, the above results can be used to characterize the single crossing condition in a class
of games with additively separable payoffs.
Consider a game of incomplete information which satisfies (2.1). Suppose
by M,(a,t) = g,(a,,?,)+/z,.(a,t). Then if (i) for all i, g. and h^ are
supermodular in (a.,t^, (ii) h. is supermodular in (a^,a^ and (a.,tj), i^j, and (Hi) the joint
distribution over types fit) is log-supermodular, then the game satisfies the Single Crossing
Theorem
4.3
that payoffs are given
Condition
(2.2).
Proof: Apply
Lemmas 4.1 and 4.2,
from above, which guarantees
letting k{a.,t)
=
h.(a.,a_^(t_),t), recalling
that for nondecreasing
a_,(t_,.),
Fact (3)
supermodularity of
/i,.
implies
supermodularity of k^.
While Theorem 4.3 can be used
Theorem 4.3
to estabUsh existence
of equilibrium, the games studied in
also satisfy Vives' (1990) sufficient condition for existence of
are supermodular, a pointwise increase in aj(tj) increases the returns to
game
is
supermodular in strategies under the pointwise order.
applies theorems about existence of equihbria in supermodular
a,,
PSNE. When payoffs
for
all
r,.,
and thus the
Under those conditions, Vives
games without relying on
the
Single Crossing Condition, and thus without reference to assumptions about the joint distribution
over types.
However, Vives'
supermodular, and
it
will not
result
rehes crucially on the
assumption that payoffs are
be apphcable to the other classes of games, including games with
log-supermodular or single-crossing payoff functions, as well as the additively separable aU-pay
Lemma 4.1
auction with independent private values (which rehes
on
monotonicity of strategies
many games, we wUl
in applying
is
of independent interest in
Lenmia 4.2 and Theorem 4.3 even
In terms of apphcations,
many of
in
dkectly).
at
Further, because
times be interested
games with supermodular payoffs.
the supermodular
games with complete information which
have been studied by economists (see Topkis (1979), Vives (1990), and Milgrom and Roberts
(1990) for examples) can also be studied as games of incomplete information.
pricing or quantity
games have
variations
For example,
where firms have incomplete information about
21
their
rivals production costs or information
Games between two
about demand.
are strategic substitutes can also be considered, such as a
firms
whose
Coumot
players
game between two
quantity
quantities decrease the marginal revenue of the opponents, but
whose choices
where the firms have
incomplete information about rivals costs.
Multiplicatively separable expected payoffs
4.1.2.
The next class of games we consider
of two
product
gi(a^,t)-H.(ai,t.).
where
H^{a.,t)
=
nonnegative
A
Pr{bid
so
terms,
game which
fits
a class where player
is
framework
wins the auction given
a,,
payoffs can be written as the
u.(si,t)=g.(a.,t.yh.(a,t)
that
into this
/'s
t.];
is
and
a private-value
C/,.(a,,a_,.(-).0
first
=
price auction,
other examples include pricing games where
firms' products are imperfect substitutes.
The theory of comparative
problems merely by taking logarithms,
separable
multiplicatively
ln(g,.(a,-,f,))+ln(/f,.(^,,?,.)),
log-supermodularity
The methods
is
for additively separable problems
statics
and applying
Lemma 4.1.
i.e.,
can be applied to
ln([/,(a,.,a_,(),r.))
=
Thus, for multiplicatively separable problems,
the right property to require in order to guarantee the SCP-IR.
for analyzing log-supermodularity in stochastic
problems are different from those
for supermodular functions; however, the sufficient conditions are in the
end
quite similar.
The
following characterization theorem follows from Athey (1996):
Lemma
and
(i)
let f.{t_.\t,)
k-(aj,t)f-(t_-\t-)dt_- is
\
supermodular,
(ii)
a probability density. For i=l,..,I, let k-:'3i''*^^—^Si^
be the conditional density o/t_, given t.. Then the following conditions hold:
\
if and
only
iff(t) is
in (a-,t)for all i=l,..,I
and all k-log-
log-supermodular.
log-supermodular in
(a^,t) for all f{t)
log-supermodular,
if and
only
log-supermodular.
Log-supermodularity
because
is
log-supermodular
k-(aj,t)f-(t_.\t.)dt_- is
ifk(a^,t) is
lj(t_,.)
where f(i)
4.4: Letk^.'3{'-^'^S{^
it is
is
an especially convenient property for working with expectations
preserved by multiplication; thus, multiplying the integrand by an indicator function
preserves log-supermodularity so long as the set
common example of a sublattice is
a cube in
We can pull these results together into
9t".
5
is
a sublattice (see Topkis, 1978); a
For more discussion see Athey (1996).
the following set of sufficient conditions for the Single
Crossing Condition to hold, in a manner similar to Theorem 4.3.
Consider a game of incomplete information which satisfies (2.1). Suppose
by u.(a,t) = g.(a,.,f,)-/z,(a,t). Then if (i) for all i, g,. and /i, are nonnegative
and log-supermodular, and (ii) the joint distribution over types f{t) is log-supermodular, then
the game satisfies the Single Crossing Condition (2.2).
Theorem
4.5
that payoffs are given
We will apply this result to private values auctions and pricing games in Section 4.2.
22
4.1.3.
Nonseparable Expected Payoffs
This section analyzes games which do not
separable
fit
into the classes of additively or multiplicatively
games analyzed above. Of course, we can always apply
of the form M,(a,t)=g,.(a„0+^.(3't) or u.(a,t)=g.(a.,t)h.(a,t),
requirements on
necessary
h.
which must be
if gi{a^,t^ is
satisfy the conditions
satisfied to apply
studied in Section 4.2.3, or
That
is,
u.
must depend on
=
M,.(a,,a_,(t_,.),t)
identical bidders using
E^
there are only
if (ii) u.{si,i)
[v,.(a;,t)|max{/'^.:;V/}=5,.]- 1,
That
,i('^<)-
^^
^
aj(-)=ai{-) for
strategies,
is,
s. is
and payoffs depend on opponent types only through the
action.
In a multi-unit auction,
appUcations,
5,
might be a sufficient
statistic for
weaker conditions on the
game
In particular, in many-player
s^.
price mineral rights auction with
all
Then,
j,l^i.
k.{a.,s^,t-,Cf._^
=
the value of the highest opponent type,
realization
might be a different order
5,.
do not
and actions through a single index, denoted
first
the
are stronger than
players, as in the noisy signaling
For example, in a
k.{a.,s^,t.;QL_^.
that
takes a very special form.
the opponents' types
symmetric
two
However,
as the mineral rights auction)
of Theorems 4.3 and 4.5. This section shows
utility fiinction suffice, if either (i)
games,
letting g.(a.,t,)=l.
Theorems 4.3 and 4.5
some games (such
Further,
constant.
the above results about payoffs
of
statistic
this
type and the associated
of the distribution.
In other
t_,..
Our theorem makes use of a weak version of the SCP-IR. Formally:
Definition 4.2 h(x,6) satisfies (Milgrom
incremental returns (WSCP-IR) in {x;6)
following condition for all
The following Lemma
Lemma
4.6:
is
proved
Consider a function
does not change with
k.{a^.,s.,t) is
6[f>6i^:
t^,
J^,.(a,.,5,.,r.)/^ (j,.|r.)rfr.
g(6i)>0 implies g(6^)>0.
in
in (a^t^)
satisfies
Athey (1996).
/:,.:9t'-^SR
and suppose
supermodular
and Shannon's) weak single crossing property of
if, for all x^yx^ g{6)=h{Xfj,6)-h(x,j6) satisfies the
f
and
SCP-IR
and a conditional
(sjit) is
satisfies
whose support
log-supermodular. Suppose further that
WSCP-IR
in (ajsj.
Then
in (a.,tj.^
Lemma 4.6 can be applied to games of incomplete information,
be used in our study of mineral
density fis^U)
and the following theorem will
rights auctions as well as noisy signaling
games.
Theorem 4.7 Consider a game of incomplete information which satisfies (2.1). Suppose
that for all i=l,..I, there exists a random variable s^ and a family offunctions ki(-;a._,):S(^—^Si
indexed by opponent strategies, such that (i) U^(a.,a_i(-),t^)=E^ [/:(a,.,5,.,?,.;a_,.)|?,.]; (ii) when
«_,.(•) is
and
with
^
nondecreasing,
f,.
Then the game
supermodular in (a.;t^ and satisfies WSCP-IR in (af,s);
(jj?,) is log-supermodular and the support does not change
k{a.,Sj,t:,(X_) is
(Hi) the conditional density
f
satisfies the Single
Athey (1996) further shows
that
Crossing Condition
(2.2).
none of the hypotheses can be weakened without strengthening the
Table 7.1 in the Appendix.
23
others; see
•
4.2.
Applications
4.2.1.
Auctions
Consider a
value
?,.,
Values Auctions
Private
4.2.1.1.
value auction, where each player
first-price, private
and values are drawn from a distribution satisfying
given by
function
Suppose players j^i use nondecreasing
V,.(a,,?,).
when
ties are
resolved randomly
where the auctioneer keeps the object
To
see
when
utility takes the
it is
form
/f(a,lf,) is
V,.
where V,(0)>0 and
will
log-supermodular.
be log-supermodular
if
strategies.
let
H,.(a,lr,.)
utility
function is
Writing the objective
moment
for the
Then,
ln(V^.) is
be log-supermodular in
=
consider the
game
Pr(a,.>a^.(r^.) for: jH\t^.
is
concave and
If
strictly increasing,
Next, consider the issue of
(f,,a,).
Since the strategies are nondecreasing (and recalling Fact 3,
by monotone transformations of
the joint distribution of types, F(t),
not necessary) condition for this
Now,
cumbersome, so
in the case of ties.
V^(a,.,r,.)=V;(r.-a,.),
that log-supermodularity is preserved
be shown using
is
Each player's
the conditions of Theorem 4.5 are satisfied, consider first the utility function.
straightforward to verify that
when
(2. 1).
own
(/=!,..,/) observes his
i
is
the variables), H{a^\t^)
wiU
A sufficient
(but
log-supermodular.
that/(t) is log-supermodular,
i.e.,
types are affihated (this can
Lemma 4.4, letting g.{zX)=\t<a.^{i))
consider the case where
likewise for players
ties are
resolved uniformly.
Then define
/i,(a,t)
as follows (and
2,..,/):
^(^)=t---iT-v^-ri(a-A)-iK<.]+A-iK=.])
Thus, the probability that player
given as Prfo, Wm^t^]
i
wins the auction when she has type
= H{a\t{) =E^
,
[/r(a,,a_,(t_,))].
?,.
If the types are affiliated, then
4.4 impUes that this probabiUty will be log-supermodular in (a,,?,)
supermodular in
symmetry)
l^i^l. This
Thus,
nonnegative
is
the following proposition (the assumption of the proposition that
always choose an action
Lemma
is
The game
log-
a.
{tj,t^
after
V,(<2,,?,.)
for
is
observing her type and thus can
which yields nonnegative payoffs).
Consider a private values, first price auction, where (2.1) holds. Suppose
utility Junctions VJ(^ai,t) are non-negative and log-supermodular,
is log-supermodular), and (Hi) ties are broken uniformly. Then:
further that (i)for all i, the
(ii) types are affiliated (f(t)
(1)
is
directly.
innocuous since the agent knows Vfa.,t^
Proposition 4.8
integrand
only to check log-supermodularity of /j(aj,a_,(t_,)) in {a^,t^ and
can be verified
we have
if the
a,,
Log-supermodularity can be verified pairwise, so that (using the
(flpt).
we need
and chooses an action
satisfies the Single
Crossing Condition
24
(2.2),
and a PSNE
exists in allfinite-
action games.
(2)
If,
in addition,
continuous,
and
there exists a
Now
PSNE
single unit. For example, in a 2-unit auction, the players with the highest
two bids win
Unfortunately, this complicates the analysis of log-supermodularity of the function
an object.
Pr(a, winsif,.).
?,.,
all
consider a generalization of Proposition 4.3 to multi-unit auctions, where each agent
demands a
on
i (iv) V, is strictly increasing in (—a-,t^), (v) V^ is bounded and
support off(t) is a product set, then (3.1)-(3.7) are satisfied and
in continuous-action games.
for
(vi) the
However,
the types are
if
drawn independently, then
and the expected payoff function reduces
supermodular
to
Pr(a,.
winsk,) does not depend
This
V^{a.,t)-Pr(a. wins).
is
always log-
log-supermodular; thus, for the case of independent types, the extension
if V,(a,,f,.) is
to multi-unit auctions is immediate.
Auction
All-Pay
4.2.1.2.
In this section,
we
consider an altemative auction format, the all-pay auction.
the highest bidder receives the object, but
model
activities
such as lobbying.
values formulation.
bidders pay their bids.
all
To keep
the analysis simple,
Let V^{a.,t^ take the form V,(?-a,).
This game has been used to
we will use an
Then a
In this auction,
independent private
player's expected payoffs from
action a. can be written as follows:
V;.(-a,)-hy,a,-a,)-Pr(fl,wins)
This game has an additively separable form, and thus by
a,.,
it
is
V;(f-a,) is increasing in
only
if it is
concave, that
Observe
that
we
t.
nonnegative and
straightforward to verify that y.(f-a,.)-Pr(a,. wins)
is
supermodular
and supermodular
is,
the bidder
is
in (a,,^,).
In turn, V,.(?-a,)
is
supermodular
Pr(a,.
and
if
strategies
Summarizing,
wins); since types are independent and bidder valuations are private,
crossing conclusion, in contrast to
Theorem
4.3.
and thus the game
we have the
is
not supermodular in strategies.
following proposition:
Consider a private values, all-pay auction, where (2.1) holds. Suppose
further that (i) for all i, the utility functions V.{t-a) are concave, and V,.(0)^0, (H) types are
independent, and (Hi) ties are broken uniformly. Then:
Proposition 4.9
The game
in
not true that a pointwise increase in player 7' s strategy leads to a pointwise increase
in player /'s best response,
(1)
if
risk averse.
this is not important for establishing the single
it is
look for the
is
to
have not discussed the interactions between the players'
determining the function
However,
we
4.1,
be supermodular. Since Pr(a. wins)
components of the objective function
nondecreasing in
Lemma
satisfies the Single
Crossing Condition
(2.2),
and a PSNE
exists in all finite-
action games.
(2) If in addition,
for
all
i
(iv) V^ is strictly
increasing, (v) V,
25
is
bounded and continuous, and
support off{i) is a product
continuous-action games.
(vi) the
in
then (3.1)-(3.7) are satisfied
and
there exists a
PSNE
Mineral Rights Auctions
First Price
4.2.1.3.
set,
We now generalize our analysis of auctions to consider Milgrom and Weber's
a mineral rights auction (their general existence and characterization results
(1982) model of
treat the /-bidder,
symmetric case and several other special cases), allowing for risk averse, asymmetric bidders
whose
utility
We
functions are not necessarily differentiable.
Suppose each agent's
begin with the two bidder case.
supermodular in
(a,,^,)
and
on both players'
conditional
I's expected payoffs
(written
utility
y,.(a.,f,,?2))
is
^^^ ^^^ ^e interpreted as the player's expected payoffs
(a,,?2)'
signals.^
When
player two uses the bidding function cc^it^, player
given her signal can be written as follows (assuming
broken
are
ties
randomly):
^,(a„a2(-),?,)=JM(«pA,?2)-4,>a,(,,)(^2)-/2(^2h)^^2
h
-\\v,{a„t„t^)-I^^^,„^,{t^)-f{t^\t,)dt^
To keep
things simple, consider the finite-action game.
incremental returns to increasing her bid from
A^
Figure 7
A^
to
(where
illustrates
m^<mu)
signal of the opponent, given that the opponent's strategy is consistent with x^.
regions to consider.
When
the
opponent bids
less than
A^
,
the
how
There are several
outcome of the auction
unchanged by the increase
Returns to player 1 from increasing
her bid from A „ to A„ given x^.
H
Li
player
1
neutral bidder, this
When
,
the
in
bid,
is
but
simply pays more. For a risk
,
A^
player I's
change with the
would be a constant.
the opponent bids
more than
outcome of the auction
is
also
not changed by the bid increase: player
loses in both cases and pays nothing.
the
Figure
7:
Player I's payoff function satisfies
WSCP-ER
in
intermediate
cases,
regions which involve
there
are
1
In
the
ties at either the
(ai;t2).
low or the high
^
To
=J
v^(z-
that
ti
see
an
example
a, )g(z|f,,fJ
and
fj
where
)^z, where z
these
assumptions
is affiliated
with
f,
and
on
/j,
and
the
v, is
payoff
(a,-,/])
and
26
are
and the region where
satisfied,
nondecreasing and concave.
each induce a first order stochastic dominance shift on G, and
(1995), supermodularity of the expectation in
function
bid,
(a„r2) follows.
v,
is
To
let
V^(a,,t)
see why, note
supermodular in (a„z).
By Athey
.
the bid increase causes player
to
1
change from losing for
winning for
certain, to
certain.
Within
each of these regions, expected payoffs are nondecreasing in the opponent's type since
increases the expected value of the object to bidder
When
realization
apply
this
1
bidder two plays a nondecreasing strategy, bidder one's payoff function given a
of
fj
WSCP-IR
satisfies
Lemma 3.2.1
to
show
in (af.^j)-
^^^
^^
i^
supermodular in
(a,;?,).
We
that the conditional expected payoffs for a bidder are
increasing in his type so long as
v.(a,.,t) is
nondecreasing in
t,
can further
always
strictly
thus satisfying (3.7). This gives the
following result:
Consider a 2-bidder first price "mineral rights" auction, where (2.1)
Suppose
holds.
fiirther that (i) for all i, the utility functions V,(a,,t) are supermodular in (a.,tj),
7=1,2, and nondecreasing in t, (ii) types are affiliated (f(t) is log-supermodular), and (Hi) ties
are broken uniformly. Then:
Proposition 4.10
(1)
The game
satisfies the Single
Crossing Condition
(2.2),
and a PSNE
exists in all finite-
action games.
(2)
If,
in addition,
continuous,
and
there exists a
for
all
(vi) the
PSNE
in
(iv) V, is strictly
i
What happens when we
try to
extend
all
signal of all of the opponents
wiU be
and Weber (1982) show
same
necessarily have the highest bid.
that
whenever
i^(a,,f,,j',)
model
=
to
V,.
is
bounded and
and
then (3.1)-(3.7) are satisfied
more than two bidders?
relevant to bidder one.
that (t^,s^) are affiliated
the opponents are using the
show
this
set,
If the bidders face
opponents use the same bidding function, then only the
a symmetric distribution, and
to
increasing in (—a^,t^, (v)
support off{t) is a product
continuous-action games.
strategies,
It is
and
is
5,
=
max(f2,-.,^„).
the distribution is symmetric.
MUgrom
Further, if
whichever opponent has the highest signal will
straightforward to extend
£'[v^(a,,f,,..,f„)|5,] is
V, is nondecreasing in t
when
Define
maximum
Milgrom and Weber's arguments
increasing in {t^,s^) and supermodular in (flp?,)
supermodular in
(a,,?,),
/=!,..,«.
Then we can apply
Proposition 4.9 to this problem exactly as if there were only two bidders.
Unfortunately,
when
players use asymmetric strategies, affiliation of the signals
sufficient to guarantee that
(f,,^,)
is
not
are affiliated, nor is their joint distribution function log-
supermodular. Thus, in some regions, a small increase in the signal received by a given player
might increase the likeUhood
competing effect
positing
that
may
the
that a given bid is higher than all opponents.
However,
this potential
not be the dominant one in particular examples; thus, one can proceed by
single
crossing
condition
holds,
characterize
hypothesis, and then verify that the single crossing condition
forms and ranges of parameters of
interest.
the
equilibrium
is in fact satisfied
under
that
for the functional
Thus, separating the single crossing condition from
the existence theorem allows us to proceed even in the absence of general sufficient conditions for
the single crossing condition.
27
s
where players are
In the case
order to highlight the difficulty. Let ^(a^,t)
This
object.
is
nondecreasing in
payoffs as follows (ignoring
[E[E[z,\t]\t„aj(tj)
Lemma
Applying
know
we
risk neutral,
t
=
E[zi
by Lemma
To
two ways
,
Then,
< a,]- a,)
•
Pr(a/f^.)
<
isolate the issue
in (?,,c), since
s
in
true value for the
can rewrite player I's expected
(4.2)
with the
We
when
first
can
by
the density
term of (4.2),
Lemma 3.2.1,
the
£'l£|z,|t]p,,a^.(f^.)
the right direction are strong
enough
is affiliated
we draw
we
Note
<
first
a,
I
that
term
-c
is
is
to
and the
strategies are
a distinction between the
by player
restrict attention to actions
less than or
1
£'[£|^z,|t]|r,,a^(rp<a]
nondecreasing in
l-c
between
a,
outweigh any competing
and
f,
log-
is
However, our
?,.
log-supermodular in (Op?,).
the single crossing property will require that the interactions
Pricing
'
4.4 to this problem in a manner analogous to the private value auction,
assumptions do not imply that
4.2.2.
player
1
a,\t,).
equal to the player's conditional expected payoff.
work in
we
is
problem another way
ties):
that a, affects this term.
supermodular
- a, where z,
3.2.1.
that Pr(aj(tj) <cii\tA is log-supermodular
nondecreasing.
rewrite the n-bidder
|t]
Thus,
which we know
effects.
Games
This section studies pricing games with incomplete information, where constant marginal costs
are the private information of the players of the
game.
Spulber (1995) recently analyzed
how
incomplete information about a firms' cost parameters alters the results of a Bertrand pricing
model, showing that firms price above marginal cost and have positive expected
model assumes
now we show
Let
ti
that costs are independently
that this
model can be
and
and
i.
Pj{tj),
max [p, - r,]
By Lemma 4.4,
affiliated
and
elasticity
of demand
the expected
D^(p^,..,p,) is
is
that values are private;
Consider a general demand function for firm
where the firms produce goods which are only imperfect
opponents use price functions
Spulber'
easily generalized to asymmetric, affiliated signals.
represent the marginal cost of firm
D'(pi,..,p,),
identically distributed,
profits.
firm I's problem can
•
J-
now be
substitutes.
When
/,
the
written as follows:
Jz)'(/7„p2(?2),..,p,(r,))/(r2,..,r,|?,)rft_,
demand
function
is
log-supermodular
log-supermodular. The interpretation of the
if the cost
latter
condition
a non-increasing function of the other firms' prices.
Milgrom and Roberts (1990b), demand functions which
transcendental logarithmic, and a set of Unear
special case is the case of perfect substitutes,
28
demand
parameters are
As
is that the
discussed in
satisfy this criteria include logit,
functions (see Topkis (1979)).
where demand
CES,
Another
to the lowest-price firm is given
by
D(P), where
P is
the lowest price offered in the market, and
Then, since [p-t-]
perfect substitutes
is
is
log-supermodular,
we have the
all
other firms get zero demand.
following result (noting that the case with
formally like a first-price auction):
Consider a pricing game as described above, where (2.1) holds.
Suppose further that (i) for all i, the demand functions D'(p^,..,p,) are non-negative and logsupermodular, (ii) types are affiliated (f[t) is log-supermodular), (Hi) in the case of perfect
substitutes, ties are broken uniformly. Then:
Proposition 4.11
(1)
The game
when
Crossing Condition (2.2), and a
continuous or when the action set is finite.
satisfies the Single
either D'
is
PSNE equilibrium
exists
where goods are perfect substitutes, if in addition, for all (iv) D(P) is
strictly decreasing in P, (v) D(P) is bounded and continuous, and (vi) the support off(t) is a
product set, then (3.1)-(3.7) are satisfied and there exists a PSNE in continuous-action
(2)
For
the case
i,
games.
This example can also be extended to problems with incomplete information about demand
elasticities.
Noisy Signaling Games
4.2.3.
Consider a signaling
2),
game between two players,
where player 2 observes the
games include games of
know
signal of player
limit pricing
1
the sender (player 1)
only with noise.
and the receiver (player
Examples of noisy signaling
(Matthews and Mirman (1983)), where an entrant does not
the cost of the incumbent, but can
draw inferences about
the incumbent's cost
by observing
a noisy signal of the incumbent's product market decision (the noise might be due to
shocks). In another example,
player gets to
move
first,
Maggi (1996)
committing herself to an action, but the
observed by the opponent. This model
mover's type
(i.e.
studies the value of commitment in a
is
used to show
move
is
demand
game where one
only imperfectly
that incomplete information
about the
first
production cost) can restore some value to commitment, in contrast to Bagwell
(1995)'s result that commitment has no value in a
game of complete information but imperfecdy
observed actions.
Consider a game where Player
1
observes his
own
type and chooses an action.
observes a noisy signal of player I's action and then chooses an action in response.
Player 2
Player
1
receives payoffs M,(a,,a2'^)' while player 2 receives payoffs M2(^i'^2'^i) ("^ the limit pricing
game, the player's action
in the
is).
not important, but the type
commitment game, player
Player 2's "type,"
/,(r2la,)
player
is
t^,
is
represents the
unknown production
cost;
I's type is not important to player 2, but the action (i.e. output)
simply a signal about a,, which does not direcdy affect payoffs.
be the density of the signal
1
f,
from choosing action
t^
a^
conditional
when
29
on
player
the action a^.
2
uses
Thus
strategy
Let
the expected payoff to
a2{t^
is
as
follows:
{u^(a^,a2(t2),t^)f^(t2\a^)dt2
and the belief
that player
hstf2(tj\t2,a^(-))
.
when
MLRP). Then,
1
t^
Then, the expected
uses strategy «,(?,)
Further, suppose that /jC^jla,)
that «,(?,) is nondecreasing.
(equivalent to the
player
given the signal
f,
uses strategy ai(-)> calculated using Bayes' rule.
1
payoffs to player 2 from choosing action a^
Suppose
be the conditional density of
is
is
as follows:
log-supermodular
the induced density /jCrilrj.^CO) will also be log-supermodular.
Sufficient conditions for the single crossing property in this scenario are then given as follows:
Proposition 4.12 Consider a noisy signaling game as described above, where (2.1) holds.
Suppose further that (i) M2(^i.«2'^i) satisfies WSCP-IR in (a2,a^) and (a2,t^), (ii)f(t2\a^) is logsupermodular and the support does not move with a,, and either (iii)(a) «[(«,, Gj'^i) '^
supermodular in fflj'^) <^"^ '" fop^i). (^f ^l^^ (iii)(b) u^(a^,a2,t^) is nonnegative and log-
Then the game
supermodular.
Proof:
(aj'.^i)
satisfies the Single
and
\u2(a^(t^),t^,a2)f2it^\t2,oc^{))dtJ satisfies
ia2,t^),
nondecreasing.
It
remains to establish that player
we can check that
Since supermodularity
supermodularity of
assumption that
in(?2'^i)-
But
\
m, is
is
CX2{-) is
SCP-IR
expected payoffs satisfy
M,(j:,«2(^2)'0/i(^2b)^^2
J
^^
(2.2).
in (a2;f2)
u^(x,a2it2),tx)fi{t2\y)dt2 in
supermodular in
(^2.^1)
(iii)(b),
we can
(jc,r,).
supermodular in
a,(r,)
(x,t^)
and
(Oi,?,)
(y,?i).
implies
nondecreasing, the
If 02(^2) i^
implies that u^{a^,a2{t2),t^)
JMj(A;,a2(^2)'^i)/i(^2|>')^^2 is
apply
when
Under assumption
preserved by arbitrary sums, m, supermodular in
u^(a^,a2(t2),t^)f(t2\a^)dt2 will
and
I's
Lemma 4.2 implies that
Under assumption
\
(2.2).
Lemma 4.6 implies that under our assumption that Uj satisfies WSCP-IR in
is
(iii)(a),
Crossing Condition
is
supermodular
supermodular in
Lemma 4.4 to establish that
be log-supermodular when
u^
and/j are log-supermodular
nondecreasing.
Numerical Computation of Equilibria: First Price Auctions
5.
In this section,
we consider the
issue of computation of equilibria of first price auctions.
the existence result considered in this paper applies to a wider class of
(arbitrary asymmetries, correlated values) than
numerically,
we can take
auction games.
a few
first
economic environments
had been previously analyzed
theoretically or
steps towards numerically characterizing the equilibria to such
This numerical exploration can potentially suggest avenues for future theorizing
about characterizations of equilibria, and
about
Since
affiliated private
values and
we
common
give
some examples of suggestive numerical
results
value auctions with asymmetries in the covariance
structure of types.
30
Prior to Marshall et
computing equilibria
existence
et al
to
al
(1994), there were no general numerical algorithms available for
asymmetric
was not known
for
first
price auctions with continuous bidding units, and in fact
some of the kinds of asymmetries considered
(1994) argue that numerical computation of equilibria in asymmetric
independent private values case
is
first
price auctions in the
They summarize some of
but can be done.
difficult,
in their paper. Marshall
the
difficulties as follows:
to a class of 'two-point boundary problems' for which
numerical solution techniques, they all suffer from major pathologies at
the origin. First, forward extrapolation produces 'nuisance' solutions (linear in our case)
that do not satisfy the terminal conditions and act as 'attractors' on the algorithm. Second,
and not unrelated, backward solutions are well-behaved except in neighborhoods of the
origin where they become highly unstable with the consequence that standard (backwards)
"shooting" by interpolation does not work.
Although these solutions belong
their exist efficient
Marshall
(1994) then devise a technique which makes use of "backward series expansions"
et al
and a transformation of the problem which
is
more numerically
stable than the original problem.
Their algorithm requires analytical input to transform the problem appropriately. Generalizing their
algorithm to the case of more than two type distributions, or to correlated values,
non-trivial extensions of their numerical
would
require
and analytic algorithms.
In contrast to the problem of computing the solution to a set of differential equations, the
algorithm for computing the equilibria to the auction
the calculation of
computation time);
it
BR(X)
is
can be broken down into a sequence of single variable optimization problems,
which the player prefers action A,
V,(a,;X,f,.)),
constructed in this
a simple exercise (both in terms of programming time and
one for each jump point of each player. That
at
finite actions
We simply want to find the matrix X so that X=BR(X),
paper involves few conceptual subtleties.
where
game with
and then proceed
to
is,
for each player
let xi
/,
A^ (computing expected payoffs
to find x^, searching
over
?,.>x|.io
be the smallest value of
to
each action according
t^
to
In mineral rights auctions,
where players must compute the expected value of the object conditional on winning against the
opponents' current strategies, the conditional expectations must be numerically computed;
looked
at signals
The more
and values which had a log-normal
difficult part
of the problem
theory of numerical analysis 12 suggests a
here
1"
we will
is
distribution. ^
The algorithm
also checks for the cases
^
solving the nonlinear set of equations
number of standard ways
only sketch a few of the numerical issues which
where a particular action
is
X=BR(X). The
to solving this problem. Thus,
arise. First, since
we have no
Judd (forthcoming) for an excellent treatment of numerical analysis
31
global
never used for any type.
To approximate the relevant conditional expectations and probabilities, we made use of the library of
made available by Vassilis Hajivassiliou as a companion to Hajivassiliou, McFadden, and Ruud (1996).
1
2 See
we
in
economics.
routines
"contraction mapping" theorem, the algorithm X''''=BR(X*)
and indeed
starting value,
it
does not appear to in numerical
+ (l-A)X*)) were
algorithm X*^'= ABR(X*^)
not guaranteed to converge for any
is
trials.
also be used.
There are potentially
large computational benefits to using an analytic Jacobian since the Jacobian is
which player
particular, the point at
/
jumps
to action
elements of the best response of opponent 7?^/: x'^.p
of changing each component of
BR()
x'
A^, denoted
jc'„,
and
j:'„+,.
on the equation X-BR(X),
it
The number of bidding increments can
number of function
the function
calls required
BR(X)
is
Thus, to determine the effect
necessary to
a simplex method, which was reliable in
thus be increased without a affecting the
M.
However, even with
we
approximations
same
of
set
^^
^^^
J^
auctions
studied in their paper, and
9)
•°
compared
the
0.3
-
calculations
4^
ar^sr**--
0.2
equilibrium
for
bid
functions
and
revenue.
Marshall
expected
-
(1994)
chose
distributional
a
alternative is
.fS^
computed
0.5
the
A final
Sequence of Auctions: Jump Points: 10, 20, and 50
2 bidders, F(tl)=tl and F(t2)-t2*4
report
0.6
We
modification,
trials.
were motivated by Marshall
et al (1994).
this
of
set
first
call the function
M-I x M-I with only
computation could be slow for the mineral rights examples considered below.
The
In
by the nonlinear equation solver, while the computation time for
increase linearly in
will
sparse.
x'^, affects only the following
only three times. This allows us to compute a Jacobian of dimension
/•3 function calls.
on the
variations
effective at generating starting values for other
Methods based on quasi-Newton approaches can
algorithms.
However,
et
set
,^—
1
1
jp
-
1
-50
jp
-
4
20
-
"
1
--so
jp
-
1
-•-•20 jp
-
4
— 10
jp
-
1
--10
jp
-
4
1
al
of
Figure
assumptions
8: First Price
Auction games between two coalitions.
motivated by the case where
five
bidders
from uniform
subset
of
draw
values
and a
[0,1],
the
with the group's value being
maximum
of
the
n
Indiv.
of the values
participants.
Our
results,
A/=100
Marshall et
al Results:
Expected Revenue
ition
Bidders
Expected Revenue
F{x)=:e
F{x)=x
Auction
Coal-
Indiv.
Auction
Coal-
Indiv
-eo-
ition
bidders
-eer
ition
bidders
bidders
collude, bidding as a group
the
Coal-
a=
\
n=4
.6664
.0334
.0334
.6668
.0335
.0333
a=l
n=3
.6508
.0344
.0374
.6510
.0352
.0371
a=3
n=2
.6085
.0405
.0487
.6089
.0406
.0488
a=4
n=l
.5055
.0574
.0840
.5057
.0567
.0860
A
comparison of the expected
32
|
revenue calculations for several auctions
presented in Figure
is
8.
The
table indicates that the
expected revenue from the auction with discrete and continuous bidding units
cases. Thus, in these auctions, the difference
Figure 6 shows
Now
how
as
discrete
more bidding
There are several potentially interesting asymmetries:
differences in means, and those arising
t),
game change
the equilibria to the discrete
within .001 in aU
games
small.
is
units are allowed.
consider an auction with affiliated private values, where the types are distributed
/n(t)~A^()X,Z).i3
of
between the continuous and
is
from differences
and those arising from differences
means and variances
in variances
are not surprising: in the trials
(which also
The numerical
in covariances.
we conducted,
those
arising
from
affect the
mean
results for differences in
types with higher
means always
bid less aggressively and get higher expected revenue. Changes in variances affect the shape of the
distribution, so the higher variance types
others, but they tend to
subtle, but
no
do
may be
better overall.
We
less intuitive.
less aggressive in
some regions and more
Differences in covariances are perhaps slightly more
will illustrate this case with
an example.
Suppose
that
bidders have types which are highly correlated, while a third bidder has a type which
Then
correlated with the other two.
aggressively and
The
win more often than
intuition is simple: the
two bidders with highly
the
Mean
two bidders whose types
.5
.25
.5
1
.25
.25
.25
1
1
Expected
are
less
is
correlated types bid
two
more
also gets higher expected revenue.
more highly
correlated are always
their values are high.
Variance Matrix
Hh)
Hh)
Kh)
Private
Player 2
Player 3
Auctioneer
0.4691
0.4691
0.5008
1.4190
0.3283
0.3283
0.3434
Player
who
the third bidder,
concerned with competing with one another, even when
Variable
in
1
Values
Auction
Revenue
Prob. of
•Bidders
Winning
1
and 2
•Bidder 3
10
Figure
9: Private values first price auction
where the
bidders have positively correlated values.
Next,
about the
we can consider pure "common values"
auctions,
where each player sees a private signal
common value. Although we do not have a theorem guaranteeing that the single
property holds,
we can
numerically verify that
it
holds in the neighborhood of the equilibrium
'^ See Wilson (1996) for a theoretical study of equilibrium in oral auctions with log-normal values.
33
crossing
we
compute for a
auctions
we
particular
game; the numerical analysis indicates
consider here. Let z be the
common
for general variance-covariance structures,
example of a signal
is
further, the errors
may
e,
also be correlated.
signal as well as the correlation of the signals.
new
effect for bidder 3
the private value auction, the competition
those two bidders; bidder 3
won
is
especially bad
that signals will
An
But
1
and 2)
which does not
between bidders
news due
in the
Variable
Mean
is
.75
.1
ln(e,)
.1
.75
auction, having
SA^SS««i-fi.i
1.2
Player 3
0.0315
0.0315
0.019
0.378
1.005
^..,
;.
^ ^
V
iW
0.242
0.6
0.4
0.2
-
Figure 10: Common values first price auction where
two bidders received positively correlated signals of the
and a
third bidder receives
3:
cov(e3,ei)=0,
= 1.2
i
"
Winning
true value,
this
In
to
two bidders
Value Auction: Two Bidders Have
Correlated Signals
Bidder
Auctioneer
Revenue
0.378
no longer
two more aggressive
to that of the
cov(e1,e2)=.1
J
« 0.8
Prob. of
was close
Common
•
Player 2
better, is
expected payoffs upon winning
-
1
of each player's
Across a range of parameter values,
1
Expected
be positively correlated,
.25
Player
where
an example where bidder 3 does worse.
1.4
Hz)
ln(z)+ln(e,),
and 2 was most disadvantageous
to the winner's curse.
.75
ln(e,)
=
arise in the private value auction.
1
common value
Variance Matrix
ln(e^)
of the
compete more aggressively,
still
the expected revenue for the bidder with an independent signal
with correlated signals. Following
in all
interesting observation is that our result of
less often, but received higher
to his less aggressive bidding.
opponents
receives a signal ln(s)
We can vary the informativeness
While the "more correlated" bidders (bidders
aggressiveness has a
due
i
"more independent" bidder (bidder 3) does
private values auctions, that the
true.
does hold
perhaps more intuitive to consider a particular
Suppose each player
structure.
it
Though our numerical algorithms allow
value.
The assumption of a common value implies
cov(z,e,)=0.
and
it
that
X
^--^
1
0.5
1.5
an independent
signal.
Another
set
of experiments
we conducted concerned the effect of the
signal about the true value of the object.
receives a signal
which
is
more
We first examined the case
precise than the others;
we
precision of each player's
of two bidders, one of
whom
then looked at the effect of a third
bidder on equilibrium strategies on expected revenue.
Notice in this auction that the player with the more precise signal, player
aggressively.
However,
since her beliefs are
34
more
1
,
sensitive to her signals,
bids slightly less
her conditional
expected value
is
more
variable than that of player 2, and so she
is
more
likely to see high expected
values. Thus, she wins just over half of the auctions. Despite the fact that she wins just over half,
much
she achieves
higher revenue in each auction.
having less precise information,
still
perhaps surprising that player 2, despite
It is
wins so frequently. ^^
Common
Mean
Variable
Variance Matrix
ln{e,)
.10
ln{e^)
.3
Value Auction: Two Bidders
Signais Have Different
Whose
Precisions
"5
Hz)
_
m
1
2.5
Player
1
Player 2
Auctioneer
.166
1.150
to
Expected
0.333
'
Bids
Revenue
Bidder
bi
Prob. of
0.528
.471
0.5
Winning
1:
In(e1)-N(0,.1)
-»
Bidder
-
2:
In(e2)-N(0,.3)
;
-
Figure 11:
Common
values auction where one
bidder receives a signal which
is
more precise than
2
4
the
other bidder's signal.
Returning to the pure
third player with a
more
common values model, we
consider a final experiment, where
The
precise signal (var(e3))=.l).
Now,
model.
different than in the two-bidder
the
strategies
we add
a
of the players look very
two well-informed players bid more
aggressively than the player with the noisier signal; this exacerbates the winner's curse for player
2.
In addition, the bidder with the less precise signal wins less often and receives very
little
expected revenue relative to the two-bidder example.
In general,
we can perform numerical computations for a variety of auctions. By
covariance structure between signals and values,
the informativeness of the signals,
we can
changing the
vary the importance of the winner's curse,
and the mean values. Further characterizations of mineral
rights
auctions with heterogeneous bidders await future research.
^^
We also examined the effect of lessening the importance of the
winner's curse in this example.
We
allowed the
bidders to have different values for the object, z, and Zj, where cov0n(zi),ln(z2))=.5, and all other parameters of the
model
1
are the
same
as the latter example.
The
qualitative nature of the equilibrium remained unchanged, with player
bidding less aggressively and winning just over half the auctions.
bidder. Player
1
one another, so they do not expect as much competition. Second, the
For the auctioneer, the first effect is most important: the auctioneer receives expected
the bidders' values are less correlated with
winner's curse has less
revenue of
However, expected revenue went up for each
has expected revenue of .65, while player 2 has expected revenue of .56. There are two effects here:
.99, less
bite.
than in the pure
common
values auction.
35
Common
Mean Variance Matrix
Variable
Ke,)
.10
Inie^)
.3
Value Auction: Bidders 1
and 3 Have More Precise Signals
.1
ln{e,)
Hz)
1
Player 2
Player 3
Auctioneer
0.130
0.030
0.130
1.335
0.369
0.260
0.369
Player
Expected
1
Revenue
Prob. of
Winning
Figure 12:
Common
bidders receive more
values auction where two
precise signals than the third
bidder.
Conclusions
6.
This paper has introduced a restriction on a class of games called the Single Crossing
Condition for games of incomplete information, where in response to nondecreasing strategies by
opponents, players choose their actions as nondecreasing functions of their types.
that
when
pure strategy Nash equilibria will exist in such games
finite,
and
games
will
have a subsequence which converges
the set of available actions is
games, a sequence of equilibria of
further, with appropriate continuity or in auction
finite-action
We have shown
to
an equilibrium.
We have
further
estabUshed that similar results can be obtained for games which satisfy our "Limited Complexity
Condition."
The formulation of games of incomplete information developed
following advantages: (1) existence of pure strategy
Nash
equilibria
in this paper has the
can be verified by checking
general and economically interpretable conditions, (2) the results for finite-action
games
require
very few regularity assumptions, and (3) the equihbria are straightforward to numerically calculate
for finite-action
auctions.
The
games, and these approximate the continuous equihbria for continuous games and
application of these results to
existing Uterature
first
price auction
on the existence of equihbria
correlated signals and/or
common values. The
crossing property, can be characterized for
in auctions
properties have not been
fiilly
led to a generalization of the
with heterogeneous bidders with
condition for monotonicity of strategies, the single
many commonly
Athey (1995, 1996). Finally, numerical analysis can be used
whose
games
studied
games using
to analyze behavior in auction
characterized in the existing literature.
36
the results of
games
7
Appendix
7.1
Single Crossing in
The following
table
Games of Incomplete
summarizes the
the single crossing condition
results of Section 4.1,
property of incremental returns.
The
which
are necessary
in stochastic problems,
it
and
results are applications of
games of incomplete information. In order
SCP-IR
which provides characterizations of
games of incomplete information,
functions and probabiUty distributions
to
Information
will
be useful
identifying conditions
on
utility
sufficient for the single crossing
theorems from Athey (1995, 1996)
to conveniently state the set
of theorems about the
to state the following definition.
Definition 7.1 Two hypotheses HI and H2 are a minimal pair of sufficient
conditions relative to the conclusion C if the following statements are true:
(i)
HI and H2
(ii)
imply C.
C must fail
If H2 fails, then
(Hi) If HI fails, then
C
somewhere
that
HI
must fail somewhere that
H2
is satisfied.
is satisfied.
This definition summarizes a strong relationship between two hypotheses and a conclusion; not
only are the hypotheses sufficient for the conclusion, but further neither can be relaxed in the
context of the other.
To read the following
table:
In each row: HI and H2 are a minimal pair of sufficient conditions (Definition 7.1)
the conclusion C; further, C is equivalent to the single crossing result in column 4.
relative to
and Definitions: Bold variables are vectors in Si"; italicized variables are real
numbers; / is non-negative; a.e. indicates "for almost all (Lebesque) t"; spm. indicates
supermodular, and log-spm. indicates log-supermodular (Definition 4.1); sets are increasing in the
strong set order (Definition 2.2); SCP-IR (WSCP-IR) indicates (weak) single crossing of
incremental returns to a,. (Defmitions 2.1 and 4.2); arrows indicate weak monotonicity.
Notation
Observe: Suppose aj{tj) is nondecreasing for all j^i. Then
(WSCP-IR) in {a.,a) and (a,;/,), then kJiai,ti,t)=u.{a.,aj{,t),t^,t)
{a.,t^.
If M,(3't) is log-spm., then A:,.(a,,t)=M.(a,,a_,.(t_,.),t) is
for j^i
and
(a,.,/;,.)
for;=l,..,/, then
A:.(a,.,t)=M,.(a„a_,.(t_,.),t) is
37
satisfies SCP-IR
SCP-IR (WSCP-IR) in
if u^{a.,aj,t^,t^
satisfies
spm. in {a^,a^
log-spm.
If u^{si,i) is
spm. in
(a,.,rp,y=l,..,/.
Appendix
Table
7.1:
The
Single Crossing Condition in
Assume: /(t)>0 and
HI:
Hypothesis
must hold a.e..
j/(t)dt finite for all S; for 1.3
H2
C: Conclusion
Hypothesis
Holds for
on/ (a.e.)
«,(•)
Games
of Incomplete Information
and
F is
/=!,..,/,
2.2,
a probability distribution.
Equivalent Single
Crossing Conclusion:
whenever
nondecreasing, j^i.
i's
objective
IR
in
is
SCP-
(a,;?.), /=1,..,/.
Two-Player Games
1.
Assume:
h.{a^,aj,t^,t) is
h.{a.,aj,,t) is
SCP-IR
{a-a),
2.
y(t) is log-
{a.,t).
H,ia,t,)=jh,ia,ajitp„t^)fit)t)dt.
spm.
in
SCP-IR
is
same.
in (a,.;^.
{a:,t).
h.(a.,aj,t,,tj)>0
is
spm. in
log-spm.
m
is
log-
H,ia,t,)=jh,ia,ajitj),t,tj)fitj\t;)dtj
spm.
log-spm. in
is
H,ia.,t,)-g,ia.,t,) is
SCP-IR
ia.;t).
for all
g.:Si-^S{^ log-spm.
3.
H,ia,t,)=lh,ia,a^itj),t,tj)fit)t,)dtj
h,(a,,aj,t.,tj) is
spm. in (a,.,aj.),
(«„0,/=l,2.
(implied by/
log-spm.)
Assume:
k-{a^,t^,s^ is
k,(a,,ti,s,) is
SCP-IR
5.
/(5,lr,.) is
supp[Fj(5;l?,)] is
k,(a,,t,,s) is
6.
ft.(a,t)>0 is
log-spm.
Summarized with
log-
j
JA:,(a,,r,,5,)/(5,lr,)J5, is
log-
t.;
k.(a.,t.,s) is
h^(a,t) is
spm.
in(a,.,a.),7?ii,
& in (a-tj).
/i,.(a,t) is
spm.
in(a,.,ap,y?ej,
in
same.
spm.
JA:.(a,.,r,,5,)/(5,lr.)J5, is
in
(a,,r,).
SCP-IR
in
i
same.
Games
/(t) is log-
J/i,(«„a_,(t_,),t)/.(t_,k,)rft_, is
J^F(t_,;0
f,.
for all
H.ia,,t,)-g.ia.,t,) is
/f,(a,,f,.)=
spm.
S
log-
s.t.
i.(U T
t.,
by
SCP-IR
(a,.,r,.).
^,(a„?,)
J;^,(a,,a_,(t_,),t)/(t_,lgrft_, is
in
for all
gf.Si—^Si^ log-spm.
t
(implied
fit)
8.
SCP-IR
spm.
spm. in
7.
Sufficient Statistic, s,6 9t
(a,,?,).
constant in
fXs,\t-) is
WSCP-IR in
Multiplayer-Player
is
spm.
spm.
in
Assume:
spm. in
+ g,(«„g
SCP-IR for all
//,(«,,0
is
ia.;t).
g.-.Si-^Sl
Games Where Opponent Behavior
4.
spm. in
is
spm.
is
+
g,(a„^)
SCP-IR
for all
g.:Si^3i spm.
(a,.,f,.).
log-spm.)
y(t) is log-
spm.
is
& in (a-t),
SCP-IR
g,.:9?->SR
for all
spm.
j=U..J.
is
spm. in
(a,,f,.)
38
V 5 sublattice.
Appendix
72
Proofs
Proof of
Lemma 2.2
The proof continues from
Nonempty. EstabHshed
Closed graph.
It is
the text,
in
clear
showing
that the conditions for
Kakutani are
satisfied:
our above definition of r(X).
from the definition
(2.3) that V,(a,.;X,f,) is continuous in the elements
of X under our assumption (2.1), which guarantees that there are no mass points in the type
distribution.
Consider a sequence (X*, Y*) which converges to (X,Y), such that Y*e ^X*") for
We wish to show that Ye r(X). To do this, we show that for each player and almost every
and a type t.e 7^\{y'}.
type, the action assigned by Y is a best response to X. Consider player
converges to y', there must
Then there exists an m e {0,..,M} such that >'i<?,< y^+i Since
all
it.
/,
y'"*
.
exist an
K such that, for all k>K,
Y*e r(X*). But, since
since
A:>/^and
T
is
all
m\
V,{a:,X,t;) is
convex -valued. Here we show
Ae (0,1). Observe
vectors,
and thus A^
,
,
continuous in X,
analysis
one of
if
X*
r/s best responses to
V,{A^;X\t;)>V^{A^,-X,0 for
all
that the fact that af^itJiX) is nondecreasing in the strong
convex. Fix
that since
X and suppose that w,y e r,.(X).
Let
z=X w+(l-A) y
convex combinations of nondecreasing vectors are nondecreasing
ze Z,.. Now, for m=0,..,M,
we show
A^
that
imply that there exists an optimal strategy which
ze r.(X).
is
then V,{A^;X,t;)>V,{A^.;X,t;).
set order implies that r,.(X) is
for
y'^<t<. y^*
is
is
an optimal action on
consistent with z, and thus
iz„,z„+i).
by
This will
definition
A consequence of the strong set order for subsets of 91 which will be important for our
is that, if
A<^C<sD, then AnD
There are four cases to consider,
cC
w„=w„^] and y„=y„+i, then z„=z„+i and there is nothing to
show, (ii) Suppose that w^ov^+i and y„<y„+i. This implies that A^ is optimal on (w^,w„+i) and
(y„,y„+i). Then, since af^it^C) is nondecreasing in the strong set order in t., it must be that A^ is
(i)
If
A„ is optimal on (z„,z„+,). (iii) Suppose
A„ must be optimal on (z„,z„^,). If
optimal on (min(w„,>'„),max(x„^,,y„^,)). Thus,
Wm=Wn,^i and y„<y„^,. If }'„^w„^,<>'„^„ then
^m+i<>'m<>'m+i' ^^^^ a /otti
af*(r,|X) is
A„
optimal
such that an optimal action for some
nondecreasing in the strong set order in
at w„^,.
But
this implies that
If 3'm<>'m+i< ^m+1' ^^erc is
some k<m such
implies by the strong set order that
Thus,
j3,.(f,,z)
A„
is
= A„,(, ^jis a nondecreasing
to almost every type;
we can
Apply Kakutani. Since H"
fill
is
A„
is
that
?,.,
e {w^^^,y^) is A^. But since
we must have A^ optimal on (y„,y„+i) and
r,.
opfimal on (w^+py^^,) and thus on
A^
is
optimal for some
optimal on
Cy„,*v„+,),
(z„,z„^.,).
e Cy„+,,wJ, which
and thus on (z„,z„+,).
strategy consistent with z
which assigns optimal actions
in optimal behavior at the jump points, implying that
a compact, convex subset of
ze r,(X).
{M+1)I -dimensional Euclidean
is
convex-valued, then
apply Kakutani' s fixed point theorem to guarantee that a fixed point exists.
39
(iv)
t.
space, and since the correspondence is nonempty, has a closed graph, and
we can
that
Appendix
Proof of Theorem 2.2
changes of
«,(?,)
nondecreasing on
[z„z,^,], let
0<k<K such
[z^.Z^+j].
that
t,
<
x^^„ = inf{zt <
As
for all
even,
above,
t,
let j3,.(?,,x)
we know
if there is
if there
= A^,
while
if
assigns this behavior on 7^\{x}
k
n<M-m
is
odd,
that r(f,.,x)
= Af^_^. Any
that
have
at
for player
/
played
consider
played on
x^^„ = z,^,.
Let l*(t,x)
let
E(K)=2,^^'^'^'^ x--
to
more than one
= max{/|A:, < t}
= k-M+m.
function
xXf ^^'^'\
combinations of nondecreasing vectors are nondecreasing, Z1(K)
Z(K)—> 2(K)
let
is
Now
z,^,.
is
A„
that
be
j3,.(f,.,x)
.
Then,
If this
k
is
which
said to be consistent with x.
Using the representation just developed,
space. Define F:
x^^^ =
xe xf ^*^^" corresponds
l<m<M such
let )8,(r,,x)
let
such that A„
^a/-.}- Otherwise,
0<^<Arand
is
some n>m such
is
Otherwise,
some
{x}, a given
f,.e
that the player's strategy will
we can define behavior consistent with x.
7^\{x}, find the integers
t-s
even,
< z,+,|a,(0 ^
Since x does not specify behavior for
strategy.
is
z,+,|a,(0 > A^}.
k odd, k<K. Then, for each m=l,..,Af,
[z„z,„], let
k
Then, for each m=l,..,M,
x^^^ = mf{z, <
the direction
which describes the points where the strategy experiences a
(Definition 2.4),
direction change. For
on
Zf which represents
consider a strategy a.{Q and the corresponding vector zg
First,
as follows.
For each player
/,
is
if
Since convex
a convex subset of Euclidean
X represents strategies for dllj^i
most Kj direction changes, then r(X) contains the representation of all best responses
with no more than K. direction changes
(this
vector exists and
is
unique by
(2.4)).
The
arguments of Lemma 2.2 can be followed exactly to show that the correspondence has a closed
graph; convexity follows by uniqueness. Then,
we can follow the proof of Lemma 2.2 exactly to
guarantee existence of a fixed point for the correspondence F, and then to construct the
corresponding pure strategy Nash equilibrium of the original game.
Proof of Theorem 3.1
Proof: For each player
// "= {o;
+ f^(a. - O;)
j3;„:r,.-»/^''"c
there exist
Selection
bounded
:
[a,.,a;.] is
/,
consider a sequence of action sets {z^"}, where
m = 0,..,10"}
.
Let /f^i^'",..^'"). For each n, the function
of bounded variation by assumption;
two nondecreasing functions,
Theorem
fiinctions
and
„,
such that
j3.„
=
j8,
„-/?.
.
Kelly's
on J^rfK has a subsequence which converges almost everywhere
it
converges
we can find a subsequence of
such that {^.„} converges to
number of players,
j3,
statement that
(Billingsley (1968), p. 227) guarantees that a sequence of nondecreasing,
nondecreasing function (and in particular,
function). Thus,
j3.
this is equivalent to the
there
almost everywhere to
j3*
except
at points
at continuity points
{1,2,..}
Consider
this
of the limiting
and a function of bounded variation
of discontinuity of p. Since there are a
must exist a subsequence {«} such
P'(t).
to a
that {PiJ.ti),..,P,J,t,)}
p
finite
converges
subsequence.
We wish to show that for each player and almost every type, the action assigned by P*(t) is a best
40
Appendix
.
response to
rationals)
P*(t).
•
Let Z'=|r,.|3n
/,
discontinuities occur only
A,n(0 converges to
Since
^,„(?,) is
to pl,(t_,) for
and a type
on a
of measure
set
set
that
Z
is
countable
(it is
a subset of the
of all actions which are in z^" for some
\Z', such that p. (t)
r,.e T,.
is
continuous
at
f,.
(recall that
Let b=P'{t^; Kelly's Selection Theorem implies
0).
b.
an equilibrium strategy for any
f/,(a',P_,„(t_,.),?,)
almost
all
t_,.,
is
n, f/,(A>(^,)'P-,>(t-,)>^)
",(A>(O.P-,>(U.O converges
Note
C/,(Z?,pi,. (•),?,.)>[/.(«',
that there exists a
converges to a". But
(/,(&, P*,(),r,)>[/,.(a*, Pl,(-),^,)
continuity of U. in
U^ib, p:,(),f,)>t/,(a", Pl, (),?,)
P_^„(t_,.)
converges
to M,.(^.Pl,(t_,),0 for almost all
atomless, this in turn implies that
Now consider an action a'^D'.
a,,
^
Because payoffs are continuous and since
for every a'^/f".
Since the type distribution
aeD'.
and note
(x'-"}},
and thus has measure zero. Let D' be the
Consider player
finite n.
€
s.t. t-
for all k
sequence
t_,..
Pl, (),?,) for all
{a*}, a'^eD',
which
by our previous arguments. Thus, by
Q.E.D.
Proof of Theorem 3.2
Following the proof of Theorem
action spaces, indexed
by
3.1,
we will
of this game in nondecreasing strategies
it
consider a sequence of games with successively finer
{«}, where z^'" is the action space for player
is
described by
P„(t).
converges almost everywhere to a set of strategies denoted
To
simplify the exposition of the proof,
we make
the allocation rule given in equation (3.4)
is
an
initial
in
i
game n, and
The sequence {«}
is
a
PSNE
chosen so that
P'(t).
transformation of the problem, so that
restricted to the rule /n,(a,)
=
a,.
Since m.
is
assumed
we have required of our primitives in (3.1)-(3.7)
are robust to (strictly) monotone transformations of the action, we can simply rescale the action
space and redefine our functions v and v appropriately; for simplicity, we will not adjust the
to
be
strictly increasing,
and since
all
properties
,.
notation for v and
.
.
v,.
To begin, we introduce some
notation. Define:
^iM'^-i) = <?,(«,'?-/,«(*-,))
Av,(a,,t)
Further,
we
w;(a,.t_,)^<p,(«„P%(U)
= v,(a,,t) - v,(a,,t)
v(a,,f,)
when
J
define several events, taken from the perspective of player
opponents' equilibrium strategies. For each of the following,
that,
=
players {/}\/ use strategies
P_,,„(-).
v,(fl,,t)
i,
/(t.,|?,)a_,
playing against
we introduce notation for the event
the realization of t_,. and the
breaking mechanism (described by piCj) in (3.4)) are such that the action
outcome of the
a.
tie-
produces the stated
outcome:
W^„(a,): Player
T?^(a,): Player
/
/
wins using
ties for
a. (either
winner
at
a,,
by winning a tie or winning
and player i wins the
41
outright).
tie.
Appendix
T,^„(a,):
Thus,
Player
if e„ is the
W'(a),
Tj^'Ca,.),
/
ties for
winner
minimum bidding
and
at a,
and player
increment,
T/-*(a,) represent the
W,,„(a,)
loses the
/
=
{
tie.
W,.„(a -e„)u
when
corresponding events
T,:^„
i
-e„)u T,"^ (a.) }
(a
.
Let
plays against the
opponents' limiting strategies.
In our analysis,
it
will
probability events.
sometimes be useful
We
to define
expected payoffs conditional on zero-
will use the following convention:
4Av,(a,,t)|?,,l^(a,)]^Um4Av,(a,,t)|f,,\^(aJu{t_,G[L,,t^
In order to establish that "mass points" are inconsistent with equilibrium,
preference to
show
that
In particular, each type
strategies
each player could do better than use actions which support the mass point.
who
uses an action which
is
a mass point in the opponents' limiting
might also see nonnegative expected payoffs from increasing her action a small amount,
"jumping over" another player's mass
which win with probability
A difficulty may arise if there is a set of actions
point.
for a given player
/
in a given finite
game, which might happen when
playing against a more aggressive opponent. In such a case, player
indifferent
action
a,,
between a
set
of actions,
all
f's
lowest types
may be
of which generate zero expected payoffs, even though
which wins with probability zero might also satisfy^j Av,. (a., t)|?.,W^ „(«,.)] <0; such a
player might not wish to increase her action a
probability of winning above zero
shows
we make use of revealed
may be
little bit
undesirable nearby action
that there exist equilibria to the finite-action
even when winning
returns to winning,
Lemma
is
which
games where
all
i,
t..
The following Lemma
actions
would give positive
we can find a PSNE of this game
satisfies the following: (i)
a nondecreasing function of
all
a..
a zero-probability event.
is
3.2.2 If /^' " is finite for
strategies P„(t))
to guarantee a win, since increasing the
(ii)
For
all
/
(described by
each player's equilibrium strategy,
and almost
all
)3, „(?,),
?,.,
E[Av,{p,Jt,)^)\t,,wM.n(h))\^o.
Now restrict attention to sequences { P„(t)} which satisfy condition (ii) of Lemma
Lemma will show that the following "no mass point" condition is satisfied:
For each player
/,
for all
a.s^,
Pr()3,.*(r,.)
=
a,.)-Pr(T}^'(a,.))
=
This condition requires that for every possible action of player
/,
3.2.2.
(NMP)
0.
either player
/
uses the action with
probability zero, or else the probability of a tie for winner with an opponent at the action
ties
occur at
a.
Lemma 3.2.3
with probability zero, then Pr( W;*(a,.)) will be continuous
will
make use of the fact that if a mass point occurs
must be something "close"
Lemma
3.2.3:
to a
mass point
in finite
at
a,..
is
zero. If
The proof of
in the limiting strategies, there
games with a large enough number of actions.
Construct P*(t) as the limit of equilibrium strategies to
42
The next
finite
Appendix
games.
P„(t),
which
satisfy conditions
The following Lemmas estabUsh
all
that
and
(i)
from
(ii)
under (NMP),
Lemma 3.2.2. Then
P'(t) describe
P'(t) satisfies
(NMP).
equilibrium behavior for almost
types.
Lemma
3.2.4:
for
A,„(0. P-.>(-),0 converges
Lemma
Together with our previous arguments,
all
(i) f/,.(^,.,P*,. (),?,) is
t.
such
that Pj„(t.)
continuous in
a. at
converges to
a~fi*{t^, and
to C/,(A'(^)'Pl,(-),0-
Under (NMP), A*(0
3.2.5:
and almost
all /
following conditions hold:
P'(t,), the
(ii) £/,(
Under (NMP),
is
a best response to
Pl,(-) for
almost
all
f..
Lemma 3.2.5 implies that a game satisfying the hypotheses
of this theorem will have a set of strategies,
P*(t),
which assign optimal actions
to almost every
type of each bidder. Since behavior on a set of measure zero does not matter to opponents,
we
complete the argument by showing that these strategies in fact assign optimal actions to every type
t-
of each bidder
3.2.3 can
such that t^ r,
/,
be used
(otherwise, types
.
Standard arguments shnilar to those in the proof of Lemma
to establish that, for
who chose
each
actions just
/,
Piil3'(t^=b)=0 for
all
6 e int{a,.: Pr(W,.'(a.))>0}
below b would prefer an action just above
£»,
as in
Lemma 3.2.3, but there must be some types playing actions just below b, otherwise b would not
be optimal for player i). Now consider c =inf{a,: Pr(W,'(a,.))>0}. Condition (NMP) guarantees
that if Pr(j3,'(?,)
each of player
=
/'s
c)>0, then Pr( W;*(c))=0. But this in turn impUes that at
opponents use
c,
most the lowest type of
so the opponents see discontinuities in their payoffs only for
their lowest types.
Proofs of
Proof of
Lemmas:
Lemma 3.2.2
We proceed by constructing such an equilibrium from the limit of a sequence of equilibria of
constrained games, where in each constrained game, the lowest types of each player must choose
We first show that, using minor modifications of our Theorem 2. 1
action Q.
game has a fixed point;
limit of the
Observe
action
Q
it
sequence of constrained games
is
by
definition
be nondecreasing, since
the lowest available action and the constraint requires that the lowest types use Q; thus
we do not need to modify our notation from Section 2
with
each constrained
an equilibrium in the unconstrained games.
that the constrained strategies of the players will
is
,
then follows from continuity of the payoffs in opponent strategies that the
finite vectors.
for representing nondecreasing strategies
We then define the constrained best response of player
(constrained or unconstrained) strategy by opponents represented
'af%\X)
af\t\X,5) =
(2
u af«(f,|X)
/
to
an arbitrary
by X:
t,>U+5
f,
= t, +5
(7.1)
t<t;+5
[Q
43
Appendix
.
Using
we modify
this notation,
f,.(X,<5)={y:
It is
3
a.(^,)
our correspondence, as follows:
which
a,(0^
consistent with y such that
is
straightforward using our definitions to
show
that an equivalent,
«f''(^,|X,5)
}
(7.2)
.
second definition
is:
f,(X,5)={y: 3 zer,(X) such thaty^=max(t,+5,zj}.
We now show that the conditions for Kakutani are satisfied for each r(X
We know from Theorem 2.
Nonempty:
elements of r.(X,5), as can be
Closed graph: Recall
consistent with
Suppose
consider tp>ti+5 such that
A„e
y''*
We argued in Theorem 2.1
all k.
Consider player
converges to
y',
r,.e
Then
T}.{y'}.
there
must
>'|'*>f,
exist
af^(r,.|X,5) since Y*er(X*). Since
y,.(a,;X,f,) is
continuous in X,
y,.(A„;X,r,.)>y,(^„-;X,f,.).
i.
Then
first that t-Kt^+S.
of the correspondence implies that that
Since
Let
a,.(^,;y')
since y'*
for all k,
an
e
and thus
there exists an
r,(X'',5) for all k, the definition
(x^(tf,y')=Qe
m g {0,..,M}
A„e afi^X) and
thus
such that y^ <
y'^
k>K and
A„e
of r(X) except that
ce
df'^(t."\X,5).
af'^(t.\X,S)
from Theorem
Fixed point
The
Thus,
2.1
show
By
exists:
and be
tj+l/k.)
Such an X*
Since each X*
is
af'^(t"\X,5), then
df'^(tj\X,5) is
that r,.(X,5)
/
is
m,
then
is
nondecreasing in the
Since the definition of
all
we
5f (r,'IX,5)<s5,^^(f."IX,5).
we
also
is
If
If
must have b e
df'^(t.\X,5)
and
the lowest available action, Q, this
nondecreasing in the strong set order, and our arguments
must then be convex.
Consider a sequence X* such that
X*"
uses actions consistent with X*, he will use action
e r(X*,l//:) for each
an element of a compact subset of finite-dimensional Euclidean space,
Without loss of generality,
let {^}
we
all
d>0.
we can find
simply need to establish that
denote that subsequence.
44
k.
Q for types less than
guaranteed to exist for each k since r(X,5) has a fixed point for
a subsequence of{fe} such that {X*} converges to a matrix X, and
Xe r(X).
all
Kakutani' s fixed point theorem, r(X,5) has a fixed point.
Construction of Equilibrium:
(Thus, if each player
,
definition of the strong set order asks us to verify that if c>&,
But, since the only element of af''(?/IX,5)
is satisfied trivially.
y'^_^_^
j^^,
and thus
used instead of af^it^),
l+5<t;<t;', then af\t\X,5)=af''{t\X) for t=t;,t;', and thus
e
<
<
nondecreasing in the strong set order. Consider tl<t".
that af'^{t^,5) is
ti+5>t', then df'^(tf\X,S)=Q.
t-
t.
af^(r,.|X,5), as desired.
T is convex- valued.
af''(r,.|X,5) is
<
Now
a^'^(t.\X,5).
af''(?,. |X*). But, since
?,>£,. +5, this in turn implies
if y,.(A„;X*,r.)>y,.(^„,;X*,r,.) for all
This implies
f,.(X,6) is identical to that
then whenever c
that lim^(X*,Y'')=
be a nondecreasing strategy
K such that, for all k>K,
strong set order implies that the correspondence
is
construct
that y,.(«.;X,r.) is continuous
Convexity: The proof of Theorem 2.1 showed that the fact that af^(rjX)
need to show
we can
Take a sequence such
that r,.(X) has a closed graph.
af^(r,.|X,5).
y'.
nonempty. Thus,
.
We need to show that for each player and almost every type, the action
in the elements of X.
Y is in
that r,(X) is
5)
easily seen in (7.3).
(X,Y), such that Y*g f(X\5) for
assigned by
1
,
(7.3)
Appendix
We show that, for each player and almost every type, actions consistent with x' are in a^''(t.\X).
Consider such that t.s 7^\{x'}. Then there exists an m e {0,..,M} such that x'^ < < jc^^,. Since
< a;^*, and t.>ti+l/k.
x'* converges to x', there must exist an K such that, for all loK, x^* <
t.
t.
t.
A„e
Find such a k>K. Then
A„e
But, since
af^it-lX').
k>K and
Finally,
all
V,(a,.;X,f,.) is
continuous in X,
that the equilibrium
definition
and since t,>i+l/k,
if V,.(A„;X\f,.)>y.(A„.;X*,r.)
m', then V.(A^;X,t)>Vi(A^.;X,t). This implies
we show
By
af''(t.\X'',l/k) since X'^e HX*^).
A„e
for all
af'CrJX), as desired.
we have constructed satisfies
the desired property, that
conditional expected payoffs always converge to a nonnegative number. In our sequence above,
V,iA„X,t.)>V,(A„.-X,t,) for
for each k,
A^>Q wins
all
foK, and
in particular, V,{A^-X,t;)>V,(Q-X,t).
We know that
with positive probability. Thus by (3.3) and revealed preference,
4Av,(A„,t)|?,, W^(A„;X*)]>0 for
all k,
which
in turn implies lim,4Av,(A„,t)|r,,
V^(A„;X*)]>0, as
desired.
Proof of
Lemma
Consider player
second player,
players use
Observe
3.2.3:
and suppose
1,
let this
be player
b with probability
that since
each
0,
that Pr(/?,'(?,)
&)-Pr(T,^'(£»))
> 0. This
implies that there
such that FriPjiQ - b)>0. For simplicity, suppose
2,
all
is
a
other
though the argument can be easily extended.
„(?,) is
)8,
=
measurable and converges almost everywhere to P*(t), the
sequence converges uniformly to P'it) except on a set of arbitrarily small measure (Royden,
1988, p. 73). Thus,
if P'(t.)
measure
such
IA
n(^/)
less than
rj
- ^1 < ^ on 5^,.
=
Z?
on an open
that, for all
interval
for every 77>0 there exists an E^ with
d>0, there exists an
Further, since each Pi„it.)
an open interval S'czS^ such that
S-,
|A,„(^,)
- ^I < ^ on
is
A'^
such
that, for all
nondecreasing in
t.,
n>Nj,
we know
that there is
S'.
A little notation, for /=1,2:
a^{d,n)
Find a
= min{a,.
set
and such
S,=
5,
of types
e /i^ and a^>b — d\
E such that
converges uniformly to P*() except for types te E,
Define C,=sup,^ {Pr(t_.eE_,.lr.)}.
= b)
=int{r,6 5,\E,}.
= inf{sup{t_,
Now, pick
a
7=1,..,/, all
n>N„ and
e
5,.,
and find a
all
:
?/<f.
p.^tj) <
A,(0 y/ ^ /}}
such that f/e
5,..
Pick any d, choose N^ such
t^^E., |^.„(?.)-j3/(f.)|<d, and further,
V7=3, ..,/)< i^Pr(T;^'(^)). (Recall
limit).
P„(-)
that sup,^ {Pr(t_,.eE_,.l?,.)}<iLpr(T.^*(Z7)).
{t:.p:{t:)
t_,(?,,iV)
t.
a,,
:
that players 3,..,/ play action
Then, the following series of inequalities holds for
45
all
Pr(i8^.„(f.)€
that, for all
(b-d,b+d)
b with probability zero
in the
n>Nj:
Appendix
mjj4Av,(6-rf,t_,,4;,W^.„(A>(0)]}
(7.4)
<£[Av,(Z;-J,t_,,r,)|r,,t_,<t.,a;,iVj]
The
inequality holds since
first
second inequality holds since Av
because by (3.6) and
(3.7),
Lemma 3.2.2, each member of the set is nonnegative; the
by
is
nonincreasing and \p.^{t')-b\<d; the third inequaUty holds
expected payoffs are continuous and nondecreasing in the
must correspond
defeated, and so the smallest expected payoffs
potentially defeated,
h_. <
The relation
i_.it ,Nj)\.
the player might tie rather than
win against these
is
an inequality rather than an equality since
types, but
winning against the highest of these
follows by (3.7), since expected returns to winning are strictly increasing in
follows from
and since
e\ Av-(b
iV^ is
definition that t_^(t.,N) is nondecreasing in N; but then,
its
decreasing with d,
+ i/,t_,.,f,.)
f,.,t_,.
is strictly
K(t,,d)=E[Av.{b
Since
Av
K(t.,d) is
is
by continuity of Av
positive for
r,.e
5,
Y^t.,d)
d (recalling
that
by
t_,(r;,iVj]
>
v,(fl',f,)
Sill
=
^L K(t,d)- (Pr(T,^'(^)|r,)
that, for
t-
e
S-
,
- C,)
(7.5)
and n>N^, the following inequalities hold:
a'e[b-d,b+d\.
(7.6)
\
W.Ja,{d,n))]
> \ ^Av.ib + J,t_,,0|?„t_, <
possible to satisfy (7.6) since
(3.7),
also
- Ub + d,t,) + J[Av,(a', t) - Av,{b + rf, t)J/(t.,|r,)rft_,
E[Av,ib + d,t_,,t,)\t,,W,„ib + d)
small and
is
definition {].<-j^Pr(Tj^'(Z7))):
d>0 small enough such
for
assumption
which based
and d small enough:
nonincreasing in d. Then, define the following positive constant, which
Finally, pick a
d gets
t..
as well. Thus, define the following constant,
+ d,t_„t,)\t,,t_, <
%it,d)
as
last inequality
nonincreasing in actions and following the arguments from the previous paragraph,
nonincreasing in
It is
The
we can find a d small enough such that
< i_i(t',NM>0
on the preceding analysis
of types
to the smallest set of types
rather than having a tie will increase expected payoffs under assumption (3.7).
It
set
Av
is
which
Jl^t.^,d)
is
t_,(?;,
Arj]=i
<?,,J)
(7.7)
nonincreasing in d, and since b+d-a approaches zero
continuous in actions. The inequality in (7.7) can be satisfied by
states that the
expected value of payoffs
46
is
nondecreasing in the set of
Appendix
we
types over which
condition, since
who
of opponent types
v,(Z7
^
J
//^)
must by definition be
less than the highest vector
use action b+d.
Pick an n>Nj. The action
J
t_,.(f/,
b-\-d will
be preferred to a'e (b-d,b+d)
if:
+ d,t)- /(t_,|r,)rft_, + j Av,(^ + J.t) w,Jb + rf,t_,)
v,(«',t) /(t_,|r,)rft_,
+
Av,(a',t) w,„(a',t_,) /(t_,|r,Mt.,
•
•
J
/(t_,|r,)rft_,
This can be rewritten as follows:
E{Av,ib + d,t)\t,,W,„{b + d)
\
W,„(a')]-Pi[w,„(b
+ d)
\
W,Ja%)
>Ua,h)-Ub + d,t,)
(7.8)
+ 4Av,(a',t) - Av,(Z; + d,t)\t,, W,Sa')]
Consider
first
Pr(w^.„(^
+ ^)
a=a^(d,n), the lowest feasible action chosen by a type in
\
W^„(a,(rf,«))|?,)>i(Pr('r;^'(Z?))-Q for n>Nj, since all
action b in the limit
b+d rather than
or
tie
must choose actions on [a.{d,n),b+d)
a.(d,n), player
/
with using action 5,(<i,«);
chooses a
and player
by increasing her action
b-\-d.
/
is strictly
of (7.8)
is strictly less
enough, type
than
will not
t.
that Vr{Pjitj)e
who choose
by choosing action
higher action than those types she would lose to
of the types
who choose
d small enough. Thus,
for
'Yi{t^,d)
d small enough. On
choose action a.{d,n). But
this
action
b
in the limit
would have otherwise
the other hand,
for the
choose
tied with
LHS
the RHS
by
all t.G
that (recalling that
letting e„
(7.6),
chosen d and for n large
then implies that
That will in turn imply
/=1,2.
(b-d,b+d) y/=3,..,/)<^ Pr(T,^'(^)), and
increment for the
be the
5,.
will
choose
d was chosen
so
minimum action
game with action space indexed by n):
Pr(w.„(b + d)
\
W.^{a'
+ ej\t,)>^(FriT,'\b))-Q.
Thus, the arguments of the previous paragraph apply again: no types
a'+e„.
The argument can be repeated
/=1,2,
no types
r,.e
Lemma
S-
first that
for both players, "unraveling"
t^e
5,.
back
will
choose action
to the point
where
for
choose actions on (b-d,b+d), a contradiction.
3.2.4:
C/,KP:,O,0 =
Suppose
that
opponent types
for n>Nj. Thus,
defeats those players she
greater than Yft^,d) for
on {a^{d,n) ,b+d) for
Proof of
Observe
S-.
the latter argument with equations (7.7) and (7.5) allows us to conclude that the
of (7.8)
actions
to
strictly
at worst, all
action d^{d,n) with grid n,
Combining
Pr(l^,(a')|r,)
Part
(i):
v,(a,,r,)
Consider
Recall that:
a,.e/^..
+ 4Av,(a,,t)|r,,l^*(a,)]Pr(w^*(«,)K)
Pr(Tl'\b))=0. Then,
probability of a tie for winner at b
is
by our assumption
zero for player
47
/.
(3.5),
(7-9)
Pr{T^(b))=0 as well, that
This will in turn imply that Pr( W;.*(a,))
Appendix
is,
is
the
continuous at a=b. Since
v,.
and
v, are continuous, (7.9) will
be continuous as well.
Now suppose that Pr(T,.^'(Z7))>0. But then, by (NMP), PT{p'{t))=b)=0, and the set of types for
whom payoffs are discontinuous at b has measure zero. Since the number of opponents is finite,
the set of actions b such that T'r(r.'''(b))>0 is also countable, and so the set of types for whom
payoffs are discontinuous has measure zero.
Part
Consider
(ii):
such that
t.
jS,
„(f,)
converges to P'(t^ and Pr(W,*(a,))
a=l3'(t.) (the latter condition is true for
almost every
w;(o.p:,(-),o-f^.( A>(0>
=
RHS
term of the
first
A>(0.
of
in the relevant region (part
(7.
and find an
A'^
and a
j=l,..J, except for
+
t
to
The
first
p:,(-),o-f/,(
So
it
(i)). First,
if f/,(^.'P-i,n()'0
f/,(^,,
(7.10)
A.(0. P-,>(-),g].
n gets large by continuity of
Pl,(-).0-
To
converges uniformly (across
Pl,() ,t) in a.
RHS of (7.10).
a. in
a
see that uniform convergence holds, pick a
than
t]
such that for
all
n>Nj,
rj,
d>0,
\p*(tj)-Pj J^tj)\<d,
max(|Av,(s,,t)|,|Av,(a;.,t)|)/(t_,|r,)rft_,
J^ _^^
max
>
max
r^.^
(7.11)
I
Av,(a,,t)[w,„(a,,t)
- w;(a,,t)]/(t_,|0^t_,
^
. Jt^<(«<'P-.-,«(-),o
- f/,(«„p:,o,o|
term of KJ^'t) can be made arbitrarily small by decreasing
small by choosing
f/,(a,,
Then, for such n>Nj, the following inequalities hold:
in set E.
>
bounded and since
note that
remains to consider the second term of the
E of measure less
set
-
=
part
at
P-,,„(-),o
10) goes to zero as
(i)).
This term will converge to zero
neighborhood of P'it))
by
continuous
[t/,(A'(0,p:,(-),0-t/,(A>(O.Pl,(-),0]
+[c/,(
The
f,
is
Pr(W,.*((3,)) is
rj
d,
since/ and
continuous at a/=P*(t). The second term can be
small enough. This establishes that there
is
Av,.
made
are
arbitrarily
a uniform bound on
I^KP-,,„(-),0-f/,(«.PH(-),Ol for a,e [A'(^,)-^.A'(^,)+^-
Proof of
Lemma
3.2.5:
We know
^.(A>(0'P-,.„(-),0^f/,(a',P-,,„(-),0 for every
actions a so that for large
n and for almost all t.,
a'e^". Thus, if a'eD' (Where D'
that for every
enough N,
a' is an available action for «>A0, then
48
is
the set of
Lemma 3.2.4 (ii)
Appendix
all
implies that U,(P:(t,),fi:,(-),t)>U^(a',fi:,(-),t,).
Now consider an action a'^ D'.
which
action Q,
is
If C/,(a',pi,.(-),?,)=0.
always available by assumption.
then
it
compare
suffices to
If U.(a, P!.,(-).0 is
can find a sequence {a*}, a'^sD', which converges to a, so that
P*{t.) to the
continuous
f/,(a*, P_,,„ (•),?,)
at a',
and
we
converges to
t/,(fl',p:,(0.O, as desired.
Suppose
that Pr( j3J(rp=a')>0 (a
can be true only
exists
mass point
if £[Av,(a',t_,.,f,.)|^.,
exists at a') for somey^t/,
M^*(a')]>0- Then,
and
t/,.(a',p*,.(),r,.)>0.
by continuity of Av,. and by
This
(3.7), there
an &>0 such that £|Av,(a' + S,t_.,t.)\t., W*ia' + 5)]>0, where 5 is chosen such that a'+5 is
used on a
set
of measure zero by the other players in the limit and such that a'+5 g D'. But, a
wins against, instead of ties with, playerj
that player
/
+5
This leads to a discrete increase in the probability
at a'.
wins. Thus, for small enough 5, U^{a+5, Pl, (•),?,)>[/,(«', Pl,(-),0- Since a'+5 e D',
by our earlier arguments we know
that
(/.( )8*(?,)
,
P',(0>O - Uf^a+5, P',(0,O
49
•
Appendix
MIT LIBRARIES
3 9080 01439 1129
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Appendix