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Working Paper Series
AGGREGATE COMPARATIVE STATICS
Daron Acemoglu
Martin Kaae Jensen
Working Paper 09-09
February 22, 2009
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Aggregate Comparative
Statics"
Daron Acemoglu^ and Martin Kaae Jensen
February
22,
1
2009
Abstract
depends on her own actions and an aggregate of
the actions of all the players (for example, sum, product or some moment of the distribution of
actions). Many common games in industrial organization, political economy, public economics,
and macroeconomics can be cast as aggregative games. In most of these situations, the behavior
of the aggregate is of interest both directly and also indirectly because the comparative statics
of the actions of each player can be obtained as a function of the aggregate. In this paper, we
provide a general and tractable framework for comparative static results in aggregative games.
We focus on two classes of aggregative games: (1) aggregative of games with strategic substitutes
and (2) "nice" aggregative games, where payoff functions are continuous and concave in own
strategies. We provide simple sufficient conditions under which "positive shocks" to individual
players increase their own actions and have monotone effects on the aggregate. We show how
In aggregative games, each player's payoff
framework can be applied to a variety of examples and how
stronger comparative static results than typically obtained in the
this
"We would
this enables
more general and
literature.
thank Roger Hartley, Jean-Francois Mertens, Alex Possajennikov, and Burkhard Schipper
remarks and suggestions. Thanks also to seminar participants at the University of Copenhagen,
University of Manchester, and at the 2008 Midlands Game Theory Workshop in Birmingham. All remaining errors
like to
for their helpful
are of course our responsibility.
department
of Economics, Massachusetts Institute of Technology (e-mail: daron@mit.edu
'Department of Economics, University of Birmingham, (e-mail: m.k.jensen@bham.ac.uk)
)
Introduction
1
In aggregative games, each player's payoff depends on her
players' actions.
all
its
number.
and some aggregate
own quantity and
Numerous games studied
in the literature
of
an aggregative
is
market.
total quantity supplied to the
be any mapping from the players' action
generally, the aggregate could
1
actions
For example, the Cournot model of oligopoly competition
game; each firm's profits depend on
More
own
profile to
can be cast as aggregative games.
a
real
These
include the majority of the models of competition (Cournot and Bertrand with or without product
differentiation),
models of
(patent) races, models of contests
aggregate
how
demand
externalities,
the aggregate
total public
(e.g., total
In this paper,
In
on. In
many
games. Our approach
in the
is
2.
for
comparative static analysis
First,
we
aggrega-
focus on aggregative games with strategic substitutes.
strategic substitutes, each player's payoff function
in
in
applicable to diverse environments that can be cast as aggregative
games with
own
parameters that only
affect the
is
supermodular
in her
own
strategy and the strategy vector of other
strategic substitutes,
Cournot game) always lead to an increase
and the
environment. In addition, comparative
to provide sufficient conditions for a rich set of comparative static results.
players. For aggregative
Changes
naturally on
supply to the market, the price index, probability of innovation,
strategy and exhibits decreasing differences in her
1.
is
can often be obtained as a function of the aggregate. 2
present two sets of results.
games with
applied problems, the focus
we provide a simple general framework
games and enables us
We
and so
models of public good provision, models with
fighting,
good provision) responds to changes
statics of individual actions
tive
and
we
establish the following results:
aggregate (such as a shift in
in the aggregate
—
demand
in the
in the sense that the smallest
largest elements of the set of equilibrium aggregates increase.
Entry of an additional player decreases the (appropriately-defined) aggregate of the existing
players.
3.
A
"positive" idiosyncratic shock, defined as a
parameter change that increases the marginal
payoff of a single player, leads to an increase in that player's strategy and a decrease in the
other players' aggregate.
'We discuss games with
The fact that a game
multi-dimensional aggregates in Section
2
aggregates.
When
they do
is
so,
7.
aggregative does not imply that players ignore the impact of their strategies on
we say that the equilibrium
results are generalized to Walrasian
Nash
is
equilibria in Section
"game-theoretic" interactions, the analysis
is
Walrasian Nash or that the play is Walrasian. Our
9. Because in this case there are more more limited
more straightforward.
The comparative
static results
mentioned above are
But
intuitive.
it
surprising that for
is
aggregative games, they hold at the same level of generality as the monotonicity results for
supermodular games
Nevertheless, not
we
no quasi-concavity or convexity assumptions are needed).
(in particular,
own
economy and other areas
We
refer to as local solvability.
theorem similar
Another example
Under
monotonicity results
in spirit to the
more
increases the aggregate
An
informal
comparative statics are "regular"
arise in simple
Example
+ 02 +03<O,
&i)~
Assume
and Ql
1
Si
illustrates
=
i
t
no guarantee that such intuitive results
how
counterintuitive, "perverse," results can
1,2,3 with payoff functions
>
1,
=
7Tj(s)
—0.5s?
+
a,(l
—
1
> a2 >
0,
<
Q3
0,
0\
<
0,
2
>
0,
Qi
+ a 2 + Q3 >
1,
1.
1
.
Q=
Ylj s j
:
= -O.5s^+a (l-a
^k{suQ)
l
response functions in this
game
are: ri(s- t
Solving for the pure strategy Nash equilibrium (s*
= a Q* + 0i, where Q* = s{ + s$ + S3 =
aggregate. Now consider a (small) increase
s*
is
defined locally in a sufficiently large neighborhood of the equi-
that a\
+ a3 <
and the aggregate
at)"
strategic substitutes
an aggregative game since we can write the payoffs as a function of players' own
strict concavity, best
—
games with
example, a reduction marginal cost increases output and so
(for
The next example
+ ft(l —
librium found below.
strategies
static" results.
under a variety of reasonable economic conditions,
that,
is
1 Consider three players
is
in the smallest
games, even in simple aggregative games.
(X^7=H s j) s i
This
an increase
also prove that entry of an additional player
should be emphasized that there
should hold generally.
This theorem
of our results from both aggregative
and from nice aggregative games
it
supermodular games.
of) the players leads to
We
local invertibility
establish a general monotonicity
and derive more extensive "individual comparative
summary
on). Nevertheless,
for
which ensures the
we
further below,
largest equilibrium values of the aggregate.
0i(l
contest (or fighting) games,
is
provided by games in which players
is
this assumption,
mapping described
implies that a positive shock to (one or
By
An example
contributions to the provision of a public good.
of the "backward reply"
_1
(in particu-
prove parallel results for nice aggregative games under an additional technical assumption,
which we
and
smooth
effort levels of all players; contests typically exhibit neither
strategic substitutes nor complements.
make voluntary
sufficiently
analyzed in industrial organization, political
satisfy these conditions.
where payoffs are determined by the
and
strategies,
Many games
twice continuously differentiable).
lar,
01
set of results
present focus on "nice" aggregative games, which are games with payoff functions that are
continuous, concave (or pseudo-concave) in
a'j)
The second
aggregative games exhibit strategic substitutes.
all
(0i
in
+
2
+ 03)~
a 2 This
.
is
1
(l
=
l
)~ 1 (Q-s l )s l +0 l {l-a l )~ l s i
)
=
aj(l
^(s^),
i
- ai — 02 - a 3
—
—
)
cti)~
1
(J2 1 jH s j)
1,2,3)
is
.
+
we obtain
the equilibrium
a "positive shock" to player
2:
holding
opponents' strategies fixed,
2's
best response function,
3
it
increases player 2's marginal payoff
>
dr2(s-2)/doi2
optimal strategy notwithstanding, an increase
But
yet
0.
in
and therefore
this positive direct effect
ao leads to a decrease in player
"lifts"
player
on player
2's
strategy in
2's
equilibrium:
9s*i
%-*-
As can
also
= „-,
Q +
dQ*
,
Q2T,
ft
=
+ ft + ft
ft
+ CX2J,
-,
+ 02 + 03
ro
be seen, the positive shock to player 2 leads to a decrease
^
<
n
in the equilibrium aggregate:
dQ*
<0
7roa2
-
In summary, a parameter change that unambiguously increases the marginal payoff for a player,
which should,
all else
equal, lead to an increase in that player's strategy and the aggregate, in fact
leads to a decrease in the player's strategy in equilibrium as well as a decrease in the aggregate.
Part of the contribution of our paper
results,
is
to provide
minimal conditions to ensure that "perverse"
such as those described in the previous example, do not apply. In particular, such results
are ruled out in nice aggregative
games by the
assumption mentioned above. In
local solvability
addition to providing minimal conditions for general comparative static results and significantly
weakening the conditions that are available
races, contests,
the
common
in the literature (for
and public good provision), our approach
is
example,
we use
These
highlights
it
results are
made
comparative static results (the more
on the implicit function theorem and lattice-theoretic
familiar approaches in the literature rely
tools in the case of
for deriving
models of patent
also useful because
features that ensure "regular" comparative static results.
possible by the alternative approach
for
supermodular games).
Our approach can be explained
as follows.
Consider a general comparative static problem
written as
A
where
in the
dt G
R
is
•
ds
= — b-dt,
the change in some exogenous variable, ds G
endogenous variables,
important question
is
A
is
an
MxM
matrix and b
to specify the conditions under which
without specifying numerical values
this generality, the conclusion
is
for the
somewhat
ascertain the sign of the elements of
A
-1
.
R^
is
designates the induced change
an Af-dimensional vector.
we can
sign ds "robustly"
elements of the matrix
A
and vector
depressing: to obtain such results
But even when
A
is
it
is
symmetric negative
An
—meaning
b.
Cast
in
necessary to
definite,
we
In other words, player 2's payoff function exhibits increasing differences in S2 and Q2 (Topkis (1978)). This is
an equivalent way of defining a "positive shock" when strategy sets are one-dimensional and payoff functions are
can do this only when one of the following two additional (and stringent) conditions hold:
when
A
a Metzler matrix, that
is
Morishima matrix, that
variables.
4
The only
it
has non-negative off-diagonal elements, or
becomes a Metzler matrix when we reverse the
general case where these conditions are satisfied
many games
games. Since
literature
is,
is, it
comparative statics
results.
The
many
not be robust and in general there are no systematic investigations of
is
a
more
sign of one or
much
in the context of specific
discussion above highlights that
A
provided by supermodular
is
that arise in applications are not supermodular,
imposes additional parametric restrictions
when
(ii)
(i)
of the applied
games to derive
of these conclusions
when
may
the specific conclusions
enjoy such robustness.
Our
tive
alternative approach
is
not to impose parametric restrictions, but to exploit the aggrega-
nature of the games in question and note that what
is
often of economic interest
entire vector ds, but the behavior of the appropriate aggregate, or just
latter corresponds to deriving robust results for a single player as
this perspective, robust
and general comparative
weaker conditions. Our contribution
static results
its
opposed to
not the
coordinates (the
all
players).
With
can be obtained under considerably
to suggest this perspective
is
one of
is
and show how
it
can be made
operational.
Our paper
results in
related to a
is
number
of different strands in the literature.
most games are obtained using the
implicit function theorem.
Comparative
static
The main exception
is
supermodular games (games with strategic complements). Topkis (1978, 1979), Milgrom and
for
Roberts (1990) and Vives (1990) provide a framework
for deriving
comparative static results
in
such games. These methods do not extend beyond supermodular games.
More
closely related to our work,
and
in
many ways
its
(1994) provides comparative static results for aggregative
only under fairly restrictive conditions, which,
libria.
These conditions are typically not easy
we provide
Corchon (1994). Corchon
games with
strategic substitutes, but
other things, ensure uniqueness of equi-
to check in
many
applied problems. In contrast,
general comparative static results for aggregative games with strategic substitutes
without imposing any additional assumptions.
nice aggregative
Corchon (1994)
(1970).
among
is
precursor,
games without
is
that both
We
also provide parallel but stronger results for
strategic substitutes.
make use
Another similarity between our paper and
of the so-called backward reply correspondence of Selten
In an aggregative game, the backward reply correspondence gives the (best response)
strategies of players that are compatible with a given value of the aggregate. 5 In a seminal paper,
4
See Bassett et
The
first
al
(1968) and Hale et
al
(1999).
systematic study of aggregative games (German: aggregierbaren Spiele) can be found
in
Selten (1970).
After defining aggregative games, Selten proceeds to define what he calls the Empassungsfunktion (Selten (1970),
154), that is, the backward reply function of an individual player.
As Selten proves, the backward reply
p.
correspondence
is
single-valued (a function) provided that the player's best-response function has slope greater
Novshek (1985) used
this
correspondence to give the
general proof of the existence of pure-
first
strategy equilibria in the Cournot model without assuming quasi-concavity of payoff functions
(see also
Kukushkin
(1994)). Novshek's result has since been strengthened
larger class of aggregative
games with
games
Dubey
(e.g.,
strategic substitutes utilize
et
and generalized to a
(2006) and Jensen (2007)) and our results on
al.
Novshek (1985)'s construction
in the proofs.
6
Our
results
on "nice" aggregative games blend the backward reply approach with the equilibrium comparison
results reported in
The
Milgrom and Roberts (1994) and Villas-Boas
rest of the
3 presents a
paper
number
of
is
organized as follows.
common
(1997).
7
Section 2 provides basic definitions.
examples, which can be cast as aggregative games.
Section
Section 4
provides the general comparative static results for aggregative games with strategic substitutes.
Section 5 generalizes and strengthens these results for "nice" aggregative games, which feature
and (pseudo-) concave
payoffs that are continuous
results
from Sections 4 and
in
own
strategies.
Section 6 shows
can be used to obtain general characterization results
5
plications. Section 7 discusses
how
these results can be extended to
how
the
in various ap-
games with multi-dimensional
aggregates and Section 8 provides additional generalizations of the results presented in Section
Section 9 briefly discusses Walrasian Nash equilibria
(cf.
footnote
2).
5.
Section 10 concludes and
the Appendix contains the proofs omitted from the text.
2
Basic Definitions
In this section,
Let
r =
(iri,
we introduce some
basic definitions.
Si,T) l€ j denote a noncooperative game with a
and finite-dimensional strategy
sets 5,
rameters with typical element
€ T.
response to changes in
Throughout the
t
C RN
We
.
In addition,
will focus
finite set of players
T C RM
on how the
is
1=
{1,...,/},
a set of exogenous pa-
set of equilibria of
T changes
in
t.
rest of the
paper we assume that the joint strategy
set
than —1. The assumptions imposed by Corchon (1994) imply that the slope of players' best-response functions lie
between -1 and 0, so that the backward reply correspondence is both single- valued and decreasing. Neither
is necessarily the case in many common games and neither is imposed in this paper.
6
Novshek's explicit characterization of equilibria is similar to the characterization of equilibrium in supermodular
games that uses the fixed point theorem of Tarski (1955). Both of these enable the explicit study of the behavior
of "largest" and "smallest" fixed points in response to parameter changes. Tarski's result is used, for example, in
the proof of Theorem 6 in Milgrom and Roberts (1990).
More specifically, our proofs repeatedly use that the smallest and largest fixed points of a continuous function
from a subset of real numbers into itself will increase when the curve is "shifted up" (see Figure 1 of Villas-Boas
(1997) or Figure 2 of Milgrom and Roberts (1994)).
strictly
compact
is
(in
the usual topology) and the payoff functions
TTj
are upper semi-continuous for each
i
:
S
x
T
R
->
E X. Let
Ri{s-i,
t)
=
max ^(s,,
arg
s_ 2
,
£)
•s,e5,
denote the best response correspondence (with the standard notation S-i 6 S-i
=
Given
Yijjti Sj)-
the compactness and upper semi-continuity assumptions, the correspondence Ri
is
non-empty-
and compact- valued, and upper hemi-continuous.
We
next define the notion of an aggregator.
Definition
(Aggregator) A mapping g
1
:
S — R^
continuous, increasing and separable across the players,
H
function
:
RK —
R^" and increasing functions hi
>
:
St
K
(with
»
i.e.,
if
— R^
>
< N)
an aggregator
is
if it is
there exists a strictly increasing
i£l)
(for each
such that:
(1)
Throughout
paper
this
K
is
referred to as the dimension of the aggregate.
we impose
analysis (in particular, until Section 7),
on N. In
K=
1,
— g cannot be
sets).
The requirement that g
aggregators for the same game.
individual strategies (thus working with
Common
restrictive.
(with hi(si)
=
examples, such as the
=
z).
Two
^=1^,
S C
and H(z)
Si
—s
l
Sj for
=
respectively, a
CES
function and a Cobb-Douglas function. 8
Definition
Gorman
A =
r
K
(Differentiability
1
1
(1968)).
=
1,
and Aggregation)
and
/
>
separable
is
3,
it
i
and
j
is
D
D
In the first case /ii(s s )
exp(z) (again with
Si
>
(for
0).
=
s
Q^sf (with
—
requirement
this
X^?=i
—
s j> satisfy
(ajSj 4-
each j) and
.
=
when g
and only
if
is
is
>
0)
=
(one-
not very
the definition
+
0,
oms'n ) 1 ^
which
,
are,
z
l/a
.
e.g.,
twice continuously differentiable,
the "marginal rate of transformation"
hij(si, Sj) for all s
and H(z)
is
strictly increasing (see,
6 S
In the second hi(s
is,
(2)
g(s)
Si
..
>
independent of the other players' actions; that
,g(s)
S3
if
1
In the case of one-dimensional strategy sets,
can be easily established that when g
It
between any two players
8
>
the standard definition of separability
is
i),
other simple examples are g(s)
g(s)
Remark
some
of strategies g(s)
R^_ + where ay
—
we can change the order on
Naturally, since
S C R N and
,
N
increasing in s ensures that both g and
is
instead of
sum
but throughout there are no restrictions
none of our results require
particular, except Corollary 3 in Section 8,
dimensional strategy
For most of the
t
)
=
a;log(s,) and H(z)
=
.
where h%j
when
it
g
Si
:
—
x Sj
>
ii is
each player
and only
Sj,
g
is still
implied provided that g
is
convenient
way
fi{si,g{s)) for all s
an aggregator g
is
for each player
5.
Game)
S — K^ and
7r,(s,, s_,-, t)
\\
r
The game Y =
:
Si x
R KK
}
7Ti
positive matrix for
Si,
Here
Sj
s
V
s t (s t
A
s
Sj
x
may be verified
R K — R N such
>
that
that
(3)
is
the case in Definition
is
1,
an alternative and often very
an aggregator.
5.,,T),gi
is
aggregative
there exists
if
T
)
must be a
=
ni (si,s- it t)
is
j
^
games with
((11*,
strategic substitutes.
Sj
and s-
z
if
Si) te j,g, T).
We
9
>
for all s\
say that the payoff
Sj,
the "difference"
a decreasing function in each of the coordinates of s_j G
The
i.
7r,(sj,
When
if
and only
payoff function m(si,s-i,t)
s_;,
t)
+ 7Ti(si, s_j, t)
V
sr
is
if
D^. s 7Tj(si,s_i,
super-modular in
and
s,
A
Sj
t) is
s,
5_
for all s z Sj
,
if 7r 2
e Si (and s_ G 5 -,,
t
of the vectors
are contained in
S whenever
t
s,
t
C
a non-
1
maximum (minimum)
denotes the coordinatewise
lattice.
(4)
summarized by the tuple
fully
Naturally, this definition requires that s t
i.e.,
(tt,,
twice differentiable, this will hold
all
?
is
will use is
+ 7Ti(sj A Sj, s-i, t) >
s-i,t)
x
exhibits decreasing differences in
7Tj(sj,s_i,t)
h en
t
generally,
such that
i
Another key concept we
fl£iv(j-i)_
S
:
More
a reduced payoff function
>
:
an aggregative game
-
3, it
i,j.
S.
Clearly, equation (3) also gives
Yli{si,g{s),t)
t)
fi
^
also twice continuously differentiable. This observation will
IL
Clearly,
e
of verifying that a strictly increasing function g
Definition 2 (Aggregative
s_j,
>
increasing (and not necessarily strictly increasing), as
play an important role in Section
7Ti(s^,
with q
sq
6l:
i
When
function
but not of any
there exist increasing functions
if
D s,g(s) =
(3)
and
Sj
twice continuously differentiable, strictly increasing, and /
is
satisfies Definition 3 if
for
a function of
t
(s 2
G T).
and
Sj, s,
V
st
.
£5,.
strategy sets are one-dimensional, supermodularity as well as the
lattice-structure of strategy sets are automatically satisfied, so only decreasing differences remains
to be checked. For multi-dimensional strategy sets, supermodularity holds for twice differentiable
payoff functions
if
and only
if
D^n s miTi(si,
S-i, t)
>
nth and mth components of of the strategy vector
9
When
m
s r of player
^n
(where s" and sj denote the
1
i).
feature strategic substitutes, we will impose additional conditions, and in particular,
on "nice games" that satisfy certain concavity and differentiability conditions. Since we do not make
use of these until Section 5, we do not introduce this concept until then.
we
a
will focus
game does not
for all
Definition 3 (Strategic Substitutes) The game F
and
Si
is
and
exhibits decreasing differences in s,
Equivalently,
we
will also
say that a
game has
Notice that decreasing differences usually
when
the aggregator g
is
is
game
a
with strategic sub-
supermodular
TTi(si,S-i,t) is
s_,.
A game
(or features) strategic substitutes.
both aggregative and has strategic substitutes,
fact,
Sj)iex
(7iv,
and each player's payoff Junction
stitutes if strategy sets are lattices
in
=
is
an aggregative game with strategic substitutes.
straightforward to verify in aggregative games.
is
+ D^Ri <
opponent
j, it
Remark
2
take
0.
must hold
(as in
Then
for all
=
Q=
1,
g(s)
define
= ^q=i
s ^ ar>d
.
.
.
,s*j(t)) is
an equilibrium
Examples
In this section,
case,
there
is
several
we show how they can be
s sufficiently
i.e.,
s
Si
D^
s
tt{
s
I1j(sj,
J2 q =i
Unless players
differences in
Si
and
this section,
Q
and the aggregate Q"). For example,
s q)
=
D^^-iisi, Q)
+
differentiable.
^22^»( s *i Q) -
Df 2 ^i{si,Q) <
®
requires that
.D^IMs^Q) <
0,
then our strategic substitutes condition
l
s,
game with
in a
0.
Clearly
strategic substitutes.
standard fashion.
,Si) ie i,g,T)
be
an aggregative game.
argmaxn
i
(s !
i
Then
s*(t)
=
6 X,
,g(s J ,sl l ),t).
Games
games that have been widely studied
cast as aggregative games. For now,
we
in the literature. In
focus on
each
games with one-
dimensional aggregates, returning to the those with multi-dimensional aggregates in Section
Throughout
=
Q
in the
£
and Q)
assume that payoff functions are twice
D^
of Aggregative
we present
*
no tight relationship between
a (pure-strategy Nash) equilibrium if for each player
s*{t)
3
i
sj
and
Definition 4 (Equilibrium) Let ((U
(si(£),
n
Y2 J= i
decreasing differences holds for some
9),
of as "strategic substitutes in
]T\=i s q- Decreasing differences in
we
if
and Decreasing Differences in
weaker and we can have D^2 ^U(si, Q) >
Finally,
^
5 -ii *)
7r i( s ii
=
opponents.
neither condition implies the other. If
is
—
J2j=i s j'
equivalent to nonpositive cross-partials,
is
strategic substitutes requires that
where
Sj,
Walrasian Nash equilibria in Section
may be thought
suppose that AT
(
and the condition that Yli(si,Q) exhibits decreasing
strategic substitutes
(the latter
11^
This immediately implies that
(Strategic Substitutes
Q as given
that
sets, so
smooth, then decreasing differences
Dj 2 n,'
In
a symmetric function there will only be one condition to check for
each player. For instance, consider an aggregative game with linear aggregator g(s)
and one-dimensional strategy
that
we drop dependence on the parameter vector
8
t <E
T
7.
to simplify notation.
,,
.
Cournot Competition
3.1
As already mentioned
in the Introduction,
Cournot games are aggregative games. Consider the
following generalized-Cournot game. There are / firms, each choosing their quantity
The market
price faced
by firm
m € X{ c K+.
i is
Pi
=
Q Xi
x
Pi{qi,Q)
where
:
is
an index of aggregate quantity.
of
its
arguments (hence
q
is
(qi)
x Xj .-> R+
(ft,
Q) =
Q
assume that
an aggregator). Firm
it is
n,
where
We
...
iTi
fe, q-i)
=
i's
is
separable and strictly increasing in each
profits are
Pi (qi,
Q)
an increasing function representing firm
q%
i's
-
c (q r )
?;
,
that this
It is clear
costs.
is
an
aggregative game.
The most
Q—
Y^i=j
familiar case
anc^ Piilii
qi
function of Q, this
for q\
>
qi, it is
is
among
In
game with
-
n
= P{Q + q[-
P (Q)
- q
(<?0
add a technology choice to
this
model. This extension
j^
also straightforward to
-
<?,
+
c,
(q t
)
i.
is
interesting,
other things, for the analysis of technological investments in oligopolistic industries.
this case,
is
is Cj (qi, a,i)
for
some measure
In
of technological
a cost of technological investment given by C,
(a,).
In
we have
U
l
(q l ,a l
,Q)
=m
(q l ,q- l ,a l ,a- i )
= P
l
(q
return to an analysis of this model in Section
l
,Q)q -
Ci(qi,a,i)
l
-
C, (a t )
6.
Bertrand Competition
Similarly,
models of Bertrand competition with differentiated products typically correspond to
aggregative games. In such models, firms choose their prices, p £
z
is
a nonincreasing and concave
qi) q[
[qu q-i)
investment a t € A,. In addition, there
3.2
is
strategic substitutes (as defined in Definition 3). Indeed,
particular, suppose that the cost function of each firm
We
P
this case, provided that
straightforward to verify that
decreasing in qj for each j
It is
that of Cournot competition with homogeneous products, where
= P (Q)-
Q)
also a
m {& q-i)
is
is
able to
sell
quantity q x given by
qi
=
Qi (Pi,P)
A',
C R+. Suppose
that firm
i
.
where
P Xi
x ..... x
:
is
i
P
a price index (and again
profits are given
is
K+
->
separable and strictly increasing in each of
its
arguments). Firm
by
7i"t
Clearly, this
is
Xi
{Pi,
P-i)
=n
t
(pi,
= piQi
P)
demand
a
(Qi
P))
(pi,
Perhaps the most commonly studied case
again an aggregative game.
constant elasticity of
P) -
(pi,
is
the
model of monopolistic competition, where
(Dixit-Stiglitz)
cr-l
1
/
\
P
and a corresponds to the
sumers'
elasticity of substitution
a
= mp~ P
while Qi(pi,P)
utility function,
between products
a~
l
,
The reduced
spending of the representative consumer.
with
m
>
in the representative con-
corresponds to the total
0,
payoff function of firm
i
can then be
written as
IT fa, P)
= mpl-op*- 1
Cl
(mp-°P°-
1
)
.
Patent Races
3.3
The
how
following model of patent races due to Loury (1979) highlights
various different models of
innovation can be cast as aggregative games. Suppose that / firms are competing for an innovation
and the value
of the innovation to firm
firm chooses a constant
R&D
i
at the rate
R
>
0.
It
The
first
3.4
0.
R&D. The
The model
cost of
by
is
Vi-h
{Si)
•
i's
R&D
straightforward to verify that this
is
continuous time, and each
Firm
i
generates a flow rate
is c; (s,).
Both h and
x
Ci
are
All firms discount the future
expected profits can be written as
-
exp
in
s,.
firm to innovate receives the prize.
can be shown that firm
/
It is
l
intensity at every date, denoted
of innovation of hi (sj) as a result of this
increasing functions.
V >
is
Ir + ]T h,{ Sj It) dt)
Cl (s
an aggregative game with the aggregator given by
Models of Contests and Fighting
Loury's model, of patent races described in the previous subsection
of contests
and
fighting, for
is
a special case of models
example, as developed by Tullock (1980) and Dixit (1987).
10
These
models often
economy models
arise in the context of political
or in the analysis of conflict
and
wars.
Suppose that
guns or armies
/ players exert effort or invest in their
in order to
or fight. Again denoting the strategy (corresponding to effort or guns) by
Si,
player
win a contest
i's
payoff can
be written as
TTi
(Si,S-i)
=
Vi
-
r-
-.
C t [Si)
,
* + ff(£j=iM*j))
where again
R>
and
0.
c,.
:
— R+
5,
>
is
The aggregator
the cost function, hi,,.,, hi, and
in this case
=H
and reduced payoff functions can be written
=
ITi
l^h^)
as
=
S_j)
(Si,
—
Vi
it
It is
are strictly increasing functions,
is
g(s)
^(Si, Q)
H
model
also useful to note that neither the
-
'
+ (4
Ci(si).
of the patent race discussed in the previous
subsection nor the more general contest models discussed here are games of strategic substitutes
(or
complements). For example, suppose that
that
Q—
strategies
X^7=i
Then,
s j-
it
h's
can be verified that player
by increasing the strategy
(strategic
decrease her strategy (strategic substitutes)
H
and
are given by the identity function, so
will
i
respond to an increase
complements)
if S{
if S{
< R+ J2j^i s
> R+
in
opponents'
5Z jjSj, while she will
7
j'
Private Provision of Public Goods
3.5
Consider the public good provision model of Bergstrom
making a voluntary contribution
et al. (1986).
There are / individuals, each
to the provision of a unique public good. Individual
i
maximizes
her utility function
U,
subject to the budget constraint
and
sz
is
agent
i's
Cj
I
+ ps =
t
Ci^Sj
+
s
m
m,. Here
t
>
is
contribution to the public good, so that 5Z,=i
good provided. The exogenous variable
level of the public
good that
Substituting for
c,,
it
is
will
s
>
income,
sj
+s
is
total
private consumption.
amount
of the public
can be thought of as the baseline (pre-existing)
be supplied without any contributions.
easily seen that this
an aggregative game with reduced payoff
is
function given by
IT
c,
Is^^s^m^p,^] =u,
11
m
t
-ps u ^2sj 4
The
Section
strategic substitutes are discussed in
6.
Models with Externalities
3.6
Many models
in
macroeconomics, environmental economics, search theory, team theory, and other
areas feature externalities from
problems the payoff of player
these
Si
game with
conditions under which this will also be a
games
and g
(s)
are aggregative.
i
some average
can be written as
(s u
g (s)) for
increases the knowledge available to
all
also
effort.
10
Most
generally, in these
some aggregator g
supermodular
generally not the case.
is
corresponds to research
st
7Tj
While some such games are
are "complements"), this
a situation in which
of the actions of others.
Naturally,
(s).
example, when
(for
For example, we can think of
Over a certain range, research by others
researchers, so that strategies are complements.
However,
over a certain other range, various researchers might be competing against each other, so that
strategies
become
substitutes. It
is
also possible that there
such that some of them are "imitators,"
are "innovators,"
who
We
fined in Section 2).
N
is
Games with
from aggregate research, g
harmed by an
(s),
while others
increase in g(s).
Strategic Substitutes
present comparative static results for aggregative games with strategic substitutes (de-
first
section
benefit
are, directly or indirectly,
Aggregative
4
who
might be different types of researchers,
we assume
11
Recall that strategy sets are not one-dimensional, but throughout this
that the aggregate
is
one-dimensional
(i.e.,
in
terms of Definition
1,
K=
arbitrary). In addition, recall also from Section 2 that throughout payoff functions are
semi-continuous and strategy sets are compact
are imposed.
in such
The main
result of this section
is
1
(Existence) Let T
Nash equilibrium
(i.e.,
it
be
and
upper
— but no quasi-concavity or convexity conditions
that "regular" comparative statics can be obtained
games under no additional assumptions.
Theorem
1
We
begin by noting the following:
an aggregative game with strategic substitutes. Then T has a
has at least one pure-strategy Nash equilibrium).
Proof. See Jensen (2007).
10
See, for example.
John (1988)
for
Diamond
(1982) for an early example in the context of search externalities and
Cooper and
a discussion of a range of macroeconomic models with such externalities.
u Note
that instead of Definition 3 (superrnodularity and decreasing differences), we could equivalently work with
quasi-supermodularity and the duai single-crossing property of Milgrom and Shannon (1994). In fact our results
will be valid under any set of assumptions that ensure that best-response correspondences are decreasing in the
strong set order
(e.g.,
Topkis (1998)). Quasi-supermodularity and dual single-crossing are ordinal conditions and
In particular, a payoff
satisfies the dual single-crossing property in s, and S-i if for all s^ > s, and s'_, < s_,, (i)
so hold independently of any strictly increasing transformations of the payoff functions.
function 7ri(s,,S-,,
7ri(s'i,s- t
)
t)
> m(si,S-i)
=*• 7r,(s^, s'_,)
>
7rj(si,s'_i),
and
(ii)
12
m(s'i,s-i)
>
ir,( J i. s
-i)
=*• 7r t(Si,s_
1
)
>
Wi(si,sLi).
Pure-strategy equilibria are not necessarily unique. In general there will be a (compact) set
C S
E(t)
of equilibria for each parameter
When
e T.
t
we
there are multiple equilibria,
on the equilibria with the smallest and largest aggregates. The smallest and
focus
largest equilibrium
aggregates are defined as
QM) =
min
g(s).
(5)
seE(t)
Q*{i)
= max
and
g{s),
(6)
se£(()
The
following theorem establishes certain important properties of the smallest and largest
aggregates, which will be used in the subsequent results.
Theorem
defined
T — K
»
when
2
(i.e.,
is
there
(Smallest and Largest Aggregates) For
a unique equilibrium aggregate for
Proof. See Section
Our
rameter
is
first
all
t,
Q*{t)
—
Q
r
:
upper semi-continuous, and thus
is
>
Furthermore the function
Q*(t)
is
continuous on T.
substantive result, presented next, addresses the situation where an exogenous pa-
teTCK
"hits the aggregator,"
Theorems 4 and
aggregate)
:
T — R
£ T, Q*(t) and Q*(t) are well
11.1.
both of substantive interest and
(in
t
smallest and largest equilibrium aggregates exist).
lower semi- continuous, the function Q*
is
all
5).
when we can
More
is
formally,
t
+ £j=1
.sj)
we
3.1),
parameter
refer to
(s)
,
=
7Ti(.s, t)
The
t))
=
ni(s,t) all
t
g.
This result
i,
as a shock to the aggregator (or
where g
simplest case would be
with G(g
amples of shocks to the aggregator include a
model (Section
only affects the function
:
continuous, increasing, and separable in s and
the relevant definition of separability).
so that n, (si,
it
also enables us to prove the subsequent characterization results
write Hi(.Si,G(g
aggregator, and G(g(s),t)
meaning that
(s) ,t)
shift in the inverse
a change in the discount factor
in the baseline provision level of the public
good
R
s in
in a
»
t
designates the
(see Definition
when the aggregator
= + £J =1 8j
t
S — R
and g
demand
(s)
is
= EJ=i
1
for
linear,
*j-
Ex-
function in the Cournot
patent race (Section 3.3), or a change
the public good provision model (Section
3.5).
Notice that
when
t
is
a shock to the aggregator and
t
is
increased, the marginal payoff of
each player decreases (provided that marginal payoffs are defined). 12 Hence we would intuitively
expect an increasing shock to the aggregator to lead to a decrease in the aggregate.
theorem shows that
12
G
is
By
this
is
indeed the case.
strategic substitutes, agent
increasing in s and
t,
The next
i's
an increase
marginal payoff must be decreasing in opponents' strategies and hence, since
in t must lead to a decrease in marginal payoff.
13
Theorem
3 (Shocks to the Aggregator) Consider a shock
game
in an aggregative
and
the smallest
decreasing in
T C RM
i.e.,
in
t
to the aggregator
leads to a decrease in
and Q*(t) are
the functions Q*(t)
(globally)
t.
Proof. See Section
Though
G
Then an increase
with strategic substitutes.
largest equilibrium aggregates,
t
11.2.
Theorem
the result in
3
we have already seen
intuitive,
is
Example
in
1 in
the Intro-
duction that such results need not hold in simple games, even in simple aggregative games. In
Section 6.3
we present an example
of a
game with
strategic substitutes
where a shock leads to a
counter-intuitive equilibrium change in the aggregate.
The proof
of the theorem exploits the constructive proof of existence of
ably generalized to
fit
Novshek (1985)
(suit-
the present framework). This approach provides an explicit description of
the largest (and smallest) equilibrium aggregate, allowing us to determine the direction of any
We
change resulting from a shock to the aggregate.
parative statics results
is
that
it
should also add that this approach to com-
to the best of our knowledge, new.
is,
A
major advantage of
approach
this
provides global results that are valid independently of any differentiability and convexity
assumptions.
Theorem
3 also allows us to derive a general result on the effect of "entry",
comparison of equilibria when an extra player
when
the original
before entry
game has
(e.g.,
I players,
when Sj+\ =
is
[0, s],
is
i.e.,
enables a
added to the game. The entrant, player
minS/ +
(by definition) assigned the "inaction" strategy
inaction corresponds to "zero", sj + \
=
0;
7+1
i
for instance,
zero production or zero contribution to the provision of a public good). Define the aggregator as
g(s)
=
g(si,
.
.
.
,sj, sj + i).
Then we have a
entry; before entry there are I players
player aggregative
game
increasing or constant".
well-defined aggregative
and sj+i
in the usual sense.
just a constant, after entry this
Thus the entrant may choose
In an aggregative
before
As elsewhere, here increasing means
model, say) and thus the equilibrium after entry
Theorem 4 (Entry)
is
game both
game
and
is
an I
+
1
"either strictly
"inaction" (zero production in the
may remain
after
Cournot
the same. 13
with strategic substitutes, entry of an additional
player will lead to a decrease in the smallest and largest aggregates of the existing players in
equilibrium (and a strict decrease
if
the aggregator g
is strictly
increasing and the entrant does
not choose inaction after entry).
Proof. This
result follows
from Theorem 3 by observing that the entry of an additional player
corresponds to an increasing shock to the aggregate of the existing players.
13
To strengthen the
results to "strictly increasing," one could
14
In particular,
impose additional boundary conditions.
let
.
g(s\,
.
,
.
.
7+1
sj,si+i) be the aggregator where
have g(s\,
.
.
sj,
,
s/+i)
=
H(g(s\,
.
.
.
,
sj), s/+i)
is
the entrant. Since g
is
separable,
we
necessarily
H and g satisfy the above requirements for
where
a shock to the aggregate (see, for example, Vind and Grodal (2003)).
Note that Theorem 4 only shows that the aggregates of the existing players decrease. 14
It is
intuitive to expect that the aggregate inclusive of the entrant should increase. Nevertheless, this
is
Remark
not true without imposing further assumptions (see
detailed description of
that,
when entry
9 in the proof of
Theorem 3
for a
next section, we will show
will decrease the aggregate). In the
under additional assumptions, entry can also be shown to increase the overall aggregate
Theorem
8).
The next theorem
presents our most powerful results for
games with
strategic substitutes.
These can be viewed as the strategic substitutes counterparts of the monotonicity
well-known
for
supermodular games. One
the results apply only
player,
when shocks
difference, however,
are idiosyncratic,
£ I. More formally, a change in U
i
(see
is
that with strategic substitutes,
to shocks
i.e.,
t{
that affect only a single
an idiosyncratic shock to player
is
results that are
i
if
payoff functions
can be written as
n(s,ti)
=
Ui(si,g(s),ti),
7Tj(s,ij)
=
Uj(sj,g(s)) for
and
all
j
jt
i.
Let us also introduce the notion of a positive shock.
Definition 5 (Positive Shock) Consider the payoff function
crease in
It is
s'i
>
is
Si,
increasing in
if
only
t
ir
>
for all s
and
t.
if
player
Moreover, as
t.
will exhibit increasing differences in s,
Dg
Then an
ni(si,s-i,ti).
Sj
and
in-
t.
straightforward to verify that Definition 5 gives the correct notion of "positive shock";
Ki exhibits increasing differences
for
=
a positive shock ifni exhibits increasing differences in
is
ti
in
The
and
t
i's
is
if
"marginal payoff',
well
known, when
and only
may
the previous definition without changing any of our results.
equilibrium strategies for player
Theorem
5
i
ti
s_i,
t)
-
sufficiently
is
7r,;(s,,
s_j,
smooth,
nonnegative,
t)
it
i.e.,
replace increasing differences in
We
also define smallest
and largest
analogously to the smallest and largest equilibrium aggregates.
(Idiosyncratic Shocks) Let
an increase in
is
the cross-partial
if
single-crossing property
-n^
7r;(s£,
ti
be a positive idiosyncratic shock to player
leads to an increase in the smallest
14
and
i.
Then
largest equilibrium strategies for player
This is the reason why we do not explicitly write Q'(t) and Q»(t). Instead, we could have defined Q*(t) =
max( si
s;) and Q,(t) = min (si
,si), and the statement would be for Q'{t)
S( 6 b(() j(si
si)eB(t) 9{si,
and Q.(t). But this additional notation is not necessary for the statement or the proof of the theorem.
)
15
i,
and
to a decrease in the associated aggregates of the
are, respectively,
and smallest such aggregates).
the largest
Proof. See Section
A
remaining players (which
11.3.
simple corollary to
Corollary
Theorem
5 also characterizes the effects of a positive
(Payoff Effects) Assume
1
in addition to the conditions of
shock on payoffs.
Theorem 5
that all payoff
functions are decreasing [respectively, increasing] in opponents strategies and that player
'
function
ti
increasing [respectively, decreasing] in the idiosyncratic shock
is
decreases] player
increases [respectively,
i 's
payoff
Then an increase
tj.
in
payoff in equilibrium and decrease [respectively,
i 's
increases] the payoff of at least one other player.
Proof. For player
some player
TT
5
{s'^g{s"))
i,
j (for j
< Tr^,
7Ti(s^, g(s'), t')
^
i)
decreases,
We now
and concave
1 in
The
(or
As
condition.
=
ir^gW).
we
^2k-tj hk{s'k )
pseudo-concave) in
own
in the previous section,
terms of Definition
Hfc^i
strategies.
we
focus on
15
We
also
Consequently,
^fc( s l-)-
impose an additional
games where the aggregator
if the
aggregator g
as part of this definition,
Games) An
is
own
strategies.
boundary of the strategy
i
is
when
is
aggregative
game T
is
said to be a nice aggrega-
all
D
St
(v
is
compact and
twice continuously differentiable, and pseudo-concave in
Si
and
Although
formally.
(see Definition 8).
TD
set Si)
games
strategy sets are one-dimensional
Furthermore, we have that whenever
the first- order conditions
for player
local solvability
real valued (i.e.,
twice continuously differentiable, each strategy set
convex, and every payoff function ni
the player's
is
differentiable
1).
following definition introduces the notion of "nice" aggregative
game
is,
<
Since the strategy of
games where payoff functions are
can be replaced by a weaker "regularity" condition
That
7Ti(s",g(s"),t").
m
focus on "nice"
Definition 6 (Nice Aggregative
tive
we must have
<
Games
we include a boundary condition
this
ni(s'i ,g(s'i ,s'[_ i ,t")
extend the framework of the previous section to aggregative games without strategic
substitutes. For this purpose,
K
<
g{s';,s'_ ))
3
Nice Aggregative
5
<
—
Si)
ni(s, t)
=
Sl
iXi(s, t)
<
for
£ dS% (with dSz denoting the
v £ Si, then
D
Si
n
z
(s, t)
=
0.
are required to hold whenever a boundary strategy
a (local) best response.
15
Weinstein and Yildiz (2008) use a similar definition of a "nice game," except that they also impose onedimensional strategies and concavity (rather than pseudo-concavity).
HI
Remark
3 (Pseudo- concavity) Recall that a differentiable function
gasarian (1965)) in
{s'i
-
Sl
)
TD
St
TT l
Naturally, any concave function
D
conditions
Si
n
l
£ Sl
S* if for all Si, s't
(s,t)
=
<
=>
maximum
is
is
will
if
TV
7Tj
=
and
1
Remark
4 (Inada-Type
maximum
D
Si
Instead,
it
first-order
Pseudo-concavity
iTi(s, t)
=
=>
maximum
D^
s
is
not a
.7Tj(s, t)
<
0,
(and in fact, that
strict quasi-concavity)
does not imply
in general.
Boundary Conditions) Note
Definition 6 does not rule out best responses on the
That
given s_i and L
in the proofs.
be unique). Quasi-concavity (or even
the sufficiency of first-order conditions for a
is
7rj(Sj,S_i,i).
easy to see that the first-order condition will be sufficient for a
maximum
the
maximize
what we use
necessary condition for this to hold. For example,
it
<
7Tj(s-,S_j,t)
pseudo-concave. Pseudo-concavity implies that the first-order
are sufficient for s t to
conditions are sufficient for a
pseudo-concave (Man-
:
{s i .S- l ,t)
is
iTi is
also that the
boundary of a
boundary condition
in
player's strategy set, dSi-
simply requires first-order conditions to be satisfied whenever a local best response
on the boundary. Consequently,
this
boundary condition
type" conditions ensuring that best responses always
best responses never
lie
on the boundary,
lie
is
weaker than the standard "Inada-
in the interior of strategy sets (since
when
first-order conditions vacuously hold for best responses
on the boundary). 16
As
is
well-known, the concavity or pseudo-concavity conditions ensure that best response
correspondences are convex valued (they are also upper hemi-continuous as mentioned at the
beginning of Section
2).
The
Nash equilibrium
existence of a pure-strategy
Kakutani's fixed point theorem.
17
therefore follows
by
For the record, we note this result in the following theorem
(proof omitted).
Theorem
strategy)
16
TV
1
of this
(Existence) Suppose
Nash
However,
>
6
all
that
T
is
a nice aggregative game.
Then T has a (pure-
equilibrium.
boundary conditions cannot be dispensed with. To
see this, consider an iV-dimensional
(with each player having TV-dimensional strategy sets) without any interior best responses.
A -dimensional game
r
can then be
mapped
bijectively into
an
N-
game,
The boundary
1-dimensional game. But since first-order
conditions never have to hold in the TV-dimensional game, the local solvability condition below (Definition 7) would
never have to hold.
In effect, the A'
— 1-dimensional
"reduction"
is
therefore unrestricted and consequently,
no
general results can be derived for such a game.
17
Without convex best response correspondences, a Nash equilibrium may
(unless the
of
game
fail
to exist in an aggregative
also features strategic substitutes or complements). See Jensen (2007).
an aggregative game where a pure-strategy Nash equilibrium
dimensional, convex, and there are only two players.
17
fails
to exist even
Example
5 for
though strategy
game
an example
sets are one-
A
game does not
nice aggregative
solvability condition introduced
necessarily have a unique equilibrium. Similarly, the local
possible multiplicity of equilibria as in the previous section
and
and
and
still
are, respectively,
t.
next introduce the local solvability condition, which will play a central role in our anal-
ysis in this section.
DoU-iisi, Q,
t)
=
DQUi(si, Q,
Using the fact that g
t).
D^is^t) = D n
1
fi{si,g{s))
Equation
(7)
=
Let us simplify notation by defining D\Yli(si,Q,t)
gator, the marginal payoff for player
Where
and study the behavior of the smallest
largest equilibrium aggregates continue to exist
lower and upper semi-continuous, in
We
therefore deal with the
and Q*(t). Theorem 2 from the previous section
largest equilibrium aggregates, Q,(t)
applies and the smallest
We
below does not guarantee uniqueness.
= D Si g(s)
(cf.
i
a twice continuously differentiate aggre-
is
can be expressed
i
(s ll g(s),t)
Remark
+ D2 n
and
JD Si IL,(sj,Q, i)
as:
i
(s l ,g{s),t)fl {s l ,g{s)).
(7)
1).
shows that the marginal payoff
a function of the player's
is
the aggregate g{s). Let us also define a function ^,
:
SixRxT — R N
>
own
strategy
S{
and
that makes this dependence
explicit:
9i(8i, Q,
Note that
=
t)
DiUiisi, Q,
this function contains the
separate the direct and indirect effects of
Q
same information
st
(the
first
,
Q).
(8)
though
as (7),
also enables us to
it
one corresponding to a change
in
^ holding
constant). Naturally,
<&i(s h
Differentiating
$! l
\D Si ^i(si, Q, 0|
We now
Q)
=
with respect to
determinant of this matrix
is
0^
s,
[D Si TTi{s,
x
and g(s)
N
matrix
If
=
D
St
Q].
^i(s l ,Q,t) e
R NxN
.
The
strategy sets are one-dimensional,
= DMsi,Q,t)eR.
introduce a key concept for the study of comparative statics of nice games.
*
t
(s,,
i
I
G
Q,t)=0
is
\Pj
=
0.
The term
refers to the fact that
would not be a unique solution to a
the subspace where $,
=
0),
said to satisfy the local solvability condition
(for Sl
local solvability condition requires that the
space where
=
denoted by \D Si $>i(si,Q,t)\ G E.
\D3 .^i(si, Q,t)\^0 whenever
The
N
yields an
Definition 7 (Local Solvability) Player
if
+ D 2 Ik{si,Q, t)fi{ Si
t)
G
Sit Q G
determinant of
if
this
set of equations of the
where a and
b are
18
{g(s)
D
St
s
:
^
G S}).
is
nonzero on the sub-
determinant were equal to zero, there
form
Z? Si 5' l
•
TV-dimensional vectors.
a
=
b
(again defined
on
This type of equation
.
when constructing
arises
the backward reply correspondences for the purposes of comparative
This discussion also motivates the term "local solvability condition"
static analysis.
We may now
boundary condition that can
also state a weaker
N=
when
replace,
the one
1,
in Definition 6.
N
Definition 8 (Regularity) For
Q
not exist a
X
e
D ^
such that
Si
=
(s l
l
Q,
,
we say
1
D
Given the relationship between ^i and
can be viewed as "perverse".
The
>
t)
that ^i
for
e
all s,
in (8),
Si iXi
is
D
Si
regular
{s,
^i
is
straightforward to check.
Remark
5
(Ordinality)
holds for $(7r,(s,
t))
l
>
(s l
,
Q,
where $
:
VE',
=
for
any
is
ordinal:
M— K
>
any
is
is
is
strictly increasing transforif
and only
and twice continuously
strictly increasing
it
differ-
needed here to ensure
is
also twice continuously differentiable). In particular, for
and Q' we have that
,
=
& $'(n,-(
solvability
condition
$i(4Q',
Ordinality
\D s ,[&{Il
i
the
of
{s'
i
,Q
|^[$'(II i (a'f)
Q
,
local
,t))V i {s'i ,Q',t)]\
/
t)
+
1
where
ipi
:
,
follows
This
0.
Q, t))Vi{ Si Q,
is
if
true
t)
=
0.
|D Bi ^(Sj,
when
since,
^
Q')\
implies
=
*,(^,Q',i)
0,
t))*i(5i,Q ,t)]|==,*'(IIiW,g',t))l^ i *i(s{ Q')|.
)
we can
particular,
Si)
,
Secondly, the local solvability condition
is
also independent of the coordinate system.
replace each strategy vector
RN — R
is
>
a diffeomorphism for each
1
K i( 1P^ {si)i4'Zl{s-i)it), where ipz}
=
s,
i.
any such (smooth) change of
by transformed variable vector
The
1
{'<l'7
It
)j^i-
solvability condition in s (in the original variables)
for
0}.
Firstly, local solvability holds for the payoff function 7Tj(s, t)
that the transformed payoff function
'
=
on the subspace where
entiable function, with derivative denoted by $' (where differentiability
all s-
t)
G T, there does
t
useful to note that local solvability condition
It also
independent of the choice of the coordinate system and holds
if it
*
for any
regularity conditions in Definition 8 rules out such perverse
cases and
mation of the aggregator.
:
if,
variables.
if
payoff function of player
can be verified that
and only
if
i
s%
the same condition
In particular, local solvability in the
4> l
In
{si)
then becomes
satisfies
7r,
=
,
is
the local
satisfied
new coordinate
system requires
Z^Since Dxp-'
1
1
^)*^- ^)^)
1
)^ (^'
l
(s l
l
{s l ),
Q)
=
=
=» \D Si \Dip;
1
<=>
*,•
(i>~
(.§,),
have that
19
Q)
=
l
(^ (vl
{Dip;
1
1
(s l )-Q)}\
^)
is
a
±
full
0.
rank matrix), we
This establishes the desired
That
local solvability
is
result.
independent of any
strictly increasing
transformation of the aggregator
we may use any aggregator
implies that instead of the aggregator g(s)
=
g(s)
f{g{s)) where
E —> R is a strictly increasing and differentiable function. To see this simply note that
Ui{si,g{s)) = Ui{siJ- {g{s))) where g{s) = f(g(s)). Denoting the new aggregate by Q = g(s),
Evidently then \D Si ^i(si,Q)\ = |Z? * (s Q)\ and the
it is clear that $i(si,Q) = $i(si,Q).
/
:
l
Sl
l
l
,
conclusion follows.
Remark
(Weaker Conditions) Some
6
Example
dispensed with.
statics
when the
1
version of the local solvability condition cannot be
shows the possibility of perverse comparative
in the Introduction
local solvability condition does not hold (see also Section 6.3).
Nevertheless,
the results presented in this section continue to hold under weaker conditions. In particular, the
following generalization would be sufficient, though the condition in Definition 7
and
The
verify.
neighborhoods
such that
for
alternative condition
J\fs
each
and A4q of
,
Q
Mq,
e
bl
Sj
{Q)
is
and Q,
is
respectively,
t)
=
0,
<
i' l (.Si
1
=
Q,t)
fy
in J\fSi
admits a local solution
easier to state
there should be open
and a continuous map
the unique solution to
that that the first-order condition $i(si,Q,
Q.
when &i[si,Q, t) =
as follows:
is
AAq —
»
:
J\fSi
This implies
.
a function of
in Si as
Naturally, in view of the implicit function theorem, this weaker condition follows from our
local solvability condition in Definition
discussed in Section
We
7.
Other alternatives to the local solvability condition are
8.
next introduce the appropriate notion of positive shocks for nice games. This generalizes
the notion of a positive shock introduced in Definition
can be written as g(s)
It is clear
H (Xw=i ^j( s j)
=
where
)
hi
:
that the term hi(si) fully "captures" agent
Because the aggregator
5.
Si —* M.
and
H
M.
:
—* M
separable
is
Definition
(cf.
it
1).
on the aggregate. Intuitively, our
i's effect
generalized definition of a positive shock requires that an increase in the exogenous variable leads
to an increase in the
term
hi(si)
and thus increases the aggregate given everyone
In comparison, our previous definition, Definition
(smallest
and
i
if the largest
spondence /if(Ri(s_,,
hi o r .(s-j,
t)
be the
shock to player
for
all
made
largest) strategy should increase with the
Definition 9 (Positive Shocks)
player
5,
i
if
and smallest
t))
A
are increasing in
t.
from
if hi
exogenous variable.
That
is,
or i (s-
l
,t)
consider
s_, G 5_j.
20
<
t!
of ^^(/^(s-i,
< h or i (sl
t
l
,t')
t
is
a positive shock to
"composite" best responds corre-
this player's
maximal and minimal elements
and only
the stronger requirement that a player's
change in the parameter vector
selections
else's strategies.
6
T
and
t))
C
R.
and h
x
let hi o r t (s-i, t)
Then
o r j(s-j,
t)
t
is
< hj
and
a positive
o r j(s_,;,
t')
Note that
if
a player
and exhibits increasing
i's
strategy set
is
and
differences in s;
a lattice, his payoff function
t,
then the shock
be
will
lattice or
Our
when
it
supermodular
But
positive.
result in this section characterizes the
first
t is
a
comparative statics of the aggregate and
Theorem
whether or not the shock
hits the aggregate,
and
it
also applies
3, this
whether or not the
hits one, two, or all of the players.
Theorem
7 (Aggregate
Comparative Statics) Consider
an increase in the smallest and
Q*(t) are (globally) increasing in
The
\l/j
Then a
largest equilibrium aggregates,
i.e.,
positive shock
t
E
T
leads
and
the functions Q*(t)
t.
remains valid without any boundary conditions on payoff functions when
result
is
game where each
a nice aggregative
player's payoff function satisfies the local solvability condition.
and
in general, a
supermodularity conditions).
results applies
to
si,
does not affect the player's payoff function (regardless of any
strengthens Theorem 3 from the previous section. In particular, in contrast to
shock
in
For example, a shock
positive shock need not satisfy these "supermodularity" conditions.
positive shock for a player
is
N=
1
regular.
Proof. See Section 11.4
Our next
Theorem 4
result extends
Theorem
4 in the previous section.
In particular,
to a statement for the overall aggregates (after the entry of the
new
instead of the aggregates of the strategies of existing players). Let us define
As
"inaction".
in the previous section, the
convention
is
that /
+
it
strengthens
additional player
£ Si to stand
for
1th player takes this action
before entry.
Theorem
in a
game
6S,
for
8
(Entry) Let Q*(I) and Q*{I) denote
with I £
all
i
N
G I.
the smallest
and
players that satisfies the conditions of Theorem 7 and where
Then for any
I
£ N, Q»(7) < Q,(I
+
1)
and Q*(I) < Q*{I
entry increases the smallest and largest aggregates in equilibrium.
g
is strictly
largest equilibrium aggregates
Moreover,
if the
S C R^\
t
+
1),
i.e.,
aggregator
increasing and the entrant chooses a non-zero strategy following entry, the above
inequalities are strict.
Proof. See Section
11.5.
Finally, our third result strengthens
Theorem
the comparative statics of individual strategies.
only) result
among
5
from the previous section and characterizes
It is
useful to note
Theorem
9
is
the
first
(and
those presented in this and the previous section that uses the implicit function
21
theorem. As such
is
it is
a purely
local result
and
also requires that the equilibrium strategy studied
interior.
Theorem
and consider player
rium aggregate
the interior of
•
Comparative Statics)
9 (Individual
at
i's
some
S and
equilibrium s*
that
is
t
=
s*(t) given
a positive shock.
Suppose that the shock
-[D Si '$>i(s*,g(s*))}~
t
be satisfied
t
1
£ T. Assume that the equilibrium s*
D
t
s*(t)
DQ'fyi(s*,g(s*)).
is
lies in
the following results hold.
provided that
*^**.^*).*) >
does not directly affect player
t
of each element of the vector
1
t
Then
s*(t) is (coordinatewise) locally increasing in
decreasing in
Theorem 7
equilibrium strategy s*(t) associated with the smallest (or largest) equilib-
-[zV^OCO-Or
•
Let the conditions of
i
(i.e.,
o
—
iTi
Then the sign
m(s)).
equal to the sign of each element of the vector
In particular, s*(t) will be (coordinatewise) locally
whenever:
Proof. By the implicit function theorem, we have
D s .%( Si
The
results follow
from
,
Q, t)d Si
=
-DQ$i{si, Q, t)dQ -
this observation
and the
fact that
D 2Sit tti(si, Q, t)dt.
Q
increases with
t
(where
Q
is
either
the smallest or largest equilibrium aggregate).
Applying the Theorems
6
In this section,
of the last
we return
to
some
of the examples presented in Section 3
two sections can be applied to obtain sharp comparative
private provision of public goods and contests, and then turn to a
an
oligopolistic setting,
which
also illustrates
how
and show how the
static results.
game
We
first
results
discuss
of technology choices in
these results can be applied
when
strategies are
multi-dimensional.
6.1
Private Provision of Public
In Bergstrom et
al.
utility function
was given
(1986)'s public
Goods
good provision model described
by:
22
in Section 3.5, agent i's
reduced
n,
As mentioned
is
Si,
=u
'^Sj,m,p,s
in Section 3.5, the aggregator
an equilibrium, we
refer to g{s*)
=
t
m,i
-
psi,
y^ Sj +
—
following condition holds for
all s
- pD\ 2 u
x
I
m
%
C
[0, s,]
The
R.
- ps^
\
is
(cf.
\] sj + s
^
Definition 3).
=
(s*)i € j
Let us
smooth and that strategy
is
private good will be normal
=i
1
+ D\ 2 Ui
- ps
lt
equal to Z?^
}] Sj +
;
i=i
V
/
is
m-i
s
11^.
The
s
if
and only
<
0.
sets
if
the
(9)
J
/
'
Therefore, whenever the private
normal, payoff functions exhibit decreasing differences and this
substitutes
s*
6 5:
Notice also that the left-hand side of (9)
good
When
^2j=i s j-
J2i=i s i as the aggregate equilibrium provision.
simplify the exposition and notation here by assuming that Ui
are intervals of the type Si
=
here simply g(s)
is
s
is
a game with strategic
following result therefore follows directly from the results in
Section 4 (proof omitted):
Proposition
Consider the public good provision game presented in Section 3.5 and assume that
1
the private good is normal.
1.
An
Then
there exists a (pure-strategy)
Nash
equilibrium. Furthermore:
increase in s leads to a decrease in the smallest and largest aggregate equilibrium provi-
sions.
2.
The entry of an additional agent
and
leads to a decrease in the smallest
largest aggregate
equilibrium provisions by existing agents.
3.
A
positive shock to agent
i
will lead to
an increase in that agent's smallest and largest equi-
librium provisions and to a decrease in the associated aggregate provisions of the remaining
I
—
I
players.
Proposition
results
is
1
holds under the assumption that the private good
from Section
(strictly)
4.
Alternatively,
normal. This
is
we can use the
illustrated next.
(which was not assumed for Proposition
only
1).
results
is
normal and exploits the
from Section
5
when the
Suppose that the payoff function
Then the
is
public good
pseudo-concave
public good will be (strictly) normal
if
and
if
2
Ds^iist^) = p Diiu (m -ps u Q) - pD 2 \u (m l
l
l
23
l
ps u Q)
<
(10)
6 5\ 18 Moreover, whenever
for all s
a positive shock,
i.e.,
D^. m Ili
>
an increase
(10) holds,
and
Z)^
II,-
<
in m,i (or a decrease in p) constitutes
respectively
0,
proposition then follows from Theorems 7-9 (proof omitted).
Proposition 2 Consider
the public good
is (strictly)
the public good provision
Any
Then
positive shock to one or
more income
game presented
Nash
there exists a (pure-strategy)
more of the agents
(e.g.,
The
next
and assume
in Section 3.5
own
normal, that payoff functions are pseudo-concave in
that strategy sets are convex.
1.
Definition 9).
(cf.
19
that
and
strategies
equilibrium. Furthermore:
a decrease in p, or increases in one or
mi,... ,mj) leads to an increase in the smallest and largest aggregate
levels,
equilibrium provisions.
The smallest and
2.
largest aggregate equilibrium provisions are increasing in the
number
of
agents.
The changes
3.
in 1
and 2 above are associated with an increase in the provision of agent
the private good is inferior for this agent,
private good
It is also
is
and with a decrease
normal and the shock does not
is
i 's
if
provision if the
directly affect the agent.
useful to note that Proposition 2 could be obtained even under weaker conditions
using Corollary 3 presented in Section 8 below. In particular,
good
in agent
i
can be verified that
it
if
by
the public
normal (condition (10) holding as weak inequality) and payoff functions are quasi-concave
(rather than pseudo-concave), the conditions of this corollary are satisfied
remains
valid.
We
used Theorems 7-9 here since Corollary 3
is
and Proposition 2
not introduced until Section
8.
Models of Contests and Fighting
6.2
Recall that the payoff function of a participant in a contest can be written as
TT,
(Sj,
S_,)
1—-
=V
fl
where
Sj
denotes agent
i's effort,
hi
:
K_|_
r-
j
x
Ci (Si)
(11)
,
+ ff(Ej=iM«i)
— R+
>
-
for
each
i
6
T and
H
:
M.+
—
>
M.+
.
As mentioned
in
Section 3.4, contests generally feature neither strategic substitutes nor complements. Therefore,
the results in Section 4 do no apply, nor do any of the well-known results on supermodular games
mentioned
18
The equivalence between
a{pDl 2 u,
1
in the Introduction. In this case, the
Note
(strict)
most obvious strategy
for deriving
comparative
normality of the public good and (10) follows since dsi(m,,p, 53, =« Sj)/dm
=
-p 2 £>iju,).
in particular that (strict)
normality implies local solvability as well as regularity so the statements
Proposition 2 are valid without any boundary conditions on payoff functions.
24
in
static results
does.
is
to use the implicit function theorem. This
is
indeed what most of the literature
Nevertheless, the implicit function theorem also runs into difficulties unless
we make
ad-
For this reason, previous treatments have restricted attention to
ditional, strong assumptions.
special cases of the above formulation. For example, Tullock (1980) studied two-player contests,
More
while Loury (1979) focused on symmetric contests with (ad hoc) stability conditions.
cently, Nti (1997)
races,
re-
provided comparative statics results in the (symmetric) Loury model of patent
assuming that
H=
id (the identity function), hi
Using the results of Section
5,
we can
—
h for
establish considerably
all
and concave, and linear
i
more general and robust
costs.
results on
contests and fighting games.
These
results are provided in the following proposition (and also generalize the existence result
of Szidarovszky
and Okuguchi (1997)).
Proposition 3 Consider
convex, hi and
Then
1.
Ci
the contest
are strictly increasing,
there exists a (pure-strategy)
The smallest and
(e.g.,
games introduced
Nash
and
in Section 3.4
R
equilibrium. Furthermore:
or an increase in
V
Entry of an additional player increases the aggregate equilibrium
3.
There exists a function n
R
-^>M such that the changes in parts
with an increase in the effort of player
Q* provided
dominant",
that
i.e.,
is
i
T
g
and
"dominant" in the sense that
h (s*)
%
i
<
any positive shock
for one or more players).
%
2.
:
is
that the following condition holds:
largest aggregate equilibrium efforts are increasing in
a decrease in
H
and suppose that
effort.
1
or 2 above are associated
the corresponding equilibrium aggregate
hi (s*)
>
n {Q*), then the changes in parts
provided that the shock does not affect this player directly
n (Q*). Conversely,
1
if
and 2 decrease player
(e.g.,
corresponding
H
not convex.
to a
i
"not
is
i 's
effort
decrease
in another player's costs).
Proof. See Section 11.6
This proposition can also be extended to the case
H
ensures that the first-order condition
D
Si ir l
(si,S- l )
in
=
which
is
is
sufficient for a
Convexity of
maximum, but
it
is
not necessary for this conclusion. Observe also that the conditions of Proposition 3 are satisfied
if
H
is
the identity function,
°The proof
of Proposition 3
q
is
convex, and
shows that the function
rj
/i t
in
25
is
concave."
part 3
is
given by
Szidarovszky and Okuguchi
(1997) prove that these conditions imply uniqueness of equilibrium provided that
Such uniqueness
is
not necessary or assumed in Proposition
covers important cases where hi
>
and studies the special case where
0),
this case,
h'[ (si) //iJ(sj)
=
So
ki-
H=
Technology Choice
As a
final application,
in
In addition, Proposition 3 also
3.
if,
=
ki
id (the identity function),
k for
all
and
hi(si)
logit
=
e
under additional assumptions.
i
in addition, costs are also exponential, c z {si)
conclusions of Proposition 3 continue to apply provided that
6.3
in (ll). 21
not concave. For example; Hirshleifer (1989) proposes the
is
specification of the contest success function, with
(ki
R=
ki
=
e
applied
statics
when
may
Oligopoly
we consider an important
class of
games
in
will also illustrate
how our
which oligopoly producers make
results with one-dimensional aggregates
such games and how
it
=
i
P
is
can be
"perverse" comparative
=
Q —
(qi,
...,qj)be the output vector
Yjjs=iQj as aggregate output.
is
IL
where
technology vector. Let us define
(ai,...,a/) the the
Profit of firm
how
3.1.
can be ruled out.
Consider a Cournot model with / heterogeneous firms. Let q
and a
the
,
<U-
strategy sets are multi-dimensional and also clarifies
arise in
In
liSl
technology choices (as well as setting output). These games were mentioned briefly in Section
Our treatment here
fc,s '
{qi, at,
Q)
=
tt, (<?,
=
a)
qt
P (Q) - a
the (decreasing) inverse market demand,
2H'(H-
m
n(Q
)
(R
+
1
(Q-))
c,
is
H"{H'
{qi,a t )
-
C
t
(a,)
the cost of firm
l
i
is
a function of
its
(Q~))
H'(H-HQ'))
Q<)
H
= h, = id (the identity function), and R = 0, we have n(Q") = Q" /2, and so
"dominant" if and only if s" > Q*/2. In the standard interpretation of a contest, this means that
she is dominant when her probability of winning the prize is greater than 1/2 i.e., when she is a favorite in the
terminology of Dixit (1987). However, this favorite-to-win interpretation does not necessarily apply for more general
games covered by Proposition 3. We therefore use the term "dominant" rather than "favorite".
21
More recently, Comes and Hartley (2005) have proposed a very nice and simply proof of this result based on
what they refer as "share functions". Although Cornes and Hartley do not consider comparative statics, their "share
function" approach could be used to establish results similar to the results in Proposition 3 under these stronger
assumptions if, in addition, one also imposed that R =
in (11). R =
amounts to assuming no discounting in
patent races and "no wastage" in contests, and is thus quite restrictive.
When R > 0, the "share function" approach cannot be used to derive robust comparative statics. The reason
for this is that the "share function" approach uses the fact that this function is decreasing everywhere, whereas
when R > 0, it may be increasing. To see this, apply the implicit function theorem to the condition $/(si,Q) =
imposing h {s,) = Si for all i. Rewrite this in terms of "shares", z, = Si/Q, so that [-V, — (R + Q) 2 c"]dzi =
[VRQ- 2 + (R + Qfc" + c[ Q~ 2 {2(R + Q)Q - (R + Q) 2 )]dQ. The coefficient of dz, is clearly negative. When
R = 0, the coefficient of dQ on the right-hand-side is unambigiously positive, hence dz,/dQ < 0, i.e., agent i's
share function is strictly decreasing. But in general, this may fail when R >
is allowed. In particular, the term
Q~ 2 {2{R + Q)Q - {R + Q) 2 ) will be positive if and only if Q > R. Clearly, nothing prevents c\ from being
c
sufficiently large for this term to dominate so that the share function becomes increasing when Q < R.
Therefore, when, for example,
player
i
is
—
t
'
'i
26
quantity and technology choices, and Ci
and Ci
Q), Ci
each
(for
is
i)
the cost of technology adoption.
is
P
are twice differentiable,
convex, and dot
/dq
{qi,a,i)
t
<
da,i
Assume that P, a
strictly decreasing decreasing (P'
is
each
(for
(Q)
<
for all
so that greater technology investments
z),
reduce the marginal cost of production for each firm.
The
maximization are
first-order necessary conditions for profit
—
dlXi
=
dci{qi,ai)
v'in\
Din\
P{Q)q
+ P(Q)
,
l
_
dju
dci(qi,a,i)
dCi(a,i)
da,i
dai
da,i
we
Naturally,
D\
;
_
also require the second-order conditions to be satisfied,
being negative semi-definite.
,7T 7
=0
oqi
dqt
of technology investment by one of the firms
a positive shock.
The
results
now
Let us
from Section
(i.e.,
which here amount to
consider the effect of a decline in the cost
a shift in
C
z
),
which clearly corresponds to
we should check the
5 suggest that
local solvability
condition. In particular, consider the matrix
D
V
(q,,o t
p
*-(
)^i-
{Q)
d2 c
dq da
\
each
i.
When
Cj (qi,a,i) is
l
® 2c}
_
_
Therefore, whenever each
d 2 Ci
t
9a
and d 2 Ci/daf <
<
negative semi-definite. Since P' (Q)
0.
daf
_ S 2 c,2
d 2 c,
dq da i
l
<
a2 C
~da[
dq da x
dq;
|-D\&i|
l
convex, the matrix
_
is
dl c
,
\
for
~&
-w
'
Cj (g,,
az )
is
0, this is sufficient to
guarantee that
convex, the local solvability condition
is
satisfied.
Hence, a decline in the cost of technology investments for one of the firms will necessarily increase
total output.
Similarly, the effects of
an increase
in
demand on output and technology
choices
can be determined robustly.
Now, the oligopoly-technology game
that
d 2 Ci(q
l
,a,i)/dq l dai
<
0.
22
will
be a game with strategic substitutes provided merely
So when technological development lowers the marginal cost of
producing more input, which of course
is
perfectly reasonable to assume, the results from Section
4 apply whether or not the local solvability condition
functions are pseudo-concave in
any positive shock
strategies (q*,a*)
22
to
own
satisfied
strategies, etc.). In particular,
an individual player
and a decrease
is
This condition ensures that payoff functions are supermodular
functions also exhibit decreasing differences in
opponents' strategies
in
own and opponents'
27
an equilibrium
will exist,
and
an increase in that player's equilibrium
will lead to
in the aggregate of the
(and whether or not payoff
own
strategies. It
strategies.
is
53,-_y 9?
(note here
easy to check that payoff
.
that the aggregator
one-dimensional,
is
=
g(s)
i.e.,
IZjQj' so tne statement on the aggregate
The
of the opponents' strategies concerns only quantities).
determined also (Theorem
(Corollary
so are the effect of entry
(Theorem
4)
is
and the impact on payoffs
provide a specific example with strategic substitutes where the local solvability condi-
violated.
is
and
shocks to the aggregate
1).
We next
tion
3)
effect of
As
will
become
Theorem 7
comparative statics conclusions
in
is
violated. Concretely, a positive shock to a player will
in the aggregate.
But we repeat that the above mentioned conclusions
the sense that the conclusion of
be seen to imply a decrease
clear, this leads to "perverse"
on, for example, individual shocks' effect
on individual strategies remain
valid.
45°-line
Backward reply correspondences Figure 2: A decrease in marginal costs for the
(dashed). Aggregate backward reply correspon- second firm leads to a decrease in aggregate out-
Figure
dence
1:
put from Q' to Q"
(solid).
Suppose that there are only two firms and
P (Q) = K — Q
by Q.5qf(ai —
also that firm l's costs are given
+
aj)
The
= \K-Q)qi -
first-order conditions for firm
1
K — Q — qi —
Since C\
strictly
is
can be written
q-\
{C[)~
strictly increasing,
(a\
l
is
convex (and conversely, when (CJ)
us define G\{z)
=
(Cj)
_1
(0.52
2
)
0.5g?(ai
which
—
a{)
=
is
-
oi)
its
be
28
0.
Suppose
payoff function
is
-Ci(oi).
0,
and
0.5(75"
{C[)~ l
strictly increasing,
will also
>
as,
well-defined.
_1
K
Ci(aj) for some differentiable, strictly
increasing and strictly convex function C\. This implies that
7Tf(g,a)
some constant
for
=
Cl( a ])-
is
strictly increasing since
C\ must be
strictly increasing.
C\
strictly convex).
Choosing G\
is
is
Let
equivalent
.
to choosing C\. Let
Gi(qi)
where
71,
>
<5i
and
0,
<
(3\
2,
= — Jig? +7i 9i +
0i>
so that the best responds choice of quantity for firm
becomes the
1
solution to the following cubic equation:
K-Q+
Figure
plots q\ as a function of
1
Q
(/3i
-
1
- qi) 9 i -
6 iq f
+
for a particular choice of
liq \
=
line)
for firm 2 (92 as a function of Q). Concretely, firm 2's cost function
+
0.5/3292
K
=
—
Q-
which shows
A
72( a 2)^
~~
The
iiiqi-
q\
+
172
2
]
+
as & function of
is
shows the same relationship
is
assumed to take the form:
Q
This yields a simple linear relationship between
#2&2-
solid line in the figure
Cournot equilibrium
(12)
parameters (the dashes "inverse-5"
shaped curve). The second dashed curve (the negatively sloped
[»2
0.
Q
(the
is
sum
of the two dashed curves).
given by the solid curve's intersection with the 45°-line in Figure
only difference between the two's parameter values
1
is
that for the dashed curve fa
is
Naturally, a reduction in fa corresponds to a reduction in the marginal cost of firm
shows that such a decrease
verified for the
though the cost function of firm
the conclusion of
Theorem
in
marginal costs reduces aggregate output Q.
parameters here, the two firms' payoff/profit functions are
"nice" aggregative
games with
Theorem
7:
:
23
(solid),
together with a similarly constructed aggregate backward reply correspondence (dashed).
figure
</2
the aggregate backward reply correspondence
Figure 2 depicts the same aggregate backward reply correspondence as in Figure
1.
and
1
is
It
strictly
2.
Unless the shock hits the aggregate
{e.g.,
24
The
can also be
This example thus illustrates that even
not convex).
may
lower.
concave (even
strategic substitutes, the local solvability condition
3 applies, a positive shock
The
is
in
critical for
an increase in K) so that
lead to a decrease in the equilibrium aggregate
when
the local solvability condition does not hold.
Multidimensional Aggregates
7
We
g
:
have so far focused on aggregative games with one-dimensional aggregates,
S—
>
E.
Many
i.e.,
games where
important examples, including the Bertrand game with homogeneous products
discussed in Section
3,
require
more than a one-dimensional aggregate, g
Another game with multi-dimensional aggregates
is
the technology choice
K
:
— RM
M
>
1.
game considered
in
S
>
,
= 4, /?i — 1 — ot\ = —4.4,
parameter values yielding the configuration in Figure 1 are:
and 02 = 40. Note that given these parameter values G\ will be strictly increasing (C\ will be
strictly convex) whenever q\ < 3.125. It is also straightforward to verify that any perturbation of these parameters
leads to the same comparative static results, so that this perverse comparative static is "robust"
24
Concretely, 62 = 10 for the dashed curve and 02 = 40 for the solid curve.
23
71
The
=
2.5,
specific set of
<5i
=
0.4,
2!
I
when technology
Section 6.3
other firms,
C —C
e.g.,
t
x
In this section,
we
(a*,
depend on some aggregate of the technology choices
costs also
A)
for
some aggregate
how some
discuss
of technology choices A.
of our results can be extended to this case.
=
the exposition we focus on the case where the aggregator takes the form g(s)
this case, naturally, g
:
S — RN
>
we denote the
players and
and
define D\Tl (s l ,Q,t)
= D Si JJi(si, Q, t) and
for player
i
Now
We
N.
s ,7r t (s, t)
Z^II^Si, Q,
Then, proceeding as
Si ,
denoting the vector of aggregates by
]T sj,
Q=
s j^
7
Q,
t)
+
sh
we again
J2 »j.
on the ^, functions. These
both
we
in Section 5,
D2 Ik(si, Q,
restrictions, together
13 )
l
(
define:
t).
Parallel with the local solvability condition (Definition 7 in Section 5),
restrictions
is
DQUi(si,Q,t). The marginal payoff
+ D 2 Ui
t
X^ =i
= Dinifa
t)
=
t)
^n
as:
= DiU,
%(su Q,
25
Ylj=i s j-
continue to assume that there are I
also exhibits strategic substitutes.
can again then be expressed
£
=
To simplify
by 1. In addition, we assume that the game
set of players
"nice" (Definition 6)
l
M
hence
,
of
(14)
we
will place certain
key
with our focus on nice games with
strategic substitutes, are collected in the following assumption.
Assumption
player
•
i,
The game F
1
is
an aggregative nice game (Definition
(Strategic Substitutes) Si
a
is
compact
Si
(Strong Local Solvability) Every
Remark
and
in addition, for each
we have:
exhibits decreasing differences in
•
6)
and
s
lattice,
3
and
(for all j
iXi{si,
^
real eigenvalue of
s_j,
its
determinant
is
non-zero (this
is
s,
and
i).
D
Si
^i \Si,Y^, 1= s j^)
\
7 (Strong Local Solvability) That every real eigenvalue of
negative implies that
super-modular in
t) is
because
D
Si
D
Si
^/i
i-
s negative.
[si,Y2, = i s j<t)
$i must have non-negative
^
s
off-
diagonal elements, see the proof of Theorem 11 for further details). Consequently, local solvability
(Definition 7)
is
Assumption
implied by strong local solvability.
1
is
straightforward to verify because of the following two relationships linking
the usual second-order matrices of
7T,
and the gradient of the ^i functions:
N
All of the following results remain valid if we assume instead that g(s) = ((^(sj,
s}),
.sf))
,g (sj',
with each function g n separable. See the beginning of the proof of Theorem 7 for details on how one can transform
such a game into a game with a linear aggregator.
25
.
30
.
.
,
.
.
.
.
.
.
DiiS .TTi
= DiiSj Hi
(s,
Since by (15),
Dq^i = D^iS
DqYI
requires simply that
D ^
matrices
only
if
Si
l
and
the matrix
Dq ^
<
t
D^
s
1
(s,,
Si ,
^
lit
for all j
in
/
i,
^
Si,
s fcl
is
D
Si
(cf.
noted
When D Si i>
}
own
strategies
Theorem
Nash
=
and
symmetric (which
is
is
Nash equilibrium
1
also
mean
that
when
this
1
(see the
proof of
verification of strong local
negative definite.
is
The
assumption holds, the existence
follows immediately by Brouwer's fixed point theorem. This
theorem (proof omitted):
10 (Existence) Suppose that T
satisfies
Assumption
1.
Then Y has a
(pure- strategy)
equilibrium.
We
Si
l
implied by Assumption
is
games with multi-dimensional aggregates the
in the following
if
the same as £> Sl ^; being a negative definite matrix. Note also
concavity implications of Assumption
is
(15)
Finally, strong local solvability
solvability "replaces" the very similar task of verifying that the Hessian
of a pure-strategy
and
Supermodularity holds
(15)).
are negative.
'i/ l
i,
Next we can sum the two
a non-positive matrix.
s
±
for all j
«
decreasing differences (strategic substitutes)
TIj has non-negative off-diagonal entries.
that concavity of payoff functions in
Thus,
*
Z?^
=
i
in order to obtain Dg. s .nt
often the case in practice), this
11).
,
53;= i s j'0
requires that the real eigenvalues of
Theorem
s fc
next define the backward reply function of player
bi(Q,t)
<=>
$i(si,<5)
=
0.
Assumption
1
i
again using the first-order conditions:
by ensuring that each
simplifies matters here
leads to a unique backward reply function (rather than a correspondence), ^(Q.t)
given vector of aggregates Q, the gradient of b-i(Q)
D Q bi(Q,t) = -[DSi V
i
is
also well-defined
(bi{Q,t),Q,t)]-
1
D Q $i(b
i
and
(Q,t),Q
is
1
26
Q
For any
given by:
t),
and thus
/
D Q b(Q,t)
=
^D Q
b ] (Q,t).
3=1
26
N x TV matrix),
D
Fixing Q, it is clear that the gradient of \Pi(-, Q),
Sl 9i(si, Q) (which is a
stationary point. In particular, from strong local solvability, the determinant of
D
never equals zero. This immediately implies that there exists a unique
point
theorem; Milnor (1965)).
31
critical
Sl
'i/ t
is
non-singular at any
(s,,Q) never changes sign and
(e.g.,
from the Poincare-Hopf
.
Let us also recall that an
off-diagonal entries.
Theorems
and
3
27
We
M -matrix
is
a matrix with positive real eigenvalues and non-positive
are then ready to state the following multi-dimensional version of
7.
Theorem 11 (Shocks to the Aggregates) Suppose
N be a shock to the aggregate, that is, let TTi(s, t) = Hi
M.
that the matrix I
— [Dq6(Q +
(Sufficency)
•
in
t
£
T
If the
leads to a decrease in each
£
T
exists t"
>
ters
t'
when
what
satisfies
Sj, t .+
J2j=i
i
is
an
component of
j
)
for °^
Then
if
I—
*
e
tgTC
1.
Let
-E,
and assume
Then:
non-singular.
is
s
Assumption
M -matrix (for allQ andt),
an increase
the equilibrium aggregate vector.
[DQb(Q(t')
+ t')}~
1
is
some vector of paramenot an M-matrix, there
such that at least one component of the equilibrium aggregate vector increases
t'
Proof. See Section
In
(
V
Q(t') be an equilibrium aggregate given
let
that hits the aggregate.
raised
is
t
and
matrixl- [DQb(Q + t)]~
(Necessity) Conversely,
•
exists
t)]~
that
follows,
from
t'
to t"
11.7.
we
will
use the sufficiency part of Theorem 11 to present direct parallels to
the other theorems presented in Section
4.
Nevertheless, the necessity part of this theorem
is
also
noteworthy, perhaps even surprising.
Given Theorem
11, the proofs of the
analogous theorems
for the
next three theorems closely follow the proofs of the
one-dimensional case and are thus are omitted. 28 For the next theorem,
suppose that the default inaction strategy of the entrant now
is
a vector of zeroes
(or,
more
generally, the least element in the entrant's strategy set).
Theorem
Theorem
12 (Entry) Suppose that T
11.
satisfies
Assumption
Then entry of an additional player
1
and
the sufficiency conditions in
leads to a decrease in the aggregates of the
existing players. In addition, at least one of the aggregates of all players
and
strictly so unless the entrant
Theorem 13
27
Recall that an
Then a
M-matrix and an
entry,
chooses inaction.
(Idiosyncratic Shocks) Suppose that T
conditions in Theorem 11.
must increase with
satisfies
Assumption
positive idiosyncratic shock to player
M-matrix
i
E
T
1
and
the sufficiency
leads to
an increase
[i.e., all of their principal minors are
an M-matrix if and only if it is also a P-matrix.
28
The only new feature is the second statement of the entry theorem (that at least one of the aggregates must
increase upon entry). This is a direct consequence of the fact that the backward reply function of the existing
b
players, b, is decreasing (this is proved as part of Theorem 11). Indeed, let Q be the vector of aggregates before
a
entry, Q the aggregates after entry, and s/+i >
be the strategy chosen by the entrant. Since b is decreasing,
a
b
b
a
b
a
b
a
Q < Q implies: < Q - Q = b{Q ) - b(Q ) - si+\ < --si+i which in turn implies si+i = Q - Q = 0.
positive). Moreover,
if
inverse
are also P-matrices
a matrix has a non-positive off-diagonal,
32
it is
and
in this player's equilibrium strategy
to a decrease in the associated aggregates of the existing
players.
Remark
S — R2
>
exists
is
the sufficient conditions are particularly easy to verify. In particular,
),
and
when — [Dq6(Q +
a non-singular M-matrix
is
l
t)\~
+
1),
.
(Am +
.
,
.
where
1),
Ai,
Dqo(Q +
is
t)
positive matrix, its trace
is
.
.
,
Am >
-[Dq6(Q +
In this two-dimensional case,
determinant of
.
To
positive.
f)]"
when
must be negative (when they are
real; if
the determinant
they are not
is
t)]~
and only
if
positive,
is
then there
real,
follows that
(real) eigenvalues,
Now
since
positive determinant
T
(x DQbi(Q
is
+
t)
Ylj=i
x
for all
if
the
both eigenvalues
nothing to check because
then
It
a non-singular M-matrix.
it is
=
.
a matrix with non-positive off-diagonal elements and positive
is
^Q^jiQ +
^
0).
This
is
Dqb(Q +
0> a sufficient condition for
that each of the matrices
is
<
t)x
t)\~
and thus
Dqb(Q +
l
l
a non-
is
t)
the definition of an M-matrix above requires only that the real eigenvalues be positive).
— [JDq6(Q +
+ £)]
:
_1
are equal
]
DQb(Q +
see this first note that since
non-positive. So
r)]
— [Dqb(Q +
a non-singular M-matrix
is
[Z?q&(Q
_1
— [Z?q6(Q +
[I
are the real eigenvalues of
1
I—
2 (g
a non-singular M-matrix. This
is
generally the case (regardless of N), since the real eigenvalues of
to (Ai
N=
Two- Dimensional Aggregates) When
8 (Sufficient Conditions for
DQb
(Q+t),
l
because the
=
i
sum
1, ...
,1
is
to have a
t)
quasi-negative definite
of quasi-negative definite matrices
quasi-negative definite, and a 2 x 2 quasi-negative definite matrix has a positive determinant.
The next
corollary exploits this observation.
Corollary 2 (Symmetric
Assumption
satisfies
Dq^
1
(s 1
,
^2 ,=i
sj
+
1
t)
and
Games
N = 2.
with Two- Dimensional Aggregates) Suppose that T
Consider a shock
has a positive determinant for
the aggregate.
to
all s
£ S and
the aggregates will lead to a decrease in both of the aggregates in
addition, the results in
strategies before
and
Theorems 12-13 continue
after entry
(Theorem
12),
when
to hold
and
t
Then
if
the matrix
G T, a positive shock
to
any symmetric equilibrium. In
the existing players choose identical
the players that are not affected by the shock
choose identical strategies before and after the arrival of the idiosyncratic shock (Theorem IS).
Proof. The aggregate
any of the
aggregate
a symmetric equilibrium
From Theorem
(identical) players.
if
only
to verify that
this holds
in
if
if [I
- [DQb(Q +
-[D Q b(Q +
and only
if
t)]~
l
_1
i)]
is
the determinant of
s
Q =
Ib l (Q
-1
/
1
DQbi(Q + t)
is
is
t)
Remark
where
8,
a non-singular M-matrix.
positive. This
is
the case
has a positive determinant and Dbi
33
+
6 lis
i
shock to the aggregate decreases the
a non-singular A/-matrix. From
z
D ^i
given by
11, a positive
= -[D Q b (Q + 01
a positive determinant, because
is
we only need
N = 2,
When
when Dq <S> has
= —\D Si
1
}~
'$>
1
1
Dq'1!
;
1
.
.
Games without
Nice
8
we extend the
In this section,
assumption of Section
Differentiability
local solvability,
5,
games presented
results for nice
in Section 5. Recall
that the main
presupposes that payoff functions and the aggregator
we show
are twice continuously differentiable. In this section,
that robust comparative statics can
be derived without differentiability as long as a non-differentiable version of the local solvability
condition
We
imposed.
is
limit attention to the case of one-dimensional strategy sets (hence the
aggregate must be one-dimensional also). Recall that an aggregator always has a representation
Therefore, for any
Intuitively, this
also
know
H
=
of the form g(s)
Q
H
(]Ci=i hj(sj)), where
in the
means that
range of
g,
and
h\,
Q=
g(s)
we have
we know the aggregate
if
Q
.
.
hi are strictly increasing functions.
<=> s,
=
and the
h~ l
if,
for all
Q'
> Q and
f(Q',y)
The main
>
y'
>
y,
H~
satisfies
l
(>) f(Q,y)
=
(G (Q,
t
Q,
y),
t)
Recall
y).
the single-crossing property
shows that an appropriately-chosen
Statics for Nice
Games
single-
and thus extends our
without Differentiability) Consider
sets, a
payoff functions that are upper semi- continuous and quasi-concave in
t
-
f(Q\y') > (>) f(Q,y')-
an aggregative game with one- dimensional convex, compact strategy
n
(Q)
differentiability.
Theorem 14 (Comparative
that
1
- J2j& ^j( s j)
- 1 players, we
we have
result in this section, presented next,
games without
(Q)
= h~ [H~
crossing property can replace the local solvability condition (Definition 7)
results to nice
1
strategies of /
the strategy of the last player. Let us also define Gi(Q, y)
from Milgrom and Shannon (1994) that a function f{Q,y)
in (Q, y)
,
(for each
i
6 T)
satisfies the single-crossing
own
separable aggregator,
strategies.
property in (Q,y).
Suppose
Then
the
conclusions of Theorems 7 and 8 continue to hold. Moreover, provided that payoff functions are
and
twice differentiable
Proof. See Section
the equilibrium
interior, the conclusions of Theorem.
9 also hold.
11.8.
Notice that differentiability
conclusions. Clearly, the
The next
is
more
is
needed
in
Theorem
interesting part of
corollary uses the insights of
9 in order to even state this theorem's
Theorem
Theorem
14 concerns
Theorems
main
7-8.
14 to provide another useful and simple
alternative to the local solvability condition for nice games.
Corollary 3 Consider a nice aggregative game with
strategy sets
(i.e.,
N—
1),
and assume
D
Si
linear aggregator g(s)
that for each player
$?i(si,Q)
<Q for
34
all Si
= y\ Sj,
one- dimensional
i:
andQ
(17)
Then
Proof.
i
e
Theorems
the conclusions of
1).
Since g
is
The condition
and Q. This
is
D
5i
= Q —
Gi(Q,y)
linear,
*,(s l Q)
,
and 9 continue
8,
7,
<
is
y
to hold.
and ilj(G ;(Q
equivalent to
-D
y)
,
s
,
,*j
Q
,
= U (Q —
t)
L
y,Q,t) (for each
= -D^TU - D^II, >
for all
turn equivalent to IT^Q - y,Q,t) exhibiting increasing differences in
in
Since increasing differences implies the single-crossing property, the results follow from
Q
and
s,
y.
Theorem
14.
Note that Condition
(17) in Corollary 3 requires that
D
contrast, the local solvability condition requires
such that $i(si,Q)
=
Thus
0.
inequality throughout,
weak
implied, though the
^i(si,Q) <
^i(si,Q)
^
for
all Si
but only when
0,
makes
inequality
and Q, then
for all s,
and Q. By
and
s,
local solvability
this condition easier to check
of examples (recall the discussion in Section 6.1).
Q
are
and apply
would be
in a variety
29
Walrasian Play
9
When
the aggregate
may
it
plausible to
the aggregate.
n
s
neither condition generalizes the other. If (17) holds with strict
D^^^Sj, Q) <
i.e.,
Si
D
i
(.s l
,
Q,
t)
Q
results
from the "average" of the strategies of a large number of players,
presume that each player
In this case, each player
with respect to
taking
Si
Q
?'
1
will
number
games where players ignore
and the associated Nash equilibrium
will
ignore the effect of
its
strategy on
maximize the reduced payoff function
as given. This
equilibrium theory with a finite (but large)
situation in aggregative
e
T
e
i
is
=
the behavior assumed in standard general
of households.
their
as a Walrasian
II,
With
analogy,
we
refer to the
impact on aggregates as Walrasian play
Nash Equilibrium.^
For such games, our
results can
be strengthened (and the proofs in fact become more straightforward). Here we briefly
outline the
main
results in this case, focusing
Definition 10 (Walrasian
The strategy profde
rium given
t
<£
T
if
s*(t)
=
on one-dimensional aggregates
Nash Equilibrium)
(s\(t),
holding Q(t.)
s*i(t)
=
.
.
.
,s*j(t))
is
Consider an aggregative game Y
called a (pure-strategy)
=
arg
max 11,(5,, Q(0,0
•
K
=
=
((IT,
1).
S
l )
ie j, g, T).
Walrasian Nash equilib-
g(s*{t)) fixed, we have for each player
e IU{Q,t)
{i.e.,
i
=
!,...,! that,
(18)
steS,
29
can also be noted that (17) with strict inequality makes up "half of what Corchon (1994) calls the "strong
The other "half of Corchon's strong concavity condition requires payoff functions to exhibit
strictly decreasing differences in own and opponents' strategies. This is not assumed in our analysis.
30
Such "aggregate-taking" behavior has been studied extensively within evolutionary game theory, see for example
Vega-Redondo (1997), Possajennikov (2003), and Schipper (2004).
It
concavity" condition.
35
Notice that under Walrasian play, a player's best responses i^(Q,f) will depend on the aggregate
Q
and the exogenous variables
smallest and largest selections from Ri{Q,
substitutes
Si
if
each Si
N
When
and Q.
is
=
to be a positive shock
strategic substitutes
As
a
lattice, 11^
1
and Hi
is
is
1.
El. Then
i
Theorems
in Si,
The game
t.
t)
>
Q and t),
Q and t).
(for all
<
(for all
and a
may be
we continue
multiple equilibria,
Hi
and Q*{t)
(cf.
equations
(6)
Nash
8,
and 9 hold for Walrasian Nash
will lead to
and
and entry increases
=
IfT features
Proof.
quasi-concave in
is
S{
G
Si for
In particular, a positive shock to
equilibria.
the smallest
and
Hi(si,Q)
(i.e.,
is
interior.
Then for each
provided that:
t
the shock does not directly affect player
is
largest equilibrium
largest equilibrium aggregates. In addition,
o,
then the sign each
i),
equal to the sign of each element of the vector
strategic substitutes, then
1
D 2SiQ U
Theorems
3, 4>
i ( S *(t),Q(t)).
an d 5 continue
to hold for
Walrasian
equilibria.
(Sketch) For part
1,
simply define Z(Q,t)
=
{g{s)
G
K
view of our compactness and upper semi-co'ntinuity assumptions,
upper hemi-continuous correspondence. Then we can proceed as
using
and
(5) in Section 4).
an increase in the smallest and
-{Dl Si Ui ( S *(t),Q(t)T
Nash
t
Equilibria) Consider an aggrega-
that the (reduced) payoff function IIj(sj, Q,t)
element of the vector Dts*(t)
2.
in
condition for
to focus on the smallest
s*(t) is locally coordinatewise increasing in a positive shock
if
features strategic
sufficient
-[p^iiiWW, Q(t)\ t)]- l Li Q ni(8W), Q(t), t) >
and
the
and exhibits decreasing differences
suppose that payoff functions are smooth and the equilibrium
£ I,
if
we maintain the compactness and upper semi-continuity assumptions.
one or more of the agents
i
i
a Walrasian Nash equilibrium exists. Moreover, we have that:
7,
aggregates,
a positive shock for player
are both increasing in
15 (Comparative Statics for Walrasian
r and assume
Ms
twice continuously differentiable, a sufficient condition for
is
largest equilibrium aggregates Q*{t)
each
increase in
supermodular
is
that D^.QlLi(si,Q,t)
Also, since as before, there
tive
t)
that D^. t Hi(si, Q,
in previous sections,
Theorem
An
t.
Z (Q,t)
instead of the function
argument remains
valid
if
this result, the proofs of
q.
8,
s,
Z
q,
€ Ri(Q,t)
(Q,
in the
Figure 10 in the Appendix makes
instead of the function
Theorems
:
t)
is
In
a convex-valued,
proof of Theorem 7 but
it
clear that the general
we use a convex- valued correspondence. Given
and 9 apply with minimal modifications.
36
for all ?}.
For part
2,
note that a shock to the aggregate
and Q), hence
in s,
from part
1.
The
a negative shock (by decreasing differences
is
leads to a decrease in the smallest
it
conclusions of
Theorems
4
and
and
largest aggregates
5 are established
by the conclusion
by straightforward modifications
of the original proofs.
A
noteworthy implication of Theorem 15
tinue hold for a Walrasian
Nash
is
that
all
our results for nice aggregate games con-
equilibria without imposing local solvability or differentiability
and boundary conditions (only quasi-concavity
is
imposed to ensure that best response corre-
spondences are convex- valued). This highlights that the challenge
static results in aggregative
on the aggregate.
are not
It
games
lies in
limiting the
magnitude of the
should also be noted that part 2 of the theorem
assumed to be quasi-concave, though the
results
do hold
strict strategic substitutes (i.e., if strictly decreasing differences is
differences in Definition 3).
tions
in deriving robust
31
The proof
of
Theorem
if
effect of
comparative
own
strategies
false if payoff functions
is
the
game
instead features
assumed instead
of decreasing
15 also shows that any separability
on the aggregator g are unnecessary: the conclusions hold provided that g
is
assump-
an increasing
function (without further restrictions).
Conclusion
10
This paper presented robust comparative static results for aggregative games and showed
how
these results can bo applied in several diverse settings. In aggregative games, each player's payoff
depends on her own actions and on an aggregate of the actions of
sum, product or some moment of the distribution of actions).
all
players (for example,
Many common games
in industrial
organization, political economy, public economics, and macroeconomics can be cast as aggregative
games. Our results focused on the effects of changes
in various
parameters on the aggregates of
the game. In most of these situations the behavior of the aggregate
and
also indirectly, because the
as a function of the aggregate.
comparative
is
of interest both directly
statics of the actions of each player
can be obtained
For example, in the context of a Cournot model, our results
characterize the behavior of aggregate output, and given the response of the aggregate to a shock,
one can then characterize the response of the output of each firm
We
focused on two classes of aggregative games:
substitutes
ferentiable,
and
(2) "nice" aggregative
and (pseudo-)concave
strategic substitutes,
31
we showed
in
(1)
in the industry.
aggregative of games with strategic
games, where payoff functions are twice continuously
own
strategies.
For example,
for
dif-
aggregative games with
that:
However, an equilibrium is not guaranteed to exist in this case because of lack of quasi-concavity. To apply the
mentioned in the text one must thus first (directly) establish the existence of an equilibrium.
result
37
1.
Changes
parameters that only
in
affect the
aggregate always lead to an increase in the
aggregate (in the sense that the smallest and the largest elements of the set of equilibrium
aggregates increase).
2.
Entry of an additional player decreases the (appropriately-defined) aggregate of the strategies of existing players.
3.
A
"positive" idiosyncratic shock, defined as a parameter change that increases the marginal
payoff of a single player, leads to an increase in that player's strategy and a decrease in the
aggregate of other players' strategies.
We
provided parallel, and somewhat stronger, results for nice games under a local solvability
condition (and showed that such results do not necessarily apply without this local solvability
condition).
The framework developed
in this
paper can be applied to a variety of settings to obtain
"robust" comparative static results that hold without specific parametric assumptions. In such
applications, our approach often allows considerable strengthening of existing results
clarifies
the role of various assumptions used in previous analyses.
and
We illustrated how these
also
results
can be applied and yield sharp results using several examples, including public good provision
games, contests, and oligopoly games with technology choice.
Our
results
on games with multi-dimensional aggregates (Section
7) are
only a
first
step in this
direction and our approach in this paper can be used to obtain additional characterization results
for
such games.
to future work.
games with
We
We
leave a
more systematic study
of
games with multi-dimensional aggregates
also conjecture that the results presented in this
infinitely
many
players
paper can be generalized to
and with infinite-dimensional strategy
the appropriate definition of a general aggregator for a
game with
sets. In particular,
infinitely
many
along the lines of the separability definitions in Vind and Grodal (2003), Ch.
results
and
in fact
even our proofs remain valid
in this case. Similarly,
solvability condition in infinite dimension, all of our results also
infinite-dimensional strategy sets.
The
with
players (e.g.,
12-13), our
main
with the appropriate local
appear to generalize to games with
extension of these results to infinite-dimensional games
another area for future work.
38
is
Appendix: Proofs
11
Proof of Theorem
11.1
For each player
i,
2
define the correspondence GrfA,]
= {seS-.
Gr[Ri](t)
Sl
:
T—
>
2
e Ri(s- U
s
by,
,teT
t)}
r
m
This correspondence is upper hemi-continuous and has a closed graph: if s™ G
(s ^l ,t
m
m
for a convergent sequence (s ,t ) —> (s,t), then by the fact that R^ itself has a closed graph.
T — 2 s is thus given by the
Si £ Ri(s-i, t). Moreover, E(t) = DiGr[/i']. The correspondence E
intersection of a finite number of upper hemi-continuous correspondences, and so is itself upper
hemi-continuous. In particular, E has compact values {E(t) C 5, where 5 is compact). Therefore,
the existence of the smallest and largest equilibrium aggregates, Q*{t) and Q*{t), follows from
the continuity of g and from Weierstrass' theorem. Upper semi-continuity of Q* T — R follows
directly from the fact that g is upper semi-continuous and E is upper hemi-continuous (see
Ausubel and Deneckere (1993), Theorem 1). Lower semi-continuity of Q* follows by the same
argument since Q*(t) = — max s€E ^ —g(s) and g is also lower semi-continuous. Finally, when
R
l
>
:
>
:
=
the equilibrium aggregate is unique for all t, Q*(t)
semi-continuous and thus continuous in t on T.
)
Q*{t) and so
is
both upper and lower
Proof of Theorem 3
11.2
Recall that
7Tj(s, t)
separable in
(s,t).
=
IT(si,
G(g
(s)
,
G(g(s),t)
where g
(s)
=
where g
t)) all i,
:
S — R
>
is
separable in s and G(g (s)
,
t)
is
This implies that,
M (X^gz ^
(
=
H
the aggregator of Definition
s i)) ls
1,
and
function. Moreover, recall that best response correspondence Ri (s_j,
for each i 6 J. Let h l (Sl ) be the image of the strategy set 5j under /i t
best response correspondence R^
I
hx{t)
+
#
r
]Ci
hji s j)
)
= ^( s -?i
M
is
a strictly increasing
upper hemi-continuous
and define the "reduced"
t) is
(•)
f° r
eacn
*•
We can then
define
the following upper hemi-continuous (possibly empty-valued) correspondence:
B
t
for
each plaver
i.
(Q,
t)
=
[i]
G hi(Si)
:
r?
e
hi o
Ri(h T {t)
+ Q-77)}
Let
/
Z(Q,t)
=
Y^B
:
(Qj)
j=i
be the aggregate backward reply correspondence associated with the aggregate
Clearly, the "true" aggregate g (s)
Q=
^2j hj(sj). Therefore,
=
M
I
J3 hj(sj)
we may without
)
is
Q —
Yli hj(sj).
monotonically increasing in the aggregate
loss of generality focus
on J2j
^-j( s j)
instead of g(s)
in the following.
Let q(Q,
t)
G Z(Q,
f)
be the aggregate "Novshek-selection" shown as the thick segments in the
figure below.
39
Q
Figure
Constructing the aggregate "Novshek-selection"
3:
constructed follows. 32 The first step is to pick Q max > 0,
max
,t). To ensure that such a "point below the diagonal"
for all q 6 Z(Q
such that q < Q
exists, we can extend each of the correspondences hi o R^ along the lines of Kukushkin (1994) (see
p. 24, 1.18-20). The details of this procedure need to be modified as follows. Let Di denote the
A
sketch of
how
this selection
is
max
R upon
and only if hi 0/^(7) ^ 0. Since
hi o Ri is upper hemi-continuous, D{ is closed. It is also a bounded set since R4 C 5, and each
5j is compact. Consequently, D has a maximum, which we denoted by di. Then extend hi o R^
from
to
U {di,Q max by taking fo o Rd (d) = _L all a! G (a( ,Q mo:r ]. Here 1, can be any
max
small enough element (for each player iel) such that Y^i -Li < Q
J-i < min /i, o R (di), and
Qmax _ j_ g ^^QTTiaij^ Defining the aggregate backward reply correspondence Z as above, it is
i
max t) =
± t < Q max
clear that Z(Q
t
1
Q""
C
*]
< max
denote the subset of R, where Z(-, t) is defined, i.e., those
Next let D
(-00,
subset of
which h %
o R4
defined,
is
i.e.,
write 7 G Dj
if
t
A
A
(
]
2
t
,
£
,
for
which Bi(Q,
t)
7^
0,
all
.
Q
The
i,
D
set
is
compact by the same argument
Q
as in the previous
denote a closed "interval" in D by [Q ,Q max ].
= D n {Q Q < Q < Q
Given such an interval [Q',Q max ], we say
That is, \Q',Q
}.
max
—
x {t}
that a function q
R is a maximal decreasing selection (from Z) if for all
[Q\ Q
Q G [Q', g maa: ], the following are true: (i) q(Q, t) > Z for all Z G Z(Q, t) (c 2 R ); (ii) q{Q, t) < Q;
and (iii) the backward reply selections bi(Q,t) G Bi(Q,t) associated with q (i.e., backward by
max (i.e., Q" > Q'
selections satisfying q(Q, t) =
bj(Q, t) all Q) are all decreasing in Q on \Q' Q
paragraph. Now, abusing notation
max
1
slightly, let us
max
1
:
]
:
>
]
V
,
]
=>bi(Q",t)<bi(Q',t)).
Denote by ft. C 2 R the set of all "intervals" \Q' Q max upon which a maximal decreasing
max
selection exists. Notice that {Q
} G fi so f2 is not empty. f7 is ordered by inclusion since for
any two elements u>',w" in Q, w" = [<?", Q mai C [Q',Q mQ:r = a/ «=> Q" < Q'. A chain in SI is a
totally ordered subset (under inclusion). It follows directly from the upper hemi-continuity of the
backward reply correspondences that any such chain with an upper bound has a supremum in fl
(i.e., O contains an "interval" that contains each "interval" in the chain). Zorn's Lemma therefore
max G f2 that is
implies that Q. contains a maximal element, i.e., there exists an interval [Q mm
not (properly) contained in any other "interval" from fl.
The "Novshek-selection" is the maximal decreasing selection described above and defined on
the interval [Q min ,Q max ]. See the figure.
The main step in proving the existence of equilibrium consists in showing that Q mm is an
equilibrium aggregate (it is easy to see that if Q min is an equilibrium aggregate, then the associated
backward replies form an equilibrium). Since we have q(Q min t) < Q min by construction, this
can be proved by showing that q(Q min ,t) < Q min cannot happen. This step is completed by
showing that if q(Q min t) < Q min holds, then q can be further extended "to the left" (and the
\
,
]
]
,
]
,
,
32
The
construction here
is
slightly different
from the original one
the same. Aside from being somewhat briefer, the present
suffer
from the "countability problem"
Lemma
in
way
in
Novshek (1985), but the basic
intuition
is
of constructing the "Novshek-selection" does not
Novshek's proof pointed out by Kukushkin (1994), since we use Zorn's
for the selection.
40
extension will satisfy (i)-(iii) above). This would violate the conclusion of Zorn's Lemma that
jQnnn^ Qmaxj j s max imal, thus leading to a contradiction. The details of this step are identical to
those in Novshek (1985) and are omitted.
Note that since hx is an increasing function, q{Q, t) will be locally decreasing in Q if and only
if it is locally decreasing in t (the effect of an increase in t or Q enter qualitatively in the same way
in the definition of Bi, and so the effect of an increase on q must be of the same sign). Figures
4-7 illustrate the situation for t' < t" The fact that the direction of the effect of a change in Q
and t is the same accounts for the arrows drawn. In particular, any increasing segment will be
shifted up and any decreasing segment will be shifted down because the shock hits the aggregate
(this feature does not necessarily hold for other types of shocks).
.
Z(QJ)
("
rf
Z(Q,t)
(
Qrj,
WO.'
x„^
1
^
^*Sls**«fcw
"^e
Q
Figure
Figure
4:
6:
Case
Case
Figure
I
Figure
III
5:
7:
Case
II
Case IV
There are four cases: Either the equilibrium aggregate varies continuously with t, and q is
decreasing (Case I) or increasing (Case II). Otherwise, the equilibrium aggregate "jumps" when t
is increased (Cases III and IV). [The figures depict the situations where the equilibrium aggregate
jumps from Q' to Q" This is associated with an increase in t. If instead t decreases, case III
.
reduces to Case I and Case IV reduces to Case II.]
Importantly, if the equilibrium aggregate jumps, it necessarily jumps down (and so is decreasing in t). The reason is that an increase in t will always correspond to the graph of q being shifted
to "the left' (that is another way of saying that any increasing segment will be shifted up, and
any decreasing segment shifted down). Hence no new equilibrium above the original largest one
can appear, the jump has to be to a lower equilibrium. See the figures. We now consider Cases I
-
and
II in
Case I:
the new
turn.
In this case
we have
Q<Q
equilibrium aggregate
follows that
Q" <
Q"
such that q(Q,
t)
—
Q
>
and q(Q,
t)
—Q <
0,
and such that
the interval [Q, Q\. Since q is decreasing in t, it trivially
Q'. Note that this observation does not depend on continuity of q in Q, but
lies in
merely on the fact that a new equilibrium aggregate Q" exists and lies in the neighborhood of Q'
where q is decreasing (in other words, it depends on the fact that the aggregate does not "jump").
41
Figure
Slope below
8:
1
is
Q' be-
impossible:
Figure
ing largest equilibrium aggregate violates that
q{Q,
t) is
9:
The "Novshek
selection"
leading to the smallest equilibrium ag-
decreasing in Q.
gregate.
we have Q" £ [Q. Q]. When q is (locally) increasing, we must have Q < Q
such that Q — q(Q, t) >
and Q - q{Q, t) < 0. Intuitively, this means that the slope of q is greater
than 1 as illustrated in Figure 5. Indeed imagine that for the largest Nash equilibrium Q', there
(this means intuitively that the slope is below unity,
exists Q° > Q' such that Q° — q(Q°,t) <
see Figure 8). Then since q(Q°,t) > Q° > Q', no decreasing Novshek type selection could have
reached Q' and there would consequently have to be a larger Nash equilibrium Q* This yields a
Case
II:
As
in
Case
I,
.
contradiction.
We now
Q" < Q'
we prove this without explicit use
used directly as seen in Figure 7). In
particular, let us establish the stronger statement that hf{t) + Q is decreasing in t (since hx{t)
First define C = hr(t) + Q,
is increasing in t, this implies that Q must be decreasing in t).
argue that
of continuity (the proof
.
As
in the previous case,
straightforward
is
if
continuity
is
and consider the function f(C,t) = C - h T (t) - q{C - h T {t),t). Let C = h T {t) + Q and C =
and f{C, t) = Q - q{Q, t) < 0.
h T {t) + Q. By what was just shown, f(C, t) =
q{Q, t) >
Since B t (C — h.T{t),t) is independent of t (t cancels out in the definition of the backward reply
correspondence), q(C — h-T(t),t) must be constant in t, i.e., q(C — hr{t),t) = q{C) for some
function q which is increasing (since we are in Case II). So / is decreasing in t and Q, and we
Q-
conclude that in equilibrium f(C*(t),t)
=
0,
C*{t)
=
hx(t)
+
Q*{t)
is
decreasing in
t.
Remark
9 The fact that the term hx(t) + Q is decreasing in t implies that, in this case, when
entry of an additional player, the aggregate of all of the agents (including the entrant)
decreases. To see this, compare with the proof of Theorem 5 and use that the aggregate of interest
there
is
is
M-Sj) +
XZj
^/+i(-s /+i)
(
in other words, take h T {t)
Combining the observations made so
=
hj+i(si+i)).
shows that the largest equilibrium aggregate is deof the previous conclusions depend on continuity
of q in Q, and it is straightforward to verify that the same conclusions hold regardless of whether
Q lies in a convex interval (strategy sets could be discrete, say). 33 The statement for the smallest
creasing in
t
as claimed in the theorem.
See Kukushkin (1994) for the details of
far
None
how the backward
cases.
42
reply selection
is
constructed in such non-convex
equilibrium aggregate can be shown by an analogous argument. In particular, instead of considering the selection q{Q, t) one begins with Q sufficiently low and studies the backward reply
correspondence above the 45° line, now choosing for every Q the smallest best response (Figure
This completes the proof of Theorem
9).
11.3
3.
Proof of Theorem 5
Let R4 denote the "reduced" backward reply correspondence defined by
J^(s_j,i) for each
To
i.
simplify notation, let us set
hits the first player, in particular then R,
strategy
j?i
(
Nash equilibrium
53 j =^i
^j'( s j)> 'i
)
will also
ari d hi(si)
G
i
=
1
t
—
t\ for all i
^
1).
Any
be a fixed point of the set-valued equilibrium problem:
hi o
Ri
I
Ylj^i h( s j)) for
=
(assume that the idiosyncratic shock
independent of
is
R (J2j-nhj(sj),t)
z
=
2,
.
.
.
,
pure-
S
si
Consider the last I —
/.
1
inclusions, rewritten as
hiis^ehioRi
Vftjfo)
+ M*i)
fori
=
2,...,/.
(19)
s\ £ Si, Theorem 3 implies that there exist a smallest and largest scalars y*(si) and
and solutions to the I — 1 inclusions in (19) which y*( s i) = Ylj^i^j( s j,*) an<^ V*{ s i) —
For given
y*{s\)
y*
Yljdki hj{s*,), respectively. In addition, y*,
Now combining y* and
and replacing
s\
with
sj
:
S\
— K
>
are decreasing functions.
£ Ri
y" that solve (19) in the sense described, with s%
= — si, we
(
53i^i hj{sj),
t\
J
obtain a system with two inclusions:
Si
e -Ri(y,ti)
and
ye
{y*(-Si),2/*(-si)}.
ascending in (sj, y) in the sense of Topkis (1998), hence its smallest and largest fixed
points are decreasing in t\ (since the system is descending in t\ in the sense of Topkis). Therefore,
the smallest and largest equilibrium strategies for player 1 are increasing in ii, while the associated
aggregate of the remaining players y is decreasing in t. That the smallest and largest strategies
for player 1 do in fact correspond to the smallest and largest strategies in the original game is
straightforward to verify: Clearly, y*(si) and y*(s\) are the smallest and largest aggregates of the
remaining players (across all strategy profiles compatible with an equilibrium given sj), and since
This system
is
R\ is descending in
the smallest.
y,
the corresponding equilibrium values of s\ are, respectively, the largest and
Finally, it follows by construction that the corresponding aggregates of the remaining players
must be the smallest and largest for the original game. This completes the proof of the theorem.
11.4
Proof of Theorem 7
m
begin by noting that there is no loss of generality in using the aggregator g{s) = J2i ^i( s i)
for all i. To see why, recall that the local
the following, and assuming that min SiS s /^(sj =
solvability condition is independent of any strictly increasing transformation of the aggregator
We
t
(Remark
5).
mation f{z)
where
has
all
hi(si)
If
the original aggregator
is
M
= H~ {z) - Si minfi6Si
= hi(s — min Si6 s, hi(si).
l
z
)
g(s)
s »)
= H(^ji
t0 g et tne
Clearly,
43
we can
new aggregator
min^gs,
the desired properties.
hi(si))
hi(si)
—
therefore use the transforg(s)
for all
i
=
f{g{s))
so this
=X)iM s t).
new aggregator
Let Ri S_j x T — Si be the best response correspondence of player i and Ri the reduced best
response correspondence defined by Ri(J2j^i^j( s j)^) — Ri{s-i,t). Define ^(^-^ hj(sj), t) =
>
:
^t °
^idZi=4t ^j( s j'))0> anc^ write
77i
G Bi(Q,t) whenever
77;
6 fj(Q -
r)i,
t).
It is clear
that
if
6 Bi(Q,t) then there exists Sj G i^i(Q — h t (si),t) with 7ft = hi(si). So Q is an equilibrium
aggregate given i 6 T if and only if Q G Z(Q,t) = Y2 i B l (Q,t) (the correspondence Z is the
aggregate backward reply correspondence already studied in the proof of Theorem 3).
r)i
JXQ.0-Q
Figure
The main
10:
Q*{t)
is
the largest equilibrium aggregate given
t
G T.
Theorem 7 is to study the behavior of the smallest and a largest
below by q{Q,t) and q(Q,t), respectively. Take the largest selection
idea in the proof of
selections from Z, denoted
q(Q,t), and consider q(Q,t)
— Q
as a function of
Q
(Figure 10).
Any
intersection with the
first
and so
axis corresponds to an equilibrium aggregate since there q(Q, t) =
G Z(Q, t). Since q
is the largest selection from Z, the largest such intersection corresponds to the largest equilibrium
Q
Q
aggregate Q*(t) as defined in (6). The desired result follows if we show that the equilibrium takes
the form depicted in Figure 10 and in addition that q and q are increasing in t. Indeed if this
is
the case,
it is
clear that
any increase
in
t
The same conclusion applies to
t) — Q on the vertical
theorem. The general features of q
will increase Q*[t).
Q*{t), the smallest intersection in an analogous figure, except with q(Q,
axis rather than q(Q,
— Q. This
is the conclusion of the
that ensured his conclusion are as follows. First, we must show that q is in fact well-defined on
a compact interval in Q (for all t G T), below denoted [#,Q]. This step is non-trivial and is the
t)
most difficult part of the proof. Second, we must show that q(Q, t) is continuous in Q. Third, we
must show that q(Q,t) — Q "begins above the first axis and ends below it", i.e., we must show
that q(0,t) >
and q(Q,t) < Q. Finally, we must prove that q is increasing in t. In order to
prove these claims, it turns out to be very convenient to introduce an index concept.
f,(Q-r\i) (solid),
./'
T;(Q+A-ri,) (dashed)
Figure
11:
77-
G
B
Z
(Q)
if
and only
if
the graph of fi(Q
—
r/,,)
intersects the diagonal at
77'.
Consider (Q, 77^) with 77^ G Bi(Q). We say that the index of such [Q,r]i) is equal to -1 (+1) if
and only if ft {Q — z) — z is strictly decreasing (strictly increasing) in z on (r/i — e) U (77, + e), e >
(on the two one-sided open neighborhoods of 77,). If the index is not equal to either —1 or +1, it
is set equal to 0. These indices are well-defined whether or not r% is single-valued at Q — r\i since
44
34
the local solvability condition implies that fi can be multivalued only at isolated points.
Figure 11 illustrates the situation including the indices just defined. For Q given, 77, G
(Q)
if and only if the graph of f t (Q - •) meets the diagonal. The dashed curve illustrates an important
B
t
When the aggregate is changed from Q to Q + A, A > say, this corresponds
exactly to a parallel right transformation of fi(Q — -)'s graph, i.e., every point on the curve in
the figure is moved to derive by exactly an amount A > 0. This simple observation has several
important consequences, (i) First, if any two points on the graph of B x lie on a 45° line, then the
whole interval between these two points must lie on the graph of Bi. In particular, if rji G Bi(Q)
feature of B{\
and T]i + A G B (Q + A), then rh + S G B^Q + 5) for all 6 g [0, A], (ii) Second, if r?< E B^Q')
and the index of (77^, Q') is not zero, there exists a continuous function /, (Q' — e, Q' + e) —» 5t
such that fi(Q) G B {Q) and fi(Q') = ?],' In the following we shall refer to such a function as
a continuous selection through (jf^Q ). This extension procedure can be repeated as long as the
index does not become equal to zero at any point or the boundary of 5,; is "hit" by the extension.
(Q' — e,Q' + e) — Si is the continuous selection from before. By the
Indeed, imagine that fy
upper hemi-continuity of S, (which follows straight from upper hemi-continuity of B4) and the
continuity of fi follows that fi can be extended to the boundary points, /, \Q' - e, Q' + e] — Si
in such a way that it is still true that fi{Q) G B{(Q) for all Q. As long as the index is non-zero
on these boundary points, we can then take each of these as our starting point and start over in
order to get a continuous selection /, \Q' — e', Q' + e'j — 5 where e' > e. An important thing to
note about such a continuous selection that extends from an initial point [rn,Q) is that the index
never changes along the selection: If the index of (rji, Q) is —1 (+1), the index of (fi{Q), Q)) is
— 1 (+1) for all Q at which the continuous selection is defined, (iii) Thirdly, because f (Q) >
t
:
%
1
>
:
>
:
>
:
t
l
at or above the diagonal. Hence it
the curve in Figure 11 always begins at rji =
follows from the mean value theorem that if n'i G Bi(Q') for some Q' > 0, there will exist some
for Q' >
then B{ {Q) # for all
> Q'
Vi e B t {Q) for all Q > Q' In other words: If Bi{Q') ^
[When Q is raised to Q' the point (r/^Q') is shifted "to the right" and so will lie below the 45°
for all
Q >
0,
Q
.
line/diagonal.
r)i
=
where
By
it
continuity the curve must therefore intersect the diagonal in order to get from
is
above the diagonal to
where
n^
it
is
below
it],
(iv)
Also from the fact that
follows that the index of the first intersection in the curve (corresponding
f° r all Q >
fi{Q) ^
whenever fi{Q) > 0. The case
to the smallest element in B l (Q)) will be equal to either —1 or
35
(v) Finally, imagine that for some
where fl {Q) =
is special and is considered in a footnote.
= max Bi(Q')
(the largest elements in Bi{Q) C K + ), and consider a continuous selection fi
Again because the graph of fi is shifted to the right when Q is raised, it is easy
to verify from the figure that for all Q > Q' where fi is defined, f {Q) = max B{(Q).
We are now ready to establish an important lemma.
Q', n\
through
Q').
(rji,
l
Lemma
1
For each player
each defined on
values in
[0j,
Clearly, for
Q<
Precisely,
This
0i
there exists a finite
= min Sie s
and
f,(0)
t
number of continuous functions 6 i
bi Mi
=
max{^
R+
s
and
taking
hi(s
G
G
5}
Q
ti
,
.
.
,
.
:
r )
,
m
min^gs,
Proof. By construction
of Z.
£l
such that for every Q G [0, Q], Bi(Q) = U m L 1 6j i7T[ ((5) (in particular B t (Q) =
for
In addition, for
Mi} it cannot happen that b ltTn (Q) = b ljn i{Q) for any
^ m' G {1,
K+
Q < 8j).
Q with bi im (Q) >
34
i
Q] where
is
if
Qx
.
.
G
B
l
(9 l
).
Q z there cannot exist
f,(Z)
is
,
.
hi(s,).
We
rji
G
first going to show that Bt{Q)
(Q) with n, > Q (indeed for such
are
B
not single- valued at Z.
l
it
must be single-valued on a
—
rji
for all
and
Q
Q <
6
t
.
we would
neighborhood
any solution to
sufficiently small
a consequence of local solvability because this condition implies that for
Q
fixed,
G fi(Q — r],) must be locally unique. Indeed, 17, € f (Q - 7/,) <=> [//, = h,(si) k ^i{s ,Q) = 0] and s, is locally
uniquely determined as a function of Q when \D s ^i\ ^
(see, e.g., Milnor (1965), p. 8).
35
When f t (Q) = it is possible that the curve intersects the diagonal at (r] t ,fi(Q — 77,)) = (0,0) and "stays
above" the diagonal for all r;, >
in a neighborhood of 0. If we extend our index definition above to boundary
t/i
t
t
points in the natural way, this intersection would have index
+1
(the natural
way
to extend to a
boundary point
to require that the original index definition holds on a suitably restricted one-sided neighborhood).
45
is
.
G Ri(Q - rji) where Q - rji < 0. But Ri is only defined on JZj^i hj(Sj) Q R+).
If Q < 6i
rji G Bj(Q), we must therefore have rji < Q (we cannot have equality because then
would
not
be
the minimum of fj(0)). So assume, ad absurdum that rji < Q. Let Sj be such
9i
—
= hi(si)). Unless Sj is on the boundary of Sj, we
that Si G ^i(Q
?7i) (in particular then 7ft
=
first-order
conditions
are necessary for an interior solution. By the local
since
have $i(si, Q)
which implies that there exists a "downwards"
solvability condition therefore \D Si ^i(si,Q)\ ^
— Si. At (fi(Q - e),Q - e) we again have
—
e,Q]
continuous selection through (rji,Q), fi
(Q
does not hold) that the
(unless fi(Q) = hi(si) where Sj is a boundary point and $>i(si,Q) =
local solvability condition applies and so we can extend further downwards. This process continues
until we reach a point Q' < Q such that any Sj G Si with fi(Q') = hi(.Si) is a boundary point of
S{. If the index of (6i, 9i) is equal to +1, we can begin the above selection procedure from (0j, 9i)
and will, as mentioned, eventually reach a boundary point in the above sense. But when this
still holds (^ is continuous), and so from the
selection first "hits" the boundary ^i(si,Q) =
local solvability condition follows that the index is (still) +1. Because of (i) above, this selection
is the maximal selection: There will be no rji G B (Q) with rji > fi(Q) for any Q where fi is
36
or — 1, we can either still construct a continuous
If the index of (9i,9 ) is either
defined.
downwards selection through (#j, 9 % ) as the one above (precisely this happens in the index is and
the graph of fj(0j - •) intersects the diagonal from below), or else we can begin at (rji, Q) (where
as before Q < rji) and construct a similar maximal extension. Precisely this happens either if the
and the graph of f (9i — ) intersects the diagonal from
index of (8i, 9{) is —1 or if the index is
above. In either case the index of the downward selection through (rji,Q) must be +1 so this
leads to a situation that is entirely symmetric with the previous one.
In sum, at Q' there is exactly one solution to the equation $?i(si,Q') =
and here hi(si) =
index
is
equal
to
Since
>
for
And
the
of
+1.
any
fi(Q)
Q > 0, this implies
(fi(Q'),Q')
fi(Q')that a selection from B different from fi, exists on [Q',Q' + e], e >
(recall from (iv) above
that the index of the smallest selection from Bi is — 1 unless possibly if fj(0) = 0. But fj(0) =
cannot happen unless 9 X =
in which case the claim that B t (Q) =
for Q < Bi is vacuous).
have to have
Sj
and
*
t
t
t
t
,
This selection is of course locally non-decreasing (possibly constant). Importantly, it necessarily
has index equal to —1. But then two selections with different indices "meet" at Q'. This can
never happen if ^i(si,Q') =
and z' = h (s l ) hence we have arrived at a contradiction if payoff
functions are assumed to satisfy the first-order conditions at any best response for the players. If
not, we have a contradiction with regularity which specifically says that there does not exist an
aggregate such as Q' above where all backward replies' indices are positive.
We conclude that Bi(Q) — for Q < 8i which is what we set out to do. Along the way, we
also ended up concluding that the index of (9i,9i) cannot be +1, nor can it be
if the graph
of fi(9i — •) intersects the diagonal from below. In either case, this implies that we can find a
continuous upwards selection through (8i, Bi). This selection has index —1 whenever ft (Q) is such
l
that fi(Q) = hi(si) where s^ is in the interior of 5, (in fact, it is sufficient that the first order
conditions hold, so this is always true in the multidimensional case and also in the one-dimensional
case unless fi(Q) is horizontal, in which case the statements below become trivial). Furthermore,
remains the maximal selection from Bi for all Q > 9
on all of [9 Q] because the
index is equal to —1 throughout (see also (iii) above). In term of the lemma's statement, we have
— M.+ namely the maximal selection from B % that
therefore found our first function bi i
[6i, Q]
since this selection
(this follows
from
is
maximal
at 9i
it
l
(v) above). Finally, the selection will actually exist
>
:
t
,
,
t
intersects at (9i,9{).
Even under the
local solvability condition,
we cannot conclude that
this
is
the only selection
36
Imagine such a larger element in B, did exist,
6 B,(Q). This would imply that the whole 45° segment between
{rn,Q) and the intersection of the graph of /t with the 45° line through (fj,,Q) would lie in B 's graph. But this
is a contradiction because at the point where the 45° segment intersects with the graph of /,, local solvability
^
l
would be violated
argument)
(local solvability implies, in particular, local
l!,
uniqueness of selections, see footnote 34 for a similar
Local solvability does, however, imply that the number of selections must be finite
would not be locally isolated which is a fact we have used several times above).
when $i(sj, Q) =
for s,
In addition, local solvability rules out that any selection has index
with fi(Q) = hi(si). Hence the indices must be alternating in sign when ordered in the natural
way (the greatest selection above has index — 1, the next selection below it has index +1, etc.,
until the least selection which always has index —1). Also from local solvability follows that two
selection cannot intersect each other (except possibly at the "lower boundary" and not at all
jt m! € {1,
when TV > 1). If they did intersect, i.e., if for any two selections
M{\ we had
bi,m{Q) = bi, m >(Q) for any Q with &t, m (Q) >
Si£ g. /ii(sj), these selection would either have to
be equal everywhere (and so there is only one!), or else they would at some point deviate from
each other and at that point local uniqueness of selections (explained several times above) would
from B{.
(or else they
m
.
.
,
.
mm
be violated.
What remains to be shown now is only that all selections are defined on [#,, Q]. This has
already been shown to be the case for the greatest selection b \. That it is also true for the other
selections follows from a similar argument recalling that the indices must be alternating and that
lt
the index of the least selection
=
Let M.i
{I,...,
is
—1.
Mi} where Mi
making the dependence on
t
explicit,
The
is
we
details are omitted.
the
number
of functions of the previous
Now
lemma.
define
max
q(Q,t)=
{mi,...,mj)eMi x...xMi
VXm.tQ.O—
i
Clearly q(Q, t) is only defined on [9, Q] where 9 = max, 9 % It is also clear that q takes values
in the interval [0, Q\. Prom now on we focus on q, which will prove the statement of Theorem
7 for the largest equilibrium aggregate. For the smallest aggregate one considers instead q(Q, t)
defined as above replacing max with min and proceeds as below with the obvious modifications.
.
The
following
lemma
will essentially finish the
proof of the theorem by establishing
all
of the
claims mentioned in our outline of the proof above.
Lemma
Q] x T — [0, Q] is continuous in
6 (of course we also have q(Q, t) < Q).
The function
2
T, and satisfies q{9)
>
q
:
>
[6,
Q
on [0,Q], increasing in
on
t
Proof. Continuity of q in Q follows directly from the definition of q and lemma 1. Let m'{i)
denote the index of the selection that solves (11.4) for player i e I. In the notation of the proof
of Lemma 1, m'(i) must in fact be equal to 1 for every player (the greatest selection). For this
greatest selection, it was shown in the proof of the Lemma that b j(9j, t) = 9,. It follows that
— Si °i^(9, t) > 6;',i(#, t) = 9 where i' is any player for which 6, = 9. This proves the last
q{@>
claim of the Lemma. Finally, we show that q is increasing in t. For any Q, 77, = b im '^(Q, t) if and
l
only
if ?7j is
the largest solution to the fixed point problem:
positive shock in the sense that the smallest
Q
and
and
£,
£ ?i{Q~ £i,
largest selections of fi{Q
t).
By
— £,,
assumption,
=
t)
h l oR
l
t is
{Q-^
l
,
a
t)
Moreover, the smallest (respectively, the largest)
selection from an upper hemi-continuous correspondence with range R is lower semi-continuous
37
(respectively, upper semi-continuous).
In particular, the least selection is "lower semi-continuous
from above" and the greatest selection is "upper semi-continuous from below" Hence any upper
hemi-continuous and convex valued correspondence with values in R satisfies the conditions of
are increasing in
t
(for all fixed
£j).
.
be such a correspondence, and /. and /* the smallest and largest selections, i.e., f"(x) =
and f,(x) = -[dieoc j6 .f(i)z]. Since the value function of a maximization problem is upper semicontinuous when the objective function is continuous and the constraint correspondence is upper hemi-continuous,
it follows that /* is upper semi-continuous and moreover f.(x) = maXjj-p^j z is upper semi-continuous, thus
37
Let
F
:
X
—
»
2
R
max^f^jz
implying that
/,
=
-/.
is
lower semi-continuous.
47
We
Corollary 2 in Milgrom and Roberts (1994).
in
therefore conclude that b im i^(Q,t)
(and likewise that the smallest fixed point of
t
& € fi(Q — &,£)
uses to prove the theorem's statement for the smallest aggregate).
increasing in t.
is
increasing in
is
increasing
t
which one
Clearly then, q(Q,t)
As was mentioned above, when the continuous function q(Q, t) (where t
everywhere up because t is increased, the largest solution of the equation
is
fixed)
is
is
also
shifted
Q =
q(Q, t) must
increase (and for the smallest aggregate the same conclusion applies to the smallest solution of
q(Q, i) = Q). This completes the proof of the theorem.
Proof of Theorem 8
11.5
The statement
aggregate
Theorem
and
t(I)
is
7.
proved only for the smallest equilibrium aggregate (the proof for the largest
Let bm be the least backward reply map as described in the proof of
Let us define t{I+\) = sj +1 6 K+ (i.e., as equal to the entrant's equilibrium strategy)
is
similar).
i
= 0. Then Q,(I) and Q»(I + 1) are the
= g(b(Q(I + 1)), t(I + 1)), respectively.
least solutions to,
Q(I)
=
g(b{Q{I)),t(I)) and
Since g is increasing in t and t(I) < t(I + 1),
this is a positive shock to the aggregate backward reply map. That Q*(I) < Q*(I + 1) must hold
then follows by the same arguments as in the proof of Theorem 7. Clearly, Q*(I) = Q»(I + 1)
Q(I +
1)
cannot hold unless
=
t(I
+
1) since
g
strictly increasing.
is
Proof of Proposition 3
11.6
We
t(I)
begin by verifying the local solvability condition in the contest
game
of Proposition
3.
Direct
calculations yield
Vi(si,Q)
=
Vi
h\( Si )
H' (H-^Q)) hfadhi
R+Q
(R +
c
'i
i
s i)
QY
and
£>s,*»
=
R+Q
h'(si)
[R +
Q?
H'{H-\Q))(h[(s )f
t
Vi
(R
Therefore,
when $j(s„Q) =
0,
£>.,,*<
h'ASr)
+ QY
we have
A
l
^M-Vi
(R
+ Q)
2
Dividing both sides by c[(si) > 0, we conclude that .D^^ < 0, and thus the local solvability
condition is satisfied and each 3^ is regular.
It is also straightforward to verify that c t — c t oh~ is a convex function under our hypotheses.
Next, consider the payoff function of player' i after the change of coordinates s,; h-» Zi = hi(si)
-l
(for
that
i
G J):
Zi is
a strict
= ViZi R +
H(1l,j=i zj)\
~
Ci{ z i)-
If
D
Zi
TTi(z)
=
is
sufficient to
ensure
a global maximum, then the original first-order condition are also sufficient for a global
= implies that 2 2 Tti(z) < 0. Hence any interior extremum
In particular,
Zi fii(z)
maximum.
is
7Ti(z)
D
maximum, from which
D
follows that there
48
is
at
most one extremum, necessarily a global
1
maximum
maximum
possible that this
(it is
to be unique; see also
Remark
on the boundary, though
is
in this case
it
continues
3).
Now, parts 1 and 2 the proposition follow directly from Theorems 7 and 8. Part 3 follows
from Theorem 9 by noting that the condition for s*(t) to be locally increasing in a positive shock
-
[DMalgis'lfJl^DQViifilgWt) >
0.
(20)
shown above, DSi ^i(s*,g(s*),t) < 0, (20) holds if and only if Dq*j(s*,Q*, t) > where
For the same reason, the condition for s*(t) to be decreasing in t when t does not
affect player i (the second statement of part 3), is satisfied is and only if Dq^/^s*, Q*) < 0.
Since, as
Q* =
g(s*).
directly
Since,
(R + Q*) 2
(20) will hold
and only
if
(R
+ Q *\3
[R
+ Q *\2
if
h(s*) > V (Q*),
where
H"(H-
2H'(H-\Q*))
vm =
H'{H-
+ Q*)
(R
}
l
(Q*))
(Q*))
This shows that player i will increase its effort if it is "dominant" as defined in the proposition. If
instead hi(s*) < n(Q*), i.e., if the player is not "dominant", Dq^^s^Q*} <
and by Theorem
9 follows that if the player is not affected by the shock, she will decrease her effort in equilibrium.
Proof of Theorem
11.7
We
1
begin with a technical lemma:
Lemma 3
(i)
D
Suppose Assumption
^i
St
(ii)
D2
s
(sj,
Hi
I
53- =1 Sj,£)
Si,
Yjj=i s ji *)
1
holds.
exists
*s
Then:
and
all
of
its
elements are non-positive; and
^e^aiiue definite.
A
Proof. For a matrix
with non-negative off-diagonal entries the following four statements are
equivalent (see Berman and Plemmons (1994), pages 135-136): (1) all eigenvalues of
have
negative real parts; (2) all real eigenvalues of
are negative; (3) there exists a vector x E IR+ +
A
A
-1
exists and all of its elements are non-positive.
E K^_; (4) A
It is clear from (16) that if D^ s II, has non-negative off-diagonal entries and Dq^, is nonpositive, then D Sl ^ must have non-negative off-diagonal entries. By assumption, all real eigenval-
such that
Ax
t
ues
Ds^j
of
(sj,
]Cj=i
s j>t)
are negative, hence (4) holds verifying the
the second claim, we use that (3) holds for
K^_.
Dlsflr
Clearly
DQ^i
£j=i^*
x E M.^ because
X E
tN
D $„
But then
and
St
Dq^i
is
(since
49
let
first
claim of the lemma.
For
x E 1R+ + be such that D Si ^i x E
Hence from (16) follows that
non-positive.
D2
.
s
IT has non-negative off-diagonal elements)
of its eigenvalues have negative real parts,
all
is
1
'
<
Q=
b(Q
+
dQ = D Q b{Q + t)dQ + D Q b{Q +
Dqb(Q +
therefore negative definite.
Si
the payoff function exhibits decreasing differences).
Finally, to establish the main result, differentiate
Since
it is
= —[D i>i]~ DQ'$>i, part (i) of Lemma 3 implies that the backward
decreasing in Q (since Dq^/^ is a non-positive matrix in view of the fact that
Next note that since Dcjb
reply function b
and being symmetric
t) is
non-singular, this
is
t)
to obtain:
t)dt.
equivalent to
[[D Q b(Q
+ 1)}- 1 - X\dQ = dt.
>
sufficiency part of the theorem will follow if we can show that dt
_1
- I]dQ > =>•
previous equation, this is equivalent to: [[Dq6(Q + i)]
(but again equivalent) way of writing this is,
The
[I
- [D Q b(Q +
l
t)]~
]dQ >
dQ >
=*>
=>
dQ <
dQ < 0. By the
An alternative
0.
(21)
.
=> x >
in (21) is very well known in matrix algebra: a matrix A such that Ax >
monotone matrix (Berman and Plemmons (1994)). A well known result from matrix
algebra tells us that a matrix is monotone if and only if it is a non-singular A<f-matrix (Berman
and Plemmons (1994), page 137). Since [I - \Dqb(Q + t)}' is non-singular by assumption, it is
a non-singular M-matrix when it is an M-matrix (as assumed in the theorem). Hence, it will be
monotone and so any small increase in t (in one or more coordinates) will lead to a decrease in
The statement
is
called a
1
}
each of Q's coordinates.
As for the theorem's necessity statement, assume that [I— [DQb(Q + t)]~
By
the result just used, this
which implies that dQ £
We cannot have [I — [DQb(Q
is
and
- [DQb(Q + t)]~^]dQ <
— [I— [DQb(Q + t)]~^]dQ >
11.8
We
[I
- [D Q b(Q +
t)]-
l
[I
— [DqbiQ +
]dQ <
for at least
not an Af-matrix.
is
]
t)]~
is
]
not monotone,
one vector
dQ e R N
.
=
1
since I — [Dqb(Q + t)]"
is non-singular; hence
+ t)]~ ]dQ
for some such vector dQ ^ 0. Now we simply pick t" - t' = dt =
and the associated change in the aggregate dQ will then be increasing
[I
in at least
the same as saying that
i
l
one coordinate/component, which
the statement of the theorem.
is
Proof of Theorem 14
begin with Theorem
7.
To simplify the notation
a linear aggregator, g(s)
—
(the general case
only difference
is
that
it
5Zi=i s j
in the proof, let us focus
is
on the case with
proved by the exact same argument, the
becomes very notation-heavy). Then,
for
each
i,
Gi(Q
:
y)
= Q — y.
Define
M
- {y} = FLj,(y, t), where Rt is the "reduced"
Mi(y, t) = argmaxQ^j, Ui(Q -y,Q). Clearly,
t {y, t)
best response correspondence (i.e., best response as a function of the sum of the opponents'
Given singleSi,t) — {Q} = Ri{Q — s,,t) — {s^}.
Milgrom and Shannon (1994), Theorem 4). Therefore,
B^(Q — Si,t) - {.Si} must be descending in s t Moreover, R^(Q — &i,t) - {s } is convex- valued.
>
Let Bi{Q, t) = { Sl eSi-.Sifz Ri{Q - s u t)}. B l {Q, t) £
since: z £ Ri(Q - ± t ,t) => z 38
(where _Lj = min5 ), while z <E Ri{Q - Tj, t) => z - T t < 0.
It may be verified that B (Q, t) is
strategies).
Hence, we can write
is ascending in y
crossing, M{(y,t)
Mi(Q —
{e.g.,
.
t
U
t
38
t
= {R,(0)} when z < 0, where we have here taken ±j =
can assume is 0. It is clear that with this extension Ri(Q - s,) — {si} is (still)
descending and now always passes through 0. Importantly, the extensions (one for each agent), do not introduce
any new fixed points for B = J2, Bj: Given Q, if s, e Ri(Q - Si), then either Q - Si >
or Q < Si e FU(Q — s ).
But if Sj > Q for just one i, we cannot have J\ s 3 = Q.
Here
it is
necessary to extend Rt by denning i?,(z)
for all j so that the least value
Q
t
50
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