Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/existenceofsubgaOOharr
HB31
.M415
no. S'U
f
economics
1991
THE EXISTENCE OF SUBGAME- PERFECT EQUILIBRIUM
IN GAMES WITH SIMULTANEOUS MOVES
Christopher Harris
Number 570
December 1990
massachusetts
institute of
technology
-
50 memorial drive
Cambridge, mass. 02139
THE EXISTENCE OF SUBGAME- PERFECT EQUILIBRIUM
IN GAMES WITH SIMULTANEOUS MOVES
Christopher Harris
Number 570
December 1990
ft-'..
' '
M.l.T. LiBi
apr o
1
nm
fl
"-'
Iffl
The Existence
in
of
Subgame—Perfect Equilibrium
Games with Simultaneous Moves:
a Case for Extensive—Form Correlation
Christopher Harris
Nuffield College, Oxford
December 1990^
Abstract
The
starting point of this paper
subgame-perfect equilibrium
existence would be restored
result of the
if
fails
is
to exist.
a simple, regular, dynamic
Examination of
this
game
in
which
example shows that
players were allowed to observe public signals.
The main
paper shows that allowing the observation of public signals yields existence,
not only in the example, but also in an extremely large class of dynamic games.
should like to thank Drew Fudenberg, Daniel Maldoom, Andreu Mas— Colell, Eric
Maskin, Jean—Francois Mertens, Meg Meyer, Jean Tirole, John Vickers, and
participants at seminars and workshops at Oxford, Stonybrook and Harvard for
helpful comments on the earlier paper from which the present paper is drawn.
The present paper is drawn from an earlier paper entitled "The Existence of
Subgame—Perfect Equilibrium with and without Markov Strategies: a Case for
Extensive Form Correlation".
I
1
Introduction
Consider the following example of a dynamic game. Firms set out with exogenously
specified capacities (which are
known
to
all).
In period one they simultaneously choose
investment levels (possibly on a random basis), and are then informed of one another's
choices.
The
result
is
a change in capacities. In period two firms simultaneously choose
production levels (within their capacity constraints), and are then informed of one
another's choices.
The
result
simultaneously choose prices.
their products, but so
is
a change in inventories. In period three firms
The
vector of prices chosen affects the vector of
demands
for
do certain exogenous random factors. Firms are informed of one
another's chosen prices and of the final realised demands.
second change in inventories.)
The
(The demands bring about a
three period cycle of choices the begins afresh.
And
so
on.
This
There
is
game
a finite
given period:
all
is
an example of a game of the following general type. Time
number
of active players.
There
players (both active and passive)
periods; the set of actions available to
is
is
also a passive player, Nature.
know the outcomes
any active player
is
discrete.
In any
of all previous
compact, and depends
continuously on the outcomes of the previous periods; the distribution of Nature's action
(which
is
given exogenously) depends continuously on the outcomes of the previous periods;
the players (active and passive) choose their actions simultaneously; and the outcome of
the period
is
simply the vector of actions chosen. The outcome of the game as a whole
the (possibly infinite) sequence of outcomes of
all
periods,
is
and players' payoffs are bounded
and depend continuously on the outcome of the game.
This
is
as regular a class of
dynamic games
as
one could ask
for.
A
counter-example
shows, however, that games in this class need not have a subgame—perfect equilibrium.
is
therefore necessary to extend the equilibrium concept in such a
way
that existence
It
is
restored. l
The problem
is
reminiscent of that of the non-existence of Nash equilibrium in pure
What
answer to
is
the most natural extension of the equilibrium concept?
this question is
One
provided by the following considerations. First,
if
clue as to the
we
simplify
the class of games under consideration by requiring that players' action sets are always
finite,
then subgame—perfect equilibrium always
appears to relate specifically to the fact that
we
exists.
Hence the non-existence problem
allow a continuum of actions.
Secondly,
suppose instead that we simplify the class of games under consideration by assuming that
players' action sets are independent of the
way
of approximating a
large but finite
game
numbers of
is
to consider subsets of players' action sets that consist of
it is
Moreover,
closely spaced actions. 2
increasingly fine approximations, and a
approximations, then
outcomes of previous periods. Then a natural
if
one takes a sequence of
subgame—perfect equilibrium
of each of the
natural to expect that any limit point of the sequence of
equilibrium paths so obtained will be an equilibrium path of the original game. 3
One
can,
however, find examples in which the limit point involves a special form of extensive—form
correlation: at the outset of each period agents observe a public signal. 4
strategies.
But there
is
one important difference:
then subgame—perfect equilibrium does
if
agents' action sets are always finite,
exist.
2
Hell wig and Leininger (1986) were the first to introduce such an approximation, in
the context of finite—horizon games of perfect information. They proved an
upper—semicontinuity result: they showed that any limit point of equilibrium paths of the
finite approximations is an equilibrium path of the original game. Their result can,
however, be understood in terms of the results of Harris (1985a). Indeed, in that paper it is
shown that the point—to—set mapping from period—t subgames of a game of perfect
information to period—t equilibrium paths is upper semicontinuous. Hence, in order to
obtain their result, one need only introduce a dummy player who chooses n e IN U {qd} at
the outset of the game. If n < od then the remaining players will be restricted to actions
chosen from the n
approximation to their action
choose actions from their original action sets.
sets.
If
n
=
od
then they will be free to
Convergence of equilibrium paths, as used implicitly in Harris (1985a) and explicitly
and Leininger (1986) and Borgers (1989) seems more relevant than the
convergence of strategies considered by Fudenberg and Levine (1983) and Harris (1985b).
3
in Hellwig
4
Fudenberg and Tirole (1985) consider two such games. They point out that the
obvious discretisations of these games have a unique symmetric subgame—perfect
equilibrium, and that the limiting equilibria obtained as the discretisation becomes
arbitrarily fine involve correlation. Their games do not, however, yield counterexamples to
the existence of subgame—perfect equilibrium. For both games possess asymmetric
equilibria which do not involve correlation.
These considerations suggest that,
at the very least, the equilibrium concept should
be extended to allow the observation of public signals. This minimal extension
sufficient.
For,
first, it
restores existence.
Secondly, with
correspondence of a continuous family of games
limit,
is
it,
is
the equilibrium
game whose
action sets are
an equilibrium path of the game. Thirdly, any limit point of subgame-perfect
equilibrium paths of a
The
also
upper semicontinuous. In particular, any
point of equilibrium paths of finite approximations to a
continua
is
game
is
e-
an equilibrium path in the extended sense. 5
basic structure of the proof of the existence of subgame-perfect equilibrium in
the extended sense
is
the same as the structure of the proof of the existence of
subgame-perfect pure—strategy equilibrium in games of perfect information given in Harris
(1985a).
It
breaks
down
The backwards program
into
is
two main
most
horizon T. In such a game,
it
equilibrium paths of the game.
period—T subgames. (The
parts, the
easily explained in the context of a
More
set of
precisely,
one solves
first for
(The
set of
is
the convex
for
equilibrium paths of a
a set of probability distributions over period—(T— 1) histories,
And
so
reached, and a set of probability distributions over period— 1 histories
There
is
period—T outcomes.) Then one solves
a joint distribution over period—(T— 1) and period—T outcomes.)
is
finite
the equilibrium paths of
equilibrium paths of a period—T subgame
the equilibrium paths of period—(T— 1) subgames.
is
game with
what turn out to be the
consists in solving recursively for
hull of the set of probability distributions over
period—(T— 1) subgame
backwards and the forwards programs.
on
is
i.e.
until period 1
obtained.
of course no general guarantee that the paths obtained in the course of
carrying out the backwards program really are equilibrium paths until equilibrium
strategies generating
them have been constructed. To construct such
purpose of the forwards program. The essential idea
is this.
5
Its
is
the
Pick any of the period—
paths obtained as the final product of the backwards program.
distribution over period— 1 histories.
strategies
Such a path
is
a probability
marginal over period— 1 outcomes can be used to
Chakrabarti (1988) claims that subgame-perfect e-equilibria exist.
claim, but are unable to vouch for the proof.
We
believe this
construct period— 1 behaviour strategies for the players. Its conditional over period—
be followed from period 2 onwards in the event that
histories yields continuation paths to
players do not deviate from their period— 1 strategies.
In order to obtain continuation
paths in the event that some player does deviate from his period— 1 strategy,
to choose
from among the period—2 paths obtained
in the course of the
some path which minimises the continuation payoff of that
continuation paths for
all
way
period—2 subgames are obtained. Period—2 strategies are then
obtained from the marginals of these paths over period—2 outcomes.
finally
sufficient
backwards program
In this
player.
it is
And
so
on
until
period—T strategies are obtained.
Implementing
this proof does,
however, involve two significant new complications.
First, for technical reasons, it is necessary to
mapping from period—t subgames
maintain the induction that the point—to—set
to period—t equilibrium paths
This follows from a generalisation of the work of Simon and
because players
may
are measurable. 6
randomise,
That
this
is
Zame
upper semicontinuous.
(1990).
Secondly,
essential to ensure that all the strategies constructed
it is
requirement can be met emerges as a natural corollary of the
construction employed: one can always construct a measurable family of conditional
distributions from a measurable family of probability distributions. 7
The organisation
to the existence of
game with two
of the paper
subgame—perfect
is
as follows.
equilibrium.
players in each period.
Section 2 sets out the counterexample
This example involves a three—period
In this game, each player's action set
is
compact
6
Hellwig and Leininger (1987) were the first to draw attention to the desirability of
ensuring that strategies are measurable. Their approach to the existence of
subgame—perfect equilibrium is not, however, necessary in order to obtain measurability.
Indeed, if the assumptions of Harris (1985a) are specialised appropriately (by assuming
that the embedding spaces for players' action sets are compact metric instead of compact
Hausdorff), then it is simple to show that exactly the construction given there can be
carried out in a measurable way. Indeed, in the notation of Harris (1985a; p. 625), all that
is
necessary
is
to ensure that the functions gj
this is possible follows
Lemma
can be chosen to be measurable; and that
from the argument given in the
first
paragraph of the proof of
4.1 in the present paper.
7
Had we employed a dynamic—programming approach, the problem of ensuring
measurability of strategies would have been much harder — see below for further discussion.
Indeed, we do not know how to solve the measurable selection problem that arises when
such an approach is employed.
and independent of the outcomes of previous periods, and each player's payoff function
continuous.
is
Section 3 formulates the basic framework used in this paper for the analysis of
dynamic games. Section 4 shows that when the dynamics are continuous, players' payoff
functions are continuous, and the framework
subgame—perfect equilibrium
signals,
4.1 provides a detailed
for the
exists.
is
extended to allow the observation of public
This
is
the main result of the paper.
Section
overview of the proof. Section 4.2 develops the analysis necessary
backwards program. Section
4.3 develops the analysis necessary for the forwards
program. Section 4.4 exploits the analysis of Sections 4.2 and 4.3 to prove the main
showing in particular how to deal with the infinite—horizon
develop a perspective on the main result.
types of
game
in
and zero-sum games.
It is
also
Section 5 attempts to
shown there that there are
which the introduction of public signals
existence of subgame-perfect equilibrium:
all
It is
case.
games of
is
result,
at least three
not necessary to obtain the
perfect information, finite-action games,
shown that the introduction of public
signals
accommodates
the equilibria that one can obtain by taking finite-action approximations to a game, or
by considering e—equilibria of a game.
signals
that
is
it is.
We
do not know whether the introduction of public
the minimal extension with these two properties, but
it
seems plausible to argue
The Counterexample
2
A
counterexample to the existence of subgame—perfect equilibrium in the
dynamic games described
First of
for
all, it
must have
in the introduction
at least
class of
must have a certain minimum complexity.
two periods. Otherwise the standard existence theorem
Nash equilibrium would apply. Secondly, there must be
at least
two players
in the
second period. For with only one player in the second period, the correspondence
describing the continuation payoffs available to the player or players in period one would
be convex valued.
between which he
(The player
is
in period
two can randomise over any
set of alternatives
So the existence result of Simon and Zame (1990) would
indifferent.)
apply to period one. Thirdly, there must be at least two players in period one. For the
continuation—payoff correspondence for period one will be upper semicontinuous, and a
an optimum.
single player will therefore be able to find
The counterexample presented
periods, with
two players
in
in this section
each period.
It
example would
class of
not this minimal.
has three
For, aside from being
in each period.
dynamic games considered
settle the question as to
It
would be of considerable interest to find an
example with only two periods and two players
minimal in relation to the
is
in this paper, such
an
whether equilibrium exists in two—stage games or
not.
The
cast of the counter—example,
period one two punters
two, two greyhounds
A
and B pick a
C and D each
their attitude to the race.
and
They pick
their choice variables, are as follows.
6 [0,1]
and b
€ [0,1]
receive an injection of size a
c e [0,1]
and d
6 [0,1]
,
they complete the course. In period three each of two referees
winner, picking e 6 {C,D} and
f e
respectively.
{C,D}
respectively.
+
b
,
In
In period
which changes
which are the times in which
E and F must
declare a
In each period choices are
made
simultaneously, and players in later periods observe the actions taken by players in earlier
periods.
Punter
A
obtains a payoff of
both referees, and
D
declare
—1 —
to win,
—a
—1 — b
and
and both want a
They would
= C
and
1
—
(a
+
— c)
b)(l
e
if
whether he or the other greyhound
would prefer to run the race
=D
.
That
is,
e
if
= D
and
1
—
(a
E
in the verdict of referee
winner and
c
It will
if
—
b)(l
d)
if
e
= C
E
Lastly, referee
.
D
he declares
:
both referees
wants
C
to win,
to
C
greyhound
2c
is
if
E
way he
but either
,
Also, he would prefer to be first rather
as slowly as possible.
+
A
if
also like to keep their
declared the winner by referee
is
—b
1
the form of his payoff depends on
than second provided that he does not have to run too
2d
declared the winner by
obtains
The payoff
contributions to the injection as small as possible.
e
B
is
In other words,
otherwise.
result.
C
greyhound
if
Similarly, punter
a otherwise.
to be the winner,
B wants D
1
The payoff
fast.
greyhound C
like
gets payoff d
to be the winner.
F
Referee
be seen that, in this game, each player's action
he
greyhound
is
only interested
C
he declares
is
's
,
D
to
is
to be the
payoffs are identical.
set is
compact and
independent of the actions chosen by earlier players, and that players' payoffs are
continuous.
The game
c
centres around the two greyhounds.
< d and D when
c
>
d
,
the race between
the discontinuous payoffs (2c,
c
>
d
— (a +
1
b)(l
them
— d))
is
<
c
if
weights in the convex combination can depend on (c,d)
therefore yield the following
Lemma
2.1
Suppose that a
distribution function
+
(a
,
(1
+ b)(l-x))
for
G
+
b
>
c
(a
=
+
d
-
b)(l
c),
2d)
if
(Note that the
.
Standard considerations
Then both C and D use mixed
.
given by G(x)
xe
.)
-
of timing with
lemma.
[$,1]
=
for
x
e [0,|]
and G(x)
=
actions with
(2x
-
l)/(2x
-
1
.
Notice that this mixed action
its
d
C when
chooses
game
essentially a
and some convex combination of these two payoffs when
,
E
Because referee
is
non-atomic, that
probability mass concentrates at £ as a
+
b
-»
0+
.
its
Not
support
is
[j,l]
,
and that
surprisingly, then,
we
all
obtain:
Lemma
2.2
one.
d
+
Suppose that a
The remainder
of the
b
=
argument
straightforward.
is
d with probability zero. Hence both referees
+
indifferent as to
C with
b
=
,
then c
=
d
=
choose
a
If
+
b
$
with probability
>
then c will equal
always agree. Also, each greyhound will
will
A
win exactly half the time. The payoffs to
other hand, a
D
Then both C and
.
—b.
and B are therefore —a and
with probability one. Hence each referee
\
on the
If,
is
which greyhound he declares to be the winner. Suppose that E opts
D
probability p and
D
probability q and
with probability
with probability
1
-
q
1
—p
A
Then
.
and that F opts
,
's
payoff
2pq
is
for
—
for
C with
and B
1
's is
2(l-p)(l-q)-l.
It is
game with
easy to see, however, that the
no equilibrium. Indeed, any a
>
is
strictly
So the only possibility for an equilibrium
B
b
to set
=
— otherwise A
two
is
with probability one. But
would
raise a
from
zero.
dominated
A
for
if
these payoffs between
b
to set
=
then
Similarly,
2(1
A
for
a
=
it
,
A
any b
as is
B
and
>
for
has
B
.
with probability one and
must be that 2pq
— p)(l — q) —
1 >
.
—
1 >
And
these
inequalities are mutually inconsistent. 8 This contradiction establishes the counter-
example.
The essence
of the counterexample
use strictly mixed actions.
is this.
As long
as
a
+
b
>
,
both greyhounds
This behaviour on their part generates a public signal
endogenously within the game. The two referees use this public signal to co-ordinate their
actions.
in
When
a
+
b
=
,
however, this signal suddenly degenerates, and the only
which the referees can coordinate their actions
one or both choosing
D
is
with probability one. But
way
by both choosing C with probability
if
they both choose
C
then
B
gets
A
and B is
see this, note that the sum of the payoffs of the two players
if e ^ f. Hence the expected value of this sum is non—positive. If the
expected value is strictly negative, then the payoff of at least one player is also strictly
negative. If the expected value of the sum is zero, then e and f must be perfectly
To
8
e
=
f
and —1
correlated.
In this case the payoff of the player against
whom
the decision goes
is
—1.
if
—1
.
then
So he would prefer the truly random outcome. Similarly,
A
will
if
both referees choose
D
wish to restore the truly random outcome.
In the light of this explanation,
it is
players to observe suitable public signals.
natural to try to restore existence by allowing
In the present example, this would
amount
to
allowing the two referees to toss a coin to determine the winner in the event of a tie (which
would certainly restore existence). That such an extension yields existence
case will be demonstrated in Section
4.
in the general
10
The Basic Framework
3
The Data
3.1
There
a
is
non-empty
whose index
also a passive player, Nature,
is
J=
therefore
J^i
U {0}
Time
.
J<A of active players, indexed by
finite set
is
eX
i
e
J
must choose an
and
Jn
«J=
game
is
=
j
The
.
indexed by
The
.
se J"=
t 6
=
STjd
set of actions available to her or
—
X
by a point—to—set mapping A..:
-»
Y,.
—
The vector x
.
is
{0,1,2,...}
.
each player
{1,2,...}
him depends on which
being played, and on the outcomes of previous periods. This situation
is
There
.
overall set of players
or
t
j
players play one of a family of
In each active period
.
The
action.
or
i
discrete,
For notational reasons, we assume that
games, parameterised by x
is
or
i
e
X
is
modelled
—
lists
the
parameter of the game being played, and the outcomes of any preceding periods, while
A, .(x
take
C
)
it
period
Y,
A.^x
that
is
t
is
-
—
)
=
—
Y,^
x
for all
(x
)
=
A,.(x
x
A X
,y,
)
eX~
such that x
payoff of any active player
outcomes of
vectors
t
> 1
.
x
all
=
periods.
~~
The
9
.
(x ,Xp...) e
X,0
EX
- Y.
set of
.
where Y.
,
X
And
outcomes possible
y, e
A,(x
)
modelled by a function
x
is
=
simply the
It is
and
Y..
x
in
set of all pairs
—
and
,
i.e.
depends on which game x
i
It is
—
.:
x
for all
)
—
~
X
€
nothing more than the set of profiles of actions that players can take.
modelled by the point-to-set mapping
A,(x
For reasons of expositional economy, we
the set of actions available.
such that x
Y.
x
u-:
the graph of A,
e
Xw
—
t-1
=
X
is
-» IR
.
.
Finally, the
played, and the
Here
/
X
00
is
N
(x ,x. ,..,x,_.)
n
the set of
V t-1
eX~
.
for all
(Nature does not have a payoff function.)
We make
the following standing assumptions about these data:
(i)
X
(ii)
for all
i
€
X4
and
all
(iii)
for all
i
e
J<A
,
u-
is
,
and
all
of the sets
Y..
t
> 1
are
,
,
non-empty complete separable metric
A..:
X
—
- J6(Y,-)
is
spaces;
measurable;
bounded and measurable.
This assumption does in principle involve a slight loss of generality. However, since
any case fix Nature's strategy below, there seems to be little advantage in
constraining her actions as well.
9
we
shall in
11
mean
(In this paper, 'measurable' will always
Borel measurable unless explicitly stated to
•%(•) will always denote the space of non-empty compact subsets of
the contrary; and
endowed with the Hausdorff metric.) These assumptions are designed
•
to ensure that every
player possesses at least one strategy, and that associated with every strategy profile and
every subgame there
is
a well defined payoff for every active player (see below).
Stronger
assumptions are needed to ensure the existence of equilibrium.
Strategies
3.2
and extended strategies
In the standard version of our model, the evolution of the
beginning of period
informed of x
1.
players are told which
1,
They then simultaneously choose
.
This completes period
period
1.
That
is,
sets for period 2.
game they
they are told y,
.
2,
2.
And
so
be identified with the set of subgames in period
For each
measurable function
x
t-1
,
6
f,.:
t
X
> 1
and each
~~
->
is,
9Jl^{
i
e
J
t
,
At the
they are
players are told the outcome of
it
goes on.
—
Definition 3.1
That
are playing.
can be summarised by a vector x
t
as follows.
They then choose simultaneously from
This completes period
information in period
is
actions from their action sets for period
At the beginning of period
1.
game
It
their action
follows that players'
eX~
—
X
and that
,
can
.
a strategy for player
in period
i
t
is
a
_
4
-)
such that supp[f,.(x
)]
C
A,.(x
)
for all
v t-l
X
Here
9Ji{XA
denotes the set of probability measures over Y,.
'probability measure' will always
mean
.
(In this paper,
Borel probability measure; and spaces of
probability measures will be taken to be
endowed with the weak topology 10 unless
explicitly
stated to the contrary.)
10
{A
all
In the terminology of Parthasarathy (1967), a sequence of probability measures
} C
&M(Y.)
converges in the weak topology to A
continuous bounded x-
Y
-» IR
f
.
iff
Jx dX n converges
to
/xdA
for
12
The
For each
are behaviour strategies.
f.-
thought of as the randomising device that player
him
available to
in period
when x
t
is
t
,
and x
i
will use to
i
f,
,
.(x
among
choose
the actions
As such,
the previous history of the game.
independent of the devices used by the other players in period
t
and of
,
can be
)
it is
devices used in
all
preceding and subsequent periods.
all
Strategies
and strategy
profiles certainly exist
under our standing assumptions.
Moreover, given any strategy profile and any subgame, the payoffs of active players can be
We may
calculated in the natural way.
therefore define
subgame—perfect equilibrium
as
follows.
Definition 3.2
such that, for
A subgame—perfect
—
all
x
t > 1, all
payoff in subgame x
e
equilibrium
We
{£. |t > 1}
is
£,
> 1
.
Then the information
available to
—
~~
=X
period
t
C
t
> 1,
e
i
Jj&>
G
X^"
,
player
i
cannot improve his
in his strategy.
subgame—perfect equilibrium
Suppose that,
=
them when they make
,£
)
each active period
outcome of the preceding period, but
—
(x
at the beginning of
[0,1]
,
where £
=
their choices can be
(£-,,...,
of such vectors as the set of extended
also of
t
)
We
refer to the set
subgames of the game in
.
Definition 3.3
period
i
—
x [0,1]
1
a sequence of signals, drawn independently from
summarised by a vector h
H
all
2 shows, however,
players are informed not only of the
,
1
therefore introduce the concept of an extended strategy.
according to the uniform distribution.
t
and
,
by a unilateral change
need not exist in general.
a strategy profile <f,-
—
X
As the counterexample of Section
Suppose that
is
is
A ti (x t_1
)
For each
t
and each
> 1
a measurable function
for all
(x
t_1
,^)
e
f,.:
H t_1
.
H
i
e
J
,
an extended strategy
for player
i
in
—
—
->
&J((Y..) such that supp[f,.(x
,£
)]
13
—
Once again the
the randomising device that player
in period t
,
when x
signals observed
So
are behaviour strategies.
f,.
i
will use to
choose
—
the previous history of the
is
up to and including period
t
.
But
f..(x
among
this
<f.
strategy profile
6
Jj6
,
player
i
|t > 1,
£
is
the vector of public
time this device
t
and of
all
him
is
independent,
devices used in
all
the public signals.
all
A subgame—perfect
Definition 3.4
can be thought of as
)
the actions available to
game and
not only of the devices used by the other players in period
other periods, but also of
,£
i
e
equilibrium in extended strategies
X^>
is
an extended-
—
such that, for
all
t > 1
,
all
h
cannot improve his payoff in extended subgame h
eH~
,
and
all
i
by a unilateral
change in his strategy.
A subgame—perfect
coordination: after the
equilibrium in extended strategies allows a limited amount of
outcome of a period has been
realised, the players
the continuation equilibrium to be played following that
public signal.
correlation
is
It is
can coordinate on
outcome by exploiting the next
therefore a limited kind of extensive—form correlated equilibrium.
limited in that the players observe a
privately observing a single
common
component of a vector of
signal, rather
signals.)
than each
(The
14
The Main Result
4
We need
the following assumptions:
(Al)
for all
t
> 1
(A2)
for all
t
> 1
is
given,
and
and
f,*:
J^i
for all
Taken
in conjunction
G
J^i
e
i
a strategy
,
(A3)
i
all
,
X
u-
f.*
—
-*
the
,
mapping
A.-:
X
~~
-»
J£(T..)
is
continuous;
(and not an extended strategy) for Nature in period
9M
(Y,,-.)
is
t
continuous;
bounded and continuous.
is
with the standing assumptions made in respect of the basic
framework, they guarantee the existence of a subgame—perfect equilibrium in extended
strategies.
The
present section
is
devoted to a proof of this
fact.
begins, in Section 4.1, with
It
an overview of the proof. The proof involves two main steps. The analysis necessary
the
first
of these,
namely the backwards program,
is
The
given in Section 4.2.
necessary for the other, namely the forwards program,
is
given in Section 4.3.
analysis
Section 4.4
then integrates the backwards and the forwards programs to complete the proof.
reader
may
4.1
Overview
wish to read Section 4.1 and then skip to Section
Imagine
Then the proof
from stage
T
for the
finding
,
two
The
5.
purposes of the present subsection that the
divides into essentially
for
steps. In the first step,
game has horizon T
.
one programs backwards
what turn out to be the equilibrium paths of the game. In the
second, one programs forwards, constructing strategies that generate these paths and that
constitute an equilibrium.
The
x
T— 1
e
basic ideas of the backwards
X T—
,
find the set of
Nash
program are
equilibria (including
strategies) that can occur in the final stage of the
probability distributions over
sets are
Y™
as follows.
Nash
equilibria in
game. Let C™(x
that result from these
Nash
compact and payoff functions are continuous, C™(x
each history
First, for
)
equilibria.
T—
)
is
a
mixed
be the set of
Because action
non-empty compact
15
set
9JC{Xrr)
contained in
point—to—set mapping
G
A T_
(x
T—
^(^^(Yrp))
to
,
the
upper semicontinuous. 11
is
T— 1
.
the set of continuation paths consistent with equilibrium
For any
v™
-
given by
is
o
<p
,y rp_
1
.
)]
(This set consists, in effect, of those probability distributions over
Nash
that can be obtained by randomising over
selection
c
T (x
,
•
)
^p
2
Crp(x
,
•)
(For a general c,y(x
.
<p
be at least one
with Crp(x
2
c,y,(x
existence theorem of
T—
,
•
that equilibrium
)
is
,
T— 1 when
•
)
.
,
•)
,
Let
mapping
2
,•)
which
,
it is
this set
This
not.
(1990).)
may
Crp_-. (x
T— 1
be the
is
C™,
a
non-empty compact
from
X
p
2
to
)]
,
And
is
for
For each
.)
find the set of
game
Nash
specified
is
by
essentially the content of the
each Nash equilibrium associated
Y™,
*
Y™
that results
when
and the continuation paths are given by
2
)
•
)
be empty. But there will always
find the probability distribution over
played in period
,
Crp(xrj,_,
the continuation of the
set of
such distributions obtained when both
and the Nash equilibrium associated with
T—
)
T
Simon and Zame
"p
,
T—
Cy,_|(x
for
,•)
p<_2
Crp(x
equilibria in
from the correspondence co[Cy(x
equilibria that can occur in period
Crp(x
T—
1
co[C T (x
YT
,
X T—
from
depend continuously on x
be the history of the game prior to period
x
let
)
C™
sets
o
•p
Next,
Because actions
.
subset of
is
Then
are allowed to vary.
&J£(Yrr<_i*Yrp)
^(^^(Y^ixYrp))
essentially the content of the generalisation of
it,
,
and the point—to—set
upper semicontinuous. (This
Simon and Zame's theorem given
is
in
Section 4.2.)
The relevance of the assumption that players' action sets in period T depend
continuously on the history of the game prior to period T emerges clearly in the context
11
of a
two—period game. For each n
game when the outcome
of period 1
> 1
is
,
let
y,
,
a? be a Nash equilibrium in period 2 of such a
and suppose that (y^a^)
-»
(y^a^) as n
-» <d
i 's action set is upper semicontinuous in the outcome of period 1, the
actually available to him in period 2 when the outcome of period 1 is y-,
.
Because player
action a„j
is
.
Also, because his action set in period 2 is lower semicontinuous in the outcome of period
any deviation open to him in period 2 when the outcome of period 1 is y, can be
approximated by deviations open to him when the outcome of period
really
is
an optimal action.
1 is
y,
.
Hence
1,
a^.
16
Finally, iterate until period 1
T
X...Y.
distributions over
is
obtained for
is
C.(x
reached, and a set
eX
x
all
game x
out to be the set of equilibrium paths of the
(The
.
of probability
)
co[C,(x
set
will
)]
turn
This completes the backwards
.)
program.
Consider
c,(x
)
now
co[C.(x
6
c..(x
)
we can
and each £
i
definition of
(i)
C,(x ), there
the f,-(x ,£
are given
)
by c«(x
of e, (x ,£
e [0,1]
)
,
)
given that y
1
f,-(x ,£
12
1 < t <
)
and cJx
T and
i
e
2
2
,£
J^£
intended to produce
is
)
.
when
is
for all
(ii)
And
so
,£
6
c~(x ,-,£
—
,-)
is
always c,(x
always the conditional distribution of e,(x
)
This ensures that no single—period deviation
is
profitable follows
is
-»
C,(x
over Y-.
)
)
For
.
By
such that:
,-)]
one obtains d„(x
,^
),
set of strategies
eJx
f,-
,(
),
for all
do produce the paths they are
The
essential points are that the
—
—
)
,
given
,£
with
the conditional distribution
and that
y,
.
)
constitute a
profitable.
c,
,
.(x
,y,,^
)
is
That they constitute an
—
equilibrium follows from the fact that the f,-(x
)
when the continuation paths
is
)
)]
,f
,£
are given
1.
2
strategies really
,£
2
in period 1
co[C (x
,£
we
,
,•): [0,1]
from co[C (x
)
on until one has a complete
That these
X
given Lebesgue measure.
is
y,, c«(x ,y-.,£
)
6
over C,(x
be the marginal of e,(x
)
the outcome in period
cJx
)
variable e,(x
[0,1]
primarily a technical matter.
expectation of e,(x
.
)
Nash equilibrium
Similarly, beginning with
f„.(x ,£
each x
can be viewed as a randomisation over
)]
random
exists a selection
and
;
find a
d,(x
is
)
let
constitute a
,-,£
we can
)
whose distribution over C,(x
each
for
associate a probability measure d,(x
Also, for each d,(x
.
Suppose that,
Since any path in co[C,(x
.
)]
paths in C.(x ),
the forwards program.
Nash equilibrium.
That no more complex deviation
from a standard unravelling argument. This completes the forwards
program.
Overall, the essential ingredients of the theorem are the following: players action
sets are
compact; players' payoffs are continuous; Nature's strategy
is
action sets in a given period depend continuously on the history of the
period; players observe public signals;
continuous; players'
game
prior to that
and players' payoffs are bounded. The
role played
17
by the
The
three of these should be obvious.
first
role played
by the fourth
ensure that the point—to—set mapping from subgames in period
those subgames
upper semicontinuous. The role of the
is
point—to—set mapping from subgames in period
convex valued. And the sixth
in addition,
we do
that
is
t
roughly, to
is,
to equilibrium paths of
fifth is to
ensure that the
subgames
to equilibrium paths of those
t
needed only
is,
for the relatively technical reason
not insist that Nature's behaviour strategies have compact support.
The backwards program
4.2
we develop
In this subsection
the analysis necessary for the backwards program.
To
this end:
X, Y- for
(i)
let
(ii)
for all
i
(iii)
let
f*:
X
(iv)
let
Y=
(v)
let
C
(vi)
for
The
:
X4
6
-»
all
i
e
let
,
A-:
9>Jt (Y*)
"jgjYj and
gr(A)
each
i
e
-»
J
and Z be non-empty complete separable metric spaces;
,
X
->
Jfc(Y-)
be a continuous point—to—set mapping;
be a continuous mapping;
A(x)
let
= Y^
x
^Aj(x)
x
x
for all
£
X
;
Jf(Z) be an upper semicontinuous point—to—set mapping;
J^i
,
let
u-:
gr(C)
-»
R be a continuous function.
similarity of this notation to that of Section 3
is
deliberate,
and
is
intended to be
helpful, not confusing.
With
reference to the model of Section 3 and the discussion of Section
A.
of past histories;
describes the dependence of player
f*
describes Nature's behaviour;
u-
is
the payoff of active player
what we have
is
C
i
.
i
's
4:
X
is
action set on the past history;
describes the set of possible continuation paths;
In terms of a
the set
and
somewhat more abstract perspective,
a continuous family of games parameterised by x
e
X
.
The
three
essential ingredients of such a family are: the continuity of the action sets; the
upper
semicontinuity of the continuation paths; and the continuity of the payoffs in the
parameter, the outcome and the continuation.
^C(x)
For each x
6
&Jt(Y
Z)
c
*
X
we
are interested in the set of equilibrium paths
that are obtained
when one
is
free to choose
any randomisation over
18
C(x,y) as the continuation path following the outcome y G A(x)
€
Jc(x)
the marginal
\i
(ii)
the marginal
/jg
(iii)
A possesses an
Y
of A over
of
Y*
over
\i
a product measure with supp[/x] C A(x)
is
is
•
|
C(x,y) for
y)] C
moreover the marginals
of
\i-
y
all
e
be helpful to state more explicitly what
probability measures
over A.(x) for
fj,-
supp[^(- |y)] C C(x,y) for
that
y
all
A(x)
6
is
/u-(x,y,z)dA(y,z)
improve
Lemma
Proof
=
his payoff
4.1
We
#C
,
and the
by changing
is
Then
.
i
in
game x
semicontinuous. Let
n-
=
measurable selection from P.
n C(x,y)}
.
for all closed
(By
D
definition p
Z
.
y
all
.
Suppose we are given
(iv).
and a transition probability v such
Y«Z
Then the payoff of an
.
Nash equilibrium
map from X
if
active player
no such player can
p-(x,y)
.
is
i
,
x
,
into Jf( 3>JC{Y
and y
A(x)
6
,
let
*
Z))
.
p-(x,y)
the lowest payoff that can be imposed on
Standard considerations show that
{z|z e C(x,y) and u-(x,y,z)
To
.
(x,y)
semicontinuous. Hence P~ (D)
-•
when
=
p-(x,y)}
We
.
p.
is
shall
lower
need a
Following Deimling (1985; Proposition 24.3 and Theorem
.
C
y) for
•
|
with v
24.3), the existence of such a selection will follow if
measurable
v(
is
Let A be the measure over
For each
following outcome y
P:(x,y)
;
his probability measure.)
begin with some notation.
|
player
.
an upper semicontinuous
min{u.(x,y,z) z 6 C(x,y)}
J
constitute a
\i-
A(x)
meant by
is
\£
all
obtained by combining the product of the
i
v over Z
over the Y- constitute a Nash equilibrium
(j,
the continuation path following outcome y
may
;
frt(*|x);
r.c.p.d. (regular conditional probability distribution)
such that supp[i/(
(It
A
formally,
iff:
(i)
(iv)
More
.
Z be the resulting measurable
=
P7 (D)
=
this end, define
oo
if
D
{(x,y)|p.(x,y)
selection.
=
(x,y)
p
n C(x,y)
=
=
.)
p. (x,y)}
{(x,y)|D n P-(x,y)
=
min{u-(x,y,z)|z
Like
is
p.
,
p
•
is
j-
0}
6
D
is
lower
measurable. Let
q.:
gr(A)
19
The proof now
paths of
game x
that are consistent with Nature's
the proof shows that this
J6{9Jl^i
Z))
*
mapping
B
This
It is
is
That
is,
that
A(K)
B
clear that
that
is
it is
we need only show
>
1— e
that ffl(K) >
1— e
A
for all
The second
step.
that
it is
that, for all
B(x)
e
.
£*(
B(x) be the set of
jx)
•
The
.
#C
e
is
that
X
,
part of
into
Since this graph
closed.
graph
its
is
x
this end, let
tight in order to
>
first
#C
is
is
non—empty
step accounts for the bulk of the proof.
compact valued. To
we need only show
closed,
let
,
to demonstrate that
is
non-empty valued, and
is
X
mixed action
that the graph of
that remains
all
done in the third
is
need therefore show
B(x)
,
e
an upper semicontinuous mapping from
is
The second shows
.
clearly contained in that of
valued.
For each x
divides into three parts.
closed.
e
X
show that
exists a
we
be given. Since
it is
compact.
K
Y
there exists a compact
But there certainly
All that
compact
c
K
C
*
Z such
Y* such
(V
K
to
and we may therefore
,
ieX* AjW
K
let
This completes the
.
be the graph of the restriction of C(x,
first step.
In order to begin the second step, suppose that
-»
A
(x,A)
.
For
.
From
all
j
i
#
,
let
j
J
and
be player
^C(x
(y )dA n (y,z)
j
all
=
continuous Xa
'
= jy
Y
-»
K
supp(A )cgr(C(x ,.))
n
n
i
's
(x ,A
n'
payoff from A
,
n y)
6
and
gr^C]
let
it.
,
and that (x
,A
n'
n')
be his payoff from
one can deduce that:
)
[/xi (y i )dA n (y,z)J |/x (y )dA n (y z)
j
j
>
continuous A
y.
i
IX0(y0)dA n (y,z)
for all
r-
the definition of
/Xi (y i )x
for all
i
•
:
Y.
i
-»
yR and A
(y )df (y ]xn )
J
:
.(5.1)
Y.
J
...(5.2)
;
.(5.3),
20
where C(x
(C(x
n
is
n ,y)
•
,
empty
y t A(x n ))
iff
/Xi (y i )u i (x,y,z)dA n (y,z)
for all active
r
M
and
i
and
i
Equation
a
all
.
ni
X\ :Y.
-»
R
;
...(5.4)
and
z)
A.(x
e
(5.1) follows
(5.2) follows
;
continuous
)
p
...(5.5)
)
.
n'
from the independence of the marginals of A
from the
A
fact that
is
its
(5.4) holds
follows
conditional
payoff.
(• |y)
is
supported on C(x
because the expectation of u-(x
from the
strategy
u
(It is
,y)
to a set of actions, each of
supported on A(x
6
A(x
is
ir
y
mass of player
which yields him
important to avoid the word 'support' here. For player
him r. need not be
and no matter what the
if
.
,
and
a.s.
's
)
,
Equation
.
)
i
This
mixed
his equilibrium
i
's
continuation
so the set of pure strategies
Inequality (5.5) states that player
closed.)
payoff must be at least what he would get
this deviation
for all
is
conditional on y.
,-,-)
payoff /u-(x ,y,z)di/ (z|y) need not be continuous in y
yielding
of A
\i
fact that, in equilibrium, the probability
must be confined
over the Y.
consistent with Nature's strategy.
Relation (5.3) follows from the facts that the marginal
and that
Z
to
= ^/^(y^dA^z)
/p i (x n ,y\a ni )dA n (y
>
for all active
Equation
all
Y
empty—valued) correspondence from
regarded as a (possibly
is
)
he were to deviate to a
•
i
and
's
if,
equilibrium
following
realised choices of the other players, the worst
conceivable continuation path from his point of view occurred.
idea that no redistribution of probability mass
Equation (5.4) captures the
among equilibrium
actions
is
worthwhile.
Inequality (5.5) captures the idea that no redistribution of probability mass from
equilibrium to out—of—equilibrium actions
Now
Also, since
it is
ic
is
worthwhile.
easy to see that relations (5.1), (5.2) and (5.3) are preserved in the limit.
=
/u-(x ,y,z)dA (y,z)
,
x
.
-» ir.
.
Hence
(5.4) is preserved too.
Lastly, for
21
any
A.(x)
a- e
of A.
there exist a
,
Since also w-
.
A-(x
e
.
such that a
)
lower semicontinuous, (5.5) too
is
has a closed graph will therefore be complete
x and A
(5.5) for
,
-»
.
which we
a-
,
The proof
preserved.
is
we can show
if
by the lower semi continuity
#C
that
that the analogues of (5.1)-
imply that A
shall refer to as (5.1')-(5.5'),
6
^C(x)
.
Using standard considerations we can deduce from (5.1'), (5.2') and (5.3') that the
marginal
Y*
over
/x
/j
Y
of A over
f*(-jx)
is
,
and that A has an
supported on C(x,y) for
specified
by v
a product measure supported on A(x)
is
all
y
A(x)
6
The marginals
.
\i-
of
need not constitute a Nash
equilibrium for this game. However, by (5.4'), the set Y- of
deviation by active player
from
i
to
/x.
6
,
is
game with continuation paths
consider the
over the Y.
\l
that the marginal of
v over Z such that £(-|y)
r.c.p.d.
Now
.
,
a- 6
A.(x) such that
the Dirac measure concentrated at
a-
a-
1
will
,
1
i
raise his payoff
that
remains an
it
= K-
K-
17)
^
but
i
one
i.
above
y. 6
Y-
r.c.p.d. of
a
ly)
y. 6
It is
.
for all
i
of no importance
,
and K-|y)
how we
we
therefore constitute a
Jc(x)
*
7j e
if
q (X|y)
.
v{
•
|
if
y)
y. e
Yj\Yj
set
for all
j
Y. for more than
deviation
no longer
is
profitable.
The
//•
.
non-empty valued. To
dense in A.(x)
game x
.
$C
has closed graph,
this end, fix
For each
.
in the natural
existing
,
Nash equilibrium with these continuations, and therefore A
Having proved that
N}
We
^(-|y) equal to £(-|y) in this case. Inequality (5.5') ensures that,
set
with the new continuation paths specified by v
<
define
=
way
v in such a
but such deviations are no longer attractive.
A,
Y\Y
therefore alter
(Such a y corresponds to a coordinated deviation by two or more players.) For
definiteness,
€
We may
has //.—measure zero.
x.
1 <
way from
The family
N <
x and
od
,
that remains
each
find, for
construct a finite
the existing
of
all
game x
games obtained
as
.
i
show that \?C
a sequence {y
game with
And
N
,
to
is
for
varies
N=
is
is
that
action sets
a>
continuous, and
non-empty when
{y
is
-
1
1 <
n
simply take the
equilibrium correspondence therefore has compact values and a closed graph.
standard results show that the set of equilibria
}
is
N <
od
.
its
Moreover
It follows
that
22
it is
non-empty
Lemma
N=
for
cd
too.
4.1 captures the essence of the
backwards program: we begin with an upper
C depending on
semicontinuous correspondence
the sequence of outcomes up to and
including the current period, and end up with an upper semicontinuous correspondence
depending on the sequence of outcomes preceding the current period. There
further complication.
When we come
is
however one
theorem in Section
to complete the proof of the
'I'C
4.4,
the continuation paths in Z will themselves be probability measures over sequences of
future outcomes.
= ^^(W)
,
In the present,
where
W
more
a complete separable metric space.
is itself
*C
equilibrium paths described by
by writing Z
abstract, setting this can be captured
9JC{Y
lie in
$>JC(W))
x
With
this notation, the
This means that
.
What
cannot serve directly as input for the next iteration of the backwards program.
required instead
&JC{Y
of
x
W)
9>Jl[X
those of
is
More
:
->
9JC{Y
x
W)
(The point
.
is
is
that elements
are probability measures over sequences of future outcomes, whereas
x
3>JC{yi))
PJC^&Jt^))
elements of
^C X
a correspondence
#C
Such a correspondence can be obtained by replacing
are not.)
with their expectations.
xGX
formally, for each
^C(x)
let
c
,?1(Y
x
W)
be the
set of
measures A
such that:
(i)
(ii)
the marginal
\i
the marginal
\l*
A possesses an
(iii)
A(x)
e
over
of A
of
over
\i
is
Y*
v over
r.c. p. d.
a product measure with supp(/z) C A(x)
is
W
fg(x)
;
such that
it
v{
/j.
of
\i
over the Y.
may
be helpful to expand on
(iv).
is
&JC(k.{xj) and a transition probability v such that
€
A(x)
of the
(jl-
Then
.
y) e co[C(x,y)]
for all
active player
The
//•
i
's
constitute a
payoff
is
constitute a
y
r/(-
|y)
for all
y
.
u(- |y) G co[C(x,y)]
/u-(x,y,^(- |y))d/i(y)
Nash equilibrium
Nash equilibrium when
Suppose that we are given
6
.
•
|
the continuation path following outcome y
(Once again
;
;
moreover the marginals
(iv)
Y
if
,
where
/z
is
\i-
for all
y
the product
no such player can improve
his payoff
23
by changing
Lemma
his
p.
.)
Suppose that the domain of definition of u-(x,y,z)
4.2
suppose that u(x,y,z)
X
semi continuous mapping from
It
on
linear in z
is
*C
domain. Then
Jtp(?JC(Y
into
should be noted that, when
its
we come
W))
x
convex in
is
an upper
z
and
,
.
Lemma
to apply
is
u-(x,y,z)
4.2,
defined as the expectation with respect to z of an underlying function of (x,y,w)
convexity of the domain of u.(x,y,z) in z
and the
,
natural consequences of the problem structure.
#C
roughly speaking, that
Lemma
4.2 will be
in
z
,
.
So the
will
be
proved by showing,
the image of tfC under the projection that consists of
is
&J£($ J((W)) by
>
replacing elements of
linearity of u-(x,y,z)
be
will
their expectations,
and that
this projection is
continuous.
Proof
Let A G
r.c.p.d of
j/(-|y)
fi
^JC{Y
Z)
x
over Z. Let
for all
y£Y.
,
fj,
let
=
fi
Then the
measure with marginal
,
be the marginal of A over
and
let
u{- |y)
X
projection
and conditional v
\i
of the r.c.p.d. chosen for
\i
A
The
.)
.
=
/z d^(z|y)
of A onto
(It
,
and
is
v be an
let
be the expectation of
M(Y
«
W)
is
X
should be clear that
step of the proof
first
Y
the probability
is
independent
to
show that projection
6
9JC{Y
is
continuous.
To
that
{A~
n}
this end, let
C
?JC(Y
/x(yMw)
for all
x
{A
n}
W)
dA~ (y,w)
n
C
9Jt{X
x
converges to
-.
/x(yMw)
continuous functions x-
Y"K
converge to A
Z)
Xe^(YxW),
it
x
Z)
suffices to
.
Then, to show
show that
dl(y,w)
and
ifr.
continuous linear functional defined by 1(k)
W
-» IR
.
Let t 3>JC(W)
= /^w)d«(w)
.
Then
-»
K be the
24
/x(yMw)d!n (y,w) =
Mw)di/n (w|y)
/
x(y)d** (y)
n
(by Fubini's theorem)
=
(by definition of
=
/
K\(-\v))
x(y)d/* (y)
n
€}
/
/<z)di/ (z|y) x(y)d/x (y)
n
n
(by definition of the expectation of u
(• |y))
/x(y)^(z)dA (y )Z )
n
(by Fubini again, and because
=
//
//
But the
).
latter expression converges to
/x(y)4 z )dA(y,z) by definition of weak convergence, and
this integral is equal in turn to
/x(y)V(w)dA"(y,z) by reversing the above argument. This completes the proof of
continuity.
To complete
the proof,
it
suffices to
show that gr(^C)
under the projection that takes (x,A) into (x,X)
Then the payoff from A
/Uj(x,y,z)dA(y,z)
Suppose
first
the image of gr(^C)
that
is
=
/
/u (x,y z)di<z|y)
>
d/x(y)
i
=
/u (x,y,/zd^(z|y))d/i(y)
i
(by the linearity of u-)
.
is
(x,A) G
gr(^C)
.
25
=
/u (x,y,K-|y)) d /i(y)
i
(by definition of V). But u{- |y) G co[C(x,y)] because supp[*/(- |y)] C C(x,y)
/u-(x,y,P(- |y))d/z(y)
e
gr(*C)
is
by
definition the payoff
is
To
the projection.
G
gr(#C)
We
.
standard considerations show that C(x,y)
gr(co[C])
u{
let
|
(for
y)
-»
=
c(x,y,F(
|
example, one could
y))
for all
y
set
v{- |y)
=
u-
,
Lemma
4.2
is
Then C has a
.
non-empty
is
G
(x,A~)
gr(#C) of which
9Ji{V) with
closed graph, and
z G co[C(x,y)]
iff
The standard
G
A(x)
&-,
,
and
x
,
let
•
|
such
for
y).
y)
it
to the
work
existence theorem for
each player's strategy
set is a
of
C
Let
.
to gr(co[C])
,
be arbitrary for y £ A(x)
Then
it
can be checked as
A obtained by combining the marginal
^C(x)
lies in
v{
.
the basic result justifying the backwards program.
present subsection by relating
If
need to find (x,A)
that the measure
with the transition probability v
follows.
easy to see that
it is
&JC(7j) be a measurable selection from the restriction of
above, using the linearity of
Jl
So
.
C(x,y,z) be the set of k G
this end, let
expectation z and such that supp[/t] C C(x,y)
c:
A~
and
.
Suppose now that (x,X)
(x,A~)
from
,
Simon and Zame
We
conclude the
(1990).
normal—form games can be expressed
compact metric space, and
if
there
is
as
a continuous
function associating a vector of payoffs with each vector of strategies, then there exists a
vector of (possibly mixed) strategies that forms a
(1990) extended this result by showing that
correspondence associating a convex
exists a selection
from
this
if,
Nash equilibrium. Simon and Zame
instead, there
set of payoffs
helpful to reinterpret their
an upper semi continuous
with each vector of strategies, then there
correspondence and a vector of (possibly mixed) strategies that
forms a Nash equilibrium when the payoff function
In order to explain
is
how Lemma
work
slightly.
is
given by this selection.
4.2 extends the result of
Much
as
we do
Simon and Zame,
in the proof of
consider a sequence of finite approximations to their basic game.
Lemma
Suppose these
it is
4.1,
they
26
approximations are indexed by
Nash equilibrium of game
N
N
for
,
suppose that the mixed strategies
some
selection
equilibrium payoffs of this equilibrium be
strategies
7iVr-
,
and payoffs
/a-
7iYr-
.
from
its
Then
/a^-
payoff correspondence, and
their proof
shows that,
constitute a limit point of the strategies /a^-
ir-
constitute a
the
the mixed
and payoffs
then there exists a selection from the payoff correspondence of the original game such
that the
constitute a
\i-
function
Nash equilibrium with equilibrium payoffs
when the payoff
ir-
given by this selection. In other words, they demonstrate a limited form of
is
upper semicontinuity
Lemma
first is relatively
family of games.
for a particular continuous family of
4.2 therefore extends the result of
minor:
it
games.
Simon and Zame
shows that upper semicontinuity holds
The second
is
more substantive. The
in
two ways. 12 The
little
Lemma
continuous
for a general
Simon and Zame
result of
about the behaviour of the equilibrium strategies and equilibrium payoffs, but
very
if
let
tells us
it tells
us
about the selections from the payoff correspondence that enforce these strategies.
4.2,
by contrast, concerns a comprehensive description of equilibrium. Such a
description of equilibrium
is
not without intrinsic interest.
the purposes of the present paper
problem that would otherwise
is
arise
that
it
But
its
main
significance for
allows us to avoid a measurable selection
when we come
to
program forwards. 13
As it stands Lemma 4.2 is not a direct generalisation of the result of Simon and
Zame. But it can easily be adapted to obtain one. To do so, view Z not as a space of
probability measures over a compact metric space, but more generally as a compact
metrisable subset of a locally convex topological vector space. (This perspective is more
12
,
and includes the
general,
subset of
IR
,
Z
possibility that
then one can define u-(x,y,z)
is
=
a compact subset of
z. .)
And
IR
^C(x)
define
.
If
in a
Z
is
a compact
more limited
way, to consist of pairs (/a,7t) G &J((Y) * IR
where /a is the distribution of a Nash
equilibrium and ir is its equilibrium payoff vector. (This more limited definition of $C is
the price one must pay for the more general Z .) Then a proof almost identical to that of
,
Lemma
4.2
shows that
$C
is
a continuous projection of
$C, and
therefore upper
semi continuous.
13
-» IR
It
would be tempting to conduct the backwards program
as follows.
Let
II:
gr(A)
be an upper—hemicontinuous convex—valued correspondence describing possible
continuation payoffs. Then $11: X -» IR
the correspondence describing possible Nash
equilibrium payoffs for the current period, is upper hemicontinuous; and so therefore is
co^II] the continuation—payoff correspondence for the next iteration of the backwards
program. This simple program is, however, difficult to reverse. For, in order to reverse
,
,
it,
27
4.3
The Forwards Program
In this subsection
this
we develop
the analysis necessary for the forwards program.
purpose we shall need a measurable selection c from
#C
.
This function can be
thought of as part of the output of previous iterations of the forwards program.
describes the equilibrium paths that
For
It
must be followed from the current period onwards
if
the strategies calculated for earlier periods by the previous iterations of the forwards
program are
really to constitute part of
Lemma
There
x [0,1] -»
(i)
4.3
an equilibrium.
measurable functions
exist
$>JC{W) such that,
for all (x,£) e
X
C
A(x)
themarginal^0(-|x,£)of/z(-|x,£) over
(iii)
i/(-|x,-,£)
for all
y
6
an
-»
^J({Y
*
W)
Y
r.c.p.d. of
Y
A(«|x,£) over
is
W
ity{-\x);
such that u{- |x,y,£)
v{- |x,y,£)
Moreover /A(- |x,£)d£
=
following outcome y
c(x) for all
x
e
X
c(x)
all,
conditions (i)—(iv) imply that
therefore tells us that,
on the basis of the current public signal £
if
is
Nash equilibrium
for all
y
e
A(x).
.
4.3 captures the essence of the forwards
First of
=
€ co[C(x,y)]
A(x);
when the continuation path
/A(- |x,£)d£
Y
a product measure such that
is
constitute a
explanation.
*
:
the marginals /j.(-|x,£) of /x(»|x,£) over the Y-
Lemma
and w.X
;
(ii)
is
x [0,1]
* [0,1]
the marginal /x(-|x,£) of A(-|x,£) over
supp[/x(-|x,0]
(iv)
X
A:
program.
A(- |x,£) €
the equilibrium path
e [0,1]
,
It
requires
^C(x)
is
some
The
.
condition
chosen from ^C(x)
then the overall path will be precisely
one must be able to find, for any measurable selection ir from #11 a measurable
selection $7r from II such that the game with payoff function $7r(x,-) has a Nash
equilibrium with payoff 7r(x)
This is not a standard measurable—selection problem. For
what is actually required is that we select, for each x a function $7t(x,-) and that these
choices of functions be coordinated in x to obtain a single measurable function 3>7r
(A
version of this non—standard problem did arise for Mertens and Parthasarathy (1987;
Lemma 1 of Section 6, pp 41—42). In their case, however, the functions $7r(x,-j could be
taken to be continuous.)
,
.
,
,
.
28
c(x)
Lebesgue measure
(If
.
conditional distribution over
Y
x
x
W
?JC{Y
-»
[0,1]
Secondly, the
.)
assertion
\i-\
period.
Y
lemma
x
W
,
then /A(-|x,^)d^
[0,1]
and
,
A(- |x,-)
if
the marginal distribution over
is
goes beyond the simple assertion that there exists
such that A(- |x,£) G *C(x) for
X
,
the proposition delivers a v that
(x,£)
all
x [0,1] -•
&J((Y.) are precisely the strategies of the players
Fourthly, the restriction $c of v to gr(A)
X
A:
For, while this
.
measurable in (x,y,£)
is
the
is
would automatically imply the existence of a v that was measurable in y
each (x,£)
the
W)
x
the marginal distribution over
is
for
Thirdly,
jointly.
for the current
describes the equilibrium
x [0,1]
paths that must be followed from the next period onwards.
the input for the
It serves as
next iteration of the forwards program.
Most
Proof
Lemmas
of
4.1
The
x
X
G
,
of the considerations necessary for the proof have already arisen in the proof,
and
first
step
supp(d(x))
The
4.2.
c
is
present proof will therefore be brief.
X
to find a measurable d:
\PC(x) and the expectation of d(x)
an argument very similar to one used in the proof of
measurable mapping
9JL ( $>JC(Y
-•
A:
X
x [0,1]
&J£(Y
-»
x
W)
Lemma
x
W)
is
precisely d(x)
'measurable' representation theorem,
The
third step
£(-|x,-,£)
an
is
e.g.
to find a measurable
is
x
for all
r.c.p.d. of A(- |x,£)
|x,«): [0,1]
.
W)) such
-»
that, for all
This can be done using
we need
Next,
4.2.
such that, when
measure, the distribution of the random variable A(-
^Ji(Y
c(x)
is
*
given Lebesgue
is
[0,1]
9Jl{Y
a
x
YV) over
Such a mapping can be obtained by applying a
.
Gihman and Skorohod
mapping
for all
u:
XxY
(x,£) e
X
x [0,1]
x [0,1]
.
(1979;
-»
Lemma
1.2,
&Jt(W) such
In other words,
p
9).
that
we need
'measurable' version of the standard result which asserts the existence, for each (x,£)
an
r.c.p.d.
v{- |x,-,£)
of A(- |x,£)
forwards program, and since
the literature,
Let
we
/x(-
.
Since this
we have not been
is,
,
a
of
in effect, the central step of the
able to find precisely the result
we need
in
sketch a proof of this.
|x,£)
be the marginal of A(« |x,£) over
sequence of sets that generates the Borel
a—algebra on
Y
Y
.
.
Let
{B
For each
1
N
1 <
,
let
n
<
<d}
be a
5?™ be the
29
a-algebra generated by {B
&**
Since
5?-^
Y
on
N}
<
And
.
.
give an explicit formula for an r.c.p.d. of A(-|x,£)
For each y
.
^l#(W)
v be a fixed element of
let
Y
G
and each measurable
W W
we
C
set
A(YxW|x,Q
-x^L
= -L--
-
,,;,
n
1 <
we can
is finite
a—algebra
given the
1
" (W|x,y,£)
l
N
M(Y|x,0
Y
if
z,
=
&N
of
It is
containing y and /i(Y|x,£)
immediate from these formulae that
and therefore that the convergence
,
K
lim!/v,on
and v
Certainly u
is
—
v outside
K
what
im M (-
1
is,
i/
u{- |x, •,£)
There are two
u(- |x,-,£) e co[C(x,y)]
.
.
The
To obtain
A(x)
Lemma
of u-^
is
|x,-,^)
exists
is
an
&„
|x,£)
//(•
r.c.p.d. of
difficulties
for all
y
the
.
To
A(x)
measurable. Set v
,
is
effectively a version of the
the martingale convergence theorem
—
a.s.,
and that u(-\x,',£)
given the Borel
;
stands:
it
it
may
and the marginals
is
a version of
a—algebra on
not be the case that
/z(-|x,£)
when the continuation path
first
property,
let
of A(-|x,£) need
following outcome y
to be q(x,y)
q be any measurable selection from co[C]
wherever
obtain the second: for each
4.1; let
measurable in
jointly
A(-|x,£). This completes the third step.
with u as
6
is
is
fourth and final step constructs a v that does have these properties from
redefine i>(-|x,y,£)
G
K
set
measurable. Also, since i/^(-|x,-,£)
not constitute a Nash equilibrium
u(- |x,y,£)
and
u-^(- |x,y,£)
effectively the conditional expectation of A(- |x,£)
is
Y. That
v
,
.
conditional expectation of A(- |x,£) given
implies that
>
= KW)
N (W|x,y,£)
otherwise.
(x,y,£)
atom
the
is
7r.(x,£)
=
i
,
it
let
does not
q.
lie in
co[C(x,y)] for
,
and
some y
be the function defined in the proof of
Ju-(x,y,v(- |x,y,^))d/i(y|x,^)
;
and
let
Y.(x,£) be the set of
a-
30
6
such that /u.(x,y\a.,^(- |x,y,£))d/z(y|x,£)
A-(x)
v{-
=
|x,y;0
=
(x,y)
if
qi
7
£(• |x,y,£j otherwise.
and the
Y^YjCx.O
6
for all j *
^(x,^)
but yj
i
Then we may
.
Y(x,0 and
£
,
Standard considerations show that the resulting v
A(- |x,£) e C(x)
fact that
A(-|x,£).
.
>
implies that
u(- |x,»,£)
is still
an
v{- |x,y,£)
is
measurable,
r.c.p.d. of
o
Completion of the proof
4.4
Before proceeding,
we need
to define the equilibrium correspondence for a family of
We
games. In making this definition we encounter a minor technicality.
equilibrium as a strategy profile that
is
Nash equilibrium
in
in every
strategy profile specifies players' behaviour for the entire family of
Hence an equilibrium
face.
6
set
X
We therefore
.
A such that A
of paths
equilibrium.
by x
t— 1
e
Theorem
from
define
X t_1
E..
(x
)
,
,
we can
the set of equilibrium paths of
correspondences E.
for
Suppose that (Al)—(A3) hold. Then E,
games
t >
is
for
2
each game x
game x
,
to be the set
by some
as a family parameterised
.
map
an upper semi continuous
J6{9Jt{**_ Yj).
into
.
The main
(see Section 5).
games that they might
game x
the equilibrium path generated in
define,
subgame. But a
whole family of paths, one
Similarly, reinterpreting our basic family of
X t—
4.1
is
will generate a
have defined an
interest of
Theorem
4.1 derives
However upper semicontinuity
from the
is itself
fact that
of interest:
implies existence
it
it is
reassuring to
know
that our concept of equilibrium possesses this standard regularity property.
Proof
The proof proceeds
period
t
B.
is
—
in three steps.
Let B.(x
that are consistent with Nature's strategy.
an upper semicontinuous
map from X
—
into
I
sequence of correspondences {C.
1
1 < t
<
cd}
is
)
The
be the
first
set of
paths beginning in
part of the proof shows that
®Y
JGi^JC^ S —
)).
X
s
constructed, each of which
In the second, a
is
upper
31
semicontinuous. In the third
shown that
it is
=
E.
B
For notational convenience, we show that
for all
co[C.]
t
.
To
upper semicontinuous.
is
this
1
B?(x
end, let
be the set of marginals over
)
backwards programming argument similar
and
4.1
shows that B,
4.2
Y
J6{9Jt{* _,
_.Y
This completes the
converge for
)
$, (C
for
each
T
.
,
,
)
T
1 < t <
C
let
,
T
step, let
T
^("/iY.)
So B,
,
too,
is
converges
upper semicontinuous.
be given. For each
> 1
T (x t—
> T
t
C
Now
T
=
define C.
n
t
Hence C, too
™,C.
for each
is
,
T
It is
A
[^ t (C.
6
paths from C,
[^
(C^
t
follows.
1 <
T <
+1
,
It is also
S
^
the case that
(C,
od
t_1
)
T
and
A can be enforced by continuation
definition,
T
.
T
belong a fortiori to C,
C = n
C
t
T=i*t( T+l^
Since
.
,)
,
together with C,,,
parameters x
By
.
C. ,,
in
for all
C. C *+(C.
Finally,
)
To
,) C C..
,
1
But paths
.
)](x
)](x
,
Then
,] .)
cC. whenever T
obvious that C,
t
see this, let
.
upper semicontinuous.
t > 1.
upper semicontinuous.
is
B.
T=
onwards when the continuation paths can be chosen from co[C,
t
T
6
T =
C.
let
,
the set of equilibrium paths that are obtained
is
)
.
the backwards program ensures that each C,
A
iff its
T
for period
> S.
} C
into
be defined inductively by the formula
.
C
(In other words,
.
t.
all
map from X
Lemmas
first step.
Turning to the second
And,
Then a
).
but simpler than, that developed in
to,
an upper semicontinuous
is
B,(x
of measures in
But a sequence of measures {A
)).
PJCi*
marginals over
,Y
«
To
.
'
.
Hence
the rec* uired conclusion
see this note that the correspondences
T=
for
,-,
And Lemma
C
T
.
,
for
,
form a continuous family of games with
cd
4.2
shows that the equilibrium correspondence of
—
such a family
is
upper semicontinuous.
certainly a limit point of points in
Overall, then,
{C.| 1 <
=
co[C,]
t
<
a)
}
for all
such that C,
t
.
We
We show first
c,
from co[CJ
we obtain
is
do
that
It
follows that any
[^.(C.
A
6 C. (x
), also lies in
,)](x
)
,
[* t (C,
which
,
is
,)](x
).
a sequence of upper semicontinuous correspondences
= ^ t (C.
this for
co[CJ
also a selection
t
C
,
=
E,
for all
,)
1
.
only.
t
.
1
So
remains only to show that E,
The other
It suffices
from E
It
.
let
to
C.
cases are analogous.
show that any measurable
selection
be any such selection. Then
32
repeated application of the forwards program starting with
strategies
and a measurable selection c.,, from co[C.,J such
f,.
,
the f.-(x
,£
continuation paths are given by
onward when the
f..
c,
,(x
,
,•>£
are employed in period
c.
It
.
programming argument that the strategy
follows
from
(ii)
We now
show that E,
c
E,.
C
co[CJ
any subgame-perfect equilibrium
-»
Y
3>JC(X *
)
in
f is
given.
Now
)
(i)
for all
t
when
follows from
To
game x
is
certainly
c™,,
is
the
t
and a standard dynamic
(i)
f
t
And
an equilibrium.
precisely c, (x
this end, let
describe the paths followed from period
=
<f.. |t > 1,
> 1
t
,
This
.
)
i
it
X<>
€
let
H
c ,:
be
~~
onwards when the strategy
inherits measurability
c,
in period
and the continuation paths are given by
profile f constitutes
.
,
and (") the paths obtained from period
extended strategies. For each
employed. The mapping
profile
t
that the equilibrium path in
completes the proof that co[CJ
Nash equilibrium
constitute a
)
are precisely those required by
c.,,
that:
t > 1
—
H~
,()e
(x
yields, for all
c,
from
f
.
T
Let
be
> 1
T
a selection from co[Cy,,]. Also, suppose that c,,
is
a
1
selection
Nash
at
T J
from co[C.
equilibria for period
once that
that
c,
is
for
t
some
1 < t <
is
a selection from co[C
a selection from co[C,
a selection from
co[CJ
.
Since f
is
when the continuation paths
T
c,
T.
]
.
T
,
]
.
an equilibrium, the
are given
by
c.
,
f,-
.
It
specify
follows
Proceeding inductively we therefore conclude
Moreover, allowing
This completes the proof.
T
to vary,
we obtain
that C,
is
33
5
Existence with and without Public Signals
This section provides a systematic discussion of the question of existence of
equilibrium in the dynamic model introduced in Section
namely
of the
main
games
satisfying the standing assumptions
result for this case,
when
that,
3.
It
begins with a recapitulation
public signals are allowed, any family of
(Al)—(A3) possesses an equilibrium. However,
the section also addresses two further questions.
Under what circumstances can existence
And
be obtained without the use of public signals?
subgame—perfect equilibrium obtained by allowing
is
the extension of the concept of
the observation of public signals
sufficiently rich?
5.1
Existence with Public Signals
The
Theorem
following result
5.1
an immediate corollary of Theorem
is
4.1:
Suppose that (Al)—(A3) hold. Then there exists a subgame-perfect
equilibrium in extended strategies.
Proof
Pick any x
eX
.
Then, by definition of E..(x
By Theorem
)
,
game x
.
potential drawback of this result
equilibrium in what
is
E,(x
)
is
non-empty. Pick any A
E.(x
In particular, there exists an equilibrium!
is
that, in order to obtain the existence
intended to be a purely non-cooperative situation,
extraneous element into the model, namely public signals.
to discover circumstances under
6
there exists an equilibrium of the family of games that
generates equilibrium path A in
The
4.1,
which equilibrium
exists
It is
it
therefore of
introduces an
some
interest
even without this element.
).
34
Existence without public signals
5.2
There are
existence:
games of
which public signals are not needed
in
Theorems
Then a subgame—perfect equilibrium
Oddly enough, Theorem
(Intuitively
in
and zero—sum games. The
and
5.2, 5.3,
for
5.4 respectively.
Suppose that (Al)—(A3) hold, and that the family of games has perfect
5.2
information.
Proof
game
perfect information, finite—action games,
games are summarised
results for such
Theorem
at least three classes of
it
5.2
would seem obvious
is
a
exists.
not an immediate corollary of
when only one player
that,
Theorem
5.1.
acts at a time, public signals
do not allow players to achieve anything they could not achieve using mixed strategies.
But there
a minor complication: players observe not only the current signal, but also
is
However, an equilibrium can be constructed using the correspondences C,
past signals.)
used in the proof of Theorem
and define f,.(x
selection
co[C
2]
L.(x
We
•
begin with any measurable selection
to be the marginal of c,(x
from co[C J that enforces
= C^
)
)
4.1.
Hence c„
is
also a
to be the marginal of c (x
2
c,
.
over
)
Since the
Y...
game
Next,
.
is
a
game
measurable selection from C«
)
over Y,.
.
And
so on.
.
let
c«
c,
from
C-.
,
be a measurable
of perfect information,
So we
may
define
o
Indeed, one can even show that there exists a subgame-perfect equilibrium in
which players employ pure actions only.
We
do not demonstrate this here, since the
argument does not appear to be a simple corollary of the construction that we have
employed. 14 Note that the proof of Theorem 5.2 shows incidentally that allowing public
signals in a
game
of perfect information does not enlarge the set of equilibrium paths.
14
See Harris (1985) for a treatment of the case of perfect information. The argument
used there can easily be adapted to the present context. Alternatively, one can work
directly with the apparatus used here, taking care to purify the selections as one programs
forward.
35
Let us say that a family of games
t
and
> 1
Theorem
all
1
x"
6
*
A^x 1
,
*)
is
of finite action
a finite
of equilibrium paths
Theorem
Proof
if
X
is finite,
5.3
is
compact.
is
more
exists.
or less well known,
and
for all
if,
set.
Suppose that (Al)—(A3) hold, and that the family of games
5.3
Then a subgame—perfect equilibrium
action.
it.
X1
is
Moreover,
and there
is
(See Fudenberg and Levine (1983) for a different proof.)
for
is
each x
of finite
X
6
,
the set
more than one way of proving
So we merely note that
can
it
be proved by a simple adaptation of the methods of this paper. The basic idea can be
expressed in terms of the notation of Section 4.2 as follows. Instead of defining #C(x) to
be the set of equilibrium paths that can be obtained by choosing continuation paths from
co[C]
one defines
,
it
to be the set of equilibrium paths that can be obtained
continuation paths from
x
e
X
this
,
It
that
X
new
C. Because
definition
still
yields
X
is finite,
by choosing
and because the A.(x) are
finite for all
an upper semicontinuous correspondence.
should be noted that both finiteness assumptions are needed. Indeed, suppose
but that one or more of the A-(x) are infinite for some
is finite
happen that there
is
a sequence of equilibria of
signal generated endogenously
game
x,
x.
Then
it
can
each of which involves a public
by players' randomisation over their action
sets A.(x),
and
that this sequence converges to a limiting equilibrium which does not generate any such
signal.
infinite.
there
is
Suppose, on the other hand, that the A.(x) are finite for
Then
it
at least
can happen that there
one player
coalesce in the limit.
for
If this
whom
is
all
xgX
a convergent sequence of games in
at least
one player
for
whom
at least
but that
X
,
X
and that
two actions
player randomises over his two actions, then he generates a
public signal which disappears in the limit.
It is
precisely in order to compensate for the potential disappearance of such
endogenously generated public signals that the exogenous public signals are used in our
basic existence theorem.
o
is
36
To complete our
discussion of the existence of
subgame—perfect
consider the case of a family of zero—sum games.
(A family of games
contains precisely two elements, and
=
Theorem
if
E.
j/^\
Then a subgame—perfect equilibrium
period zero—sum games.
It is
known
well
convex in such a game. But the
if
it is
J<A
.)
helpful to recall
is
of zero sum.
some
facts about oneis
regarded as the set of probability
set of equilibria,
Hence, even in a zero—sum game, the
set of
The
set of
players are allowed to observe a public signal.
equilibrium payoffs, on the other hand, will not.
to co-ordinate
if
that each players' set of optimal strategies
distributions over outcomes, need not be.
equilibria will be enlarged
sum
we
exists.
In order to understand this result,
equilibria
of zero
Suppose that (Al)—(A3) hold, and that the family of games
5.4
two players
is
equilibrium,
For the public signal merely allows the
on a choice of Nash equilibrium
have the same equilibrium payoff (namely
(v,
for the
—v)
,
game, and
where v
is
all
Nash
the value of the
game).
Proof
Note
first
that our proof of the existence of equilibrium shows, in particular, that
the extended family of games has a Nash equilibrium.
It
follows immediately from
standard considerations that this family also has a value. That
measurable mapping v,
(Indeed, since v.,(x
)
is
X
:
such that v, (x
simply player
upper semicontinuous, v,
Similarly, there exist
-> [R
is
mappings v
X
is
there exists a
the value of extended
payoff from any path in E,(x
actually continuous.
extended game beginning in period
With
l's
)
is,
~~
-» IR
We
do not need this
such that v.(x
)
is
),
game x
.
and since E,
fact,
is
however.)
the value of the
t.
these considerations in mind,
we can
construct a
equilibrium of the original family of games as follows.
subgame—perfect
The construction
follows the
construction of an equilibrium of the extended game, but takes care to avoid the need for a
37
public signal.
The
step
first
to find a measurable selection
is
measurable selection from C,
may
product measures. So we
be the marginal of c,(x
rather than co[C,]
)
define strategies
over
from co[CJ that enforces the
Y...
f,.
)
2
C9
But
.
continuation payoff, namely v (x
Similarly,
.
X
-»
^^(Y.-) by
strategies £.:
X t_1
?JC{Y.^
-»
Since c,
.
A e co[C (x
are already
setting f,.(x
to
)
c«
if
Cr,
generate the same
)]
2
a
is
generates a measurable selection
c,
a measurable selection from
is
then (L generates the same continuation payoffs as c«
Hence the
•
f,
•
can be
(The continuation paths specified by c« do not
•
necessarily constitute an r.c.p.d. of c,
Iterating this
of course.)
,
for all
>
t
2
argument we obtain
o
as well,
Approximation by finite—action games
Suppose that we are given a single game
singleton) satisfying
(Al)—(A3). Then a
roughly speaking, a
new family
corresponding to n
=
is
oo
of
X
a family of games for which
(i.e.
game
finite—action approximation to this
games indexed by n
e
IN
the original game; for each n 6
of finite action; and taken as a whole the
is
marginals over Y,
In particular,
.
enforced by <L just as well as by c«
5.3
all
its
,
f,.:
from C,
c,
new family
of
U{od}
IN
,
is,
such that: the game
the
games
a
is
game corresponding
satisfies (Al)-( A3).
to n
(Here
(Al) ensures that players' action sets in the original game are continuously approximated;
(A2) ensures that Nature's behaviour
is
continuously approximated; and (A3) ensures that
active players' payoffs are continuously approximated.)
Finite-action approximations to a single
Indeed,
we can even
To do
(Al)-(A3).
i
6
J,
{yjj|n
let
game
satisfying (Al)-(A3) certainly exist.
construct a finite-action approximation to a family of games satisfying
this,
we need
the following ingredients.
Yt
be dense in
6 IN}
Secondly, for each n €
IN
,
Y||
let
=
-
;
and
m
{y
|
1 <
{yj|n
let
m<
n}
.
each
First, for
6 IN}
be dense in
Thirdly, for each
6
IN,
o£j:X
let
-»
Y..
be a continuous function such that, for
—
—
a..(x
)
A..(x
).
YQ =
t > 1,
—
—
n
belongs to A,.(x
)
,
and ay(x
all
x
and each
> 1
t
e
X
X°.
i
J and
G
—
:
—
)
is
as close to y?.
as
any other point
—
(Such a function exists because A,.:X
-»
J£(Y..)
is
continuous. In
in
38
Nature's case the function
-.
MY
ti
be defined by
)
Fourthly, for each
such that, for
close to
f rt(x
simply constant with value y.*
^(x 1-1 =
and n
t > 1
x
all
is
e
6
X
:
as another
)
IN,
)
{aj^x*
let
^\*:
supp
[</>
measure
X
-1
—
j(x
C
)]
{y™0
^.(x
1
1 <
1-1
)
m<
n}
.
Yj = X
Finally: let
= A (x t_1
ti
)
and
;
for all
t >
In terms of these ingredients
Q
1
{y
A\
.
&J((Y.
*)
for all
t > 1
f
;
and x
we can
n} for
all
x
v
X t-l
n
a.-:
let
t_1
X t_1
e
.
&JC(Y.q) be a continuous function
-»
t(
in
m<
)! 1 <
And
.)
<
m
< n};
whose support
t_1
e
,
6
i
X t_1
and
is
^
(x
rt(x
is
)
as
contained in
Jj6 and x
let
<P t
t_1
t_1
6
=
)
X t_1
(x
f
t0
let
,
t_1
)
.
define a larger family of games, which can be
interpreted as a sequence of families of finite-action games approximating the original
ssr\
family, as follows.
First, let
Secondly, suppose that
X
^t—
X
be the union over n
t
the restriction of aj-(-) to
> 1
,
f.0
is
the restriction of
in the obvious
Now
X
—
s\
1
;
fl^-)
A.
to
t > 1
=
A
x-
X
.
•
and X.
{n}
of the sets
Then:
for all
*
Yn
ie 7, A,.
=
gr(A.)
.
Thirdly, for
all
—
.
Finally, payoff functions
u-
are defined
suppose that we are given a finite-action approximation to a single game.
is
simply a family of games parameterised by n e
is
it
IN
U {qd}
upper semicontinuous in
any subgame-perfect equilibrium path of any game n
equilibrium path of that game,
6
IN
is,
n.
in particular,
and
Hence,
an
follows that any limit point of subgame-perfect
equilibrium paths of the finite-action games
is
an equilibrium path of the original game.
our solution concept (namely subgame-perfect equilibrium in extended strategies)
includes everything that can be obtained by taking finite-action approximations to a
(or,
.
^
/\
/\
s\
satisfying (Al)-(A3), its equilibrium correspondence
And
U {qd}
way.
Since this approximation
since
IN
has been defined for some
S\i
is
6
more
precisely, everything that can be obtained
game
by taking continuous finite-action
approximations to a game satisfying (Al)-(A3)).
5.4
Approximation by e-equilibria
One can
a given single
also
game
is
show that any
limit point of subgame-perfect e-equilibrium paths of
an equilibrium path of the game, by combining the techniques of
this
39
paper with those of Borgers (1989). Indeed, suppose that we are given a family of games
satisfying (A1)-(A3).
For each
t
> 1,
x
t_1
e
X t_1
and
e >
,
let
E
(e,x
t_1
)
be the
set
t
of equilibrium paths of
except that:
(i)
subgame x
each player
i
G
Jji
of a family of
is
it is
identical to the original
replaced by a sequence {t-|t > 1} of agents;
each agent chooses actions that are at or within
Then
games
e
game
(ii)
of being optimal in each subgame.
easy to adapt the methods of Section 4 to show that E.
is
upper
semicontinuous. But this implies the required result. For any subgame-perfect
e—equilibrium path of the original game
path
is
is
also
an e-equilibrium path; any e—equilibrium
an e—equilibrium path of the game in which each active player
sequence of agents; and the 0—equilibrium paths of the
game
in
is
replaced by a
which each active player
is
replaced by a sequence of agents coincide with the equilibrium paths of the original game.
40
Conclusion
7
The
general
results of this paper concerning the existence of subgame-perfect equilibrium in
dynamic games create a small dilemma.
If
one adopts the point of view that the
observation of public signals violates the spirit of non-cooperative
interpretation of the counterexample of Section 2
continuum of actions are somehow
wholly non—pathological in
all
is
If,
theory, then the
presumably that games with a
intrinsically pathological.
other respects.
game
After
all,
that example seems
on the other hand, one thinks that games
with a continuum of actions are an indispensable part of non-cooperative game theory,
then presumably one must accept that games must be extended to allow the observation of
public signals.
Faced with
dilemma
this
There
actions as pathological.
too simplistic.
First,
are,
tempting to dismiss games with a continuum of
however, at least two reasons
why
may
this reaction
be
can be argued that a solution concept should be robust to small
it
errors in the data of the
equilibria that occur in
it is
games
games
to which
it
applies, in the sense that
arbitrarily close to the
game under
one should not rule out
consideration.
15
(In other
words, the equilibrium correspondence should be upper semicontinous.) Provided that such
errors are confined to payoffs, the solution concept of subgame-perfect equilibrium satisfies
this
requirement when applied to finite—action games.
present in the extensive form
semicontinuity
do violate the
is
itself,
to be satisfied.
spirit of
If,
however, such errors
then public signals must be introduced
Secondly,
it is
if
may
be
upper
not entirely clear that public signals really
non-cooperative game theory. After
all,
players
must
in
any case
coordinate on a specific Nash equilibrium to play; and public signals merely increase the
extent to which they can coordinate, by allowing
them
to coordinate on a
equilibrium.
15
See Fudenberg, Kreps and Levine (1988) for example.
random Nash
41
References
Borgers,
T
(1989): "Perfect Equilibrium Histories of Finite—
47, 218-227.
and Infinite—Horizon
Games", Journal of Economic Theory
Chakrabarti, S
K
(1988):
"On
the Existence of Equilibria in a Class of Discrete-Time
of Economics,
Dynamic Games with Imperfect Information", mimeo, Department
Indiana University.
A Mas—Colell
Duffie, D, J Geanakoplos,
Equilibria", Research
&;
A McLennan
(1989): "Stationary
Markov
Paper 1079, Graduate School of Business, Stanford
University.
Forges,
"An Approach
F (1986):
1385.
to
Communication
Equilibria", Econometrica 54, 1375-
Fudenberg, D, D Kreps & D Levine (1988): "On the Robustness of Equilibrium
Refinements", Journal of Economic Theory 44, 354—380.
Fudenberg, D, & D Levine (1983): "Subgame-Perfect Equilibria of Finite— and InfiniteHorizon Games", Journal of Economic Theory 31, 251—268.
Fudenberg, D, and J Tirole (1985): "Pre— Emption and Rent Equalization in the Adoption
of a New Technology", Review of Economic Studies 52, 383-401.
Gihman,
Harris,
Harris,
1 1,
and
AV
Skorohod (1979): Controlled Stochastic Processes Springer—Verlag.
,
C (1985a): "Existence and Characterisation of Perfect Equilibrium in
Perfect Information", Econometrica 53, 613-628.
C
(1985b):
"A Characterisation
of
of the Perfect Equilibria of Infinite-Horizon
Games", Journal of Economic Theory
Harris,
Games
37, 99—125.
C
(1990): "The Existence of Subgame-Perfect Equilibrium with and without
Strategies: a Case for Extensive— Form Correlation", Discussion Paper 53,
Nuffield College, Oxford.
Markov
W
M, &;
Leininger (1986):"A Note on the Relation between Discrete and
Continuous Games of Perfect Information", Discussion Paper A—92, University of
Bonn.
Hellwig,
W
Hellwig, M, &
Leininger (1987): "On the Existence of Subgame-Perfect Equilibrium in
Infinite—Action Games of Perfect Information", Journal of Economic Theory 43,
55-75.
Mertens, J—F, & T Parthasarathy (1987): "Equilibria for Discounted Stochastic Games",
Research Paper 8750, CORE, Universite Catholique de Louvain.
Parthasarathy,
KR
W
(1967): Probability Measures
&
R Zame (1990): "Discontinuous
Rules", Econometrica 58, 861-872.
Simon, L K,
on Metric Spaces Academic Press.
,
Games and Endogenous Sharing
R
QQ§
Ui
Date Due
MIT LIBRARIES DUPL
3
IDflD
00b72155
t