HB31
.M415
working paper
department
of economics
EQUILIBRIUM PAYOFFS WITH LONG-RUN
AND SHORT -RUN PLAYERS AND
IMPERFECT PUBLIC INFORMATION
Drew Fudenberg
and
David K. Levine
No.
524
June 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
EQUILIBRIUM PAYOFFS WITH LONG-RUN
AND SHORT -RUN PLAYERS AND
IMPERFECT PUBLIC INFORMATION
Drew Fudenberg
and
David K. Levine
No.
524
June 1989
Digitized by the Internet Archive
in
2011 with funding from
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http://www.archive.org/details/equilibriumpayofOOfude
EQUILIBRIUM PAYOFFS WITH LONG-RUN AND SHORT-RUN ^L^YERS
AND IHl'ERFECT PUBLIC INFORMATION
Drew Fudenberg
and
David K. Levine
May 1989
*
We are grateful to David Kreps and Eric Maskin.
NSF Grants
SES 87-088016 and SES 86-09697 provided financial support.
Departments of Economics, MIT and UCLA, respectively.
AUG 2 3
1989
Abstract
Ue present a general algorithm for computing the limit,
the set of payoffs of perfect public equilibria,
as
5
-»
1
,
of
of repeated games with
long-run and short-run players, allowing for the possibility that the
players' actions are not observable by their opponents.
We use the
algorithm to obtain an exact characterization of the limit set when the
players' realized actions, but not their choices of mixed strategies, are
observable.
Ue show each that long-run player's highest equilibrium payoff
is generally greater in this case than
when their actions are unobserved.
1.
Introduction
Fudenberg-Kreps-Maskin [1989]
(here referred to as FKM) show that the
folk theorem extends in the natural way to repeated games with long run and
short run players if the players' choices of mixed strategies are observed at
the end of each period.
They further show that the folk theorem need not
obtain when the only information revealed at the end of each period is the
realized choice of action as opposed to the intended mixed strategy.
In the
latter case, with a single long-run player, they characterize the equilibrium
payoffs in the limit as the discount factor goes to one.
Their proof expli-
citly constructs equilibria in "review strategies" whose form is simple.
This paper considers the case of multiple long run players.
Rather
than explicitly constructing equilibrium strategies, we extend the techniques of Fudenberg-Levine-Maskin [1989]
(FLM)
to study games with several
long-run players facing a sequence of short-run opponents, allowing the
possibility that players' strategies need not be observable.
As in FLM, at
the end of each period all players observe a public signal that is
imperfectly correlated with players strategies.
We present a general
algorithm for computing the limit of the set of payoffs of perfect public
equilibria, and use it to provide an exact characterization of the limit set
in the case where the players' realized actions, but not their choices of
mixed strategies, are observed by their opponents.
When all players are long-run, the limit set of payoffs is the same
regardless of whether or not the players' actions are observable, provided a
"pairwise full rank" condition is satisfied.
We show that this is not the
case in games where some of the players are short-run:
the introduction of
moral hazard may lower each long-run player's highest equilibrium payoff.
2.
The Model
In the star,e pame
(pure) action
profile
^
i
a e A
X.
1-1
We let
A,
m.
'^
r
(z
realized outcome:
-^
denote the marginal for
n (a)
Each action
elements.
n
yz
and privately observed outcomes
y
Y
with
elements.
m
y.
z
(a)
over
-
The public
Each player i's
depends on his own private information and on the
,y)
his opponents' strategies matter only in their influence
on the distribution over outcomes.
This model includes as a special case
games where the public information
y
It also includes games where
the actions.
- a.
For example,
y
y
perfectly reveals the actions chosen.
conveys only imperfect information about
can be the realized market price and the
can be choices of output levels as in Green-Porter [198'4), or
be the realized quantity of a good and
good is manufactured.
outcome
y
a
r
induces a probability distribution
i
outcomes lie in a finite set
realized payoff
with
A.
simultaneously chooses
n
to
1.
publicly observed outcomes
(z..,...,z ).
- 1
i
from a finite set
a,
•^
each player
.
z
- a.
y
z.
can
the care with which the
The key to the folk theorem is that there is an
is publicly observed, which rules out games where the players
receive only private signals.
The latter case is considered in Lehrer
[1988] and Fudenberg and Levine [1989].
Player i's expected payoff to an action profile
g^(a) -
S
yGY
zeZ
(a)r (z
ff
a
is
y),
-^
which gives the normal form of the game.
We will also consider mixed actions
profile
a -
(a..
,
.
.
.
,a
)
a.
for each player
i.
For each
of mixed actions, we can compute the induced
distribution over outcomes,
1
-
(q)
TT
y"
2
aeA
(a)Q(a), z (a) -
tt
y^
Z
aeA
y
(a)Q(a)
tt
y
and the expected payoffs
-
(q)
g
S
S
(a)r (z
Q(a)7r
We denote the profile where player
follow profile
a
by
(a,,Q
-'
)
.
-1
1
,y).
>
y€Y aGA
2GZ
;
tt
i
plays
and
(a., a .)
y^ 1
-L
and all other players
a.
p.
"i
(a., a
1
are defined in
.)
-
the obvious way.
As in
FtCM,
we label players so that
ieLR-(l,2
2)
,
i<n,
are
long-run players, whose objective is to maximize the normalized discounted
value of per-period payoffs using the common discount factor
(g.(t))
6.
If
player i's present
is a sequence of payoffs for long-run player i,
value is
(1-6)
where we normalize by
S
5^g.(t)
t-0
^
to measure the repeated game
(1-5)
payoffs in the
same units as payoffs in the stage game.
The remaining players
j
represent different types
S SR - {i+l,...,n)
of short-run players, each representative of which plays only once.
One
example of a model with long and short-run players is Selten's [1977] chain.store game, where a single long-run incumbent faces a sequence of short-run
opponents (see FKM for other examples).
In the repeated game,
in each period
c
the stage game is
- 0,1,...,
played, and the corresponding public outcome is then revealed.
history at the end of period
history for long-run plaver
t
1
is
h(t) - (y (0)
,
.
.
.
,y
(
t)
at the end of period c is
)
.
The public
The private
h.(t) -
(a.(0),z.(0),...,a.(t),z.(t)).
sequence of
A strater.v for long-run player
mapping public and private histories
ir.aps
(h( t-
1
)
i
.
A strater.v for the period-t plavers of type
mixed actions.
a map from the public
short- run player
j
information
h(t-l)
,h
(
.
j
to mixed actions.
observes the public information at
is a
1
) )
to
G SR,
is
t-
1
Note that the
but not the
t,
private history corresponding to the play of his predecessors.
the public information reveals the past choices of action,
However,
if
as in FtCM then
these private histories are extraneous.
Each strategy profile generates probability distributions over
histories in the obvious way, and thus also generates a distribution over
histories of the players' per-period payoffs.
with discount factor
Let
S
by
We denote
repeated game
tlie
G(5).
denote the space of mixed actions, and
A.
B:
A,
X ... X A.
1
i
A
-*
,
„
£+1
X ... xj^
n
be the correspondence that maps any mixed action profile
(a..
,
.
.
.
•°'n)
fo^"
the long run players to the corresponding static equilibria for the short
run players.
maximizes
That is, for each
e.Cq'. ,a
J
J
a e graph(B),
and each
> i,
j
a.
.).
-J
Our focus is on a special class of the Nash equilibria that we call
perfect public eouilibria
each time
c,
.
A strategy for long-run player
it depends only on the public information
the private information
i
is
h(c-l)
public if
at-
and not on
A perfect public equilibrium is a profile
h.(t-l).
of public strategies such that at every date
t
and for every history
the strategies are a Nash equilibrium from that date on.
h(t-l)
(Note that in a
public equilibrium the players' beliefs about each others' past play are
irrelevant:
No matter how player
i
plays, all nodes in the same information
)
set for
at the beginning of period
i
distribution over
i's
payoffs.
lead to the same probability
t
This is why we can speak of "Nash
equilibrium" as though each period began
a
new subgame
.
It is easy to show that perfect public equilibria are sequential,
that the set of public equilibrium payoffs is stationary,
of perfect public equilibrium payoffs in any period
history
h(t-l)
is
independent of
t
the set
and for any public
t
However, when the
h(t-l).
and
that is,
and
players' actions are not perfectly observed, there may be sequential
equilibrium payoffs that are not obtainable with public strategies.
(With
observed actions all equilibria are public.)
In any period
actuibs
player
Q(h(t-1))
of a public equilibria,
t
are common knowledge.
the players period-t mixed
In particular,
believes he will face the mixed action
j
a
each short-run
.(h(t-l)).
Since the
short-run players care only about their one-period payoffs, in a public
equilibrium each period's mixed action must lie in the graph of
B.
(This
need not be true of the sequential equilibria that are not public, for
subsets of the players can use their knowledge of their own past actions to
correlate their play.
See FLM for an example of the way this correlation
can generate additional equilibrium payoffs.)
3
.
Dynamic
Let
Protrrarominfr
E(5) C
IR
and Local Generation
be the set of normalized values that can arise in
perfect public equilibria when the discount factor if
Abreu-Pearce-
6.
Stachetti [1987] introduced the notion of a self -generated set of payoffs
for games with long-run players only, and showed that a sufficient condition
for a set of payoffs to be in
E(5)
is
that it be self -generated.
showed that a sufficient condition for a set of payoffs
W
to be in
FLM
E(5)
,
for all
sufficiently close to
8
is
1
be locally generated.
W
that
This section observes that those definitions and results carry over
immediately to games with some short-run players.
Definition 3.1
Let
:
5
W c
,
enforceable with respect to
i
w(y) e
IR
a.
e A.
1
L
- (l-5)g.(a.,a
v.
....
1
V.
and
v.
W
such that
y € Y,
,
IR
l
''l
> (l-5)g.(a.,a
.)
-1
.
)
v e
and
+
5
5
S
S
TT
y
TT
(a., a
L
(a., a
a
is
if there exist vectors
6
q e graph(B)
+
Action
be given.
IR
and for all
.
-r
,
)w, (y)
for
)w, (y)
for
i
and
e LR
i
s.t. a. (a.) >
a.
i
i
-^
l
s.t. a. (a.) -
a.
and
w(y) G W.
(3.2)
We also say that the
If for a given
a,
w(y)'s
a
enforce
with respect to
(a,v)
exists satisfying (3.1), we say that
v
enforceable with respect to
W
and
S
Note that each
.
w(y)
and
W
a
5.
is
specifies
continuation payoffs only for the long-run players.
Equation 3.1 says that if all players
short-run player
i
j
'^
i
play
ct
•
can increase his payoff by deviating from
if long-run player i's present value starting tomorrow on when
today is given by
w. (y)
,
then player
i
from any of the actions in the support of
yields a higher present value.
then
a.,
y
no
(i)
and
(ii)
occurs
receives exactly present value
v
and no choice of action
a.,
We can exploit the linearity of the
incentive constraints to show:
Lemma 3.2 (Enforcement With Small Variation)
i
IR
.
w - 2
Suppose
„
yeY
TT
y
(a,v)
(Q:)w(y).
^
'
-^
is
enforced by
Then there is
a
Let
C
W(y) e C
and
S.
constant
k.
:
be a convex subset of
Let
such that for all
S'
>
S
there is a
enforce
e
v'
and there are continuation payoffs
IR
<
Kd-S')
||w'(y)-w||
(ii)
Z^^Y 'ry(a)w'(y) = w
(iii)
w'(y) + iS')'^
(iv)
w'(y) - w +
in
(V
-v)
|f|^
e W.
(w(y)-w).
W (y)
This is just a matter of checking that
:
and satisfy (3.1),
C
that
Moreover,
(q,v').
(i)
Proof
e C
w' (y)
defined by (iv) are
see for example
(3.2) and the other properties;
FLM.
I
Definition 3.3
If for a given
:
is enforceable with respect to
P(5,W)
W.
v,
W
W,
and
and
S,
is the set of all points generated by
v G W
set
U c P(5,W).
containing
Lemma 3.U
exists a
Proof
with
W C
If
:
<
5
such that
is compact,
IR
such that for all
1
there exists a
S
<
1,
such that
6'
>
6
S
5'
v e U
to the discount factor
this implies
,U).
we may find
5'.
W c P((5',W).
So
of
IR
and an open
1
> S, W c E(5').
W
Since
is compact,
Since
U c P(5',W),
W
U C P(5
and since the
is compact,
together
choose a finite
,W)
that enforce
w(y) e W
Finally, since
W
of
(U)
Let
over this subcover.
S
in the subcover.
U
<
W
convex and locally generated, then there
be the maximum of
for each
3.2(iii) for each
U c P(5
5
(a,v)
is generated by
v
A subset
W.
By local generation we may find an open cover
:
subcover, and let
Then
v
exists so that
a
we say that
is locally generated if for each
U
an
5
,
v
U's
5'
>
5.
by Lemma
with respect
cover
W,
it is straightfor-
ward to use the principle of optimality to conclude that since each point in
W
can be enforced using continuation payoffs in
W,
that
M c E(5).
I
.
A.
Enforceability on Halfspaces and the Set of Limit Equilibria
This section describes an algorithm that computes the limiting value of
the set
E(5)
as
Section
tends to one.
S
uses this algorithm to
5
characterize the limit set of equilibria.
The key to the algorithm is the study of the payoffs that can be
generated using continuation payoffs that lie in half-spaces
H(A,k) - (v|A-v<k)
these are sets of the form
Definition 4.1
that there exists
and
8
5,
v e
*
k (a, A, 5)
denoted
IR
and
k G
and such that
H(A,k)
this maximum is
6
A
The maximal score attainable by action
:
with discount factor
to
for
*
H (a, A, 6);
IR
with
k - A
•
H
and
IR
a
IR
k e
:
IR.
in direction
maximum of
is the
i
of
A*v
X
such
enforceable with respect
(a.v)
The halfspace associated with
v.
*
h (a, A, 5)
its boundary is the hyperplane
A»v-k(a,A,5).
given by
This definition asks us to fix a mixed action
and a direction
a
then find the "highest" halfspace in direction
and
A,
such that a point on the
A
boundary of the hyperplane can just be generated with action
a
and
continuation payoffs in the halfspace.
Lemma 4.1
:
(i)
k (a,
A, 5)
(ii)
k*(cr.A)
< A
o
g(Q);
(iii)
k (a, A) - A
•
g(Q)
- k (a, A)
independent of
if and only if
to the hyperplane orthogonal to
A
enforceable with respect
is
g(Q)
at
5.
g(a)
•k
Proof
:
(i)
if
k (a,A,<5') - k,
continuation payoffs
(5',H(A,k)).
then there is a
w' (y) e H(A,k)
that enforce
v
with
(a,v)
A
•
and
v - k
with respect to
Now consider the payoffs
w"(y) - [(5"-5')/5"(l-5')]v +
[5
'
(l-6")/(5" (1-5
'
)
]w' (y)
.
.
6"
Since
Then since
is less than one as well.
each
W (y)
the coefficient of
less than one,
is
is
in
W (y)
H(A,k),
is
is on the
v
in
enforce
w"(y)
H(A,k)
Moreover,
(a,v)
and
it is
with
((5",H(A,k)).
(ii)
k (a, A) >
A
would imply that
g(Q)
•
of the maximal halfspace.
Since each
g(Q)
and points in
(iii)
k (a, A) - g(Q)
tion of
boundary of
as well.
H(A,k)
easy to check from (3.1) and (3.2) that the
respect to
in the above equations
v
is
g(Q)
is a strict
v* G H*
in the interior
convex combina-
this is impossible.
H*,
clearly requires that
(a,g(Q))
be
enforceable with continuation payoffs on the hyperplane orthogonal to
and passing through
g(a)
'
We should point out that
continuous in
A
k (a,
A)
is not
necessary upper semi-
Although the constraints in the definition of
a.
enforceability (3.1) have the closed graph property, half-spaces are not
compact so the set of payoffs enforceable with continuations on a given
half-space is not generally upper hemi-continuous in
a.
St
Definition 4.2
k (A) - sup
:
The maximal score in direction
,,„\
Qegraph(B)
^ (o,^)'
^
•
A,
k (A),
solves
The maximal halfspace
in direction
r
is
A
H*(A) - H(A.k*(A)).
Definition 4.3
Note that
Q
:
Q - nH (A).
is convex,
and that each point
q
on the boundary of
enforceable on all of the half spaces that are tangent to
Theorem 4.1
(ii)
:
(i)
For all
If the dimensions of
5,
E(5) C Q.
Q c
IR
p
is
i,
then
lim^
,
o-*l
Q
at
q.
E(5) - Q.
Q
is
.
,
10
Proof of (i)
We will show more strongly that
:
contained in
E(5),
is
point
V G E(<5)
v
A
v - k > k*(A)
•
and
A
H(A,k)
v < k
•
and a
for all
must be enforceable with continuation payoffs in
contradicting the definition of
E(5) C H(A,k),
the convex hull of
(5),
If not, we may find a halfspace
Q.
such that
Then
e E (5).
v'
E
k (A).
Some preliminaries are needed before proving (ii).
Definition 4.4
W C
A subset
:
i
interior with respect to
smooth if it is closed, has non-empty
is
IR
and the boundary of
W
is
C
Since any compact convex set with non-empty interior
Q
[R
,
2
submanifold
i
of
IR
.
can be
W C interior(Q)
approximated arbitrarily closely by smooth convex sets
part (ii) of the theorem will follow once we show that each such
interior of
Lemnia 4.4
in the
is locally generated.
Q
W C interior(Q)
If
:
W
smooth convex set, then
is a
Q
is locally
generated.
Proof
:
We must show that for all
neighborhood
U
of
v e interior(W).
containing
u e U
5
<
so
v
v
with
As in FLM,
let
a
5
<
1
and an open
This is easy to do for
U c P(5,W).
be a static Nash equilibrium.
Fix a
U
This implies each
W.
w e W for some
where
(l-5)g(Q) + 5w,
w(y) - w
Moreover, the continuation payoffs
(a,u)
clearly enforce
U C P(5,W)
Next we consider points
let
there is a
and with closure in the interior of
can be expressed as
1.
v € W
A
be normal to
W
at
v
v.
on the boundary of
Let
the unique halfspace in the direction
k - A
A
•
v,
W.
Fix such a
H - H(A,k)
and let
that contains
W
and whose
and
v,
be
11
boundary
that
h
H
is
is a
tangent to
W
at
Since
v.
W C interior(Q)
proper subset of the maximal halfspace
*
H (A)
strategy that generates a boundary point of
payoffs in
subset of
H
Since
(A).
H (A),
v
enforced with respect to
6'
<
and
1
€
> 0,
w(y,(5")
be a
a
which is a proper
H,
(or.v)
can actually be
H(A,k-€).
By enforcement with small variation (Lemma 3.2(iv)),
may find
Let
.
using continuation
boundary point of
is a
for some
H (A)
it follows
,
that enforce
and a
(a,v)
/c
S" > 6'
for
we
,
such that
>
w(y.5") e H(A,k-[5'(l-5")/5"(l-5')]0.
and
|w(y.6")-v| < k(l-S")
Consider, then,
W
is smooth,
for
difference between
that there exists a
payoffs
.
the ball
5
and
W
<
such that
1
w(y) e interior W.
in
of radius
is at most
U(5)
2/c(l-5").
Since
payoffs in a neighborhood
U
The Characterization of
of
E(S)
/c(l-5")
2
.
/c
Since
>
can be enforced by continuation
w(y)
are interior they may be transy
the
It follows
v
lated by a small constant independent of
5.
v
sufficiently close to one there exists a
5"
H
around
U(5")
generating incentive compatible
v.
I
for Games with a Product Structure
We now specialize to games with a product structure.
A special case of
these games is the case in which actions are observable but mixed strategies
are not.
In these games Fudenberg and Maskin [1986] show that the folk
theorem holds when there are no short-run players.
With short run players and
a single long-run player FKM show that the folk theorem fails,
theless able to characterize equilibrium payoffs.
but are never-
Using Theorem 4.1, we can
This observation originates with Matsushima [1988]
.
12
extend this latter characterization to encompass many long-run and short-run
players, and moral hazard as well.
probability one, and
=
(a)
-n
y
^1
the marginal distribution on
where tt
IS
(a.)7r
(a. ,)
'^^
^
y^
^i
y^+i
In other words, each long-run player's
(a,)
tt
y..
.
.
.
tt
1
action influences only the distribution of hiw "own" outcome
y.
are statistically independent.
condition:
for
-
i
L
1
'
and columns
y,
the matrix
tt.
-
(
1
'
m
tt
with rows
(a.))
y.^ x'
a.
6 A.
1
1
equal to the number of player
It is easy to see that under this condition any
actions.
i's
and the
y.,
In addition, we require a full rank
should have rank
€ Y.
with
y - (y-,,---,y„,y(, ,)
Formally, a game has a product structure if
a e graph(B)
can be enforced by some (not necessarily feasible) specification of the
continuation payoffs.
observable actions
so that
is
TT.
Let
2
Notice that this is a generalization of a game with
Observable actions means that
.
y.
isomorphic to
a.
the identity matrix.
be the unit vector in the direction of the
e.
is
i
coordinate
1
axis, and recall that
*
k (-e.)
and
k (e.)
corresponds to maximizing player
to minimizing it. We also define
V
i's
payoff
to be the subset of
X.
IR
generated as convex combinations of payoffs to long-run players
(g..
since
where
g (q)),
(q)
B
is,
Theorem 5.1
Q
"k
.
Note that this set is closed,
Our goal is to prove:
In a game with a product structure satisfying the full rank
:
condition,
and convex.
a e graph(B)
is
the intersection of
V
with the
2i
constraints
-k
k (-e.) < V. < k (e.
X
2
L
)
1
FLM call this the "individual full rank" condition to distinguish it
from a stronger condition called "pairwise full rank".
.
.
,
13
If,
i,
following Fudenberg and Maskin [1986], we assume that
it follows from Theorem 4.1 that
Although this gives
a
has dimension
as
E(6)
characterization of limit payoffs as
is not interpretable as a folk theorem,
best payoff for player
the limit of
is
Q
Q
as in general
k (e.)
5
is
6
-»
-
1
1
it
,
not the
in the graph(B)
i
In proving Theorem 5.1 there are two cases:
directions
parallel to a coordinate axis, and those that are not.
that are
X
We refer to the
former as coordinate directions, the latter as non-coordinate directions.
From the definitions of
for non-coordinate directions
Lemma 5.2
:
k (A) - A
(g, (a)
g
•
v,
v e V
where
is tangent to
(a)) - V
we may find
(g,(Q:)
g
if
is a non-coordinate
\
V c H(A,k (A)).
and
V
at
and a
a € graph(B)
(q)) - V
and
h(A,k')
w(y)
h(A,k').
A
V
•
with
,
v.
Consequently,
and such that the
Since
^^ initially restrict attention to
w. (y)
- w^(y^)
and such that the incentive constraints all hold with exact equality.
may be written as
(5.1)
Since
V.
n.
- (l-5)g.(a.,a
has rank
m.
,
.
)
4-
5Z
k'
By the definition
at
w(y) - k'
incentive constraints (3.1) and (3.2) are satisfied.
^i'^i+l^'
a
and
such that
k'
is tangent to
satisfying
v e V
that there is an
v,
that may be enforced on
we need only construct
y - (y^
on the
k (A)
By Lemma 4.1(iii), it suffices to show that for some
h(A,k')
V
we can obtain scores
A
In a game with a product structure,
:
such that
of
must satisfy the constraint
Q
,
v.
direction, then
Proof
k
To prove Theorem 5.1 in turn suffices to show that
k (-e.) < V. < k (e.).
boundary of
*
and
Q
n
there exist a solution
(a.)w.(y.)
w.(y.).
They
1
1)
.
14
Since
i
= 1,2.
A
is
non-coordinate,
Consequently,
A
•
X
.
for at least two players,
j^
w(y) = k'
say
provided that
i
1=3
Clearly, setting
- w.(y.)
w. (y)
tions
and
w'
w' (y
of
)
0^(y) =
for
f^
and
y
-
i
"^
^^'-^l''
1
,
w'
^2^^^ " ^2^^2^
"l^^-l^'
^^^^
"^
^2^y]L)'
preserves incentive compatibility for any func-
2
of
y
.
A.w.(y.)),
(1/A )(k'-S.
w'(y
If we choose
the resulting
-
)
-
(-^-i
)
/-^^
^1
(Y-i
)
.
completes the
w(y)
proof.
I
Now we show how to determine the constraints
some classes of games of economic interest.
Following FKM
bounds on the payoffs of the long-run player
V.
-
is the minmax,
1
a min
g.(a.,a
max
1
X
Qegraph(B) a.eA.
and
k (e.)
,
- 1,...,^.
m
k (-e.)
we define two
First,
.)
incorporating the constraint that short- run players must play
best responses to some play of the long-run players.
long-run player's payoff is at least
v.
FPCM
argue that every
in every equilibrium.
The second
bound is
*
V.
L
which
is
H max
Qegraph(B)
the most player
i
11-1.),
min
p,.(a.,a
a.6support(a.
can get if he cannot be trusted not to maximize
within the support of a mixed strategy.
i's
opponents that lead to
v.,
and let
players that solve the problem defining
FVsH
Let
a
be the strategies for
be strategies for all
a
v.
show that for games with observed actions and a single long-run
player the limit set of equilibrium payoffs
is
the interval
[y
v
]
.
FLH
,
,
15
consider games with unobserved actions where all players are long-run.
*
these games all strategy profiles are in graph(B)
any strategy profile
*i
a.
that gives player
a
to
static best response
^
a
1
induced not to deviate from
gonal to player
*i
*
which is why
'
.,
-1
*i
..
-1
k (e.)
so
v.
payoff
i
- max g. (a)
and
,
necessarily has
v.
.
This means that player
'
i
,
can be
with continuation payoffs that are ortho-
a
k (e.) - max
coordinate axis, so that
i's
some players are short run,
Q
a
,
In
*i
o.
When
g.(a).
need not be a static best response to
can be less than
max
1
a
g.(a).
"=1
An important class of games that do not have perfect observability are
hazard mixing games
moi'al
positive for all
a.
.
e A.,
These are games for which
y.
G Y.
and
- l,...,i,
i
tt
(a.)
strictly
is
so there is genuine
k
moral hazard.
then
In addition we require that if
is not a best response to
a.
a
.
.
o e graph(B)
g.(Q) - v
and
This will imply that the
incentive constraint due to moral hazard binds on player
1
at his optimal
equilibrium.
The orem 5.3
*
k (-e.
)
1
k*(-e.) -
-
In games with observable actions
k (e.) - v.
and
v..
In moral hazard mixing games
(iii)
We calculate for each
*
k (-Q,-e.).
In the
e.
case,
a e graph(B)
let
7 - +1
,
*
*
k (e.) < v.
the scores
and in the
•k
7 - -1.
and
~i
1
:
k (e.) < v.
< -V.
-L
(ii)
Proof
In games with a product structure
(i)
:
Then
k
(a, 76.)
max 7V.
subject to;
is the solution to the linear
*
k (a,e.)
-e.
and
case let
programming problem
)
.
16
(5.2)
V.
=-
(l-(5)g. (a. ,Q
.)
+ 6S
TT
(a.)w.(y.)
Q(a.) >
V.
> (l-5)g.(a.,a
.)
+
n
(a.)w.(y.)
Q(a.) -
TV
> 7W
.
.
(
y
.
)
To see that this program determines
condition that
(
for each
i
S
Suppose
:
there exists
e [0,1]
with
a.
a(a.
1
is simply
'^
'
k (e.) > v.
k (a,e.) > v. + [6/(l-S)]e
all
by the full rank condition,
11
L
)
can always be satisfied for some
1
w(y.) e H(7e.,v.)
•^
Proof of
w(y)
note that the incentive
k (a,e.),
constraints for players other than
continuation payoffs
61,
and
Then,
.
since
a
°i
e.
1
independent of
is
such that
graph(B)
t.
-
5,
It follows that for
>
w.(y.) < k(e.,Q) +
L
k
'^
L
+
.)
S
E
y eY
.
g.(a.,a
L
1
k (a,e.) - k (e.)
k*(a,e.) - (l-5)g.(a.,a
with
> 7w.(y.).
7V.
and
>
e
and that the
< g.(a:,a
.)
-1
g.(a.,a
1
'^i
< v.
.)
.
Let
e.
for all
.)
-1
-^
.
be such that
a.
Q(a'.)
>
0.
1
(a.)w.(y.)
n
1
a. (a.)
and
>
*
v.,
By the definition of
1
'
Consequently,
k^(Q,e.) < (l-5)v* + 5(k(e.,Q)+£)
or
k (a,e.) < v. + [5/(,l-S)]e
k (a,e.) > v* + [5/(l-5)]e,
Next, suppose
such that
.
so
This contradicts the assumption that
k*(e.) < v*.
k (-e.) > -v..
k (a.-e.) >
follows that for all
-
y.
As above,
+ [S/{l-6)]e
a.
6 A.
1
1
there is an
k (a,-e.) - k (-e.)
and
-k*(-e^,a) > (l-5)g^(a^,a_^) +
S
w.(y.) > k(e.,Q)
L
-^
1
X
-
e.
Let
a.
X
-
£.
It
(a^)w^(y^)
n
Z
-^i
y.ey.
'
'
X
with
a G graph(B)
be such that
X
g.(a.,a
^x
X
.)
-X
>
g.(a'.,Q:
°x
X
.)
-X
1
.
17
for all
eA..
a.
By the defir'.
tionof
e.(a.,a
v.,
>v..
.)
Consequently
-k*(-e^,a) > (1-5) + 5(-k(e.,a)-€),
or
k (e.,Qa) < -v. + [6/il-8)]e
Proof of (ii)
this is a contradiction.
Again,
.
Under perfect observability, the constraints (5.2) simplify
:
to
(5.3)
V.
- (l-5)g.(a.,Q
V.
> (l-5)g.(a.,Q
L
7V.
L
For
1
''l
7-1,
> 7W. (y
set
.
-^
1
+ (5w.(a.)
Q(a.) >
.)
+ 5w.(a.)
a(a.) -
1
1
-1
)
1
,
11
[v.
w.(a.) - min
1
-(l-5)g. (a. ,a.
"^1
1
1
1
g.(a)
^1^ ^
.
aeA.
)
1
g.(a.,a
„
1
with
a.
11
a. (a.)
1
a.
and
.),
-
1
for
1/5
"
for
.
Q. (a. )>0 °x
X
w.(a.) -
1
I
- min
v.
.)
with
1
Since these continuation
- 0.
a. (a.)
1^ 1'^
> 0;
1
payoffs satisfy constraints (5.3), we conclude that
*
> min
k (a.e.)
^
g.(a.,a .),
,
a.esupport(a.) ^i^ i' -i^'
*
- max
k (e.)
^
> max
k (a,e.)
^
'
and
.
J.^
i'
*
a
'
i"^
a
min
Combining this inequality with that of part
For
7 - -1,
set
V.
- max
1
a.
g.(a.,m
'^i
1
,
a.Gsupport(a.
(i)
.),
-1
)
/
g.(a.,a
^i
i
\
.)
-i
*
-v..
i
k (e.) - v..
we have
where
.
m
is the
minmax
1
profile against player
all
a..
player
i,
and set
i's
payoff to
Proof of (iii)
77-.
-1
[v.
-
(l-5)g. (a. ,m .)]/5
for
Once again these continuation payoffs satisfy (5.3), and hold
v.
regardless of how he plays.
and so combining with part (i) yields
Let
w.(a.) -
- min
Thus
k (-e.) < v
k (-e.) - v
We turn finally to moral hazard mixing games and
:
.
a.GA. ,y.ey.
n
y.
ii-'i-'i-'i
(a.) >
i
-y
- 1.
that all outcomes
from the assumption
^
have positive probability under all profiles.
18
Now for each
consider Che problem of finding
a
the constraints (5.2).
V.
subject to
max v.
At the solution to this program,
= k (a.e.) > w.(y.),
so that
*
2
Choose
so that
a.
(a.)w.(y.) < k (a,e.) +
TT
Then
a. (a.) > 0.
11
1
w.(y.).
min
rr,
*
k (a.e.) < (1-5)2. (a. a
1
''i
+
.)
,
1
-1
<5v.
+
Stt.
~i
1
min
y.ey.
•^
-^
1
w.(y.),
-^
1
1
1
or
w^(y^) > e[
[k (a.e^)-max g^].
ia-S)/6]
Since we know from above that
satisfies
k (a.e.)
1
max
g.
> max v. > min g.
we may add this constraint to the LP problem and
,
find also
max g^ > w^(y^) ^
[min g^-max g^].
[(1-7)/5]
7l[
Consequently the relevant constraint set of continuation payoffs is bounded
independent of
and it follows that there exists
a,
k (e.,a) - k (e.);
that is,
the sup over
*
Suppose
then that
^^
g.
=1 (a. ,a
1
*
< v.,
.)
-1
i'
'
For some
*
v..
^
y.eY.
S
•'
g.(a.,Q
^x
1
and since
tt
.)
-1
< v.
and
1
(a.) >
y.
^
Since
11
a. (a.)
1
a_^) +
<v.,
w.(y.)
1-^1
for all
y.
1
a. (a.)
11
1
-^
v^ - (l-(5)g^(a
Since
with
a.
1
by the definition of
1
is attained.
a
*
- v..
k (a,e.)
^
with
a
1
>
(a^)w^(y^).
ff
-^i
1
it follows that
1
,
-'i'
> 0,
w.(y.) - v.
1 'i
1
for all
1
g.(a.,a
°i
1
e
^11
y.
Y.
.)
-1
.
-v
1
But we
then have
V. >
1
for all
11
a',
e A.,
(l-5)g.
(a'.
'^i
1
,a
.)
-1
+ 5v.
1
with exact equality
if
J
J if and only
-^
Q.(a.) > 0.
1^1
In other
19
words,
a.
is a
best response to
contradicting the definition of
Remark
:
player
a
q
.
yielding the payoff
v.
and
,
moral hazard mixing game.
An alternative interpretation of part (iii) of the theorem is that
moral hazard reduces his best equilibrium payoff relative to
i's
the case of observed actions unless the best observed action payoff
be attained in the stage game with a profile where player
an incentive to deviate.
i
v.
can
does not have
It is intuitive that moral hazard should not be
costly in this case, as there is no need to "keep track" of player
i's
action; the theorem shows that this is the only case where moral hazard has
no additional cost.
We now give an example of a moral hazard mixing game.
L
M
U
4,0
0,1
1,-100
D
2.2
1,1
0,3
"
R
Figure
The payoff matrix in Figure
the actions.
player 2's.
egy,
Player
1
1
gives the expected payoffs as a function of
is a long-run player facing a
1
If player I's actions are observable, but not his mixed strat-
then by Theorem 5.1, his maximum equilibrium payoff is
corresponds to player
1
playing
D
with probability
Now imagine that there are two outcomes
n
,
sequence of short-run
(U) -
TT
„(D) -
1
-
e.
y'
and
p
v.
r^(yj^,L) - 4-6£/(l-2£)
r^(y^,L) - 2-6€/(l-20.
1,
which
G [1/2,100/101].
y"
and that
It is clear that we may define payoffs
that give rise to the normal form in Figure
- 2,
for example
r.(y^,a
)
20
This class of games obviously has
a
product structure and for
€
^ 1/2
satisfies the full-rank condition.
These games also satisfy the pairwise full rank condition described in
FLM.
According to FLM
,
if player
2
were a long-run player the limit set of
equilibria would be the same as that with observable actions.
a short run player,
the limit set is strictly smaller when
c
However, with
This
> 0.
follows from the fact that the game is a moral hazard mixing game:
r
1-f.
l-£-J
is strictly positive,
is
< V
- 2,
and the most player
while if player
1
is
can get from a pure strategy
1
indifferent between up and down, he
must clearly get no more than one.
Note that it can be shown that as
This is quite generally true:
£
->•
,
k (e.)
converges
v
- 2.
if we fix a payoff matrix and consider a
sequence of corresponding information structures that converge to perfect
observability, the maximum payoffs converge to
6.
v..
Conclusion
We have investigated the limit of the set
E(5)
of public equilibria.
When this limit is smaller than the limit set of feasible payoffs, it raises
the possibility that a strictly larger limit set could be obtained by
considering a larger class of equilibria.
In particular,
some action by a
short- run player might not be a best response to any independent random-
ization by his opponents, but be a best response to a correlated
distribution over the opponents' strategies.
When the players' actions are
perfectly observed, this possibility is irrelevant, as all equilibria are
public equilibria.
However, when actions are imperfectly observed,
two
21
players can condition on the public information and their own past play in
such a way that the distribution over their actions given only the public
information is a correlated distribution.
Thus,
there can be equilibria
where the short- run players choose responses that are not in the range of
B.
The characterization set of all the equilibrium payoffs in these games
remains an open problem.
22
REFERENCES
Abreu, D.,
Pearce and
D.
E.
StaccheCCi (1987),
"Toward a Theory of
Discounted Repeated Games With Imperfect Monitoring," Harvard.
Fudenberg, D.
,
D.
Kreps and
E.
Maskin (1989), "Repeated Games With Long-Run
and Short-Run Players," this journal.
Fudenberg, D.
,
and
D.
Levine (1989),
"An Approximate Folk Theorem in
Repeated Games With Privately Observable Outcomes," mimeo.
Fudenberg, D.,
D.
Levine, and
E.
Maskin (1989), "The Folk Theorem in
Discounted Repeated Games With Imperfect Public Information," miraeo.
Fudenberg,
D.
,
and
E.
Maskin (1986), "The Folk Theorem in Repeated Games
with Discounting or with Incomplete Information," Econometrica
54,
.
532-54.
Green, E., and R. Porter [1984],
"Non-Cooperative Collusiong Under Imperfect
Price Information," Econometrica
Lehrer,
E.
[1988],
.
52,
975-94.
"Nash Equilibria of n-Player Repeated Games With Semi-
Standard Information," mimeo. Northwestern.
Matsushima, H.
[1988],
"Efficiency in Repeated Games with Imperfect
Monitoring," forthcoming. Journal of Economic Theory
Selten, R.
[1977],
.
"The Chain-Store Paradox," Theory and Decision
127-59.
594U U06
1
.
9,
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3
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1
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