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HB31
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department
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THE FOLK THEOREM WITH IMPERFECT
PUBLIC INFORMATION
Drew Fudenberg
David K. Levine
Eric Maskin
No.
523
May 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE FOLK THEOREM WITH IMPERFECT
PUBLIC INFORMATION
Drew Fudenberg
David K. Levine
Eric Maskin
No. 523
May 1989
THE FOLK THEOREM WITH IMPERFECT PUBLIC INFORMATION^
By
Drew Fudenberg
David
K.
Levine
AND
Eric Maskin
MAY 1989
*
-»
This paper combines and expands two earlier ones, "Discounted Repeated
Games with One-Sided Moral Hazard" and "The Folk Theorem with Unobservable
Actions". We would like to thank Andrew Atkinson, Patrick Kehoe, Andreu Mas
Colell, Abraham Neyraan, and Neil Wallace for helpful comments.
NSF grants
87-08616, 86-09697, 85-20952 and a grant from the UCLA Academic Senate
provided research support.
**
Department of Economics, Massachusetts Institute of Technology
***
Department of Economics, UCLA; part of this research was carried out at
the University of Minnesota.
Department of Economics, Harvard University
AUG 2 3
1989
ABSTRACT
The
Folk
Theorem
obtains
repeated
in
public
imperfect
with
games
information if for each pair of agents there is at least one action profile
where
the
deviations
information
revealed by
by
the
one
of
deviations by the other,
agents
Folk
Theorem
"Nash- threats"
give
be
observed
statistically
outcome
permits
distinguished
from
Under somewhat stronger conditions, we obtain
strict
equilibria.
Without
pairwise
full
rank,
a
Folk Theorem obtains if the observed outcomes statistically
independent
applications
for
to
publicly
and the dimension of the set of feasible payoffs
equals the number of players.
a
the
of
information
our
results
about
to
each
repeated
player's
actions.
agency models,
oligopoly model and repeated mechanism design.
the
We
give
Green-Porter
Introduction
Folk
The
asserts
Theorem
infinitely
for
repeated
that any feasible payoff vector
discounting
with
games
the minmax
dominates
that Pareto
point of a stage game can arise as the average discounted payoffs in a Nash
(or
perfect)
equilibrium of the
sufficiently close to one.
repeated game
This means
if
discount
the
factor
is
can obtain a
that patient players
large set of payoffs without using formal, contractually enforced agreement
by playing an equilibrium with the same payoffs.
The proof of
the
Folk
Theorem exhibits strategies that yield the desired payoffs and that punish
deviations
important
sufficiently
hypothesis
strongly
of
that
theorem
the
all
is
players
that
wish
to
actions
players'
the
An
conform.
are
observed at the end of each period, so that deviations outside the support
of the equilibrium strategies are sure to be detected.
This paper provides conditions under which the conclusion of the Folk
Theorem holds when players do not directly observe their opponents' actions,
In keeping with
but receive imperfect statistical information about them.
previous
work,
we
assume
this
information
takes
the
of
form
publicly
observed "outcomes" at the end of each period, so that all players receive
the
same
information about
their
opponents'
play.
Obviously,
the
Folk
Theorem fails if the outcomes are completely uninformative about players'
actions;
the
relevant question is how much information the
outcomes must
convey.
^
In each period t of a repeated game equilibrium, each player's action
maximizes a weighted sum of his payoff at period t and his
payoff from period t+1 on.
the
realized
period
t
continuation
If the continuation payoffs are independent of
outcome,
maximizes his period t payoff,
each
player
will
choose
an
action
that
so the action profile will correspond to
a
equilibrium
Nash
constructing
of
the
strategies
stage
which
in
the
key
The
game.
to
Folk
the
continuation payoffs
Theorem
is
vary with
the
outcomes in such a way as to induce players to choose sequences of actions
can be
It is easy to determine which actions
that have any desired payoff.
enforced with arbitrary continuation payoffs, but such considerations leave
open the question of how these continuation payoffs
enforced.
themselves
are
be
to
This is why we look for a "perfect Folk Theorem", meaning that we
require the continuation payoffs themselves to be sustainable by a perfect
equilibrium.
To induce the players to choose a specified profile of actions,
must
exist
continuation
response.
In
that
payoffs
we
case,
that
say
make
that
player's
each
profile
the
action
is
there
a
best
enforceable.
Enforceability is ensured by the "individual full rank" condition introduced
by
Fudenberg-Maskin
which
[1986b],
requires
the
that
probability
distributions over outcomes corresponding to different choices of action by
a single player be
given player,
linearly independent.
When this condition fails for a
different probability mixtures over the player's actions can
yield the same distributions over outcomes,
distinguished.
As
we
will
see,
and so the mixtures cannot be
individual
full
rank
is
essentially
sufficient for the Folk Theorem in principal-agent games (see Section
is not, however,
8)
it
;
sufficient in general.
Rather, we require the stronger condition of pairwise full rank:
every pair of players
i
and
j
,
For*
there must exist a strategy profile for which
the probability distributions over outcomes corresponding to deviations by
and
j
are
linearly
independent.
actions of a single player can be
Individual
full
(statistically)
rank
ensures
that
i
the
distinguished from each
other; pairwise full rank extends this condition to the actions of any pair
Pairwise
players.
of
rank
full
implies "enforceability on hyperplanes"
because
important
is
Any profile of pure actions can be
:
The Folk
enforced by continuation payoffs lying on (almost) any hyperplane.
Theorem
follows
enforceability
from
it
and
hyperplanes
on
the
,
"full-dimensionality" condition introduced by Fudenberg-Maskin [1986a].
develop
To
"work"
"shirk"
or
Assume
"bad".
for
significance
the
on
enforceability
of
consider a two-player partnership game where each player can
hyperplanes,
either
intuition
and there are
payoffs v
that the
-
two observed outcomes,
(v^ ,v^)
"good"
when both work are
and
on the
efficient frontier, but that shirking is a dominant strategy in the stage
can be approximated in a repeated- game equilibrium
Now ask whether v
game.
where both players work in the first period.
Fix such an equilibrium,
(v^ (B)
,
V (B))
be
-
and let v(G)
continuation payoffs
the
(v^ (G)
after
v„(G))
,
good
and v(B)
and bad
-
outcomes
If we let v denote the average payoffs in this equilibrium,
respectively.
then
V - (l-£)v* + 5[pgV(G) + PgV(B)],
(1.1)
where p
are the probabilities of the good and bad outcome when both
and p
player works and
Since
differences
is the discount factor.
S
each
player
v. (G)
-
v
has
(B)
a
and
short-run
v„(G)
-
incentive
v_(B)
must
not
to
be
work,
positive
sufficiently large to induce the players to work in the first period.
S
is
has
close to
large
together.
1,
the*
and
When
even a slight difference between the continuation payoffs
incentive
However,
effects,
and we
can
take v(G)
and v(B)
to
be
close
the distance between the efficient first-period payoff
V
and the equilibrium payoff v cannot converge
reason
The
difference
[5/(1-5)
]
is
that
continuation
in
(v(G)
order
in
-
v(B)),
must
maintain
to
payoffs
be
of
value
1.
the
utility,
magnitude
of
to
of
stage -game
of
order
same
the
tends
6
the
incentives,
units
in
zero or
to
the
as
one-shot gains the players could obtain by deviating, and so the "cost" of
the difference
(This
the wedge between v and v
essentially
is
paper
their
--
for
the
argument
remains constant as
--
Radner-Myerson-Maskin
complete
the
*
counterexample;
[1986]
that
6->l.
equilibria
are
see
uniformly
inefficient for all discount factors.)
The example above does not satisfy pairwise
enforceability on hyperplanes,
bad
outcome
occurs,
and
full rank,
as both players must be
rewarded
after
the
good
and it fails
"punished" when the
outcome,
so
that
only
hyperplanes with positive slope can be used (see Figure 1.1).
The
idea behind enforceability on hyperplanes
that
is
construct
to
approximately efficient equilibria the continuation payoffs must be arranged
to avoid punishing all players at once:
other's
payoff
outcome
"provide
should
increase.
statistical
If one player's payoff fails,
Such
information
about
that
requires
discrimination
player
which
some
the
deviated":
The
Otherwise everyone must be punished, which will lead to inefficiency.
pairwise
full
rank condition ensures
that
deviations by player
(statistically) distinguished from deviations by player
j
,
i
can be
and thus permits
^
enforceability on hyperplanes,
To
illustrate
this
idea modify
the
there are two "bad" outcomes, Bl and B2.
(1-2)
V - (l-5)v*
.+
partnership
game
above
so
that
The analog of equation (1.1) is
5(PqV(G) + Pg^v(Bl) + Pg2v(B2)
SXr:,
fi
yc
For
- 1,2, suppose that,
i
sustains v
if player
i
deviates from the action profile that
there is a higher probability of outcome B
,
The modified game
satisfies
than of Bj
j'^i.
,
and so we can
enforceability on hyperplanes,
arrange the continuation payoffs to provide the necessary incentives while
lying in a line that is parallel to the Pareto frontier (see Figure 1.2).
earlier argument that the equilibrium payoffs are bounded away from v
loses
its
force,
as
and,
we
will
Folk Theorem obtains
the
show,
in
The
now
the
modified game.
So far we have stressed the role of full rank condition in ensuring
enforceability
on
efficient vector
hyperplanes.
However,
actions
enforceable
of
individual full rank holds.
is
we
that
show
will
whether
of
regardless
any
show that,
We use this observation to
Pareto
or
not
in the
class of games where enforceability implies enforceability on hyperplanes,
all payoffs that Pareto dominate a Nash equilibrium of the
stage game can
arise in an equilibrium of the repeated game.
A case of particular importance is when the distribution over outcomes
has
"product
a
identified with
structure",
the
actions
with
of
different
distribution depending only on the
these
games,
enforceability
different
action
implies
components
and
players,
of
the
of
each
the
outcome
component's
associated player.
enforceability
on
hyperplanes.
For
We
exploit this observation to explain how "full insurance" can be provided in
Green [1987]
's
model of repeated insurance contracts for privately observed^
risks.
Another class of games where enforceability implies enforceability in
hyperplanes is games where the actions of all but one player are observable,
including
the
familiar
Fudenberg-Levine-Maskin
principal-agent
[1989],
it
model.
often
applies
Indeed,
to
as
games
we
with
show
in
several
?\Y^
^^
OoLor^^
"agents"
(whose
actions
not
are
observable)
provided
,
exists
there
a
"principal" who can make observable transfers.
There has been considerable previous work related to the Folk Theorem,
including Aumann-Shapley [1976], Friedman [1971], Porter [1983], Green-Porter
[1984], Fudenbcrg-Maskin [1986a], Rubinstein [1979], Radner [1981],
[1985],
[1986], Rubinstein-Yaari [1983], and Radner-Myerson-Maskin [1986].
In the
observable action case,
Aumann-Shapley
considered the Folk Theorem in games with no
[1979]
Friedman
proved
[1971]
"Nash
a
version
threat"
and Rubinstein
[1976]
discounting at
result
the
of
all.
the
for
discounting case, and Fudenberg-Maskin [1986a] established the general Folk
Theorem for discounted games with observable actions
under
the
condition
that the dimension of the feasible set be equal to the number of players.
They
showed
by
example
that
with
non-degeneracy condition is required.
case
of
unobservability
to
Folk
the
more
two
than
introduced one
also
(They
special
considering
by
literature
Theorem
such
some
players,
mixed-strategy punishments.)
Porter
[1983]
and Green-Porter
games with unobservable outputs
[1984]
considered repeated oligopoly
and a stochastic price.
They showed that
there are equilibria that Pareto-dominate the static equilibrium.
Radner [1981], Rubinstein-Yaari [1983], and Radner [1985] established
results
similar
to
the
where only one player's
two
of
these
papers
of principal-agent
Folk Theorem in a class
(the agent's)
treated the
case
action is not observable;
of no
We
discounting.
Radner [1985], who assumed discounting, in Section
8.
the
games
first^
generalize
Radner [1986] showed
that a Folk Theorem obtains in a class of repeated partnership games with
time-averaging
informational
(i.e.
no
conditions.
discounting)
The
payoffs
no-discounting
that
does
not
assumption
is
satisfy
our
essential:
Radner-Myerson-Maskin [1986] construct an example to show the result fails
with discounting.
and Rubinstein-Yaari used
The papers by Radner
techniques based
on
These strategies avoid the
statistical inference and "review strategies."
Instead, at the
need to provide period-by-period incentives not to deviate.
end of each "review phase" a test is conducted to determine whether it is
likely that some player has deviated a non-negligible fraction of the time.
This approach may give
Theorem is
the
impression that the key to obtaining the Folk
the
improvement
information over time.
in
that
feel
We
this
intuition is misleading, as increasing the number of periods during reviews
increases
also
the number
of profitable
opportunities
players
the
for
to
deviate.
In contrast to this statistical approach, we use dynamic programming
arguments
and
geometry,
and
focus
on
the
relationship
between
information revealed by outcomes and the enforceability conditions.
permits
us
to
provide
more
general
theorems.
provides
also
It
the
This
an
alternative understanding of when and why the theorems obtain.
Several authors have exploited the link between discounted repeated
games
and
dynamic
programming,
including
[1986a], and Radner-Myerson-Maskin [1986],
of
Abreu-Pearce-Stacchetti
"self-generation,"
which
[1988],
extends
the
who
Abreu
[1988]
,
Fudenberg-Maskin
Our methods are closest to those
develop
optimality
the
principle
technique
of
of
stochastic"*
dynamic programming to pure strategy equilibria of games with unobservable
actions.
We apply their technique to a class of mixed-strategy sequential
equilibria
that
we
call
"perfect
public
equilibria,"
where
the
strategies depend only on the publicly observable information,
private information about their own past actions.
players'
and not on
The set of perfect public
equilibrium payoffs is stationary, which greatly simplifies the analysis.
After a draft of this paper was
prior contribution of Matsushima
ideas
In particular,
.
we became
completed,
that anticipated
[1988]
aware
the
of our key
some
idea of enforceability on hyperplanes
the
of
used,
is
and differentiability of the boundary of a set of equilibrium payoffs plays
a
key
role
Our work
proof.
the
in
several important ways.
differs
from
that
of
Matsushima
in
First, where Matsushima makes assumptions about the
endogenous limit set of equilibrium payoffs,
instead impose conditions
we
directly on the structure of the stage game, which enables us to develop the
link
between
revealed by
enforceability
outcomes.
the
continuum of actions,
for
conditions
consider
games
an
tangent
and defines
finitely
hyperplanes
Matsushima
Second,
optimum,
with
on
and
assumes
that
many
second-order
the
actions,
incentive compatibility constraints.
and
Finally,
study
there
the
a
We
conditions.
full
consider a
we
is
first-order
of
an equilibrium by means
ignoring
information
the
set
"full"
of
Folk
Theorem, whereas Matsushima shows that a limit of points efficient relative
to
the
set
of
equilibria
is
efficient
relative
to
the
set
of
feasible
payoffs leaving open the question of the "width" of the equilibrium set.
This
conditions
paper
for
uses
the
enforceability
Folk Theorem.
By
on
hyperplanes
modifying
the
to
give
techniques
sufficient
developed
here, and considering enforceability on half -spaces rather than hyperplanes,
one can obtain an exact characterization of the limit set of perfect public*
equilibrium payoffs for any information structure.
use
this
approach
to
analyze
repeated games with
players, where the Folk Theorem fails.
Fudenberg-Levine [1989b]
long-run and short- run
Section
Section
2
introduces our model and defines perfect public equilibria.
the way
reviews
3
programming,
that
one-player dynamic
self-generation extends
and establishes several useful lemmas.
introduces
Section 4
enforceability on hyperplanes and shows how it can be used to prove a Folk
Section
Theorem.
that
imply
that
5
investigates the conditions on a player's
enforceability
on hyperplanes
satisfied.
is
treats games where the outcomes have a product structure.
the results of
Section
6
to adverse selection models,
considers
8
principal-agent
general remarks in Section
information
Section
Section
7
6
applies
such as that of Green [1987].
games.
We
conclude
with
more
some
9.
The Model
2.
In the stage game
action
a.
1
each player
,
- l,...,n simultaneously chooses
i
from a finite set A. with m. elements.
1
i
We let a e A ™ X A,
.
an
A
i
profile of action a e A = xA. induces probability distributions on a public
outcome y e Y and on n private outcomes
Z,
has m.
elements.
We
let Z = X Z.
G
z.
,
js^i.
Until Section
7
,
where Y has m elements and
Each player
.
r.(z ,y) depends on his private outcome z
on z
Z.
i's
realized payoff
and the public outcome y, but not
below, the private outcome z. can be identified
with player i's action a.; we require the extra generality only to deal wi th
repeated games of adverse selection. Let
and z induced by
a.
We let
tt
(a)
-
J n
tt
yz
(a)
(a)
be the distribution over y
be the marginal distribution'
zeZ
of y.
Player i's expected payoff from an action profile a is
Si(->
-[
I
yeY zgZ
Ty,(a)r,(z..y).
-
We will wish to consider mixed actions a
profile
a
-
(a^
,
.
.
.
.
,a
mixed
of
)
for each player
actions,
we
compute
can
For each
i.
induced
the
distribution over outcomes,
'r,,^(a)
yz
- y\,^(a)a(a); t (q)
^ yz
y
aeA
-)
(a)a(a)
tt
,
y
aeA
and the expected payoffs
^i^"^
"III V^'^^i^'i'^^
yeY zeZ
aeA.
We denote the profile where player
follow
profile
a
by
^
(a,, a
i
.);
-1
"^"^
and all other players
plays a
i
n (a., a ,)
-i
y 1
and
p.,
(a,, a
i
'^i
.)
-i
are
defined
the
repeated
analogously.
Our
formulation embraces
games literature.
First,
the
several models
considered
All but the last example identify
z.
in
with the action a
.
if the public outcome is simply the profile of actions, we have
conventional
Fudenberg-Maskin
framework
[1986a].
of
Second,
perfect
as
in
interpretation of
the
observability
with a judicious
ex
variables, we can obtain the standard principal-agent model.
post
.
In this model;
one player, the principal, moves first and selects one of a finite number of
monetary transfer rules, where the transfer to the other player,
depends on output.
Next,
the agent selects an effort level (unobservable to
the principal), which stochastically determines output.
one-sided moral hazard.
the agent,
If we designate player
2
can identify his action a„ with a transfer rule.
This is a model of*
as the principal,
then we
Because we have modeled
the players as moving simultaneously- -whereas actually the agent moves after
the principal-- we must interpret a^
a
as a contingent effort level,
function dependent on the principal's move.
10
that is,
A public outcome is then a
.
realized
output
together
level
with
the
principal's
monetary
transfer
rule. (By including the principal's action as part of the outcome, we make it
observable to the agent)
Third,
we
can think of the players
player provides an unobservable input a
[1986]).
as
a
in which
partnership
to a production process
each
(see Radner
Output y depends stochastically on the quantities of all inputs
and is divided among the partners in a predetermined way.
This is a model
of multi-sided moral hazard.
Fourth, we can imagine that the players are Cournot oligopolists whose
outputs a
market
are not publicly observable.
price,
y,
is
a
stochastic
Demand is stochastic,
function
of
output
(see
and so the
Green-Porter
[1984]).
Finally,
if a.
is no
longer identified with z., we can imagine that,
each period, "agents" receive private information about their endowments that
their actions are "reports"
(possibly untruthful) of their information,
and
Although this interpretation of our model has the same normal form as
standard principal-agent model, it has more subgame-perfect equilibria
because of the simultaneity of moves.
For example, one subgame-perfect
equilibrium of our model (but not of the standard model) entails the agent
To
playing the weakly dominated strategy of refusing all transfer rules.
ensure that the equilibrium sets in the two models are the same, we could
either (a) invoke solution concepts such as trembling hand-perfection (Selten
[1975]) or sequential equilibrium (Kreps-Wilson [1982]) (once we repeat the
game, however, we would then have to modify these concepts to deal with the
infinity of strategies) or (b) perturb the game slightly so that the
principal literally makes small "mistakes", in which case no strategy for the
agent that is ruled out by backward induction in the sequential (standard)
principal- agent game can be a weak best response in our simultaneous
formulation.
We should note that many of our results guarantee strict
equilibrium, so that the issue of refinements does not arise.
the
11
that
mechanism
a
Here
reports.
r
chooses
is
a.
transfers
report,
a
z.
'^
1
1
between players
the
is
as
a
endowment
true
function
the
of
and the
level,
public outcome y is a profile of reports.
In the above examples each player's payoff depends on other players'
actions
through
only
the
outcome
and
y,
for y with n (a)>0,
Formally,
outcome z..
public
not
through his
we can define
private
conditional
the
expected utility
r^(z^.y) n^^(a)
r,(y.a)
-I
n (a)
^
zeZ
One
case
particular
of
interest
when
is
following
the
condition
is
satisfied.
Definition 2.1
Player i's payoff satisfies the factorization condition if
:
for all (a,y) with 7ry(a) > 0,
that is,
it depends
outcome y.
on the play of other players only through the public
When the factorization condition holds, we can write
Naturally, g^(a) - 2
r^(y,Q).
only for values of y for which
tt
(Q)r^(y ,q:^)
,
provides
provided by
condition for
through
8,
the
the
information
public
time
(a) - 0.
tt
about
outcome.
being.
(y,a.) -
where r^(y,a^) is undefined
Note that when factorization is not satisfied,
payoff
r,
his
We
We will
opponents'
will
not
assume
a player's
actions
impose
the
factorization
stage-game
that
is
not*
factorization
in
Section
5
to obtain results for games that do not satisfy the informational
conditions we develop in Section
5.
12
Turning now to the repeated game, in each period
-0
t
,
1
,
.
.
.
The public
game is played and the corresponding outcome is then revealed.
history
at
history
end
the
for
player
mapping
(t)
cr
period
public
strategy
histories of the players'
private
and
.
.
.
,y(t)
period
histories
)
t
is
i
(h(t-l),
private
The
.
a
-
h (t)
is
sequence of
h (t-1))
to
.
probability
generates
histories in the obvious way,
of
,
A strategy for player
.
profile
-=(y(0)
end
the
probability distributions over A
Each
h(t)
is
t
at
i
a. (t) ,z. (t))
(a,(0),z.(0)
maps
of
the stage
,
over
distributions
and thus also generates a distribution over
per-period payoffs.
Player i's objective
to
is
maximize the average discounted expected value of per-period payoffs using
the
common discount
payoffs for player
i,
factor
5.
Thus,
if
{g.(t))
is
sequence
a
of
stage
player i's overall average payoff is
CO
(1-5)5^ 5^g^(t).
t=0
(We use average payoffs,
obtained by multiplying total discounted payoffs by
in order to measure
1-5,
the
repeated game payoffs
in
the
same units
as
payoffs in the stage game.)
Let V be the convex hull of the set of vectors g(a) for a e A.
Sorin
>
[1986]
shows that all payoffs in V are feasible in the repeated game if
1-1/d,
where d is the number of pure strategy profiles of the stage game.
The
extremal points
other
points
payoffs.
in
V;
of V are
extremal
those which are not
actions
are
those
which
Note that all extremal strategies are pure.
13
convex combinations
Let
generate
5
of
extremal
*
- rain raax g.(a., a
V.
in
be player i's mininax value
)
.
and let
.
best response by
to m
.
it
i;
is
v
individually
is
individually
strictly
V
rational
rational
>
v
if
all
for
inequality
this
if
v
is
-k
•k
Let
Strict.
the minmax profile against player 1.
We call m
.
payoff vector
The
players
i
ma
and
be strategies for i's opponents that attain this miniinuin,
,
-=(v
e
V|
>
v
v
for
all
i)
be
set
the
of
feasible,
The Folk Theorem for games with observable
individually rational payoffs.
actions asserts that any strictly individually rational payoff in V
can be
supported by an equilibrium of the repeated game if the discount factor is
sufficiently close to one.
Our focus is on a special class of the Nash equilibria that we call
perfect public equilibria
each time
t,
i7(t)
.
A strategy [a(t)) for player
i
is public if,
at
depends only on the public information h(t-l) and not on
the private history h.(t-l).
A perfect public equilibrium is a profile of
public strategies such that beginning at any date t and for every history
h(t-l)
the
definition
strategies
allows
the
are
a
players
Nash
equilibrium
from
to
contemplate
deviating
strategy, but such deviations are irrelevant:
public strategies, it follows that player
strategy.
i
date
that
a
to
This
non-public
If player i's opponents use
cannot profit from a non-public
Note also that in a public equilibrium the players' beliefs about
each others' past actions are irrelevant: No matter how player
nodes in the same information set for
the
on.
1
same probability distribution over
i
plays,
at the beginning of period t lead to
i's
payoffs.
This
is
why we can
speak of "Nash equilibrium" as though each period began a new subgame.
14
all*
perfect
Every
equilibrium.
2
public
equilibrium
a
perfect
Bayesian
It is easy to see that the set of perfect public equilibrium
payoffs is stationary,
meaning that the set of perfect public equilibrium
continuation payoffs in any period
independent
clearly
is
of
and
t
lie
and for any public history h(t-l)
Note
h(t-l).
equilibrium payoff must
t
V
in
also
that
every
Consequently when
.
perfect
the
is
Bayesian
Theorem
Folk
obtains it exactly characterizes the perfect public equilibrium payoffs as
«->l.
However,
not in V
play
.
without
there can be perfect Bayesian equilibria whose payoffs
The reason is that the players may be able to correlate their
using
an
external
"correlating
public and private information.
lead
i
equilibria where
some
by
combining
their
can be strictly lower when his opponents
allowed to use correlated strategies,
to
device"
(This was pointed out to us by A. Neyman.)
Since the minmax value for player
are
are
players
this
have
endogenous
lower
payoffs
correlation can
than v
,
We
provide an example of this possibility in the Appendix.
3.
Dynamic Programming and Local Generation
Let E(5) C V
be the set of average payoffs that can arise in perfect
public equilibria when the discount factor is
S,
and let V(5) denote the set
of average payoffs that are feasible when the discount factor is
S,
that is,
the values that can be generated by some sequence of repeated- game strategies
that depend only on public information.
2
Here we use the weakest notion of perfect Bayesian equilibrium: actions
are required to be sequentially rational given beliefs, and beliefs
after
any history that the equilibrium assigns positive probability are obtained
using Bayes rule.
15
,
*
*
Lemma 3.1
Proof
:
E(6), V(5)
are compact sets, and V and V
V and V
,
are convex.
The compactness of E(5) is well known, see for example Fudenberg and
:
*
definition, and V
Definition 3.1
are obviously compact; V is convex by
The sets V(6), V and V
Levine [1983].
:
Let
S
i
Q
VI
,
with respect to v^ W and
such that for all
intersection of two convex sets.
is the
S
and v e K
R
be given.
Profile a is enforceable
if there exist vectors w(y)G R
,
for each y e Y,
and a.e A.,
1
1
v^ - (l-fi)g^(a^,Q_^)
+fij^
TT
(a^,a_^)Wj^(y),
for all a^ with a^i^^) >0
v^ > (l-5)g^(a^,a_^) +5^
;r
(a^ ,a_^)w^(y)
for all a^ with a^(a^) -0
(3.1)
_
,
and
w(y) e W.
(3.2)
We say that the w(y)'s enforce a with respect to v,W and S.
If for given a,
there exists v satisfying (3.1), we say that a is enforceable with respect
to 5 and W,
V
Formula 3.1 requires
that
if
the
players
other
than
i
play a
and
,
"
-»
player i's continuation payoff contingent on outcome y is w.(y), then player
i
obtains the same average payoff from all actions in the support of a
no action yields a higher average payoff.
16
,
and
Definition 3.2
If,
:
for given v,
with respect to v,W and
5,
W,
and
there exists an a enforceable
6,
we say that v is generated by
the set of all points generated by
S
6
B(5,W) is
and W.
and W.
The following lemma exploits the stationary nature of the set of perfect
public equilibria.
Lemma 3.2
:
(Self -Generation)
Sketch of Proof
Pearce
Abreu,
If W C B(^,W) and W is bounded then W C E(6).
The idea behind this observation and its proof are due to
:
and
,
Stacchetti
who
[1987],
considered
pure-strategy
equilibria of games with a continuum of outcomes; our finite-outcome model
avoids questions of measurability.
The proof mimics that of the sufficiency
First, decompose w e
of the Principle of Optimality in dynamic programming:
W as in (3.1).
The decomposition defines the players' first period actions.
Next decompose each continuation payoff w(y)
period actions, etc.
,
By construction and the boundedness of W, the sequence
of actions thus defined has average payoff w,
verify that no player can gain by deviating.
case,
this
is
a
thereby defining the second
consequence
of
condition
so w
is
Next we
feasible.
As in the dynamic programming
(3.1),
which
implies
that
no
one-period deviation is profitable, and the fact that payoffs are uniformly
bounded.
^
We can now prove a simple consequence of (3.1) and (3.2).
Lemma 3.3
:
(Monotonicity on convex sets)
[B(5,W) n w] c
[B((5' ,W)
n W].
17
If W is convex and S <
S'
,
then
Remark
:
A related result is proved by Abreu, Pearce, and Stacchetti [1987];
in their model the set of equilibria is itself convex.
Proof
If V e
:
[B(5,W)
n W]
then since v can be enforced by continuation
,
We will
payoffs in W there exist a and w(y) G W satisfying (3.1) and (3.2).
construct a
G W that enforces a when the discount
w' (y)
factor is
and
6'
such that the corresponding average payoff is still v.
Set
w'(y) - [(5'- fi)/6'(l-5)]v + [fi(l-5')/5'(l-5)]w(y).
Since
8'
>
is a convex combination of
w' (y)
S,
For each player
(1
-
i
and profile
a'
6')g^(a^' ,a.) +
i' s
,
v and w(y)
average payoff from
I n^(a'^,a_^)
5'
so w' (y) G W.
,
(q'.
,
a
.
)
is
-
w'^(y)
y
[(5-.
(1
-
5')/(l
-
5)[(1
i
satisfied for a,
be
6
and w(y)
,
convenient
-
5)]v^ +
.
It then follows that since (3.1) was'
it is satisfied for a,
for
]
plays a., he obtains
+ (1-5' )/(l-5)]v. - v^.
[[(5' -5)/(l-5)]
It will
5)/(1
5)g^("'.,a ^) + SIt: (a'^.a ^)w' (y)
-
In particular, if player
-
the
subtracting out the expected values.
analysis
below
Thus define
18
S'
,
w' (y)
as well.
rewrite
(3.1)
and
to
by
)
g^(a^,Q) - g^(a^,a_^)-gj^(Q),
w^-
(3.3)
5^
T^(a)w^(y),
yeY
and w^(y) - w^(y)-w^.
Formula (3.1) can be rewritten as
-
g^(a^,a)(l-fi)/5 - ^
TT
(a^,a_^)w^(y)
for a^ with «^(a^) >
yeY
(3.1')
A
-
A
g^(a^.a)(l-fi)/5 >
,r^(a^,a_^)w^(y)
for a^ with a^(a^) -
0,
J^
yeY
and the requirement that w(y) e W is
A
w + w(y) e W.
(3.2'
establish that a set W is
To
factor,
must
one
show
every
that
continuation payoffs lying in W.
self-generating
point
in
W
can
for
be
a
fixed discount
generated
using
We will find it convenient to work instead
with the weaker property of local generation.
Definition 3.3
exists
5
<
1
:
A subset W of
R
is locally generated if for each
v G W there
and an open set U containing v such that UoW C B(5,W).
19
5
that
Note
generation allows
local
different points v G W.
this multiplicity of
However,
if the
a
different
set W is compact,
we can replace
More precisely, we
with a single discount factor.
5' s
correspond to
to
6' s
have
i
Lemma
exists a
Proof
If W £ R
3 .4
:
<
5'
1
compact,
is
then there
convex and locally generated,
such that W C B(5, W) for all
S
>
6'
.
The open neighborhoods U of the definition of local generation form a
cover of the compact set W, and thus there is a finite collection (U
cover W.
let
S'
[B(5
Let
k
5
be a discount factor such that Z
be the largest of the
S
n W] C [B(5,W) n W].
,W)
.
k
From monotonicity,
Thus Z^ C B(5,W),
- W n U
k
k
C B(6 ,W)
(lemma 3.3),
that
)
if 5
,
S:
and
5'
,
Since the union of the Z
equals W, we conclude that W C B(5,W)
Lemma
3
(Enforcement with
.
subspace of R
.
Small
Suppose
Variation):
H
is
If a is enforceable with respect to H for some
there exists a constant
k.
such that for all v e
and
R
5
>
S'
0,
continuation payoffs w(y) that enforce a with respect to v + H and
a
>
linear
,
then
there are
5
,
such
that
(«)w(y) - v, and |w(y) -v| < /c(l-6)/«.
^r
"l
yeY
Remark
:
If H - R
shows that by taking
5
,
we say simply that a is enforceable
.
The lemma
sufficiently close to one, an enforceable profile can
be enforced with continuation payoffs that are arbitrarily close together,
and that if a profile is enforceable in a hyperplane
20
,
it is enforceable in
any translate
of
that hyperplane.
As
we will
payoffs can be supported by equilibria when
S
reason that many
the
see,
near one is precisely that
is
Conversely,
the variation in continuation payoffs can be made small.
clear
that
required variation
the
if
large,
is
there
may be
it is
way
no
to
enforce certain actions with continuation payoffs that lie in the feasible
set.
A
Proof
Let
:
w' (y)
A
y
w'
and (3.2')
satisfy (3.1')
for H+v'
and
6'
Since
.
A
(a)w' (y) - 0,
TT
and
e H for each y, and w'-O.
w' (y)
One can check that
yGY
A
A
A
w(y) -[5' (l-5)/5(l-5')] w' (y) satisfies (3.1') for S, and clearly w e H.
A
Take w(y) - v+w(y) e v+H.
Then
||w(y)-v||
<
[5' (l-fi)/5 (1-5'
So the conclusion of the lemma holds for k —
6'
/(l-S'
)
)
max
y
]
max |w'(y)-v'||.
y
|w' (y)
-
v'
I
.
||
Enforceability and Local Generation
4.
This section introduces a condition that is sufficient for a set of
payoffs to be locally generated.
Section
5
proves a version of the Folk
Theorem by providing assumptions on the information structure that imply the
sufficient
condition
is
satisfied
by
sets
that
arbitrarily
are
good
St
approximations
payoffs.
to
the
set V
,
the
set
of feasible,
We offer a counter example in Section
5
individually rational
to illustrate the need for^
the informational assumptions.
Recall that V is the convex hull of socially feasible payoffs in the
A
one-shot game.
Let V be an arbitrary closed convex subset of V.
Such a set
satisfies enforceability on tangent hyperplanes if for every extremal point
A
A
A
V 6 V and every n-1 -dimensional subspace H such that v + H
21
A
is tangent to V
at V there exist v' weakly separated from V by v + H, a mixed strategy a with
g(Q) - v'
and continuation payoffs w(y)eH for each yeY that enforce a.
A
somewhat weaker condition is
A
Condition 4.1
A closed convex subset V c V satisfies weak enforceability on
:
A
A
tanpent
hyperplanes
if
extremal
every
for
point
v
-dimensional subspace H such that v + H is tangent to V at v,
sequence of mixed profiles [a
V +
H,
with lim g(a
k
)
-
v'
k
)
and a point
and for each a
^k
associated w (y) e H that enforce a
Note that Lemma 3.5
v'
k
n-1
A
A
A
every
and
V
e
there
is
a
weakly separated from V by
continuation payoffs with
,
k
.
(enforcement with small variation)
implies that,
under Condition 4.1, each extremal point can be enforced with continuation
payoffs that vary little across actions if
S
is near 1.
Note also that the
A
a
hypothesized
Condition 4.1
in
need not
have
payoffs
that
are
weakly
A
separated from V,
Theorem 4.1
If V is a closed convex subset of V
:
that satisfies conditiort
A
4.1,
then for every closed set W in the interior of V there is a
that for all
>
S
5
,
\J
Q E{S)
5
<1 such
.
To prove Theorem 4.1, we will show that each closed W in the interior oi
A
W
V is contained in a smooth set
Definition 4.1
:
A subset W C
R
that is locally generated.
is smooth if it is closed and has
interior with respect to k", and if its boundary is a C
22
non-empty
submanifold of
K
.
Lemma 4.1
:
there exists
If W is a closed convex subset of the interior of V,
A
a smooth convex set
W
such that W C
W
C interior V.
Standard.
Proof:
Theorem 4.1 now follows once we show that each smooth convex W in the
A
interior of V is locally generated.
A
A
Lemma 4.2
If V satisfies condition (4.1), and W C interior (V) is a smooth
:
convex set, then W is locally generated.
Outline of Proof
Fix V on the boundary of W, and consider how it might be generated by
A
A
W,
We
know
that
must be
there
payoff v
extremal
some
of
strictly separated from W by the tangent plane v+H to W at v.
V,
which
is
Moreover, we
can clearly pick the extremal point v so that a translate of v+H is also
A
tangent
to
v there.
Consequently,
condition 4.1,
from
"
weakly separated from V by T
k
and a sequence {a
for any translate of H,
a
lie on that translate.
Since W C interior
such that g(a
)
exist
there
k
)
->
v'
a
v'
and;
can be enforced with continuation payoffs that
A
from W by T
from
W
,
by v+H.
and. we may pick k so
(See
Figure
that g(a
(V),
)
v'
is
is
also strictly separated
4.fe)
If a is enforceable with respect to v, then
v - (l-5)g(a) + 5[v
-
(V
T (a)w(y))]
-
J^
V + (l-5)(g(a)
-
23
V)
-
fi(v
-
strictly separated
£),
Vn
V,
for
(\
-fpr
i.
Figure 4.1: Selection of Extreme Points
^
where
I
- w
-
Y
(Q)w(y).
TT
y
L.
of the continuation payoffs from the boundary at W.
is the average distance
Because W is smooth, the distance from w to the boundary of W along the
hyperplane w+H is on the order of
(see figure
/1|£||
tf)
From Lemma 3.5,
.
we can make do with continuation payoffs whose variation is proportional to
(l-6)/5.
But
<
for
||£||
-/llfll
near
5
also
is
||£||
1,
proportional
to
and
(l-fi)/5
because
so,
we conclude that the continuation payoffs belong
to W for such discount factors.
The
role
of
Condition
4.1
in
obtaining
local generation is to ensure that we can find continuation payoffs lying on
a translate of H.
If,
instead, the continuation payoffs were constrained to
lie on a hyperplane orthogonal to the boundary of
example of the
introduction)
,
then the distance
W
at v (as in the first
||£|
would be of the same
order as the variation in the continuation payoffs, so that the continuation
payoffs could lie outside W even for
Proof of Lemma 4.2
Step 1:
S
near
1.
:
Suppose v e interior (W)
.
Let
C W be an open ball of radius r
U'
that contains v, and let U be an open ball of radius r/5
By condition 4.1,
.
there exists a profile a that is enforceable by some continuation payoffs
{w(y)).
By lemma 3.5, we may assume |v-w(y)
[(l-5)/5]
implies
[v'
||v-g(Q)|
that
there
-(l-5)g(Q)]/5.
||w'(y)-v||
-
||w'
and k(1-S)/S < r/5.
< r/5,
exist
This
< k.(1-S)/S.
|
For given v' e U,
continuation payoffs
implies
that
v'
(y)-w'+w' -v'+v' -v||< |
r.
B(5,W).
24
-
Choose
(w' (y)
(l-5)g(a)
Hence
w' (y)
5
so that*
Lemma 3.5
satisfying
)
5w(y)
+
S
U'
,
.
w'
-
Moreover,
and
_
so
U £
a.' A<
'^
• v-\-
tr^
O^Vtt
(L0o4-^''^^^
H
W
wr H
^.->-
.
Step
2:
Consider v e boundary (W)
hyperplane v+H tangent to W at
Since W is smooth,
.
there is an action profile a such that gCo)
strictly separated from W by
is
and such that a can be enforced with continuation payoffs that lie on
v+H,
Lemma 3.5 then implies that there exists a k such that for any w e
v+H.
g(a)
is
Step
3.:
,
choose the vertical
and the remaining axes
smooth
and
convex,
transformation of coordinates,
all norms on
0?
For X G K
/c
7
- max {D f(x
f(x") >-i||x"||^.
[il-5)/S]^.
x
,
)
,
in v+H.
by
changed
is
where,
since
k'
as
in the
Thus if
Set w
H
2
a C
H
||x
)
||<c)
.
|
H
n
|
||x
||<c,
concave function x
|<c
OH
||x
-f (x
),
)
By Taylor's Theorem,
- (-/c"[(l-5)/5]^,0)
w
e W if
5
is
to be
a point
that lies along
sufficiently close to
1.
5
,
is
the
< K'{l-S)/S. f(x") > -s[/c'(l-6)/5]^ -
||x"||
the first coordinate axis;
Since W is smooth and convex
within the set X - {x
S
lemma 3.5 g(a)
this
is the component on the
where x
the component in v+H.
boundary of W can be represented by
s
lie
to
- v-w.
t
there is a c>0 such that,
Let
norm
Note that w - (-|W|,0),
we now write x — (x
vertical axis and x
the
axis to
may be replaced by a new constant
are equivalent.
outline of the proof,
while
and,
(1st)
In the new coordinates, W
Since g(a) C v+H this transformation is possible.
-/c"
As shown
It is now convenient to choose a new coordinate system.
be the line connecting v and g(a)
Step 4:
,
.
in Figure 4, we take v to be the origin,
still
K
enforceable with respect to w +H using continuation payoffs w(y)
that differ from w by at most k(1-S)/S
is
a unique
is
from condition 4.1,
We argued above that,
v.
there
enforceable with continuation payoffs w(y) such that
25
Fr om
|[w(y)-w
II
W
< k'(1-S)/S, w(y)e w+H, and
implies that w(y)e W.
By construction,
(a)w(y)- w.
this
The corresponding average payoffs are:
v^ - (l-5)g(Q) +
[ ^y(a)w(y) - (l-6)g(a) + 5w^
S
yeY
[(l-5)(g(«))°-/c"[(l-6)/5]^
v
where
is
defined
by
the
final
- (v°,
0]
0),
Since
equality.
(g(a))
first
the
,
o
component
5
<
such that
1
also w
o
there
in the new
of g(a)
<v
< c/4.
Choose a
is
an
e U, set
e
>0
such
v'
- V
-v'
that
not
:
for
any
Then
.
w' (y)
a
> 1/2 satisfying that condition and
v'
e
K
,
-v'||<
||v
£
implies
Let U be a ball about the origin of radius
by continuation payoffs w(y)
w' (y)
5
there is
> -c/4. We conclude that (w(y) -v /5) G interior(W) for all y.
(w(y)-v')e interior(W).
v'
positive,
coordinate system, is
= w(y)
-v'/5 e W.
for discount factor
5
,
c.
Thus
that
For
Since a is enforced
a is
also enforced by
subtracting off a constant from all of the continuation payoffs does
change
the
incentive
to
deviate.
payoff from playing a followed by
(l-5)g(a) + S^
TT
w' (y)
And by
Thus U C B(5,W)
26
the
average
is
(a)w'(y) - (l-5)g(a) +6^
from the definition of v
construction,
tt
(Q)w(y)
-v^ + v'
-
V
)
5.
Informational Conditions
In
section we develop sufficient conditions
this
that
Theorem to hold,
is,
Recall that
Condition 4.1.
under which we may
conditions
ra.
is
for
the number of actions
a
"full"
Folk
V - V
take
in
and that m
in A.,
denotes the number of outcomes in Y.
Definition 5.1
full rank at a for
Given a pair of players
:
(
i
.
i
.
if the (m + m
^
)
H
i
j
(•)
tt
,
satisfies pairwise
vectors
^
a.GA.
'^
-^
a,GA,
considered as rows of a matrix, (where the columns correspond to outcomes y)
have rank
+ m.
(m.
-1)
.
J
1-
This is a full rank condition because the vectors must always admit
one linear dependency
^
)
L,
-^
—
)
a,(a.)7r (a., a
1^ i' y^ i'
-i'
.
tt
(a) —
y^
'
a.(a.)7r (a.,ot
)
^
J
j
y
J
The
,).
-j
pairwise full rank condition obviously requires that the number of outcomes
m be at least
+ m. -1).
(m.
1
Condition 5.1
all
pairs
:
i,j,
J
for
The game satisfies the pairwise full rank condition if,
i
>
j,
there
exists
profile
a
a
such
that
n
satisfies
pairwise full rank at a.
*
Lemmas 5.1 and 5.4 show that condition 5.1 is
satisfy weak enforceability on tangent hyperplanes.
is
much weaker than the requirement that
at all profiles
or.
Indeed,
tt
sufficient for V
to
Note that the condition
should satisfy pairwise full rank
in a "strongly symmetric" game, where permuting
27
the actions of the players does not change the distribution of outcomes,
the
outcome function n cannot satisfy pairwise full rank at any profile where
Nevertheless, in such games, pairwise full
all players use the same action.
may
rank
below)
the
satisfied
be
at
certain
asymmetric
profiles
(see
Example
5.2
This explains why there are many strongly symmetric games in which
.
Folk Theorem
applies,
but
in
which
the
set
payoffs
of
symmetric
to
equilibria are bounded away from efficiency.
Lemma 5,1
sequence a
Proof
Under condition 5.1,
:
k
->
a such that
tt
every profile a can be approximated by a
satisfies pairwise full rank at each a
k
.
We defer the proof until later in this section.
:
[1988],
who
studies a static partnership problem with a balanced-budget constraint.
He
Remark
:
This
lemma
is
an
extension of
a
result
of
Legros
shows that while there is no balanced-budget sharing rule that induces both
partners
to
work with probability one,
if
one
player
randomizes
working and shirking while the other works with probability one,
between
then the
distributions over outcomes will be different if one player shirks than if
the other does,
and so work incentives can be provided while satisfying the
balanced budget-constraint.
In addition to condition 5.1, which requires that pairwise full rank
be satisfied by some profile,
we will require that the weaker property of
individual full rank hold at all pure strategy profiles.
28
Definition 5.2
the m. vectors
tt
:
[n
1
individual full rank at a if for all players
(•) has
(a., a .))
yi-ia.eA.
.
Note that in order for
independent.
are linearly
'
to have individual full rank,
(•)
tz
i,
the number of
y
observable outcomes m must be at least max
Lemma 5.2:
If
This
Proof:
tt
(•)
follows
inequalities (3.1'
Condition 5.2
m
.
has individual full rank at a, then a is enforceable.
immediately from inspection of the
linear system of
).
The game satisfies the individual full rank condition if
:
tt
has individual full rank for all pure action profiles.
Condition 5.3
The dimension of V
:
is equal to n,
Obviously the dimension of V
however be less.
•k
is
no greater than that of V;
it may
Fudenberg-Maskin [1986a] give an example of a three -player
game with observable actions and dim V
obtain.
the number of players.
Fudenberg-Maskin [1986b]
two player game with dim V
"k
- 1 where the Folk Theorem does not
construct a similar counter example in a
•k
- 1 where one player's actions are unobservable.
Note that Theorem 5.1 holds vacuously for sets V of dimension less than n,
for such sets have an empty interior.
Theorem 5.1
.
(Folk Theorem)
Under conditions 5.1, 5.2 and 5.3 for every
closed set W in the relative interior of V
all
5
>
5,
W £ E(5).
29
'k
there is a
5
<
1
such that for
Proof:
The proof invokes a series of lemmas developed below to verify that
*
V
satisfies
weak enforceability
sufficiently large
5
consists of equilibrium payoffs
by theorem 4.1.
interior and relative interior of V
(condition 4.1).
tangent hyperplanes
any closed set in the interior of V
Thus,
for
on
that
implies
Condition 5.3
the
coincide.
It will be convenient to distinguish hyperplanes that are orthogonal to
a
coordinate axis,
hyperplanes.
"coordinate" hyperplanes,
the
Any non-coordinate hyperplane
separates an extremal point g(a) from V
.
that
is
tangent
From lemma 5.1,
5.1 any profile can be approximated by a sequence a
full rank,
"non-coordinate"
from the
k
to
weakly
V
under condition
that satisfies pairwise
and lemma 5.5 shows that profiles satisfying pairwise full rank
can be enforced on non- coordinate hyperplanes, hence, weak enforceability is
satisfied for the non-coordinate hyperplanes.
Next consider the hyperplane H
orthogonal to the ith coordinate axis
at a point where player i's payoff is maximal in V
and tangent to V
hyperplane weakly separates V
all points in V.
from a point that is best for player
The corresponding profile a
best response to a
,
in the stage game,
This
.
i
has the property that a
and so a
is
among
is a
by
enforceable on H
lemma 5.3 below.
Finally,
consider the tangent hyperplane H
ith coordinate axis and tangent to V
where i's payoff is minimized over V
From lemma 5.4, there is a sequence (a k
*
->
[a
V.
k
)
k
k
and a. a best response to a
)
of enforceable profiles with
in the stage
game.
This
has a subsequence whose payoffs converge to g(m i ,m i ,),
separated from V
on H
.
that is orthogonal to the
*
by H
i
,
and lemma 5.2 implies that the a
.
30
k
g. (a
implies
which
.*
k
)
that
weakly
are enforceable
)
An important step
Remark:
proof above
the
in
inference
the
is
then a
profile a maximizes player i's payoff over the feasible set,
static best response to a
a best response
(that is,
.
that
if
is
a
stage game).
in the
This property fails in repeated games where some players are short-run (that
is,
play for only one period), as in Fudenberg-Kreps-Maskin [1989], because
there the definition of the feasible set must incorporate
that the short-run players play static best responses.
player
payoff,
response to a
Theorem
.
fails
with
unless
well,
publicly
are
an action a
players'
the
observed,
observed probabilities,
that
is
and
choices
simply
not
not
best
a
but the Folk
randomizing
of
their
paper
to
with
and
probabilities
unobserved
of
choices
5
and
->
a
Fudenberg-Levine [1989b] extend the techniques of
single long-run player,
this
play
Fudenberg-Kreps-Maskin characterize the equilibrium payoffs as
actions.
1
to
in order to
Thus,
Not only does our proof fail in this case,
.
as
probabilities
may need
i
restriction
that maximizes player
induce the short- run players to play the profile a
i's
the
obtain
characterization
a
the
for
remaining
case
of
unobserved probabilities and several long-run players.
Proof of Lemma 5.1
:
For any profile a and each pair of players
ir^j
,
let
tt
''
be the matrix
whose rows are the m.+m, -1 vectors,
,
'
1
1
(a.
n-
y
^
,
a
.
)
,
-^ a.eA.
11
(a.
TT
y
,
y
a
a'
that
satisfies pairwise full rank for
(a
)
a.EA.
J
Fix a profile
TT
,
-J
and a pair of players inj
31
,
J-
and let a
i
and j.
'-'
be a profile such
Then for each
e
e
.
[0,1],
define q(£) by a(e) ~ (l-£)a'
polynomial in
and the rows of
e,
the determinant of any m.
Thus
itself a polynomial in
+
- w
'-^(q(1))
tt
its determinant is not zero for some such submatrix.
identically zero or has
either
is
determinant
n
'-'(q(£))
of
n
'^(a(€))
full
has
neighborhood of
nonzero
is
rank,
set of zeros
a
By considering
Q;(f).
for
has
(a)
tt
in
square submatrix of
+ m.-l
and since
e,
are polynomials
'^(a(£))
tt
Thus q(£)
for all k.
ta^'^'
(a
3
a
as well.
e
'^(a(£))
tt
is
has full rank,
'-')
Now every polynomial
of measure
almost
all
e.
rank
for
all
full
is
zero
so
the
Moreover,
if
a
in
open
an
near zero, we conclude that there is
t
an open neighborhood of the fixed profile
that satisfies pairwise
q'
full
rank for player i,j, and since a was arbitrary we conclude there is an open,
dense set of profiles that satisfy pairwise full rank for
is
i
and
j
As this
.
true for each pair i,j, and the intersection of open dense sets is open
and dense,
we conclude
that
there
is
an open dense
set of profiles
that
satisfies pairwise full rank for all pairs i,j.
Lemma 5.3
:
If a,
a static best response
is
to a
and a is enforceable,
then a can be enforced with continuation payoffs such that w. (y) is equal to
any
constant.
That
the
is,
can
w(y)
be
chosen
to
on
lie
any
(n-1)
dimensional hyperplane orthogonal to the ith coordinate axis.
Proof
:
Since
payoffs w(y).
a
is
enforceable
Let w. ^
)
^
tt
y
it
(a)w. (y)
,
is
enforceable
and define
w' (y)
1
by
some
by
w' (y)
J
continuation"*
-w (y)
,
ji^i,
J
yeY
We
thank
polynomials
Andreu
Mas-Colell
for
32.
suggesting
this
argument
using
.
and
w'
(y)- w.
Clearly a
.
hyperplane orthogonal
(Leirana
the
ith
axis.
a
can
be
implies
then
3.4)
to
enforceable
is
by
the
w' (y)
which
,
lie
on
a
Enforcement with small variation
hyperplane. H
on any such
enforced
We now show that each player i's minmax value can be weakly enforced on
a hyperplane orthogonal to the ith coordinate axis.
(Weak Enforceability of the Minmax)
Lemma 5.4;
each player
such that
Proof:
there is a sequence of enforceable mixed strategy profiles a
i
g. (a
Let m
k
->
)
v
*
and a
k
is a static
i
to m
Define a - (m.
,
m .),
matrices
the
.
,
be a best
and let m
i,
and for each player
(a,,
tt
y
of
k
that is not weakly dominated by an alternative
.
the matrix whose rows are the vectors
combination
best response to a
be the minmax actions against player
.
response for player
response.
Under Condition 5.2 for
whose
j
rows
a
,).
-j
are
j
r*
Then
tt
i
let
,
(a
,)
tt-^
is a convex
(a., a
tt
'.
)
.
for
in the
proof of Lemma 5.1,
the
G
a',
-j
-J
J
individual
support(a .),and each of these matrices has full rank from the
As
be
.)
-J
y
full rank assumption.
(a
fact
that each
matrix has full rank implies that almost every convex combination of them
Moreover,
does.
i
m.
there is an open dense neighborhood of U of a
.
is a best response to all a'
a such that
i
tt-"
(a
k
.)
in U.
Therefore,
has full rank for each player
response to each of the a
k
.
Therefore each a
k
is
.
such that
there is a sequence a
j
i-"
i,
and m
enforceable,
i
is
and
k
->
a best*
g, (a
k
)->
gj^Cm ).
The remaining case
involves hyperplanes that are not orthogonal to a
coordinate axis, which we call non-coordinate hyperplanes
33
•
Lemma 5.5
If n satisfies the pairwise full rank condition at a,
:
then a is
enforceable on non-coordinate hyperplanes.
Proof
to
:
Consider an n-1 dimensional linear subspace H that is not orthogonal
Without loss of generality, we may assume this has
a coordinate axis.
A
the
-^
coordinate
A
-
form w (y)
n
- S. ^3 .w. (y)
1=1 1
axis,
Moreover,
.
•'
since
H
is
1.
may
we
assume
fi^
y*-
0.
lemma
By
not orthogonal
°
3.4,
it
to
suffices
a
to
A
consider the case
5
- 1/2.
Because a is enforceable,
(3.1') and (3.2') exist for players 1-2
A
,n-l.
solutions
Let w
- s"^^ ^^^^Cy)
A
We must find w (y) and w (y) such that:
(i)
Zy7r^(a)w^(y) -
(ii)
2
" (a^, a_^)w^(y) - -g^(a^, q) for
a^^
e support(a^)
^y %^^1' "-l^^l^y^ - '^\^^V °^ ^°^ ^1 ^ support(a^)
^
(iii)
A
A
,
- yS^w^Cy) + w (y)
^n(y)
(iv)
Sy^^(a)w^(y) -
^^^
^y'^y^^n' "-n^^n^^^
"
"^n^^n'"^
^°^
support(a^)
^n ^
support(a^)
^y^y^^n' "-n^^^n^^^ " "Sn^^"'*^ ^°^ ^n ^
34
{w (3'))
to
Constraints
(i)
,
(ii)
,
and
(iii)
that
require
(iv)
satisfy
w. (y)
A
condition
Constraint (iv)
constraint
and
i-l,n,
for
(3.1)
redundant given (i),
is
ensures
(iii)
w(y)
and the fact 2
(iii),
tt
e
H.
(q)w (y) -
0.
Substituting (iii) into (v)
(vi) ^i2y'fy(aj,.".n)wi(y) -
,
we have
-&J\'^)
Vy^^n'"-n^''
A
fl.S n
^1 y y
(a
n
,a
-n
< -g (a ,q)
"^n
n
)w^ (y)
1
-'
-
S
tt
(a
y y
^
,a
""
^°''
^y^
^n^
support(a^)
A^
)w (y) for a C support(a
n
n)
A
Thus
it
suffices
to
satisfying (i),
find w (y)
given any solution w (y)
constraints
to
(ii)
and
(ii)
,
and
(vi)
,
(vi)
we
Moreover,
.
can
obtain
S
(q)w^ (y)
a
A
solution that satisfies (i) as well by setting w
(y)
- w (y)
-
tt
.
Finally, observe that a solution to (ii) that solves (vi) with equality for
all but one
a'
n
eA
n
solves (vi) for
'
as well,
a'
n
^
since
A
A
a(a') ;9j7r
(a' ,a
n' ^IL y^ n'
a(a
a
,
n
,
r^a'
n)
- -fiS
)w- (y)
,"(a
(a ,a
^iLa Ha' ^ n') Y^
L y^ n' -n"^ 1^-^'
'
n n
)w- (y)
-n' l^-''
g (a
''n
n
,a
-n
)
+
)
L
tt
y
(a
n
,a)w (y)
-'
n
^^^'n>^nK-^^
That is, one of the
- (-;.")w*(y)
-^K^l
(ra^+
m
)
constraints in (ii) and (vi) is redundant.
The
assumption of pairwise full rank is precisely that for each pair of players
In
this system is has rank (m,+ m -1).
exact
Hence (ii) and (vi) can be solved with
equality.
35
of Theorem
It may be helpful to compare examples where the hypotheses
5.1 are and are not satisfied.
Example 5.1
(a
-0).
Players
:
i
-=
1
both players work,
work
either
can
-
(a.
not work
or
1)
—
output
of
(although
y
is
satisfied).
Indeed,
the payoffs
(1,1)
Pareto dominate
-^y
3a
-
Because
.
Player i's
if neither does.
and
has
game
this
rank
full
condition
--
condition
only
must satisfy v +v
<
1
any payoffs
is
although
are feasible and
(corresponding to both players working)
(0,0),
two
fails
--
5.2
the Folk Theorem does not apply to this game:
the minmax point
2
^ of
is
12
the pairwise full rank condition (condition 5.1)
individual
the
The probability of output
or 12.
if only one does,
-^
observable outcomes,
E(5)
2
The work decisions are unobservable and jointly result in output
(which is observable) y -
utility
and
(v ,v
regardless of the value of
5.
)
belonging to
(This
can be
shown using arguments similar to those in Radner-Myerson-Maskin [1986]).
By contrast, consider the following game:
Example 5.2
Now, however,
work,
Players have the same choices and payoffs
:
there are three output levels:
the probabilities of these outcomes are w,
only one works,
probability of
and
12,
the
other
the probabilities are
is
does
1.
-^,
^,
and 0.
6,
and
5-,
and ^,
as
-p-
in Example 5.1.
If both players
respectively.
If
and if neither works the
Notice that at the profile where one player works,
not,
the
probability
definition of pairwise full rank is
36
matrix
corresponding
to
the
:
Because
.
this
matrix
Theorem 5.1 applies.
>
0,
belong to E(5)
has
full
condition
rank,
5.1
.
is
satisfied,
and
so
We conclude that any feasible payoffs (v ,v„) with v
for
5
near enough
1
(despite the fact that the action
profile where both players work violates pair-wise full rank)
We can modify the proof of lemma 5.5
theorem 5.1.
to
obtain a sharper version of
Call an equilibrium strict if each player strictly prefers his
equilibrium strategy to any other.
Theorem 5.2:
Suppose
every action profile,
that
that
positive
every outcome has
tt
probability under
satisfies pairwise full rank for all
(i,j)
at
every pure action profile, that, for each player, there is a minmax profile
in pure actions, and that condition 5.3 holds.
in the
interior of V
there exists
S
<
1
Then for every closed set W
such that for all
6
>
S,
W is
contained in the set of strict equilibria of E(5)
R emark
Even if some minmax profiles entail mixed actions, we can obtain as
strict equilibria all payoffs above the pure-action minmax levels as long as*
the other two hypotheses hold.
37
)
When pairwise full rank holds,
Proof:
action
When
profile.
outcomes
all
(ii)
solved to hold strictly for any
in the proof of lemma 5.5 can be
and (vi)
pure
inequalities of conditions
the
have
probability,
positive
any
deviation then induces a strict loss in payoff if the continuation payoffs
are as specified.
In finite games, strict Nash equilibria are stable as singleton sets in
the
sense
of
Kohlberg-Mertens
established definition of stability
there
Although
[1986].
infinite-horizon
for
games,
likely that strict equilibria would satisfy any such concept.
5.2 casts doubt on conjectures
the Folk Theorem.
not
is
it
Thus,
that stability might restrict the
an
yet
seems
theorem
force of
strict equilibria are robust to small changes
Moreover,
in the information structure of the game.
To obtain strict equilibria requires both that players do not randomize
and that,
for each player,
every information set that could be reached if
some opponent deviated is assigned positive probability by the equilibrium
strategies.
These
observable actions.
conditions
However,
cannot
be
met
because
games.
to
a
equilibria.
totally mixed equilibria,
infinite-horizon
with
and (vi) with
their
We conclude that in such
individually rational payoffs can be approximated by the
totally mixed
(However,
(ii)
games
and thus induce players to play all
actions with positive probability in every period.
payoff
repeated
in such games we can solve
exact equality for all actions,
games all feasible,
in
it
games
is
less
like
clear
stability
This
strict ones,
that
will
stable.
38
observation
all
make
are
extensions
the
totally
of
is
stable
of
interest?
in finite
stability
mixed
to
equilibria
3
Theorems 5.1 and 5.2 show that any point w in the interior of V
obtained as an equilibrium for large
on the efficient frontier.
That
can be
but the theorems do not cover points
S,
under the hypotheses
is,
*
theorem
of the
efficient payoffs can be approximated by equilibria but may not be exactly
This contrasts with the Folk Theorem for repeated games with
attainable.
observable
where
actions,
efficient payoffs
discount factors sufficiently close to one.
why
general
in
theorem
cannot
5.1
be
exactly
can be
attained
for
The following theorem explains
strengthened
obtain
exactly
to
efficient payoffs.
Theorem
there
Let v be an extremal payoff of V.
5.
exists
support n
(a'.
Remark
The
:
player
a
,
.)
i
and
action
such
a'
If for each a with g(a) - v
that
Q support n (a), then for all
inclusion
of
the
supports
clearly satisfied if the support of
hypothesis fails, player
i
tt
(•)
g. (a'
5
<
,
>
)
v 6 E(S)
1
hypothesized
is
a
in
a'
and
.
theorem
independent of
can be deterred from playing
g, (a)
5.2
is
When the
a.
by continuation
payoffs that are very low outside the support % (a).
Proof:
Because v is extremal,
the only sequence of feasible payoffs with
average value v is the sequence where v is repeated each period.
an
equilibrium
strategies must
giving
rise
to
this
sequence.
specify a profile a with g(a)
Then
- v,
payoff w(y) must satisfy w(y) - v for all y e support
^.^) > gj^(a),
a',
to a
.
and support n
(a'.,
a
)
the
39
first
period*
and the continuation
tt
Since
(a).
C support n (a), player
.
Consider
i
g.(a'.
,
will prefer
6
.
Games with a Product Structure
So
far
focused
we
have
of
actions.
enforceability
on
full
Radner
hypotheses
rank
positive
obtained
however,
[1985],
ensure
to
results in repeated principal-agent games without considering the kinds of
informational conditions developed in Section
structure
can
that
take
place
the
These games have special
5.
of
informational
our
of
some
conditions .The distribution over outcomes has a "product structure", meaning
that the outcome y is a vector
(y..
,
.
.
.
,y ),
independent and the distribution of each y
of other players
sufficient
for
than
"Nash- threats"
a
does not depend on the actions
Theorem 6.1 below shows
i.
are statistically
where the y
that
this
pertaining
Folk Theorem,
to
condition is
payoffs
that
Pareto-dominate a static Nash equilibrium.
We first show that profiles corresponding to Pareto-eff icient payoffs
are enforceable regardless of the
assumption
payoffs
that
information structure under the standard
"factor"
as
in
Definition
have
we
As
2.1.
In
emphasized, enforceability alone is not sufficient for the Folk Theorem.
games
with
product
a
enforceability
enforceable
on
on
structure,
hyperplanes
hyperplanes.
necessarily Pareto optimal,
assumptions.
hyperplane,
Of
so
we
so
the
course,
Pareto-efficient
the
minmax
equilibrium is
6.1
can use
covers
payoffs
are
are
not
profiles
theorem 4.1
trivially enforceable on any»
to
conclude
Pareto -dominating a static equilibrium can be attained for
Theorem
implies
enforceability
and will not be enforceable without additional
But any static
and
and
though,
the
principal-agent games
that
S
all
close to
payoffs
1.
considered by Radner,
where the principal's only action is a transfer to the agent.
Section
7
applies theorem 6.1 to a class of repeated mechanism problems that also have
40
.
a product structure.
Section
extends the results to games with a weaker
8
kind of product structure when the actions of some players are observable.
We will now restrict attention to games that satisfy the factorization
that each player's payoff is completely
condition given in definition 2.1,
determined by the public outcome y and his own action a
.
If payoffs satisfy the factorization condition,
Lemma 6.1:
then all external
payoffs on the Pareto frontier of v are enforceable.
Proof
A
:
If V is external and Pareto efficient then there are positive weights
such that v solves max A o v.
V e V
for all
i,
Choose a profile a such that g(a) - v, and
set
w.(y) - [a-S)/S] I
^
j
/A
[A
r (y,
]
^
J
i
J
a
)
J
Then
(l-5)g^(a'^.
(6.1)
a_^) + S I
rt
a ^)w^(y) -
(a'^.
y
[(l-5)/A^]A^ig(a'.,a_.) + [(l-5)/A.]
I
I
y e Y
js-!i
l-S)/X.]
I
J-1
A
A
o
a .).
1-1.
g(a'.
a'.eA.
1
1
enforced by
{w. (y)
,
(a'.,
tt
-^
^
a_^)r (y,
a.
-^
(a'..a_^).
g
-^
-^
Because g(a) - v, a solves max
a e A
solves max
A
A
Thus
o
g(a'
a.
1
)
41
)
is
,
and thus for all players
maximized
(6.1),
i,
implying a
a.
^
is
Definition 6.1:
The game has a product structure if we can write
y - (y
where, for all
y
),
distributions of
y.
-^1
,
n
(a)
i
and a - (a., a
.)
and a', the marginal
satisfies
,
y.
'i
(a.,
n (a., a .) -tt
-1
1
y.
y. 1
-^1
(6.2)
'i
ff
(a) - n
n
for all a ..a,
-1
-1
eA -1,,
and
(a.)
"
^i
i-=l
a.)
-1
The first condition says that the marginal distribution of y,
alone,
depends on a
and the second that the joint distribution over y is the product of
the marginals.
When the game has a product structure, one might expect that
an enforceable profile can be enforced with continuation payoffs {w(y)
that w. (y) depends only on y.
,
that is,
)
such
that the incentive constraints for
The following lemma shows that this
the different players are not linked.
expectation is justified.
Lemma 6.2
:
If the game has a product structure, then any pure action that is
enforceable is enforceable on non-normal hyperplanes.
Proof of Lemma 6.2
:
Fix a,
and as in the proof of Lemma 5.5,
pick an n-1
dimensional linear subspace H that is not orthogonal to a coordinate axis,
A
having the form
I._-|/9.w. (y)
generality we can assume that
- 0,
fi^y^O
with
r^P^.
42
y3
S
K
,
and where,
without loss of
Also set 5-1/2. We must solve
(a)
l^ n^(a) w.(y) -
(b)
I
A
(6.3)
a_^) w^(y) < -g^ia'^, a_^)
(a'.,
,r
Since profile a is enforceable,
payoffs w (y) satisfying (6.3)
for all
for each player
and (b)
(a)
there
i
Moreover,
.
a'.
are
continuation
since the game has a
product structure, we can write these formulae as
(a')
n
y
L
y
^-i
(6.3)
(^'?
Z
y
y,
^i
^>^i
%y -1.(^-i)L.
-X
'i
'
.
n (a.) w.(y,,y
(a .) y
-1
L,Y.
-^
y
y-i
.
.
L.y.
.
1
-
.)
i -^i •'-I
%W ^^^i-y-i)^
y
0.
.
-^i^^'i-
^-i>
-'i
Thus if we set
*
w^(y^) -
(6.4)
(a_^)
TT
w^(y^,y_^),
J^
y-,'-^
then the w.(y.) are continuation payoffs satisfying (6.3) (a) and (b)
Define
(6.5)
w^(y) - w^(y^)
^2(7) - w*(y2)
w
(y)
- w (y
)
-
[
-
^^^^.(y.)/^^.
^^/^2 w*(y^),
for
j
^ 1,2
43
^'i'-^i
1
.
satisfy 6.3 (c) by construction:
The w.
1-1
for
Moreover,
i
>^
1
,
J
-J
j-=2
j=3
*
and
from w.
differs
w,
2,
For players
clearly satisfy (6.3) a and b.
the w.
2
J
by
expected value
addition of terms whose
the
under n (a) is zero, so 6.3 a is satisfied.
and y
since y
Further,
1
are
A
independent,
for each
a'
the
expected value of w (y) under n
the same as the expected value of w (y)
Theorem
6
In
.
game
a
with
product
a
& j)
,
i-^
satisfies (6.3) b.
so that w
,
{&',
satisfies
that
structure
the
factorization condition, let V be the convex hull of the payoffs of a static
Nash
equilibrium
Pareto-dominate
discount factor
Proof
on
and
this
equilibrium.
such that for all
6
any v g
For
>
5
5
strictly
that
payoffs
Pareto-eff icient
all
interior
(V)
there
is
a
v e E(6)
From lemma 6.1 and 6.2, extremal efficient profiles are enforceable
:
all
profile
non-coordinate
yielding
hyperplanes
maximal
the
where
enforceability
hyperplanes
on
i' s
payoff
tangent
Both
.
payoff
is
for
player
constant.
hyperplanes,
and
Thus
the
enforceable
are
i
and
equilibrium
static
the
the
set
conclusion
V
a
on
satisfies
follows
from
theorem 4.1.
One
games
class
of games with a product structure
considered by Radner
[1985]
publicly observed transfer payment,
output
depends
only
on
the
where
is
the
the principal's
principal-agent
only action is
a
and the distribution of each period's
agent's
effort.
44
Theorem
6.1
thus
extends
Radner'
results
s
general
to
specifications
of
the
and the agent's
the principal can be allowed to be risk-averse,
particular,
utility function need not be
in effort
separable
and
In
functions.
payoff
Section
income.
extends the conclusion of theorem 6.1 to games where the principal can,
8
in
addition to making transfers, also influence the distribution of output.
7
Incomplete Information
.
We now apply theorem 6.1 to a class of games of incomplete information.
We then observe that this class includes mechanism design problems.
We
receives
private
the
Each player
following class of games.
information determining his
type
z
G
Z
type has distribution ;r.(z.), which is common knowledge.
their types,
Player
where
,
first
Z.
is
The distribution of types is independent across players, and player
finite.
i' s
first consider
i' s
players choose actions y
e Y
,
After learning
which are publicly observed.
payoff r.(z.,y) depends on the profile of actions and on his type
but not on the types of others.
A pure strategy for player
information
is
Y
G
y
-
i
is
In
Y.x.
.
.xY
a map a
and
,
the
corresponding to strategy profile a is
n'?
T
yz
(a)
i_i 'fi(z)
-
y - a(z)
y H a(z)
Because the marginal distribution of y
\
/
^1
/
". (z.
-
z.e a
)
satisfies
,
-1
(y.)
45
from Z
to Y.
distribution
The public
of
outcomes
)
it is clear that
,
has a product structure in the sense of definition 6.1.
tt
It is also clear that the payoffs satisfy factorization.
types in each period are
In the repeated version of the game, players'
independent
than
draws
fixed
the
distribution
tt
players observe the outcome y but not their opponents'
games
these
that
Note
satisfy
not
do
each
In
(z.).
types.
full
individual
the
period,
rank
condition, and a fortiori fail the stronger condition of pairwise full rank:
Let
z'liZi'
y'
a
(z
satisfy
')
- y,
"
(z'
tt
.
)
>
tt
(z")
A
TT^
probability
(z
'')
since
)/7r
the
(z
'
)
.
with a^(z') -
a-
This strategy yields the same distribution over outcomes
tt
A
A
as (mixed) strategy a
with
and fix a pure strategy
,
,
(z
where o (z") ")/7r
(z
'
)
,
y'
and a
,
and
(z'
with
y'
takes on the value y
)
probability
(tt
''
(z'
Thus many strategy profiles cannot be enforced. However,
theorem 6.1 applies,
outcomes have a product structure,
and we
obtain the Nash threat version of the Folk Theorem.
An example of a repeated game in this class is mechanism design under
repeated adverse selection.
y.
are reports z.
of their types.
and fix a mechanism m:
z,).
T
imagine that the players'
To see this,
->
S
,
Let
Player
actions
be a finite set of social states*,
S
i' s
payoff is r (y,
z
)
- u (m(z)
Each player's payoff depends on his own type and on the social state,
but not on the types of the other players.
One
example
of
this
kind of repeated adverse
been considered by Green [1987].
observed)
endowments,
and
the
provide insurance against fluctuations.
and
Scheinkman-Weiss
[1988]
are the players'
The types z,
mechanism
consider
theory.
46
m
selection problem has*
redistributes
Bewley [1983],
related
models
(privately
endowments
and Levine
to
study
to
[1988],
monetary
Call a vector of payoffs efficient relative to m if it can be attained
by some reporting strategy a and no other strategy profile a yields payoffs
Theorem 6.1 shows
superior.
Pareto
are
that
that
that
payoffs
some
are
efficient relative to m can be approximated by equilibria of the repeated
game
for
close
S
enough
to
might
payoffs
such
However,
1.
be
Pareto -dominated by payoffs that are attainable under a different mechanism
ra'
motivates
This
mechanism
"universal
a
which
game"
define
we
as
A
First, a status-quo state
follows.
action y
i' s
is a pair
each player
i' s
S
(Note that since
.
type and m is a
is a report of his
and T are finite,
S
the
same map m,
payoff is r,(y,z.) - u. (m (z),z.); otherwise, the status-quo
A
State
Each period, player
is specified.
If all players announce
the space of mechanisms.)
is
S
(z.,m) where z
mechanism, that is, a map from T to
so
e
s
A
occurs, and player
s
payoff is r (y,z
i' s
)
In any perfect public equilibrium of this
- u (s,z
what he would obtain under the status quo rule.
player i*s expected
game,
continuation payoff, public history h and current type
).
z
must be at least
,
Furthermore,
the efficient
payoffs of the universal game are first-best payoffs: No choice of mechanism
and
reporting
incentive
is close to one,
S
compatible
or
not,
Pareto-superior
yields
Consequently if these payoffs can be approximated by equilibria
payoffs.
when
rule,
we can conclude that repeated adverse selection has
negligible efficiency costs when players are sufficiently patient.
Since^
Theorem 6.1 applies to these games, this is indeed the case.
To
help
insurance
agents,
game
interpret
discussed
one good,
must equal
2,
3,
this
result,
above.
consider
Imagine
that
Green's
that utility of consumption is u (c
or
4,
and
that
each agent's
47
are
there
)
endowment
repeated
[1987]
- ln(c
z
identical
two
)
,
takes
that
c.
on
the
values
2
and 4 with equal probability.
The best symmetric allocation is for
each agent to consume his endowment when endowments are identical,
each to consume the average endowment of
and for
when one endowment is high and
3
Theorem 6.1 says that the corresponding expected utility
the other is low.
1/4 ln(2) + 1/2 ln(3) + 1/4 ln(4) can be approximated by an equilibrium when
close to
is
S
The conclusion is not that risk and/or risk aversion per
1.
become unimportant,
se
if they had
or that the players can do as well as
access to a productive storage technology: No mechanism can approximate the
utilities
corresponding
to
each
Repeated play does not reduce
total endowment.
player
consuming
risk corresponding
social
the
repeated play reduces,
Rather,
endowment.
average
his
to
and for large
a
random
virtually
6
eliminates, the incentive costs of inducing truthful revelation.
8
.
Principal -Agent Games
In this section we consider the class of games where the actions of at
least
player
one
are
observable.
This
class
includes
the
standard
principal -agent model, where the agent is subject to moral hazard but the
principal's actions
which take the form of transfers to the agent
--
--
can
be observed.
As we noted in the
observability
the
efficient points
when
our
particular,
full
in
of
to
rank
one
be
player'
s
freedom provided by
the additional
introduction,
actions
often
permits
two-player
games
are
not
(with
This
satisfied.
one
player's
(nearly)
repeated game
sustained as equilibria in the
conditions
many
is
actions
true,
evert*
in
observable),
where any payoff vector that Pareto dominates a static equilibrium can arise
in
equilibrium provided
that
the
part
of
the
boundary
equilibrium is downward-sloping. Let the agent be player
48
1
lying
above
that
and the principal
be player
support
We say that player
2.
(tt
2
'
s
actions are observable If for all a the
(a^,a„)) n support (n (a^,a')) -
observed outcome
for all a
r*
so
a'
consistent with exactly one action by player
is
2.
from a static
let V be the convex hull of payoffs
As in theorem 6.1,
that any
equilibrium and the Pareto efficient points that Pareto-dominate it.
Theorem 8.1
payoff vector
For any
>
S
Because
:
payoffs
the
,
actions are observable.
exists
there
outcome reveals player
corresponding to
long as player
shows
v e interior(V)
2' s
such
5
that for all
v € E(6).
§_,
Proof
In a two-player game where player
:
that
the
2
2' s
deviations by player
2
actions,
the
continuation
have probability zero
so
An inspection of the proof of theorem 4.1
never deviates.
condition of enforceability on tangent hyperplanes
can be
weakened to the condition that all continuation payoffs that have positive
probability under the profile being enforced lie on the hyperplane, and that
all continuation payoffs either lie on the hyperplane or correspond to the
Static
equilibrium.
Now fix a static equilibrium,
with payoffs v
,
and
consider a smooth set w in the interior of the convex hull of the payoffs v
of the static equilibrium and the Pareto frontier.
We will show that for
A
all v e w there is a
v'
e V and an a with g(a)-v'
from w by the tangent line H(v)
such that v'
is separated
A
A
^
to w at v,
and such that the continuation^
A
payoffs w(y)are in H(v) if y e
w(y)-v
u
a.
support
otherwise.
49
(tt
y
(a.,,a ))
L
Z
- support 7r(a„)
i-
and
Consider v e W that maximizes player
"maximax"
(v
actions
,v )-=g(a
(a^,a.) is
a
)
sustain
that
player
v
)
a
payoff
maximal
1' s
Clearly H(v) separates w and (v
.
Let (a
payoff.
1' s
be the
)
and
let
Moreover, because
.
maximal payoff, ve can take w(y)-v for all ye(support 7r(a.)).
1' s
A
A
Next consider any other v e W.
separates w and the static
If H(v)
A
equilibrium payoffs
then we can take w(y)-v for all y (support (a.).
v,
then because w lies
not,
in
the
interior of
Pareto frontier, there exists a Pareto optimum
v'
and
convex hull v
the
If
the
which is separated from W
A
by H(v).
there
Let
exist
(a'
a')
the actions
be
continuation
payoffs
associated with
v'
player
1
{w (y)
)
for
From Lemma 6.1
.
enforce
that
a'
A
Because H(v)
is
not parallel
(w^(y) .w^Cy)) e H(v) for all y
A boundary point of V
belongs
to
ray
a
to
Theorem 7.1
Corollary
can
(strict)
choose
:
If
we
boundary to Theorem 8.1,
from
the
>
5,
that
Q.E.D.
with respect to v
equilibrium payoffs
(v
,
.
.
.v
if it
)
with
The outer boundary is downward- sloping if
Pareto optima.
An immediate
add
the
hypothesis
of
a
implication of
downward- sloping
outer
then for any vector v in the interior of V
Pareto dominates a static Nash equilibrium there exists
5
such
w (y)
the following "Folk Theorem-like" result.
is
8.1
we
is on the outer boundary
emanating
only of
axis
e (support (a )).
positive slope in all directions.
it consists
2' s
v e E(5).
50
6
such that, for
that
all"*
corollary
The
applies,
particular,
in
where
games
to
player's action is a monetary transfer to the agent,
observed
the
in Radner
as
[1985].
Both theorem 8.1 and its corollary clearly extend to multiplayer games where
the actions of only one player are unobservable. In fact, we can extend them
to games with several players whose actions are unobservable,
as long as the
observable outcomes have a product structure.
Theorem
8.2
Consider
:
n-player
an
game
where
actions
the
Suppose that we can
n are observable and those of l,...,l are not.
l+l
write
outcomes
and
y
-
y
>
(.V-,
•
•
•
lYp)
,
where
players
of
all
for
i-l,...,t,
n
m
depend only on
a.
and
(a.
a
,
t+1
i
n
)
and
-
(a)
tt
y
IT
i
a.
tt
1
Then the conclusion
y^
of Theorem 8.1 obtains.
When the product structure hypothesis fails, not all efficient payoffs
Pareto dominating a static equilibrium can,
in general,
be approximated by
repeated game equilibria, even when one player's actions are observable and
From Lemma 6.1, any Pareto optimal
the outer boundary is downward sloping.
actions are enforceable;
enforceable
player's
any
on
axis.
The
hyperplanes.
In
profile
the
(for
hyperplanes.
hyperplane
hyperplane
that
difficulty
is
not
enforcing
is
to
the
observable
them
on
the
coordinate
the
maximax
parallel
proof of
theorem
8.1,
we
used
unobservable
player)
for
one
of
the
the
two
But such a profile may not be enforceable on a
once
there
are
they are
from the proof of Theorem 8.1,
indeed,
several
players
51
subject
to
action^
coordinate
coordinate
moral
hazard.
Nonetheless,
as
show
we
Fudenberg,
in
Levine
Maskin
and
,
the
[1989],
efficient payoffs Pareto-dominating a static equilibrium are attainable as
separated game equilibria in a large subclass of games where the observable
player can make transfers of money or some other private good.
9
,
Discussion
.
method
Our
constructing
of
equilibria
based
is
on
dynamic-
the
programming decomposition of equilibrium payoffs into "payoffs today" g(a)
and "continuation payoffs" w(y)
the point being that
,
if
continuation
the
payoffs are known to be equilibrium values, and profile a is enforceable by
the w(y)
then (l-5)g(of) +
,
S
This
approach
prompts
question
the
is an equilibrium' payoff as well.
n (a)w(y)
y
}
L.
how
of
equilibrium
the
set
E(5)
is
related to the payoffs that can be enforced by some continuation payoffs in
the
feasible
set
equilibria.
whether
V(5),
This
latter
first-period payoffs
that
set,
or
not
these
we
denote
which
players
the
payoffs
C(5)
could attain
themselves
are
consists
,
they
if
of
the
could commit
themselves (say, through binding contracts) to the continuation payoffs w(y)
e
V(5)
as
a
function
equilibrium payoffs;
equilibrium payoff
profile a.
Since
of
outcome
the
Let
y.
Nash equilibrium requires
if
the
be
N(5)
that
definition of C(5)
imposes
set
each w(y)
of
Nash
a Nash
be
first
period
fewer constraints
on the
positive probability under
y has
the
the
continuation payoffs than that of N(5), it is clear that E(5) C
N((S)
Q C(5)
.^
From the study of repeated games with observable actions we know that for
fixed
5
less
reflection
than
should
one
the
convince
first
the
inclusion
reader
strict as well.
52
that
can
the
be
second
strict,
and
inclusion
brief
can be
Let us compare the behavior of these sets as
notice
that
is
if
too
is
6
feasible, whereas V(6) = V if
many
small,
5
> 1
6
1
->
payoffs
The first thing to
.
may
V
in
even be
not
1/d, where d is the number of vertices
-
This discrepancy reflects the fact that one effect of
of V (Sorin [1986]).
repeated play with patient players is to allow "Intertemporally transferable
utility" in the interior of V, even if the stage game does not provide much
flexibility for utility tradeoffs.
A second effect of sending
5
towards one
is to make the possible gain from a one-period deviation smaller relative to
a given change
intuition
This is the basis for
in the expected continuation payoff.
that
the
Folk
Theorem
obtain
"should"
repeated
in
with
games
observable actions.
Let
E(l),
and
N(l)
corresponding sets as
5
(A point
1.
-^
denote
C(l)
the
limiting
sufficiently
large
interior(V
in the Nash version N(l) 2 interior(V
condition
);
obtains
for
satisfies
Theorem
either
every pure
if
The
6.)
perfect
Folk
C(l)
2
action profile
is
is
of
in E(5)
for all
if it is
in E(l)
is
values
asserts
Theorem
interior
that
the
E(l)
2
Clearly, a necessary
).
This
(V ).
enforceable,
and
individual full rank at all pure strategies a;
it
thus
inclusion
n (a)
if
fails
if
for
some individually rational vertex v of V the corresponding profile a is not
enforceable.
Even when
all
the
pure
Theorems may still fail.
For
actions,
the
perfect
Folk
action
profiles
example,
Theorem
are
enforceable,
in repeated games with
fails
without
the
the
Folk
observable*
full-dimensionality
condition (5.3). Radner-Myerson-Maskin [1986] consider a game with two-sided
moral hazard where C(l) 2 interior (V
)
yet even the Nash Folk Theorem fails.
Although individual full rank implies
and condition 5.3 is satisfied and
that any Pareto-eff icient profile can be enforced with nearly continuation
53
.
payoffs when
5
close
is
equilibria without
the
to
continuation payoffs
these
one,
stronger condition of pairwise
full
may not be
rank
some
(or
alternative assumption such as a product structure or the existence of an
observable player)
^
The Folk
To conclude, let us mention some limitations of our results.
Theorem shows
amount"
"small
example,
are
equilibria
the
in
of
the
right
kind
of
information
is
Condition 5.1 requires that at some profile a,
have full rank,
case
limiting
of
One consequence of looking at limit results is that even
extreme patience.
a
many payoffs
that
sufficient.
For
the matrix n
(a)
but the matrix is allowed to be arbitrarily close
that is singular.
For a fixed discount factor,
S,
to
one
the equilibrium set is
smaller when the outcomes reveal less information in the sense of garbling,
(Kandori
Intuitively,
[1988]).
the
as
information
required variation in the continuation payoffs grows, and for a fixed
necessary variation may not be
uninformative
near to
,
the
S
feasible.
When
the
the
deteriorates,
outcomes
S
the
almost
are
required to approximate efficient payoffs may be very
1.
Furthermore, as Abreu, Pearce, and Milgrora [1988] have stressed,
misleading
to
interpret
interval between periods
5->l
in
our
model
growing shorter.
as
Our
a
consequence
results
hinge
of
on
it is
time
the
S
being
large relative to a given information structure while quite plausibly,
the
information revealed by the outcomes is a function of the period length. t.
Abreu,
Pearce,
and Milgrom give an example where the Folk Theorem does not
obtain in the limit of shorter time periods because the informativeness of
the outcomes decays too quickly.
54
REFERENCES
Abreu,
Theory of Infinitely
Discounting", Econometrica 56, 383-396.
(1988),
D.
"On
the
with
Games
Repeated
.
"Information
and Timing
in
Abreu,
and P. Milgrom
Repeated Partnerships", mimeo.
Abreu,
Pearce and E. Stacchetti (1986), "Optimal Cartel Monitoring
with Imperfect Information", JET, 39, 251-69.
Abreu,
"Toward a Theory
Stacchetti (1988),
Pearce and E.
Discounted Repeated Games with Imperfect Monitoring", Harvard.
D.
D.
D.
Pearce,
D.
,
(1988),
D.
,
,
of
D.
Aumann, R.
and L. Shapley (1976), "Long-Term Competition: A Game-Theoretic
Analysis", mimeo, Hebrew University.
,
Bewley, T. (1983), "A Difficulty
Econometrica 51, 1485-1504.
with
the
Optimum
Quantity
of
Friedman, J. (1971), "A Non-Cooperative Equilibrium for Supergames",
of Economic Studies 38, 1-12.
Money",
Review
.
Fudenberg, D.
D. Kreps
and E. Maskin [1989], "Repeated Games with Long-Run
and Short-Run Players," mimeo.
,
,
Fudenberg, D.
and D. Levine (1989a).
"An Approximate
Repeated Games with Private Information," mimeo.
,
Fudenberg, D.
and D. Levine (1989b).
and Short Run Players, mimeo."
,
Folk Theorem for
"Perfect Public Equilibrium with Long
Fudenberg, D.
and D. Levine (1983), "Subgame-perfect Equilibria of Finite
and Infinite Horizon Games", Journal of Economic Theory 31, 251-268.
,
D.
and D. Levine (1988), "The Folk Theorem in Repeated Games
with Unobservable Actions", mimeo.
Fudenberg,
,
Fudenberg, D. and E. Maskin (1986a), "The Folk Theorem for Repeated Games
with Discounting and Incomplete Information", Econometrica 54, 533-54^
.
Fudenberg, D. and E. Maskin (1986b),
One-Sided Moral Hazard", MIT.
Fudenberg, D.
and E. Maskin
Randomizations," mimeo.
,
[1988],
"Discounted
"On
the
Repeated
Dispensability
with
Games
of
Public
(1987), "Lending and the Smoothing of Uninsurable Income", in
Prescott and N. Wallace (eds.). Contractual Arrangements for
Intertemporal Trade Minneapolis: University of Minnesota Press.
Green, E.
.
55
E.
Green,
Imperfect
E. and R. Porter (198'^),
"Noncooperative Collusion under
Price Information", Econometrica 52, 87-100.
.
Kandori, M. (1988), "Monotonicity of Equilibrium Payoff Sets with Respect to
Observability in Repeated Games with Imperfect Monitoring", raimeo.
Kohlberg, E. and J.-F. Mertens
Equilibrium," Econometrica
.
Strategic
"On the
1003-1034.
(1986),
54,
"Sustainability
(1988),
Legros, P.
Institute of Technology
Partnerships,"
in
Stability
of
California
mimeo,
"Asset Trading Mechanisms and Expansionary Policy", mimeo,
Levine, D. (1988),
UCLA.
Matsushima,
H.
(1988),
Monitoring", mimeo.
"Efficiency
Repeated
in
Porter, R.
(1983), "Optimal Cartel Trigger-Price
Economic Theory 29, 313-38.
,
Games
Imperfect
with
Strategies",
Journal of
.
Radner,
R.
"Monitoring Cooperative Agreements
in a
(1981),
Principal-Agent Relationship", Econometrica 49, 1127-1148.
Repeated
.
Radner, R.
"Repeated Principal
(1985),
Econometrica 53, 1173-1198.
,
Agent
Games
with
Discounting",
.
Radner, R.
(1986), "Repeated Partnership Games with Imperfect Monitoring
and No Discounting", Review of Economic Studies 53, 43-58.
,
.
Radner, R.
R. Myerson and E. Maskin (1986),
"An Example of a Repeated
Uniformly
Partnership Game with
with
Discounting
and
Inefficient Equilibria", Review of Economic Studies 53, 59-70.
,
.
Rubinstein, A.
"Equilibrium in Supergames with
(1979)
Criterion", Journal of Economic Theory 21(1), 1-9.
,
the
Overtaking
Rubinstein, A. and M. Yaari (1983), "Repeated Insurance Contracts and Moral
Hazard", Journal of Economic Theory 30, 74-97.
.
Scheinkman, J., and L. Weiss (1988), "Borrowing Constraints
Economic Activity", Econometrica 54, 23-45.
and Aggregate
^
Sorin, S. (1986), "On Repeated Games with Complete Information", Mathematics
of Operations Research 11, 147-160,
.
56
APPENDIX
We show that there can be sequential equilibria with payoffs,
V
.
Consider the following three person game.
the first element being Up
Tails
(T)
Player
.
3
(U)
or Down (D)
,
Players
the
chooses Right or Left.
otherwise
he
gets
Notice
zero.
2
and
that
3
2
choose pairs,
second being Heads
Players
and
1
regardless of what happens, while player 3's payoff is
chose Up and he went Right, or if
and
1
-1
not in
(H)
or
receive zero
2
if 2
and
3
both
both chose Down and he went Left;
the
choice
of
H
or
T
is
payoff
irrelevant.
The payoffs to player
may be suinniarized as
3
L
lU,
2U
lU,
2D
ID,
2U
ID,
2D
R
-1
-1
3's payoff
The minmax for player
50-50 between Up
correlate so as
player
3
3
is
-1/4,
and Down.
to
If,
which is achieved by
however,
players
1
1
and
and
2
2
randomizing
could jointly
both play U 1/2 of the time both D 1/2 of the
would only get -1/2.
57
time,
Now suppose player 3's choices are observable,
that whether 1 and
2
play Up or Down is observable, but that the comnion public information about
their choice of H or T is only the total number of H chosen by both players.
we have y(t)
Formally,
-
I(a.(t)),
J^
where !(•)
is
indicator function
the
i
1(H)
-
1,
I(T)
-
actions of players
0.
1
Notice
and
2
period-t actions of players
(since
it
player
is
that
if y(t)
is
2
or
are public information.
and
1
2
period-t
1,
then the
are common knowledge for players 1 and
common knowledge that they each know their own action)
,
2
but
does not know which player played H.
3
1/2-1/2 between H and T in every period.
play
the
If y(t) -
Now consider the following strategies for players
H,
then
0,
D,
1
and
2.
"Randomize
If y(t-l) - 1 and player 2 played
if y(t-l) - 1 and player 2 played T,
then play U.
If y(t-l) -
or 2, play U with randomize 1/2-1/2 between U and D." Since players 1 and 2
are indifferent between all their strategies, they are willing to, follow the
strategies just described.
3's play,
As
these strategies are
independent of player
player 3's best response is to play in each period to maximize
that period's expected payoff.
when y(t-l) -
1.
Doing so yields an expected payoff of -1/2
Thus player 3's expected normalized payoff facing these
strategies converges to -3/8 as
SSfeO
U0
5
->
1,
which is less than v. - -1/4.
5
58
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