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REPUTATION AND EQUILIBRIUM CHARACTERIZATION IN
REPEATED GAMES WITH CONFLICTING INTERESTS
Klaus M. Schmidt
No. 92-7
March 1992
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
REPUTATION AND EQUILIBRIUM CHARACTERIZATION IN
REPEATED GAMES WITH CONFLICTING INTERESTS
Klaus M. Schmidt
No. 92-7
March 1992
M.l.T.
AUG
LIBRARIES
4 1992
RECtlVtO
Reputation and Equilibrium Characterization in
Repeated Games with Conflicting Interests
by Klaus M. Schmidt^
Department of Economics, E52-252C
Massachusetts Institute of Technology
Cambridge,
MA
Tel: (617)
USA
02139,
253-6217
March 1991
revised: February, 1992
A
two-person
game
is
of conflicting interests
player one would most like to
minmax
payoff.
Suppose there
a "commitment type"
get at least her
game
if
who
will
commit
is
if
the strategy to which
herself holds player
two down to
his
a positive prior probability that player one
always play this strategy.
commitment payoff
in
Then
is
player one will
any Nash equilibrium of the repeated
her discount factor approaches one.
This result
is
robust against
further perturbations of the informational structure and in striking contrast
to the message of the Folk theorem for
games with incomplete information.
Keywords: Commitment, Folk theorem. Repeated Games, Reputation.
^This paper
is
based on Chapter 3 of
Doctoral Programme at
my PhD
thesis
which was completed within the European
Bonn University. I would like to thank David Canning, In-Koo Cho, Benny
Moldovanu, Georg Noldeke, Ariel Rubinstein, Avner Shaked, Joel Sobel, Monika Schnitzer, Eric van
Damme and in particular Drew Fudenberg for many helpful comments and discussions. Financial support
by Deutsche Forschungsgemeinschaft, SFB 303 at Bonn University, is gratefully acknowledged.
1
Introduction
Consider a repeated relationship between two long-run players one of
private information about her type.
A common
intuition
whom
has some
that the informed player
is
may
take advantage of the uncertainty of her opponent and enforce an outcome more favourable
to her than
what she would have got under complete information. This intuition has been
and has found considerable attention
called "reputation effect"
purpose of this paper
to formalize this intuition in a general
is
with "conflicting interests" and to show that the
effect
in the literature.
The
model of repeated games
robust against perturbations of
is
the informational structure of the game.
The
formalizations of the "reputation effect" in
first
games with incomplete
in-
formation have been developed by Kreps and Wilson (1982) and Milgrom and Roberts
amount
(1982).
They have shown
cient to
overcome Selten's (1978) chain-store paradox.
that a small
may
faces a sequence of potential entrants
"toughness"
if
there
of incomplete information can be suffi-
An incumbent
deter entry by maintaining a reputation for
a small prior probability that she
is
monopolist who
is
who
a "tough" type
prefers
a price war to acquiescence. Recently, this result has been generalized and considerably
strenghtened by Fudenberg and Levine (1989, 1992).
peated games
whom
in
prior probability of a
all
previous play.
like to
commit
herself,
what she would have obtained
this strategy.
This result
player one's payoff in
games, and
i.e.
it is
(iii) it is
They show
that
if
there
"commitment type", who always plays the strategy
and
all
is
if
player one
if
can enforce at least her commitment payoff
at least
class of all re-
which a long-run player faces a sequence of short-run opponents, each of
plays only once but observes
one would most
They consider the
a positive
which player
sufficiently patient, then she
any Nash equihbrium,
in
i.e.
she will get
she could have committed herself publicly to
very powerful, because
Nash
is
to
is
equilibria, (ii)
it
(i) it
gives a tight lower
bound
for
holds for finitely and infinitely repeated
robust against further perturbations of the informational structure,
independent of what other types
Fudenberg and Levine's analysis
is
may
exist with positive probability.
restricted to
However,
games where a long-run player
faces a
sequence of short-run opponents. Our paper provides a generalization and qualification
of their results for the
two long-run player
condition for this generalization to hold
is
case.
We
that the
show that a necessary and
game
is
sufficient
of conflicting interests.
The
who
first
studied reputation effects in repeated games with two long-run players
were again Kreps, Wilson, Milgrom and Roberts (1982). They demonstrated that
sequential equilibria of the finitely repeated prisoner's
will
is
be achieved
committed
if
there
is
dilemma the cooperative outcome
a small prior probability that the players are of a type which
to always play the "tit-for-tat" strategy.
In the beginning of the 1980s
repeated games with incomplete information have been very popular to solve
of puzzles in
game
in all
all
But the
theory, industrial organization, macroeconomics etc..
kind
initial
enthusiasm has been considerably dampened by a Folk-theorem type result of Fudenberg
and Maskin (1986). They have shown that
exists
an e-perturbation of this game
(in
for
any
finitely or infinitely
game
repeated
there
which each of the players has a different payoff
function with an arbitrarily small but positive prior probability) such that any individually
rational, feasible payoff vector of the
unperturbed game can be sustained as the outcome
of a sequential equilibrium of the perturbed
and
if
there are enough repetitions. But
if
game,
if
the players are sufficiently patient
any outcome can be "explained" by just picking
the right perturbation then the predictive power of the theory of repeated games
is
very
limited indeed.
However, before this message
is
we should have a
accepted,
closer look at
Fudenberg and Maskin mean by an e-perturbation. They assume that with an
small but positive probability each of the players
may be
"crazy",
arbitrarily
she
i.e.
what
may have
a completely different payoff function as compared to the original game. For any given
payoff vector Fudenberg and Maskin then pick one very specific type of "crazyness" which
is
used to sustain this payoff vector as an equilibrium outcome.
informational structure of a
game should capture
A
perturbation of the
the idea that the players
may be
slightly
may
include
uncertain about what the exact payoff functions of their opponents are. This
the possibiHty that the opponent
it
is
actually crazy as in Fudenberg and
Maskin
should not be restricted to only one (very particular) type of crazyness.
the modeller should allow for a broader class of perturbations, such that
payoff functions
may have positive
robustness yields a result which
No matter what
types
may
is
prior probability. Surprisingly, insisting
in striking contrast to the
possibly be
if
the
game
is
argue that
many
on
different
this kind of
message of the Folk-theorem.
drawn by nature (including those considered by
Fudenberg and Maskin) and how Hkely they are
patient,
We
(1986), but
of "conflicting interests",
to occur,
and
if
there
if
is
player one
is
sufficiently
an arbitrarily small but
"commitment" type, then we can give a
positive probability of a
equihbrium outcome of
To make
all
more
this
which player one would
equilibria.
precise, consider a repeated
commit
like to
action", in every period.
"commitment
Nash
informational structure of this
repeated
game always
says that
if
if
the
herself to take an action aj
game
that the
that playing aj holds player two
down
game
is
to his
is
,
called her
in
"commitment
to play aj
and
call
of "conflicting interests" in the sense
minimax
payoff.
Now
whom
it
is
may be
one of
a dominant strategy in the
her the "commitment type".
commitment type has any
Our main theorem
arbitrarily small but positive probability
player one's discount factor goes to one then her payoff in any
bounded below by her commitment
suppose that the
perturbed such that player one
Consider a type for
several possible "types".
game with complete information
player two responds optimally to aj player one gets her
If
Assume
payoff'.
tight prediction of the
This result
payoff.
is
the other possible types and their respective probabilities.
we show
the case of two-sided uncertainty. Furthermore,
and
Nash equihbrium
is
independent of the nature of
We
generalize the theorem to
that "conflicting interests" are
a necessary condition for our theorem to hold.
Our
result highlights the
importance of the relative patience of the two
Player one has to be sufficiently patient as compared to player two,
<
discount factor ^2
commitment payoff
there exists
1
in all
Nash
a.
<
S^{62)
equilibria
if
1
interests,
at the
then
same
it is
time.
his discount factor satisfies
is
two-sided uncertainty.
most
However,
is
the
if
one of them
is
commitment type
sufficiently
is
<5i
>
^i(<52).
game
has conflicting
most preferred outcomes
more patient
sufficiently higher)
(or
if
the prior
then the reputation
works to her advantage.
A complementary
class of repeated
analysis to ours
is
Aumann and
Sorin (1989).
For a different
games, coordination games with "common interests", they obtain a
similar result. However, they have to restrict the possible perturbations to types
like
The
intuitive in the case of
If this
clearly not possible that both players get their
probability that she
effect
game with
any given
such that player one can enforce his
importance of the relative patience of the two players
a completely symmetric
for
i.e.
plaj'ers.
automata with bounded
with positive probability then
recall.
all
They show
that
if all
pure strategy equilibria
who
act
strategies of recall zero exist
will
be close to the cooperative
outcome. In
contrcist to
Aumann and
Sorin
we
allow for any perturbation of player one's
Games of "common" and
payoff function and for mixed strategy equilibria.
interests are
two polar
We
cases.
Finally, in a very recent
Nash
in
more
detail in Section 5.
paper Cripps and Thomas (1991) characterize the
equilibria of infinitely repeated
which players maximize the
them
will discuss
of "conflicting"
set of
games with one-sided incomplete information
mean
limit of the
of their undiscounted payoffs.
in
Following
a different method pioneered by Hart (1985) they also find that in games of conflicting
commitment payoff
interests the informed player can enforce her
small prior probability of a
commitment
type. Since there
seems to indicate that the relative patience
we
will
show
The
in Section 4, this interpretation
closely
Then we
to the class of
on how
paper
rest of the
model following
results.
is
all
is
is
if
there
an arbitrarily
no discounting their result
not that important after
is
is
all.
However, as
misleading.
organized as follows. In the next section
Fudenberg-Levine (1989) and we
briefly
we introduce the
summarize
their
main
give a counterexample showing that their theorem cannot carry over
repeated games with two long-run players. This gives some intuition
this class has to
be restricted.
Section 4 contains our
main
results.
There we
generalize Fudenberg-Levine's (1989) theorem to the two long-run player case, and
show that the
games with
restriction to
this generalization to hold."
"conflicting interests"
is
we
a necessary condition for
Furthermore we extend the analysis to the case of two-sided
incomplete information. In Section 5 we give several examples which demonstrate how
restrictive the "conflicting interests" condition
is.
Section 6 concludes and briefly outlines
several extensions of the model.
2
In
most
of the paper
Description of the
we consider the
Game
following very simple
model
of a repeated
game
an adaptation of Fudenberg-Levine (1989) and Fudenberg-Kreps-Maskin (1990)
which
is
to the
two long-run player
In every period they
action sets
.4,-,
i
case.
The two
players are called "one" (she)
move simultaneously and choose an
G {1,2}. Here we
will
action
assume that the Ai are
a,-
and "two"
(he).
out of their respective
finite sets.^
As a point
of
^See Section 6 for the extension to extensive form stage games, continuous strategy spaces and more
than two players.
reference consider the unperturbed
game
denote the payoff function of player
i
(with complete information)
the unperturbed stage
in
actions taken by both players. Let Ai denote the set of
and
strategies
an abuse of notation) gi{ai,a2) the expected stage game
(in
The
game
T-fold repetition of the stage
We
or infinite.
will deal in
g
denoted by
is
most of the paper with the
game the
repeated
overall payoff for player
from period
i
q,-
where
G-^,
if
the
of player
i
payoffs.
T may
infinite horizon case
the results carry over immediately to finitely repeated games
t) is
game g depending on
mixed
all
Let 5^,(01,02)
first.
T
is
be
but
finite
of
all
large enough. Li the
onwards (and including period
t
given by
00
= E^r'9i,
v^
(1)
T=t
where
<
denotes her (his) discount factor (0
<5,-
average discounted payoffs
u,-,
8i
<
Our
1).
results are stated in terms of
where
CO
vi
(2)
=
= {i-s,)-Y.8igi
(i-<5,).v;°
T=0
After each period both players observe the actions that have been taken.
perfect recall
and can condition
on the entire past history of the game. Let
their play
be a specific history of the repeated game out of the
histories
up
game
a sequence of
is
to
and including period
maps
5'
:
A
t.
H^~^
all
pure (mixed) strategies
Let
B
game and
:
^1
1-^
is
denoted by
^2 be the
—
where
(S,-
if
ct,-
A2y
for player
:
there
H^~^
is
let
—
>
i
of all possible
in the repeated
a,-
=
(a/,(jf,
•
•
•)
Ai. For notational
no ambiguity. The
set of
respectively).
best response correspondence of player two in the stage
gl
"commitment
= max
min
payoff' of player one.
guarantee for herself in the stage
Note that the minimum over
indifferent
i,
suppressed
5,-
{Ai x
Correspondingly,
Ai.
>
5,-
=
h^
define
(3)
as the
is
H^
set
pure strategy
denote a mixed (behavioral) strategy of player
convenience the dependence on history
They have
all
game
if
51(01,02)
That
she could
is
gl
commit
is
to
the most player one could
any pure strategy
02 G B{ai) has to be taken since player two
between several best responses to
ci in
which case he
may
Oi
£ Ai.
may be
take the response
"commitment action")
player one prefers least. ^ Let aj (her
min
(4)
=
Oi(a',a2)
g^
satisfy
.
cr2€S(aJ)
Furthermore,
let
G B{al) denote any strategy
Oj
two which
is
a best response
and define
to Oj
(5)
92
So 92
Oj.
of player
52(01,^2)
the most player two would get in the stage
is
^ A2
Suppose B{a1)
payoff
=
is
her
maxmin
(otherwise the
=
^2
(6)
Then
payoff).
game is
•
game
"trivial"
there exists a 02
g2{a\,~a-i)
=
max
^
if
player one were committed to
because player one's commitment
-^(^1) such that
<
^2(01,02)
^2
•
a2gB(oJ)
Note that the
^ B{a^). So
(22
is
maximum
92
the
is
because
exists
maximum
it is
taken over the finite set of
player two can get
if
maximal payoff player two can
Clearly, in the repeated
game
t
and
Consider
(before the
€
all /i'-^
now
first
of a countable set
payoff function
f:sr-92=
T^
=
"^2
^
W-\
a perturbation of this complete information
stage
f)
now
•
must be true that
it
T=t
for all
=
game
is
(a;o,<j-'i,
•
is
a
(1967-68). In the perturbed
game such
played) the "type" of player one
•
•)
game with incomplete
game a
is
:
that in period
drawn by nature out
according to the probability measure
additionally depends on her type, so 91
perturbed game G^{fi)
action. Finally,
a2£>l2 ai€^i
<
V^
(8)
commitment
all as
= max max ^2(^1, 02)
92
(7)
get at
(pure) actions
he does not take an action which
a best response against aj, given that player one takes her
define the
all
Ai x A2
fi.
><
Cl
Player one's
-^ H.
The
information in the sense of Harsanyi
strategy of player one
may
not only depend on history
^Fudenberg and Levine (1989) refer to gl as the "Stackelberg payoff'. However, it is now customary
maxo, rnaXa^eBiai) 9ii<ii>0'2), that is for the maximum payoff player one
could get if he could publicly commit himself to any action a\ and player two choses the best response
player one prefers most. See Fudenberg (1990). The analysis can be extended to the more general case
where player one would like to commit himself to a mixed strategy or to a strategy dependent on history.
See Fudenberg and Levine (1992) and the remarks in Section 6.
to use this expression only for
but also on her type, so
cr{
//'
:
Two
xQ, —^ Ai.
^
types out of the set
Q.
are of particular
importance:
•
The "normal" type
as in the
of player
denoted by wq. Her payoff function
=
giiai,a2,uo)
many
fi{ujo)
•
is
applications
=
the same
fJ,{uo) will
be close to
<7i(ai,a2).
1.
However, we have to require only that
>0.
(1°
The "commitment" type
repeated
is
unperturbed game:
(9)
In
one
game always
is
denoted by
to play Cj. This
uj*.
is
For her
it is
a dominant strategy in the
for exarrlple the case
her payoff function
if
satisfies
(10)
5i(ai,a2,u;*)
=
for all oi 7^ aj, Ci
€ Ai, and
02)^2 € ^2-
repeated
game impHes
all
gi{a'^,a2,uj'')
=
>
fi'
will
now
tical inference
is
flj
s\
=
aj for all
which
is
and
if
/i'~^
€ H'
<
and
7f
will
<
1.
If
update
has
turn implies that
in
in
The
set of all
lemma
such histories
is
if
denoted by H*.
of Fudenberg-Levine (1989) about statis-
The lemma
says that
The
number
which player two
of periods in
intuition for this result
is
if
u' has
which player one has always played aj up to period
player two observes Cj being played in
his beliefs.
t
he
is
will believe
the following. Consider any
player two believes that the probability of aj being played in period
7f,
uj'
player two observes Cj being played in every period then there
"unlikely" to be played.
history
strategy property in the
equihbrium path. This
basic to the following analysis.
a fixed finite upper bound on the
is
t.
restate an important
positive probability
The dominant
then with positive probability there exists a history in any Nash
equilibrium with
We
5i(ai, 4,0;')
that in any Nash equilibrium player one with type
to play Cj in every period along the
fi{uj')
>
i
t
—
is
"surprised" to
1.
Suppose
smaller than
some extend
Because the commitment type chooses a^ with probability
1
while player two expected aj to be played with a probability
it
follows
bounded away from
1
from Bayes' law that the updated probability that he faces the commitment
type has to increase by an amount bounded away from
arbitrarily often because the
0.
However,
this
cannot happen
updated probability of the commitment type cannot become
This gives the upper bound on the number of periods in which player two
bigger than
1.
may
a'[
expect
Note that
to be played with a probability less than W.
argument
this
is
independent of the discount factors of the two players.
To put
it
more
formally:
probability distribution
Each
(possibly mixed) strategy profile {cri,a2) induces a
Given a history h^~^
over {Ai x ^2)°° x ^-
tt
probability attached by player two to the event that the
played in period
variable 7r'(aj)
is
7r'(aj)
t, i.e.
a
random
history h induced by
=
Along
(possibly infinite) of the
random
a random variable, so
n.
Lemma
< W <
1 Let
such that Prob{h G
H'
Prob n
(11)
Furthermore, for any
all
Fix any
1.
this history let n(7r'(aj))
\uj')
Suppose
=
(7r'(a')
^
l.
=
h{uj')
converges to
1, i.e.
^^
<¥) >
^
>
and
0,
<
to play aj.
Lemma
\
h E
is
one with type
Lemma
1
=
H'
0.
always played,
fi{uj'
it
of course to
\
ht
h^) is
1
mimic the commit-
if
is
bigger than
if
8
ht
E
h')
facing u'
he
is
W
be played.
only interested in his
is
a
tt
<
1
such that
then a short-run player
player one mimics the
her short-run opponents will take 02
is
\
he observes aj being played
Oj is "unlikely" to
completely myopic, that
is
fi{uj'
become convinced that he
says that
he cannot continue to believe that
is
ujq is
does not say that in this case
that player two will gradually
Lemma
is
1.
choose a best response against Oj. Thus,
type, then by
W. Again, since h
h such that the truncated histories
the probability that player one will play a]
will
be the number
that (cri,<72) are
payoff of the current period. Fudenberg and Levine show that there
two
a ra/.dom
and consider any
7f)
<
is
being
t.
Suppose that player two
if
/i'~^
is
logTT
he observes aj always being played. Rather
in every period
h'
1,
strategy
Then
infinite history
feasible strategy for player
ment type and always
< F <
tt,
variables 7r'(aj) for which T^\a.l)
Proof: See Fudenberg-Levine (1989),
One
Note that since
/i'~^).
|
have positive probability and such that a^
nondecreasing in
if
Oj
variable as well.
(crijCr^).
is
=
Prob{s\
commitment
be the
let 7r'(aj)
B{a'^) in at
commitment
most k
=
°^Jt,
periods.
The worst
beginning of the
happen
that can
game and
to player one
that these k periods occur in the
is
that in each of these periods she gets
(12)
g,
—1
= min
gi(a',Q2).
026.42
This argument provides the intuition for the following theorem.
Theorem
0.
Then
(Fudenberg-Levine) Let
1
there
(13)
in
If 61
least her
If
goes to
1
> (l-^f'"'')£, +
«?"*
The
result
payoff no matter what other types
is
discussed in
1 is
more
crucially based
detail in
One
if
he
may
he believes that
paper, however,
well
is
may
pay
Even
off in
may
if
may be around
to
show that
with positive
is
completely
be played with a probability arbiis
not a short-run best response
be that he might invest
this yields losses in the
in screening the
beginning of the game the
the future. This leads Fudenberg and Levine to conclude
for
the returns from an investment
point of our
well. Since player two's discount factor is smaller
may
not be delayed to far to the future.
type arbitrarily often
two long-run player games.
The main
a more restricted class of games a similar result holds
arbitrarily close to one. This idea will
for
>
n'
trade off short-run losses against
take an action 02 which
two long-run players case as
"test" player one's
1
=
'-»r
that their result does not apply to two long-run player games.
1
}i{u')
Fudenberg and Levine (1989). Note
Cj will
intuitive reason for this could
different types of player one.
in the
and
on the assumption that player two
he cares about future payoffs then he
trarily close or equal to 1,
investment
0,
the "normal" type of player one can guarantee herself on average at
long-run gains. Thus, even
against Gj.
>
any Nash equilihrium ofG°°{fi).
however that Theorem
myopic.
0, /x(u;°)
any average equilibrium payoff of player one with type
fi';u>°) is
commitment
probabilty.
=
a constant ^(/x*) otherwise indepedent of {p.,[i), such that
i-iC*,,,.-;^")
where Vi{Si,
uq
is
62
if
He
the probability that she will play a\
be used
in
Section 4 to prove an analog of
than
will
is
not
always
Theorem
A Game
3
not of Conflicting Interests
Before establishing our main result
us show that
let
repeated games with two long-run players.
We
Theorem
1
does not carry over to
commitment
the normal type of player one cannot guarantee herself almost her
all
Nash
The example
equilibria.
instructive for two reasons.
is
game
give a counterexample of a
First,
it
in
all
which
payoff in
shows how to
construct an equilibrium in which the normal type of player one gets strictly less than
her
commitment
which survives
and
payoff. This equilibrium
is
not only a Nash but a sequential equilibrium
standard refinements. Second, the construction leads to a necessary
all
sufficient condition
on the
games
cla^s of
for
which Theorem
can be generalized to
1
the two long-run player case.
Consider an infinite repetition of the following stage
game with
three types of player
one:
R
L
u
1
'
D
1
=
0.8
H'
Figure
Player one chooses between
each
cell.
to play
U
1
'commitment" type
"normal" type
fi°
1
°
1
U
— A game
and
D
=
with
U
'
D
'
'
/i'
common
is
'
10
=
given in the upper
which would give her a commitment payoff of 10 per period
commitment type
game always
to play
U The
.
it is
0.1
interests.
Clearly the normal type of player one would like to publicly
equilibrium. For the
1
"indifferent" type
0.1
and her payoff
R
L
'"
U
'
D
R
L
left
corners of
commit always
in every
Nash
indeed a dominant strategy in the repeated
indifferent type, however,
is
indifferent
between
U
and
D
no
matter what player two does.
If
0.75
<
5i
<
1
and 0.95
<
<52
<
1
then the following strategies and beliefs form a
10
sequential equilibrium of G°° which gives the normal type of player one
=
lim vi{u}°)
(14)
Normal
•
type of player one:
<
9.5
"Play U.
=
10
g^
.
you ever played D, switch to playing
If
D
forever."
type of player one: "Always play [/."
•
Commitment
•
Indifferent type of player one:
"Always play
U
•
along the equilibrium path.
has been any deviation by any player in the past switch to playing
Player two:
•
path.
If
"Alternate between 19 times
L and
R
player one ever played D, switch to
deviated in the last period, play
L
1
times
R
forever.
D
L
there
forever."
along the equilibrium
If
player two himself
in the following period. If player
the deviation by playing U, go on playing
If
one reacted to
forever. If she reacted with
D, play
R
forever."
•
Beliefs:
D
Along the equilibrium path
to be played he puts probability
to a deviation of player
beliefs don't change. If player
on the commitment type.
U
two by playing
both cases the respective two other types
up
to
Why
would
is
this
about her type
it
is
the indifferent type gets probability
may
is
is
she
U
is
the normal or the
In
add
commitment
type. Since
all
three types of
along the equilibrium path the only way to transmit information
to play D. However, playing
D
"kills"
a dominant strategy always to play U.
He expects U always
is
0.
get arbitrary probabilities which
the
commitment
type, because
But without the commitment type
impossible to get rid of the "bad" equilibrium {D,R).
which
player one reacts
an equihbirum? Consider the normal type of player one. Clearly she
player one always play
it
If
1.
like to signal that
for her
two ever observes
What about
player two?
to be played along the equilibrium path. Nevertheless he plays R,
not a short-run best response, in every twentieth period. His problem
faces the indifferent type with positive probability.
to play R, then this
If
he choses L when he
is
is
that he
supposed
might trigger a continuation equilibrium against the indifferent type
11
which gives him
far less
commitment type
than what he would have got from playing against the normal or
of player one.
this risk
It is
Note that there are very few
which sustains the equilibrium outcome.
restrictions
imposed on the updating of
The example only
information sets which are not reached on the equilibrium path.
that
if
D
played for the
is
first
time the commitment type gets probability
perfectly reasonable given that for her
it
beliefs in
0,
requires
which
is
a dominant strategy in the repeated game
is
always to play U.
To what extend does the example
Without the
indifferent type
it
gives player one less than her
is
still
rely
possible to construct a
commitment
normal and the commitment type
of player
After any deviation they switch to playing
period
L and one
period R.
If
on the existence of the indifferent type?
payoff.
Actually, this
one always play
D
plays
D
off
What
it is
is
It
requires for
is
The
very simple:
along the equilibirum path.
two alternates playing one
there has been any deviation, he plays
not sequential.
sustains both equilibria
if
only
5.
R
forever.
This clearly
is
Note
a Nash
example that the commitment type
already hold
down
is
the possibility of a continuation equilibrium which
he plays his short-run best response against a^
supposed not to do
in this case
is
the equilibrium path.''
punishes player two
he
U
forever. Player
that the average payoff of the normal type of player one
equilibrium, but
Nash equilibrium which
to his
so.
Note that
this construction does not
in periods
work
minimax payoff by the commitment strategy
if
when
player two
is
of player one, since
nothing worse can happen to him. In the next section we show that this
is
the
only case in which Fudenberg and Levine's result can be generalized to the two long-run
player case.
''Whether there exists a sequential equilibrium in which player one gets substantially less than 10 if
commitment type around is an open question. Note, however, that we
want to characterize equilibrium outcomes which are robust to general perturbations of the informational
there are only the normal and the
structure of the game.
From
this perspective
it
makes
types only.
12
little
sense to ristrict attention to two possible
Main Results
4
4.1
The Theorem
Suppose that the commitment strategy of player one holds player two down to
In this case there
payoff.
is
no
his
minimax
"risk" in playing a best response against Cj
because
minimax payoff
player two cannot get less than his
in
any continuation equilibrium. This
motivates the following definition:
A game
Definition 1
one
to player
to his
if the
minimax
g
called a
commitment
i.e.
payoff,
(15)
is
gl
=
game
o/ conflicting interests with respect
strategy of player one holds player two
down
if
g2{a'i,a2)
=
minmax5'2(Q;i,Q;2)-
"Conflicting interests" are a necessary and sufficient condition for our
main
result.
Note
that the definition puts no restriction on the possible perturbations of the payoffs of player
one.
a restriction only on the
It is
We
player two.
Section
in
5.
commitment
will discuss this class of
game with
Clearly, in a
games extensively and give
conflicting interests player
any continuation equihbrium after any history
V,'
(16)
This
is
strategy and on the payoff function of
=
several examples in
two can guarantee himself
ht at least
Y^^-g;.
crucial to establish the following result:
Lemma
and
let fJ.{oj')
tory
Qj.
2 Let g be a
/i'
=
Suppose
(17)
<
>
0.
of conflicting interests with respect to player one
Consider any Nash equilibrium
((7i,(72)
and any
his-
consistent with this equilibrium in which player one has always played
that,
scribes to take
62,
/x*
game
62
<
M
given this history, the equilibrium strategy of player two pre-
.Sj"^
I,
>
^
-^(^i) ^^^^ positive probability in period
there exists a finite integer
+
I.
M,
N = H^-^^) + H92-97)-H92-92)
In ^2
13
t
^
^^
For any
and
a positive
number
=
e
(18)
t,
(^-'^^/•(f2--g2)
^2-52
such that in
one of the periods
at least
_^M.(^_^^) ^ 0^
<
+ l,i+2, •••,f +
M
the probability
that player one does not take a^, given that he always played a^ before,
be at least
must
e.
Proof: See appendix.
Because g
Let us briefly outline the intuition behind this result.
interests player
two can guarantee himself
librium after any history
^
action s*^^
B{a'[) in period
for the rest of the
equal to
Therefore,
ht.
game.
t
+
this period. Recall that ^2
is
-
may
get in any period in
of course
and
82
<
-
\
On
M periods
(1
—
e).
^
maximal payoff player two
The numbers
M
B{a'{) in period
is
gets
if
— 92 >
in
he does not
an upper bound on what player
she plays a\. But
if
and
t -\- \
e
loss
argument
this
if
future payoffs are bounded
today must not be delayed too
are constructed such that
then
Otherwise player two would get
Note that
of at least ^2^^
which player one does not take her commitment action, and
it
less
if
player two takes
cannot be true that in each of the next
commitment action
the probability that player one takes her
contradiction.
type and takes an
B{a'[) yields a "loss" of at least g2
then the compensation for an expected
s':^^
any continuation equi-
tries to test player one's
the other hand, ^2
he cannot get more than g^
far to the future.
a strategy
^
defined as the
take a best response against a^.
two
of conflicting
player one chooses Cj with a probability arbitrarily close or
If
then choosing an action 02
1,
he
in
must give him an expected payoff
this
I
if
= r-^
Vj^^
at least
is
than
minimax payoff
his
also holds for finitely repeated
is
bigger than
in equilibrium, a
games
if
T
is
large
enough.
Lemma
in the history
in
n
•
2 holds in any proper subform of
up
to that subform.
M periods, then
in at least
not play a\ must be at least
Theorem
/z(u;°)
>
0,
e.
Thus
n
if
G
as long as player
one always played a\
player two chooses actions G2
^ B{a\) along
of these periods the probability that player one does
Together with
Lemma
1
this
imphes our main theorem:
2 Let g be of conflicting interests with respect to player one and
and
n{u}')
=
/j.'
>
h'
0.
Then
14
there
is
let
a constant k{jj.',62) otherwise
independent of{Q.,fi), such that
where Vi{6i,62,
uq
>
«,(SuS2,i>-;^°)
(19)
in
fJ.*;uj°) is
(l
-<;"*') 2. +
'"9-:
''!"'
any average equilibrium payoff of player one with type
any Nash equilibrium of G°°{pL).
Take
Proof: Consider the strategy for the normal type of player one of always playing a\.
the integer
N
where
M=
and
e
[N]
+
1,
where
are defined in
is
[A'']
Lemma
an action 02 ^ B(aJ) then there
is
the integer part of
By Lemma
2.
we know
2
one period
at least
N, and a
(call
it
real
that
ti)
if
(1
—
e).
Lemma
1
we know
among
<
In
tt)
>
In
than
more
,
Tr*^ )
is
M
smaller
~
fi'
.
W in more than
=
TT
commitment action cannot be smaller
the probability that player one takes her
'-j^ periods. Therefore, player two cannot choose actions 02
^
B{a'[)
often than
s
,
,
times. Substituting
M = [N] +
k{^',6,)
(23)
=
1
and
([7V]
+
e
In u"
,
from
Lemma
In the worst case player
2,
we
get
In ^'
1)
In (l
V
-
two chooses these actions
'^-^!'-<^J-^-)
S2-52
+
<5f'
2
+^)
J
in the first k{fx', 82) periods.
the lower bound of the theorem.
If 61
one
the next
that
n(7r*
(21)
is,
0,
1-e =7:.
<, <
That
>
So
(20)
However, by
e
player two takes
periods in which the probability that player one will play Cj (denoted by
than
number
is
—
»•
1
This gives
Q.E.D.
(keeping 62 fixed) then the equilibirum payoff of the normal type of player
bounded below by her commitment
payoff.
Thus,
in the limit
our theorem gives
the same lower bound as Fudenberg and Levine's theorem does for the case of a long-run
player facing a sequence of short-run opponents. Their result can be obtained as a special
15
Theorem
case of
then
to
A'^
2 for the class of
goes to
games with
a\
game with
if
82
goes
So
0.
92-92 J
\
In a
Note that
conflicting interests.
\92-92 J
conflicting interests a short-run player
two
will
play a best response against
if
(25)
>
gl
7r-^2
or, equivalently, if
TT
(26)
+
(l -7!") -^2
^^ =W.
>
92-92
Using (26)
It is
Lemma
in
1
immediately impHes Theorem
important to note that the lower bound given
discount factor of player
reduced.
1.
2.
If 62
compared
Theorem
increases, so does A:(/i*,^2),
Hence, to obtain his commitment
sufficiently patient as
in
payoff" in
The
to player two.
2
depends on the
and the lower bound
is
equilibrium player one has to be
following corollaries elaborate on the
importance of the relative patience of the players.
Corollary
1
For any
such that for any
player one
Proof:
Choose
8_i
Si
is at least
62
>
gl
<
I,
fJ.'
>
and
e
>
there exists a ^1(^2,^*1 ^)
1j
normal type of
81(62 , fi',t) the average payoff of the
—
<
t.
such that
g',-t^
(27)
(l-^f^'^*''^^)-£,+^?'^" ^^).,r
Solving for ^^ yields
(28)
Clearly,
8_,
if ^i
>
=
8_,{ti\82.e)
^^>^'
^^^±-^-1
<
1
.
^i(/i*,62,e) then
vi(<5a,62,/^-;u;°)
(29)
=
<''^'''^-5:
>
(l-<5f(^*''^)-5,
>
{i-i^''''')-g_,+A^''''-9\=gl-t
+
Q.E.D.
16
Corollary 2 Consider any sequence {62},
e
>
0.
<
S2
<
there exists a sequence {(5"}, 6"
Then
=
YinXn-^oo^i
1,
=
lim„_oo ^2
1,
1,
for any {^",^2} ^^^ average payoff of the normal type of player one
below by gl
—
where
fix
such that
is
bounded
t.
Proof: Take any {<5^}
all n,
and
^
and
81(82, (i',e)
is
>
fix e
Choose
0.
given by (28).
-^
{8'^]
Then the
such that
1
>
8"^
8^(82, n',e) for
immediately from the
result follows
Q.E.D.
previous corollary.
Corollary 2 shows that there
sequence {8^,82}
—
is
an area
—
in the 81
82
space such that for any
commitment
(1,1) in this area player one gets at least her
»
(up to an arbitrarily small
however, that limn~.ooYE^
e) for
—
any pair of discount factors along
*'he limit
'^^
0) '•^-
player one
is
this sequence. Note,
infinitely
more
patient than
player two. This observation helps to understand a related result of Cripps and
who
(1991)
the limit of the
mean
prior probability of a
Banach
Under
of their payoffs.
perturbations they show that
payoff as the
games without discounting,
consider repeated
if
the
game has
commitment
limit of the
slightly stronger conditions
conflicting interests
of her stage
game
Thomas
which players maximize
in
and
if
on the possible
there
type, then player one gets at least her
mean
payoff
payoffs.
is
a positive
commitment
However, the case of no
We can give examples
— 1, lirrin^ao^z^ > 0,
discounting obscures the role of the relative patience of the players.
of equilibria in
games with
conflicting interests
and player one's equilibrium payoff
is
{8^,82} along this sequence. Thus,
player two our lower
If
bound does not
player one has to be
where
<5i
—
>
1,
62
>
bounded away from her commitment payoff
if
player one
is
for
any
not patient enough as compared to
apply.
much more
patient than player two the reader might be
left
with the impression that we are essentially back to Fudenberg and Levine (1989) where
a long-run player faces a sequence of short-run players.
=
However,
this
may be
is
not the case.
First,
Fudenberg and Levine's
to
Second, we are going to show in the next subsection that whenever the
1.
not of conflicting interests
result requires 82
it is
while here 82
is
one player does not care at
all
game
is
possible to find an equilibrium which violates Fudenberg
and Levine's lower bound no matter how much more patient player one
to player two. Thus, there
arbitrarily close
is
as
compared
a fundamental difference between repeated games in which
about her future payoffs and games
17
in
which she does care
but
is
of the
than her opponent. Finally, the importance of the relative patience
less patient
two players
is
very intuitive as will be shown after we have introduced the case of
two-sided uncertainty in subsection 4.3.
4.2
Necessity of the "Conflicting Interests" Condition
The question
arises
interests. If the
equal to her
game
minimax
Proposition
ests.
whether Theorem 2
is
e
>
there
is
an
tj
game which
>
and
a 62
is
commitment
payoff
is
<
not of conflicting interI
such that the following
a perturbation of g, in which the commitment type of player
has positive probability and the normal type has probability
is
of conflicting
no:
is
1 Let g be a non-trivial
is
games which are hot
not "trivial" in the sense that player one's
payoff^ the answer
Then for any
holds: There
also holds for
(1
—
t),
and
1
there
a sequential equilibrium of this perturbed game, such that the limit of the
average payoff of the normal type of player one for
8\
—^
from her commitment payoff by
>
8_2-
at least
rj
for any 82
\
bounded away
is
Proof: See appendix.
Proposition
1
shows that the condition of conflicting interests
but also necessary for Theorem 2 to hold, in
in
is
two respects.
First,
it
says that
if
the
fact,
game
is
it
a
is
is
not only sufficient
stronger than that
little bit
not of conflicting interests, then
it
not only possible to find a Nash equilibirum which violates Fudenberg and Levine's
lower bound, but also to find a sequential equihbrium. As has been indicated in Section
3,
the construction of a Nash equihbrium using threats which are not credible
simpler.
Secondly,
Theorem
we could have established
2 only requires that y.{u°)
in the
perturbed game. So
who
credibly threatens to punish any deviation
two from the equilibrium path we want to sustain. However,
applications
^It is well
it is
known
much
necessity by constructing a perturbation which gives a high
prior probability to an "indifferent" type
of player
>
is
natural to assume that ^i{uP)
is
close to one. This
that a player can always guarantee herself at least her
equilibrium.
18
is
in
many economic
why we provide
minimax payoff
in
a
any Nash-
stronger proposition which says that even
if
/i(w°)
is
arbitrarily close to
to construct a sequential equilibrium in which the payoff of the
is
bounded away from
Note that
more
in Proposition 1
(5i
—
>
1
while 62
is
games
possible
normal type of player one
fixed, so player
Thus, Proposition
1
in
one may be arbitrarily
shows that there
whom
between games with two long run players, one of
the other, and
it is
g^.
patient than player two.
difference
one
is
an important
is
more patient than
which a long-run player faces a sequence of short-run playe:s.
the latter Fudenberg and Levine's
bound holds
games,
for for all stage
in the
In
former
it
only holds for games with conflicting interests.
4.3
If
Two-sided Incomplete Information and Two-sided Confiicting Interests
there are two long-run players
sided uncertainty.
the
game
is
Our
it is
result can
most natural
be extended to
perturbed such that there
is
what happens
to ask
measure
/x,-,
respectively.
player
i,
Without
i.e.
i
game out
two-
Suppose
incomplete information about both the payoff
of the countable set
fi,-
Finally,
player
z's
suppose that the game
commitment
Lemma
2
we
?
=
is
z's
type which
is
drawn by
according to the probability
of confiicting interests with respect to
strategy holds player j
1.
didn't say
Now
why
test player one's
matter what the reason
down
to his
minimax
pa3'off.
consider the normal type of player two.
player two might choose an action which
a best response against player one's commitment strategy.
wants to
is
G {1,2}. Let uf and m' represent the normal and the commitment types,
loss of generality let
the proof of
there
this case in the following way.
functions of player one and player two. Let w, denote player
nature in the beginning of the
if
He might do
is
Lemma
expect that player one choses
s\
probability. This holds for the
^
2 states that
Cj in
if
he takes 02
^
B{a'l),
one of the following periods with
not
so because he
type or because he wants to build up a reputation himself.
is.
In
No
then he must
strictly positive
normal type of player two no matter what other possible
types of player two exist with positive probability.
A
possible strategy of player one
the normal type of player two, then by
which player two
still is
to play Cj in every period.
Theorem
2 there are at
will not play a best response against a\.
19
most
If
she faces
k[ji',82) periods in
In the worst case for player
one
this
happens
in the first k periods of the
face the normal type of player
which
=
/i,(a;°)
3 Let g
at least
is
g
if
she does not
in every period.
This
expected payoff of the normal type of player
for the
fi°
>
be of conflicting interests with respect to player
and
ki{fi',6j) otherwise
/x,(a;*)
=
fi'
>
0,
z
Then
G {1,2}.
i
and
let
there are constants
independent o/(f2,, Qj,/i,), such that
>
r;,(6.,<5„;.-,/z;;u;°)
(30)
(l
-
type uf in any
Nash equilibrium ofG°°{fi).
is
fJ^S^'^"''^^ g:
any average equilibrium payoff of player
ti*,
fi^,uf)
+
fi^6^'^'''''%
where Vi{6i,S2,
if
the other hand,
given in the following theorem:
is
Theorem
Thus,
On
two her expected payoff
argument establishes a lower bound
i
game.
the probability of the normal type of player two
very patient, then the lower bound for her average payoff
is
close to
and
1
if
again close to her
is
with
i
player one
is
commitment
payoff.
What
like to
course,
it is
can be said
commit
if
there are two-sided conflicting interests and
The
kept fixed and
in the limit player
game with
down
them
gets his
i
i
Si
i.e.
to his
most prefered
payoff.
minimax
is
played, and the lower
goes to
1
is A:,(/x',^j), i.e.
is
her lower bound.
then this k periods become
commitment
more patient than the
payoff. This
the
(5,-
and
number
of
is
less
and
On
the other hand,
less
important, and
very intuitive. In a symmetric
if
one of the parties
other.
3 are in striking contrast to the
message of the Folk theorem
games with incomplete information by Fudenberg and Maskin
(19S6).
cis
an equilibrium outcome of the perturbed game
20
for
The Folk theorem
says that any feasible payoff vector which gives each of the players at least his
payoff can be sustained
Of
payoff.
must expect that a strategy other than the best response against
will get his
Theorems 2 and
each player would
But suppose that
conflicting interests reputation building has an effect only
sufficiently
if
both players are equally patient
if
bigger player j's discount factor the bigger
commitment strategy
if Sj is
interests,
to a strategy which holds his opponent
periods in which player
is
g has two-sided conflicting
impossible that each of
5j differ.
her
if
if
minimax
the right
perturbation
is
Theorems
chosen.®
further perturbations.
Theorem
-
and
show that
3
one of the players
If
type has positive probability then
probability
2
this result
is
patient enough and
is
not robust against
her
if
commitment
no matter what other types are around with positive
-
2 imposes a tight restriction on the set of equilibrium outcomes in
any Nash equihbrium.^
We
have to be very precise here in what
argues that the Folk theorem
is
meant by robustness. Fudenberg (1990)
robust against a further perturbation of the informal-ional
is
structure in the following sense: Consider a sequential equilibrium which has been con-
structed using an
e
probability of a "crazy" type as the Folk theorem suggests.
given time horizon of the
small as compared to
game
other types are introduced with a probability which
then this
e,
an equilibrium.
is still
We
goes to infinity the equilibrium must break down.
conflicting interests
and
if
then the commitment type
there
will
If for
is
However,
a
is
the time horizon
if
have shown, that
if
the
game
is
of
an arbitrarily small probability of a commitment type,
dominate the play
are interested in the set of equilibria for
T
as
T
becomes
large enough. Thus,
—^ oo, the Folk theorem
is
if
we
not robust against
small perturbations of the informational structure.
5
Examples
The Chain Store Game
5.1
Consider the classical chain store game, introduced by Selten (1978), with two long-run
players. In every period the enirant
may
choose to enter a market (/) or to stay out (0),
^Fudenberg and Maskin's Folk theorem for games with incomplete information considers only finitely
However, the extension to discounting and an infinite horizon is
repeated games without discounting.
straightforward.
^Note that even
set of
this
if
the
Nash equilibrium
game
suppose that the game
a type
who
mimicks
is
is
not of conflicting interests we can
payoffs, although the
is
bound
still
to his
impose some restriction on the
be weaker than Fudenberg and Levine's. To see
not of conflicting interests, but that there
committed to hold player two down
this type, then she
will
minimax
is
a positive prior probability of
payoff. If the
normal type of player one
can guarantee herself on average at least the payoff she would have got
if she
were publicly committed to this strategy. This doesn't give her her most prefered payoff but it may still
be more than her minimax payoff and thus reduce the set of Nash equilibrium payoffs as compared to the
Folk theorem.
and elaborate
I
am
grateful to
Drew Fudenberg
for this observation.
this idea.
21
A companion
paper
will generalize
while the monopolist has to decide whether to acquiesce (A) or to fight (F).
game
the payoffs of the unperturbed stage
Assume
that
are given as follows:
o
A
'
F
°
'
2
'
-1
3
— The
Figure 2
The monopolist would
commitment payoff
player two's
like to
commit
chain store game.
to fight in every period
which would give her a
and which would hold the entrant down to
of 3
minimax payoff the game
is
interests with resprect to the monopolist.
repetitions of this
ffi
-
Since
0.
according to our definition
-
is
of conflicting
Kreps and Wilson (1982) have analyzed
game with some incomplete
also
finite
information about the monopolist's type.
For a particular perturbation of player one's payoff function they have shown that there
are sequential equilibria in which the monopolist gets on average almost her
payoff
if
her discount factor
is
close
enough
to
one and
if
commitment
there are enough repetitions.
However, Fudenberg and Maskin (1986) demonstrated that any feasible payoff vector
which gives each player more than
of figure 2, can
his
minimax
payoff,
i.e.
be sustained as an equihbrium outcome
any point
it
in the
shaded area
the "right" perturbation
is
chosen.
Thus, our Theorem 2 considerably strenghtens the result of Kreps and Wilson
(1982).
It
says that the only
Nash equilibrium outcome
against any perturbation gives the monopolist at least her
that she cannot get more),
®I
am
grateful to Eric van
average payoff of player two
if
she
Damme
is 0.
is
of this
game which
commitment payoff
sufficiently patient as
compared
is
robust
of 3 (note
to the entrant.*
Theorem 2 does not imply that the
more patient than player two. So it may be
she gets less than 3 and player two gets more than
for the following observation:
Recall that player one
that in the beginning of the game, say until period L,
is
but after period L payoffs are always (3,0). For player one the first L periods do not count very much
because she is very patient, so her average payoff is 3. However, player two puts more weight on the first
L periods and less on everything thereafter, so her average discounted payoff may be considerably bigger
0,
than
0.
'?2
Furthermore
Now
it
this result carries over to the infinitely repeated
shows that
suppose that there
of the entrant.
He would
is
like to
a commitment payoff of 2 while
So the game
payofT.
theorem
is
applies. If there
one
sufficiently
two
is
commit
which would give him
to enter in every period
would hold the monopolist down to
it
minimax
her
1,
two-sided uncertainty Proposition 2 says that
is
relative patience of the
of player
also incomplete information about the payoff function
also of conflicting interests with respect to the entrant
on the
is
game.
it all
two players and the prior probability distribution.
more patient than
player two and
if
and our
depends
If
player
the probability of the normal type
close to one, then player one will get her
commitment payoff
in
any Nash
equilibrium, and vice versa.
A
5.2
Repeated Bargaining
Consider a buyer
and a
(6)
of a perishable good.
seller (s)
The
who bargain
is
= ^^^
would
and only
if
like to
seller is p,
=
of the buyer
PI.
commit
in
G {-,',''-}, and there
p,-
=
=
- in every period.
The unique
perturbed such that with some positive probability she
2 applies
any Nash equilibrium
down
to a
is
if
minimax
^, his
and the buyer
this
will get
is
Some
cooperation
example
is
trade at price
a minimal bid -
minimax payoff
best reply of the
Suppose the payoff function
always offer
will
almost her commitment payoff of ^^^
is
close to one.
not as clear-cut as the chain store game.
>
If
0.
is
the buyer could offer Pb
But
of 0.
that bargaining over a pie of fixed size
is
payoff.
her discount factor
possible prices, so he might choose p,
point
is
Consider the commitment strategy of the buyer. She
is
hold the seller
In
and the production costs of the
which gives him g^
have to assume that there
all
p,.
herself to offer p^
Note however, that
between
>
1
^,
Then Theorem
on average
if pt,
is
a sealed bid double auction in every period: Both players
simultaneously submit bids pb and p^,
p
repeatedly in every period on the sale
valuation of the buyer
Suppose there
seller are 0.
Game
>
if
he gets
the seller
=
is
We
she could
indifferent
and we end up with no trade. The
not quite a
needed to ensure that trade takes place
game
at
of confiicting interests.
all.
Schmidt (1990) we considered a more complex extensive form game of repeated
bargaining with one-sided asymmetric information, which confirms the above result that
23
the informed player can use the incomplete information about his type to credibly threaten
to accept only offers
which are very favourable to him. There, however, we took a
approach and
interesting to
it
is
not allow for
"all possible"
function,
we assumed
€
costs, c
i.e.
[0, 1].
compare the two models.
different
Schmidt (1990) we did
In
but only for "natural" perturbations of player one's payoff
that there
The problem is
may be
incomplete information about the
that in this case there
is
seller's
no commitment type since none of
the possible types of the seller has a dominant strategy in the repeated game. However,
the
game has a
finite
horizon
it
a weak Markov property the
and only
who
is
can be shown that
seller
in
any sequential equilibrium satisfying
with the highest possible type will accept an
mimicked by
all
the other possible types of the
try to test the seller's type at
most a
fixed finite
seller.
We
show that the buyer
number of times and
seller will get his
commitment payoff even
if
well,
consider there
is
is
Theorem
much
There are
not of conflicting interests.^
will
2)
we can
less patient
than
Furthermore the bargaining
the buyer, so the relative discount factors are not crucial.
game we
he
if
that this will happen
only in the end of the game. Surprisingly (from the point of view of
show that the
offer
commitment type
covers at least his costs. This type plays the role of the
if it
if
common
interests as
because players have to cooperate to some extend in order to ensure that trade takes
place.
5.3
Games with Common and
Conflicting Interests
"Pure" conflicting interests are a polar case and in most economic applications there are
both
-
common and
prisoner's
conflicting
dilemma depicted
interests, but
our theorem
hcis
in
no
-
interests present.
Figure
bite.
3.
one her minimax payoff as
in every
down
Nash equilibrium.
to his
minimax
®Note that not
all
In a formal sense this
game
down
like
most to commit
game
payoff, but to
of conflicting
to his
minimax
herself to play Z)(efect) in
payoff, but
well. So, trivially she will get at least her
In this
is
Given that player two takes a best response against
her commitment action player one would
every period. This holds player two
Consider for example the repeated
the problem
commit
is
not to
it
only gives player
commitment payoff
commit
to hold player
two
to cooperate.
possible perturbations are allowed
necessary condition for the result in Schmidt (1990).
24
for.
This
is
why
conflicting interests are not a
D
C
c
'
D
'
°
2
3
1
1
3
— The
Figure 3
Another interesting example
commit
is
prisoner's dilemma.
a repeated Cournot game. Firm one would like to
to choose the "Stackelberg-leader" quantity, which maximizes her stage
profit given that firm
two chooses a best response against
follower" payoff of firm
two
is
it.
is
not to hold player two
interest to
conflicting in the sense that each of
minimax payoff
since he
player one gluts the market. So our result does not
if
Again the problem of player one
Both players have the common
game
However, the "Stackelberg-
positive and thus greater than his
can be held down to zero profits
apply.
51
maximize
them would
down
as far as possible.
joint profits, but interests are also
like to get
more
for
himself at the expense
of the other.
6
To keep the argument
stage
Extensions and Conclusions
as clear as possible
games with only two
we considered a very simple
class of possible
players, finite strategy sets, a countable set of possible types,
and commitment types who always take the same pure action
in
every period.
these a.ssumptions can be relaxed without changing the qualitative results.
and Levine (1989) provide a generalization to n-player games
are
compact metric spaces and
In Fudenberg
one.^°
to the case
^"If
i
=
n
>
2, ...,n
3,
in
which there
is
in
sets
a continuum of possible types of player
and Levine (1992) they show that the argument can be extended
the definition of a
to their
Fudenberg
which the strategy
where the commitment types play mixed strategies and
down
All of
minimax
game
to
games with moral
of conflicting interests requires tiiat aj holds
payoffs simultaneously.
25
all
other players
hazard, in which not the action of player one
itself
but only a noisy signal can be observed
by player two. Since the technical problems involved
as in our
model we
refer to their
What happens
work
for
commit hermself
to a
more complex
strategy which prescribes to take different actions in different periods and which
conditional on player two's past play?
It
doesn't change anything in the proofs of
game with
down
has to hold player two
to her
Fudenberg and Levine (1989)
game
is
is
easy to check that replacing Oj by
and 2 and Theorem
1
The problem
1.
buyer has to decide
a
a'[{ht)
demonstrated that the assumption that the stage
is
that in an extensive form
game
player two
example a repeated bargaining game
first
whether to buy or not and then the
the seller would have produced high quality. This
commitment payoff
context.
The
in equilibrium.
definition of a
strategy of player one holds player two
two takes an action 02
more than
his
player one's
minimax
commitment payoff
player two chooses 02
Theorem
To conclude,
a repeated
If
this
g
is
any
is
defined as g^
finite
it
commitment
in every period the
has to choose whether to
will not
observe whether
fail
problem does not
is
to get her
arise in our
assumes that the commitment
paj-off.
Therefore,
if
player
a-i
7^ Gj,
then player two cannot get
=
is
maXajg/ij Tnma^^B[ai)
get less than g^.
However
9i{(^i-,o:2)-
So
if
Therefore, following
straightforward that our result holds
extensive form game.
long-run players
one of the players
the
the seller might
this
paper has shown that "reputation
game with two
is
take an action
So 02 must have been an element of B{al).
€ B{al) player one cannot
if
seller
minimax
to his
which
may
which player one's commitment strategy aj
2 of Fudenberg and Levine (1989)
without qualification
interests.
down
some other strategy
payoff.
why
conflicting interests
in equilibrium after
observationally equivalent to
is
Note however that
game with
in
buy then he
deliver high or low quality. If the buyer decides not to
in
in
any period and after any history
in
which player one has no opportunity to show that her strategy
strategy. Consider for
is
a'[{ht)
However,
2.
gained by this generalization because
minimax payoff
also
may be
simultaneous-move cannot be relaxed without an important qualification of their
Theorem
after
is
Lemmata
conflicting interests very little
same
any formal statements and proofs.
player one would like to
if
in these generalizations are the
if
effects"
and only
if
can explain commitment
the
game
is
of conflicting
very patient as compared to the other player, then
26
any Nash equilibrium outcome which
is
robust against perturbations of the information
structure gives her on average almost her
commitment
payoff.
message of the Folk theorem may be misleading. However, we
the evolution of
conflicting
-
commitment and cooperation
interests are present,
in
which clearly
future research.
27
is
games
in
This indicates that the
still
know very
which both
-
little
about
common and
one of the most important issues of
Appendix
Lemma
Proof of
2:
Consider any equilibrium
and
{a-i,a2)
a history
fix
player one has always played aj, such that
Such a history
exists because
equilibrium strategy
probability.
It will
be shown that
Note that
e
>
0.
0.
Suppose that according to the (possibly mixed)
Define
each of the periods
this can't
minimax
e is
in
be true
^
B{al) in period
+
t
t
+
l,t
in equilibrium
+
2,-
•
•
,t
+
M
with positive
I
Oj (given that
smaller than
is
because then player two would
payoff.
independent of
7r'^(ai)
along which
t
Suppose further that the probability of player one not playing
get less than his
that
>
up to any period
has positive probability given {(Ji,^^).
player two chooses s^^^
a^"^^
he always played aj before)
e.
fi*
/i'
h'
=
t
Prob{sl
and that
=
Gi
|
M has been chosen in a way to guarantee
h'^~^)
and
let V'2^(5[,<72)
be the continuation
payoff of player two from period t onwards (and including period r) given the strategy
profile (sl^a^) in period r.
is
The expected
payoff of pla\-er two from period
t
+
I
onwards
given by:
V^^\a„<T,)
(31)
+
•••
=
+ ^'+^(aI)-L2(aI,4^^)
5]7r'+^(a:)-yr^(ai,4^^)
+
+ 62-Vr''^']---\\.
It
will
be convenient to substract §2 from both sides of the equation
(Recall that §2
is
the maximal payoff for player two
best response against Oj.)
Then we
if
in
every period.
he takes an action which
get:
92
+
^'^\a',)-\[g2{al4^')-g2]
28
is
not a
+
92
Vl^\auar)-
J^tt'+^K)
8,.
1
-So
ai^iaj
(32)
+
<52-:r'+2(ai)-{[^2(«I,^r')-^2]
+
•••
+
J^tt'+^O
+6,.
92
)-
(ai,cr2
^2
-(5,
1
+ 52-7r'+^K)-{b(at,a^+'')-^2]
+
By assumption
^'
<52
}
1-6,
the conditional probability that player one does not take her
action given that she always played a^ before
1,- •• ,i
+ M,
^
and, of course,
we can use
all, it
that
7r'''"'(aJ)
V^+\a^,a'+')
,„„„, ,
.,
Furthermore, s^^^
<
any period from
t
+
e,
Since ^2
1.
-^
—
^_^^
92
<
_
is
the maximal payoff player two
Js
and
„.._
g2{a',sl+'^)
we can use that
^2(01,
-
V^'-'ia^.a^)
<'"2)
<
92
-92
-T3^
^^^''-fzf +
52
92
•
1
-92
—+
-62
•••
+6.
= e.{l+82 +
---
+
-^
,.
+ i-{{92-92)
,
.
,
+
+
•••
S2-l-{{g'2-h) + S2-'-f^]--
1
,
Oj, so
52-
82-l-{[g'2-g2)
.
92
62- 1-
M-\
<
-92
92
^
<
rrx
92
+
92
^_^^
92- Substituting these expressions yields:
+ <52-e-^?—^ +
1-62
^
e
n+M+i <
_
V^
.,
Co
supposed not to be a best response against
(35)
=
<
T'+'(ai)
i.
(36)
e in
has to be true that
(34)
Finally
smaller than
so
(33)
can get at
is
commitment
-92
—+
-82
.
92~ 92
1_62
,
'
r
62
.
-
-
X
92)
M
r-M
^^
8^'-').^-f:f1—02
29
,
(52
'^'
^"
'
^'
+ 8','.^f^
— 09
1
- 92
1-82
92
+
(i_+ 62
92
+
•
-h
Recall from the statement of
+
6^_-'). {g;
rM
92-92
•
,
Lemma
,.
-
easy to check that
92
^
-92
^•(ITr^ +
(^^)
.
,
92
-92
.(1-6,) > 0.
has been chosen such that
e
f'iQ\
- h)
{gl
2 that
-92
92
It is
- h) -
= il^iil-Mjliil-^f
e
(37)
•
rM
'^2
92
-92 _
.
,
- 32-92.
-13^
Therefore we get:
However, since ^j
we
is
player two's
minimax payoff
this
is
a contradiction to the fact that
Q.E.D.
are in equilibrium.
Proof of Proposition
The proof
is
game g such
1:
similar to the construction of the counterexample in Section 3. Perturb the
that there are three types of player one, the normal type, the
type and an indifferent type, whose payoff
probabilities (1
— e),
|,
and
|, respectively.
is
commitment
the same for any strategy profile, with
Let 62(e)
= ^^ <
1
and suppose
62
>
^2(^)-
Define
- ili^
In \l
(*°)
and
"
let
m=
[n]
+
2,
where
[n] is
straightforward to check that n
Since the
gi
,nS,
the integer part of n. Given the restriction on 62
well defined
commitment payoff of player one
there exists an action
that
is
= -
0.2
such that gi
> minmax^i and
§2
>
=
and
is
^1(0^,02)
minmax^2-^^
We
positive.
strictly greater
<
gl
will
and ^2
now
=
than her minimax payoff
^2(01,52)
<
^2-
Suppose
construct an equilibrium such
that the limit of the average equilibrium payoff of the normal type of player one for
is
bounded away from her commitment payoff by
(41)
V
at least
= --[^r-^i] >
m
it is
77,
61
—*
I
where
0.
^^If for any of the players gt < minmaxy,- the construction of the "punishment equilibria" which are
used below to deter any deviation form the equilibrium path are slightly more complex. In this case
players have to alternate between the outcomes g' and g such that both get on average at least their
minimax
payoffs.
30
Suppose
get at
>
1
>
^1
J - J-
^
Z.^\_-
)
Along the equilibrium path
all.
where
payoff player one can
types of player one play a\ in every period,
all
while player two plays a^ G B{a\) in the
maximum
the
is
g^
first
n, then starts again playing a\ for the next
m—
m—
\
periods, then he plays
\
periods and so on.
If
0.2
in period
player one ever
deviates from this equilibrium path player two believes that he faces the normal type with
probability
we
In this case
1.
where the Folk theorem
tells
are essentially back in a
game with complete information
us that any individually rational, feasible payoff vector can
be sustained as a subgame perfect equilibrium. So without writing down the strategies
we can construct a continuation
explicitly
is
(y^j-^i, 735-^2
)•
to check that
-
commitment and the
Clearly, the
no incentive to deviate since
given
m>
a\
equilibrium, such that the continuation payoff
is
at least
indifferent type of player
weakly dominant
2 and the restriction on
-
8\
both of them.
for
one have
It is
easy
the normal type of player one will
not deviate either.
Now
the
suppose player two ever deviates
commitment type
i
+1
any period
are supposed to play a\ in period
switches to another strategy
period
in
s^-^'^
7^ a\.
If
i
t.
+
In this case the
1,
normal and
while the indifferent type
player two does not observe a\ being played in
he puts probability one on the indifferent type. Using the Folk theorem we can
construct a continuation equilibrium in this subform which gives player two jzfQ'i ^rid
which would give the normal type of player one ^zj'9\a\ being played in period
i
+
1
he puts probability
^^i
however, plaj'er two observes
on the indifferent type.
In the
continuation equilibrium of this subform {a\,a'2) are always played along the equilibrium
path.
If
there
is
any deviation
type with probability one and
is
f
Yr£~5i
»
Tr5~^2
)
•
-
player one, player two believes that he faces the normal
using the Folk theorem again
Clearly, always to play Gj
always a\ and always a\
It is
bj'
easy to check that
is
is
-
the continuation payoff
a best response of player two against
a best response for the commitment type against any strategy.
it is
also a best response for the
normal type of player one, given
the "punishment" after any deviation.
We
have already shown that the strategies of the players form an equilibrium after
any deviation from the equihbrium path and that given the continuation equilibria player
one has no incentive to deviate from
strategy
is
this path.
We
still
a best response along the equilibrium path.
31
have to check that player two's
The
best point in time for a
when
deviation
is
period,
never
it
player two
will.
is
supposed to play
If it
0.2-
Suppose player two does not deviate. Then
m-l
=
(42)
However,
is
if
does not pay to deviate in this
his payofT
is
given by:
2m-l
h +Y^^-92
he deviates, the best he can do
is
-
rf^-(^2--^2).
to play Cj in period
t.
In this case his payoff
given by
v.K) =
(43)
It is
now easy
+ i,.{(i-5).^.,j +
s;
to check that
e
and
Thus we have established that
We now
6(e)
this
have been constructed such that
payoff of the normal type
is
is
this equilibrium
—
indeed smaller than g^
^2(02)-
rj
path the average payoff of the
when
6\
—^
I.
The equilibrium
2m-l
t=m+l
t=\
=
>
given by:
m-l
(44)
¥2(0.2)
indeed an equilibrium path.
is
have to show that along
normal type of player one
f ^.j,}
16"*
1
^^^-g^ - -.^.[gl-g,].
Therefore the difference between her commitment payoff and her average payoff in this
equilibrium
is
(45)
—
=
—
=
!!"!'!'S
{l-Sl)-Sr'
Y:rs^
<5r"^
Taking the limit
for 61 —* 1
we
.
-b--^-'
-
1
^T~^
get
cm —
(46)
r
r.
-1
1^1-^1]
?^. ~^
1
'
^3'i
- 9i] = --[^r-^i]
Q.E.D.
32
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