Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/foll<theoremstwodOOsmit
DEWEy
MIT LIBRARIES
3
IDflD
QD7Sb=]bS 7
working paper
department
of economics
Folk Theorems: Two -Dimensional ity is
(Almost) Enough
Lones Smith
Dep artment of Economics
M.I.T.
No.
597
Jan.
1992
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Folk Theorems: Two-Dimensionality is
(Almost) Enough
Lones Smith
Department of Economics
M.I.T.
No.
597
Jan.
1992
M
IT. LIBRARIES
MAR
9 1992
Folk Theorems: Two-Dimensionality
is
(Almost) Enough*
Lones Smith
Department
M.
of
I.
Economics
T.
original version: October 8, 1990
current version: January 22, 1992
Abstract
We
exhibit a parallel constructive proof technique for three founda-
theorems without requiring a full-dimensional payoff space.
Instead, we show that a less stringent and more natural projection condition suffices. Along the wa}', we substitute a strengthened folk theorem
tional folk
for finitely-repeated
games.
'The current version reflects helpful comments from two referees. Financicd assistance from the
and Humanities Research Council of Canada, the Jacob K. Javits Fellowship Fund,
and the Searle Foundation is gratefully acknowledged.
Socicd Sciences
INTRODUCTION
1.
The problem
known.
of multiplicity of equilibria in repeated
games
by now well-
is
Indeed, the popular moniker 'folk theorem' has been effectively hijacked
by game theorists to mean that any individually rational payoff of a stage game
can be attained on average in an equilibrium of the corresponding repeated game.
Constructing n-player folk theorems for various conceivable economic contexts has
developed into a veritable cottage industry. Most recent efforts can trace their lineage to the perfect folk theorem in Rubinstein (1979), which considered supergames
without discounting. When players do discount the future, Rubinstein's recourse
to infinitely-bootstrapped punishment hierarchies is no longer an option. It was a
technique first pioneered by Fudenberg and Maskin (1986) (henceforth simply F-M)
for infinite horizon
ment
in
interior
F-M
games that sparked the
Smith (1990)
theorem
folk
Exactly
plained by
that the feasible payoff space have full
i.e.
The
require-
a non-empty
and Krishna (1985) (hereafter denoted B-K).
exploited the full-dimensionality condition to establish a uniform
later
for
games
why a
means
Here, player
full-dimensional payoff space was
of an
example game Gi found
in
deemed necessary
is
best ex-
B-K:
3,
3,3
0,0,0
1,1,1
2,2,2
0,
1,1
0,0,0
0,1,1
0,0,0
0, 0,
0,1,1
0,1,1
chooses rows, 2 chooses columns, and 3 chooses matrices.
1
Nash
in Benoit
overlapping generations games.
0, 0,
are two
dimension —
— found immediate application in the folk theorem (without discounting)
for finitely repeated
is 0,
latest flurry in this arena.
equilibria:
(3,3,3) and (2,2,2).
Since each player's
There
minimax
level
a folk theorem would assert that any positive feasible payoflf vector should be
attainable (on average) in Gi(T), the T-fold repetition of Gi, for
Notice that the payoff space of Gi
T sufficiently large.
two-dimensional, and thus
is
is
not of
full-
dimension.^ But the choice of Gi was really rather clever! For consider the simple
proof in B-K that in no subgame perfect equilibrium of any Gi(T) does player 2 or
on average. The
3 earn less than 1
case
T=
1
is
obvious, so assume
outcome with
T+
I
3 could gain at least
1,
result proceeds
holds up to
by (backward) induction. The
some
T>
1.
Then
the prescribed
periods to go cannot be (0,0,0), for if so, either player 2 or
1 by deviating and then, by hypothesis, average at least 1 in
the following subgame. But
receive at least
it
if
(0,0,0)
is
and the claim follows
not prescribed, then players 2 and 3 each
for
Gi(r
-I-
1),
as required.
Unfortunately, this neat result relies crucially on the fact that players 2 and 3
receive positively linearly related payoffs.
^Tlie
example
in
F-M
It
turns out that such a non-genericity,
only has a 1-dimensional payoff space, and hence might not establish
that a 3-dimensional payoff space
is
needed.
which was conceded by B-K,
when
find that
is
this possibility
not without loss of generality.
In the sequel,
we
expressly eliminated, the folk theorems for n
is
>
F-M, B-K, and Smith (1990) obtain with any payoff space that is
two-dimensional. This task actually proves rather unburdensome, as a
2 players in
(at least)
We
unified constructive treatment suffices for all three cases.
also take the liberty of
strengthening the folk theorem of B-K, by admitting the possibility that one player
might not have distinct Nash
2.
The
set-up
is
standard, and
G
is
:
A
is
IN 2-SPACE
taken from Smith (1990), which we summarize
an n-person normal form game, and the (compact and
,An} consist of mixed strategies. Each player f's
— 7Z is continuous. Define U = n"_iC/,-. Elements
convex) strategy spaces {Ai,
payoff" function f/,U^^^Aj
a*6
game.
PUNISHMENT OUTCOMES
Throughout,
here.
payofi"s in the stage
.
.
.
>
are referred to as outcomes of G. Liberal use
made
is
of correlated strategies,
allowing players to condition their actions on the outcome of a public randomizing
device. This renders the feasible payoff" space
V
the convex hull in 7^" of the set of
pure strategy payoff vectors.
minimax strategy
a mixed strategy, we assume that
Let M' be a
We may
feasible
V*
also
and
assume
WLOG
against player
i
in the
game G.
Since M' could be
deviations from mixed strategies are observable.^
=
that C/,(M')
for all
z
G
N=
{1, 2,
.
,
.
.
n}.
strictly individually rational payoff set is therefore the positive
The
orthant
of V.
The example game Gi was
Even though the
atypical in the following sense:
whole payoff space was two-dimensional, its projection onto the plane of payoffs
and 3 was only an (upward-sloping) line. It was thus impossible to
decouple the payoffs of 2 and 3. We claim that so long as we avoid such a non-
for players 2
we can construct n personalized one-shot (correlated)
punishment outcomes Pi*,P2*,
,Pn*- That is, (?) pk* is a strictly individually
rational outcome, (n) providing k a lower payoff than the equilibrium outcome a*,
and (Hi) a strictly lower payoff than any other p,*,i ^ k. Observe that because we
genericity for n-player games,
do not have a full-dimensional payoff space, other players j
Pk*; however, every j ^ k still prefers pk* to pj*.
^
For the sake of definiteness, we shall (locally) parameterize
most two-dimensional. The extension
suffer
V* assuming
to higher dimensions should
N
may
from
it is
at
be obvious. Let
=
i E
be 7r,(5, t)
ai + PiS + 'jit. All subsequent analysis
on the following crucial assumption which will supplant full-dimensionality:
the payoff of each player
rests
k
(P) The projection
k
is
Vij of
V*
onto the coordinate space of any two players j and
^ Pklj for all j 7^ /:, or a line with
either two-dimensional, so that Pjfk
negative slope,
For example, with n
i.e.
=
(3j
=
OjkPk and fj
=
^jjt7it
3 players, (P) implies that
ular to the coordinate plane of
for
if
F-M
6jk
<
0.
the payoff space
any two players, then
'Abreu-Dutta (1991) modify the argument of
some
it
is
perpendic-
intersects that plane in a
to handle unobservable
mixed
strategies.
negatively sloped
Now
come
line.
suppose that we want to support the
a*, as parameterized
(51, fi),
.
.
.
,
by
(s„,i„) for pi*,p2*,
ti
i G N. Then
Moreover, if j 7^
(5*, t*).
•
•
>Pn* satisfying
•
k,
where the inequality
is
TTj
(^
7r_,(5",t*),
7rj{s',t')-e^l3j
TTj
^^
Sj ,tj
below
negatively-sloped
774
is strict
by
(P). Alternatively,
>
is
line,
+ jj
),
(P). Finally, in the process
a-
0.
one-dimensional case,
as required. Illustrated
players, with V24
>
Sj ,tj),
>
—
by
individually rational for small enough e
due to Cauchy-Schwartz, and
in the negatively-sloped
<
we can exhibit coordinates
conditions (i), (ii), and {iii).^ Let
(P),
then in the two-dimensional case,
=
since 9jk
strictly individually rational out-
= f - liSl^JW+^j
{si,ti) is also strictly
for
Under
it
was established that
'Kj{sj,tj)
the parameterization of the case of n
and
all
=
<
4
other Vjj two-dimensional.
7r4(54, ^4)
TTi
>
7ri(5i,ii)
>
T^2{S2M)
* s
TTS
>
7r3(s3,i3)
K2
{S3,t3)
^This
is
nearly the converse of the
{Hi) implies (P).
first
lemma in Abreu-Dutta
(1991),
who show
that condition
THE FOLK THEOREMS
3.
We now
turn to the folk theorems. That
we show that any u E V* can be
is,
(approximately) attained as a subgame perfect discounted average payoff vector in
each of three distinctly different repeated contexts. Throughout, a*
outcome yielding the desired payoff vector u.
Case A:
Infinitely
is
a correlated
Repeated Games
game with stage game G and discount
Theorem 2 in F-M, substituting (P) for full-
Let G(<5) denote the infinitely-repeated
factor
Below
6.
The constructed equilibrium
dimensionality.
as
it
a modification of
is
is resilient: After
any
also differs
from
F-M
in
another sense,
sequence of deviations, play always returns to the
finite
equilibrium path within finitely-many periods.
INFINITE-HORIZON n-PLAYER FOLK
3.1
=
Let u
(ui,...,u„) G V*.
==>
[(5o, 1]
THEOREM
has a subgame perfect discounted average payoff
G((5)
<
Suppose that (P) holds. Then 36q
so that 6 G
1
u.
Proof:
Along the equilibrium path, a* is played. But if someone deviates then punishment interlude follows, consisting of a minimax phase and then a 'recovery' phase.
Thus, the equilibrium strategies
1.
Play a*.
2.
Play M-' for
3.
Play pI for
We next
uj
are:
'^
player j deviates, start
[If
Q
R
periods. [If player k
periods. [If
Q
2.]
^
any player j deviates,
R so that
Then
j deviates, start 3.]
restart 2.]
set k <— j.
Return to step
1.
a subgame perfect equilibrium. Let P and
be the best and worst payoffs for any player in G, and first suppose that S — 1.
select
and
this
is
For each step, we consider the 'worst-case scenario', where the incentive to deviate
is
greatest.
First note that given continuity of discounted
by some positive margin, say
6
G
[(('o,
1],
for
some
6o
<
1.
1,
Now
they
will
choose
Q
sums
remain
in 6, if
strict for
each deterrent
any
strict
is
level of discounting
so that^
u + QU,{p-) >
13
+
I
(1)
G N. Since < Uj{pj*) < Uj by conditions (z) and (u'), (2) simultaneously
renders the punishment interlude a strict deterrent to deviations from steps 1 and
for all j
*Throughout the paper,
j,
k,
and
/
denote arbitrary players.
Moreover, for
clarity,
we use
the simple computer science k *— j to mean "assign k the value j." Also, program steps always
follow sequentially, unless otherwise indicated. Conditional interrupts within square brackets are
executed immediately.
*The inequality
(1)
—
in particular,
why
the left-hand side
is
not (Q-t- l)Uj{pj)
—
reflects the
obeying the random correlated outcome pj* might sometimes require j to play
possible outcome.
fact that
his
worst
any R. Next, step 3 deters deviations by the punishers from step 2
large enough that
3, for
for all 7,
N
G
^'
Qu +
RUkip-)
with j
^
k.
>P + RUkipl) + (Q
By
condition
(n'z),
- ^>k +
such an
R
if 7? is
1
(2)
QED^
indeed exists.
Case B: Finitely Repeated Games
we
G
with the discount
factor 6 < \. What follows is a strengthening of Theorem 3.7 of B-K,^ on several
fronts.
First, we admit payoff discounting.^ Second, we substitute (P) for fullIn this section,
let G((5,
dimensionality. Finally,
Nash
payoffs, to all
T) denote the T-fold repetition
we weaken
the condition that
all
hut one — provided his Nash payoff
is
of
players
is
done to underscore the
we simply
essential similarity
distinct
However,
strictly positive.
unlike B-K's (more intuitive) use of long deterministic cycles,
correlated outcomes. This
must have
rely
upon
among
all
we essentially only need adjust
backward induction. Finally, we remark
has the useful biproduct of permitting an exact (rather than
three folk theorems. In fact, with finite-lived players,
the strategies of Case
that correlation also
A
to avoid the axe of
approximate) folk theorem.
Let the Nash outcome e,* yield player
and
of G,
/,* the lowest, for all
i
N the highest among
G
all
Nash
payoffs
€ N.
EXACT UNIFORM n-PLAYER FINITE-HORIZON FOLK THEOREM
3.2
=
Let u
(ui,...,u„) 6 V*. Suppose that (P) holds, and that either (a) Ukie].)
G N, or (b) Ukiel) >
Ukifk)
3To < oo and Sq < I so that T
VA;
(7,(A!)
>
^
VA;
Tq and
<5
G
1,
{5o, 1]
=>
=
>
>
Then
U,{f^)
G((5, T) has a subgame
but Ui{el)
0.
perfect discounted average payoff u.
Proof:
Let e* be a correlated equilibrium according equal weight 1/n to each of the
Nash outcomes e,*. The typical T-period equilibrium outcome sequence is
e',
e*, where e* lasts S periods, and U{a) G V* will be seen to satisfy
preferred
a,
,
.
.
.
5;
.
,
.
.
the required target payoff equation
{T-S)U{a) + SU{e') = TU{a')
if
(5
=
1,
or
(1
if
(5
<
1.
(3a)
- 6^-^)U{a) +
For ease of exposition,
Q+R+S
.5^-^(1
-
6^)U{e')
=
(1
late deviations are those
periods of the repeated game;
^.A.breu-Dutta (1991) prove that (P)
is
all
-
6'^)U{a')
(3b)
occurring during the final
others are called early deviations.
also necessary for the infinite-horizon result.
'A much better and constructive demonstration is found in Krishna (1989): B-K requires a
messy iterative approximation process.
^In so doing, we can establish a uniform folk theorem, meaning that the discount factor and
horizon length can vary independently over the relevant range.
1.
Play a until period
late, start 4.]
2.
Play M-'
3.
Play
I
7^ 1
for
pfc* for
T — S.
Then
Q
periods. [If player k
R
j deviates early, start 2; if player
I
deviates
play e* until the end.
periods.
[If
deviates late, start
4.
Play //* until the end.
5.
Play
M^
[If player
4.
j deviates, start
some player
If
1
T-S+
until period
^
[If
set k <— j.
j deviates early, restart 2; if player
deviates late, start
[v^J-
Then
3.]
player
I
5.]
Then
return to step
Then
deviates, start 4.[
7^ 1
1.
play e* until the end.
We
proceed recursively. First choose Q, R, and S to ensure subgame perfection.
'Early' deviations are handled exactly as in the infinite horizon case. Thus (1)
determines Q, while (2) must be modified because a
We now
a' (unless
perchance a*
=
e*).
require that
Qu +
for all k
^
^
j.
>P + RUk{pl) + {Q-
RUkip'j)
Given that a
is
as yet unspecified,
Qu + RUkip]) >P +
RUkipl)
+ {Q-
we
l)Ukia)
+
1
(4)
R satisfy
shall instead insist that
l)(2Ht
-
U,{pl))
+
1.
(5)
This will turn out to imply
(4). Morever, a will be chosen so that the personalized
punishment vector p^.* satisfies the analogue of condition (n) with respect to a, V/c G
N. Thus, just as in Case A, the punishment interlude, steps 2 and 3, will deter
deviations from step 1.
Next, step 4 will deter
all 'late'
deviations by players
/
7^ 1
so long as
5
is
large
enough that
iQ
for
/
alH
+R+
7^ 1.
7^ 1 strictly
is
y/S)iJ
+
(5
- VS)Ujie')
This inequality obtains for
>P + {Q + R + Sall sufficiently
Qu +
and since S grows faster than
deviations by player 1 provided
RUi{p\)
+
SUiie')
> p + {S-
+
1
(6a)
large 5, because each plaj-er
prefers e* to //*,
a deterrent to late
l)Uj{fJ)
\fS. Similarly, step 5
VS)Ui{e')
+
1
(6b)
and
SUi{e')>P + {S-y/S)Ui{e') +
l.
(6c)
enough 5, inequalities (6a), (6b), and (6c) are valid.
Given Q, R, and S as defined above, the above program is feasible if To >
+ R + S. Next, since a' G V*, there is some 77-neighbourhood around u entirely
Clearly, for large
Q
contained within V*.
Then
5
Tn-5
"u
-
let
To be sufficiently large that
U{e')\\
<
min(77,min_,gNK-
-
Uj{p'j)])
(7a)
so that
^'^~
for all
<
(5
1
-6^
_ ^T-s W"" -
1
\
T>
and
T >
^(^*)ll
< min(r7,min,eNK- -
(7b)
Ujip])])
Tq.
and introduce discounting just as in Case A, so that each
all 6 G [<5o,l], for some 6q < 1. Define a implicitly by
the target payoff equation (3a) or (3b). To verify that indeed U{a) G V*, (3a) and
Finally, let
Tq,
deterrent remains strict for
(3b) can be rewritten as
T^[u-U{e')]
f
In light of (7a)
and
(7b), equation (8)
Uk{a)
I
-Uk\<
if
both says that a 6
-
min^g n[wj
6
=
1
V* and
also that
Uj{p'j)]
(9)
e N. By the triangle inequality, there are two immediate consequences of
First, we may conclude that Uk{a) < 2uk — Uk{pl), so that (5) implies (4); and
for all k
(9).
second, that
Uk{a)
-
Uk{pl)
both of which were asserted
-
-
=
Uk
>
[uk-Uk{pl)]-\Uk{a)-Uk\>0,
Uk{pl)
+
Uk{a)
Uk
QED
earlier.
Case C: Overlapping Generations Games
Smith (1990) considers a model of overlapping generations games 0LG(G;<5, T),
in which
in its most basic formulation
a player dies and is replaced every
T periods.^ Using the techniques developed for Case B, it is possible to render
that folk theorem exact too. Moreover, its full-dimensionality condition can also be
weakened to (P).
—
3.3
—
EXACT UNIFORM n-PLAYER OLG FOLK THEOREM
Let u
=
so that
{ui,
T >
.
.
.
,Un) 6 V*.
To and
6
€
[(5o, 1]
=>
0LG{G;6,T)
<
oo and ^o < 1
has a subgame perfect discounted
Suppose that (P) holds. Then 3To
average payoff u.
A
parallel proof of this result
We
is
possible, but
—
is
omitted.
—
remark that (P) is required here
to payoff distinguish
as elsewhere
between players at the stage game level: In Cases A and B, this was the only tool
at our disposal. But in overlapping generations games, because the players' tenures
in the game do not coincide, we may also distinguish between them intertemporally.
That is, we may await the death of specific players before rewarding or punishing
^Kandori (1990) describes a related model, but doesn't consider the uniform
discussed below.
folk
theorem
the others. ^° In the as yet unexplored arena of spatial games, one might distinguish
among
the players
interspatially. This insight allows us to
it
method
of (P) as just one
4.
more generally think
of awarding players separable payoff streams.
CONCLUDING REMARKS
games with n > 2 players have a full-dimensional payoff
space is unnecessarily harsh. Although it must be noted that full-dimensionality has
an obvious interpretation itself as an assumption of genericity, our new condition (P)
can be easily seen to hold generically for the smaller class of (stage) games for whose
The requirement
that
relevant
— for
—
Such a class might prove economically
instance, when the payoffs of all players or even of several coalitions
payoff spaces have dimensions
2, 3,
.
,
.
.
n
1.
has constant sum.
Not only
is
our condition (P)
less stringent,
emphasizes the fundamental basis of
any two players
to payoff distinguish
be singled out one at a time, so that
all
Nash
at once,
it is
but
it is
also
strategies, that
and not
all
pure overkill to
more
it
is
of them.
natural.
For
it
only necessary
Deviants can
insist that all
punishers
be rewarded in the stage game. Such a requirement ignores the wealth of available
dynamic strategies
a repeated game.
in
REFERENCES
Abreu, D. and
Condition,
Benoit
P.
Dutta,
Mimeo
J. -P.
A
Folk Theorem for Discounted Repeated Games:
A New
(University of Rochester, Rochester, NY).
and V. Krishna, 1985, Finitely Repeated Games, Econometrica
53,
905-922.
Fudenberg, D. and E. Maskin, 1986, The Folk Theorem in Repeated Games with
Discounting or with Incomplete Information, Econometrica 54, 533-554.
Kandori, M., 1990, Repeated
Games Played by Overlapping Generations
of Players,
forthcoming in Review of Economic Studies.
Krishna, 1989,
The Folk Theorems
for
Repeated Games, Mimeo 89-003, Harvard
Business School.
Rubinstein, A., 1979, Equilibrium in Supergames with the Overtaking Criterion,
Journal of Economic Theory 21,
Smith,
L.,
1990, Folk
Theorems
Games and Economic
^"The development of
1-9.
in
Overlapping Generations Games, forthcoming
in
Behavior.
this idea
is
the substance of the non-uniform folk theorems of Smith
(1990) and Kandori (1990).
8
5^58
79
Date Due ?-^-^v
MIT
3
TDfiO
UBRARIES DUP
DD75b1t.5 7