The Fudenberg and Maskin (1986) folk theorem for discounted re- peated games assumes that the set of feasible payoffs is full dimensional. We obtain the same ...
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R. The game G satisfies NEU if for i and j, there do not exist scalars c, d where d > such that 717(a) = c + dnj(a) for all a G A. Let Mi be the set of player i's mixed strategies, and let = x"=1 M,. Abusconsider a finite n-player
game
in
where 7T,-
:
M
ing notation, (jii,...,Hn)
(n
—
= we write 7r,(/x) for i's expected payoff under the mixed strategy £ M. For any n-element vector v = (ui,...,v„), the corresponding /j,
l)-element vector with element v
{
missing
is
denoted
v_;.
Let
=
7r* (//_,)
be player i's best response payoff against the mixed profile /i_,. Denote by m = (m\, ,m'n ) e a mixed strategy profile which satisfies m'_ € argmin _ 7r*(^_,) and m\ e argmax^.Tr^/i^mL,). In words, ml,- is an (n — l)-profile
max,,; 7T;(a;,/x_,-) 1
.
/J
3
.
.
M
t
j
In the context of
Nash equilibrium and
its
refinements, only single person deviations need be
deterred. Hence, the focus on pairs (an original deviant as opposed to coalitions of players.
and a
single subsequent deviant) of players
mixed
of
=
We
is
An
2.2
player
i,
and m\
:
is
have adopted the normalization
e M}, so that the set of feasible and = F* {w 6 F w > for all i}.
co{7r(//)
payoffs
minimax
strategies which
being minimaxed.
=
7T,(m')
for all
when
i
i.
Let
F
(strictly) individually rational
fi
:
a best response for
{
Equivalence Result
NEU has two quite powerful and equivalent representations that are developed in the lemmas below. For j ^ let F,j, denote the projection of F on the i-j coordinate i,
plane and dimF.j the dimension of
The
Definition (a) dimF.-fc all
^
k
i,j.
=
1
profiles.
Then
diva
^
or
i,
dimF,j
(b)
NEU holds
Suppose that
F
=
1 for
and
that
is
some j
^
and
i
i,
either
dimF^ =
2 for
a line with negative slope.
no player
is indifferent
over
all
action
satisfies the projection condition.
Since no player
F^ >
satisfies the projection condition if for all players
k
all
Furthermore, in the latter case, F^
Lemma Proof:
F
set
2 for
Fij.
1 for all i
^
is
j.
indifferent over all possible action profiles,
Now
suppose the dimF.y
=
1
dimi* ,*
=
1
it
follows that
some j ^ k. and k) implies
for
NEU applied to the payoffs of players i and j (and similarly players i that the payoffs are perfectly negatively correlated. This in turn implies that the payoffs of players j and k are perfectly positively correlated.
k have equivalent payoffs, in violation of
Definition
Lemma ble
The
vectors {v 1
2 Suppose that
F
,
.
.
.
,v
n }
That
is,
players j and
NEU.
Vj
3
k, there exist v].
.
some feasible payoff
For each player
i,
order the
^ k) in increasing size (break ties arbitrarily) from and assign these ordered vectors strictly decreasing weights 6h,h = 1,2, .. ,n(n — 1), summing to one. Let v' be the resulting convex combination of the payoff vectors vjk Note that in defining the u"s we use the same weights 6h for all i. Then by construction, v\ < vj for all i ^ j, establishing payoff asymmetry. ^ Finally, it is straightforward to see that the existence of asymmetric payoffs implies NEU (and rules out universal indifference for any player). In other words, NEU, the projection condition, and the existence of asymmetric payoffs are equivalent assumptions (absent universal indifference). n(n
1)
payoff vectors v
the point of view of player
(Vj i,
.
.
THE MAIN THEOREMS
3.
We
game with
will analyze the infinitely repeated
discounting that
is
associated
with the stage game G. We assume condition his action in period t on the past actions of all players. In addition, we permit public randomization. That is, in every period players publicly observe the perfect monitoring; that
random
realization of an exogenous continuous
variable
is
each player can
and can condition on
its
outcome. This assumption can be made without loss of generality in the infinitely repeated game; a result due to Fudenberg and Maskin (1991) shows explicitly how public randomization can be replaced by "time-averaging" (see also, Sorin (1986)). ,a«, Denote by o, = (a,i, .) a (behavior) strategy for player i and by 7r,«(a) his .
.
.
.
expected payoff in period function
is
.
given the strategy profile a.
t
Each player
his infinite-horizon average expected discounted payoff (1
under the (common) discount factor
— 5)
i's
objective
£2° ^ t7r «t( a )
Let V(5) denote the set of subgame perfect
5.
equilibrium payoffs.
3.1 Sufficient
We
Conditions for a Folk Theorem
will establish here that
that for a wide class of games,
NEU it is
and later on a necessary condition for the folk theorem
sufficient for the folk theorem,
is
also
to hold.
Theorem game
is
Under NEU, any (strictly) individually rational payoff in the stage a subgame perfect equilibrium payoff of the infinitely repeated game when 1
players are sufficiently patient. That
x\
j (payoff asymmetry). For small enough e
(strict individual rationality).
itj
we must have
i
define
3
the definition of w* and v\
then x x-
=
r?),/?2
G F}. Now
/W + fov* + P u
=
x*
where
(u_;,u,)
:
even
if v\
>
u,-.
0,
Finally, for small
5 6 '
Strategies Let a (respectively,
game
payoff
with m',
The (for
is
a')
denote the publicly randomized action vector whose stage
u (respectively, x
minimaxing player
i.e.
%
Further, let
).
and
i,
recall
v'
be the payoff vector associated
from section
2.1 that u)
=
0.
strategy vector that will generate the target payoff u as an equilibrium payoff
appropriate choice of parameters) can be defined in Markov strategy terminology
as follows:
1.
When a'{ 7^
2.
3.
a,-
When and
in "state" u, play a.
and
a'_ :
-
=
a_;,
go to "state"
a'_j
=
—
When
a'.
in state a'_j,
x
: ,
play
go to "state"
first
q)
If
^
m'
proceed to "state"
x'.
the observed action vector
a' satisfies
a'j
7^
but
a'j
y?. Else, stay in "state" x'.
In words, the strategy says:
the
Else, stay in "state" u.
m]_j, go to "state" v*. Else, with probability q stay in "state" v\ (1
=
v'.
a' satisfies
in "state" v', play m'. If the observed action vector a' satisfies a'
while with probability
a'_j
the observed (mixed) action vector
If
and continue to play this action till i). Then, minimax player i for one the event of no observed deviation) continue
Start with a
single-player deviation (say by player
period (with probability one) and the minimaxing with probability
minimaxing and play
a' until
(in
With
q.
the remaining probability, terminate the
further deviations.
Treat players symmetrically and
subject every single player deviation to this (stochastic) punishment schedule. 4
in
This construction together with
F* C K". Then a simple
Lemma 2
choice of x'
is
has a nice geometric intuition. Consider u as a point F* with the smallest i coordinate on the
the point in
Be (u) about u, for small enough e > 0. Indeed, Lemma 1 implies that the projection of B £ {u) n F* onto any two players' coordinate plane is either an ellipse or a line with negative slope. e-sphere
and x k lie at different locations on the ellipse, while in the second they reside opposite ends of a line segment. 5 A referee's comments helped clarify the specification of the /?,'s above. 6 x" We remark in passing that there is a minor error in the construction of the vectors x Fudenberg and Maskin (1986): They implicitly assume (as pointed out to us by Peter Sorensen MIT) that because u £ F*, it does not lie on the lower boundary of F. This is false. One way patch up their construction is suggested by a closer reading of footnote 4.
In the at
first case, x*
,
in
of to
.
.
,
Choice of Parameter The only parameter
in the
feasible payoff for player
above strategy
Choose q to
i.
rrj
>
by equation
(1),
we can
q.
Let
6,
be the best
satisfy:
< l^-x\ \-q
bi
Since
the probability
is
(4)
find such q
€
(0, 1).
Verification of Equilibrium
We show that no one-shot deviation by any player from any state is profitable. Hence the strategy proposed is unimprovable and consequently a subgame perfect equilibrium.
State
v*:
Player
(discounted average) payoff in state v\ denoted
"lifetime"
i's
Li(v'), satisfies
=
Litf)
S (qLiivj)
+
(1
- q)x])
so that
Ate')
Note that
—>
=
f^*- >
0.
(5)
as 6 f 1- Player i will not deviate in state v' since the maximal payoff to one-shot deviation is SLi(v ). Player j ^ i will not deviate for high 5, since his maximal payoffs are bounded by (1 — S)bj + 8Lj(y?), which is less L)(v')
x\,
l
than
by equation
x^,
State x
1 '.
From
(3).
the definitions
and conformity
to one-shot deviation
[{I
-
it is
5)bi
is
is
that (6)
bounded above by
is
+ SL^))- x\ =
where we have substituted from defining q
clear that the difference in the lifetime payoffs
is
(5).
{I
-
5)
^
i
l
+ 5-6q \ (6)
An immediate
strictly negative for all
immediate that players j
(
implication of inequality (4) 5 close to 1. Since x > x-j, j ^ i, it x
-
do not have a profitable one-shot deviation
either.
State u: Since, by target payoff domination (2), u, > x\, the arguments above also imply that (1 — *(a )mi(a
)+
t
+ 1-9(8),
ay.
choices ay and
iff
a'-,
is
zero
between his action minimaxing strategy mlindifferent
State
The arguments
c'-':
+p
ij
i
(aj )c
f
E &*(«*)«*(«*) -py
(«j)
XJ
(
g)
term and the two others involving p ,J'(aj), is Hence, the difference in lifetime payoffs, from the
except the
independent of the choice
)e?
kjiij a k
I
Everything in
fc
first
equation
(7) is satisfied.
In that event, player j
choices. In particular, a best response
that no one-shot deviation
is
is
is
to play the
profitable in state
7
c'-
is
arguments that have established that no one-shot deviation is proffrom state x'. (Note that these last arguments do not change at all since no mixed strategies are played in state x f ). identical to the
itable
3.2
Necessary Conditions for the Folk Theorem
We
turn
for all j}
now
to the necessity of
be player
We
f's
NEU.
Let
/,•
=
min{vi
|
v
€
F
and
Vj
>
worst payoff in the set of weakly individually rational payoff
minimal attainable payoff. 7 The necessity of payoff asymmetry is shown for games in which no two (maximizing) players can be simultaneously held at or below their minimal attainable payoff. In stage games where every player's minimal attainable payoff is indeed his minimax payoff (/, = 0), the condition stated below (which uses the term minimuing) is equivalent to the restriction that no pair of players can be simultaneously minimaxed. 8 Say that a subset of players S C {1, n} can be simultaneously minimized if there exists a strategy profile, fi such that 7r*(^,) < /,- for all i € S. vectors.
will refer to
/,•
as player
i's
.
.
,
.
Definition G satisfies no simultaneous minimizing [NSM] can be simultaneously minimized.
Under
this
Theorem
if
no two players of
G
assumption we obtain a complete characterization. 2 Suppose
NSM
obtains.
Then
NEU
is
necessary for the conclusion of
the folk theorem. 7
Note how
8
A game in
from w\. which the minimally attainable payoff is not the minimax payoff for every player is the two-player game specified by the following payoffs: co[(0, -1), (1, 0), (2, 1)] with (0, —1) a payoff at which player 1 is minimaxed and (1,0) a payoff where player 2 is minimaxed. /, differs
Proof:
To
establish necessity,
that for
all
i
^ j, x\
/,-, as 6 ->
1.
By
By
the folk theorem hypothesis,
playing his myopic best response in period one and
— 7T*(7l,(
