JOURNAL
OF ECONOMIC
THEORY
28, 320-346 (1982)
Pareto-Optimal
Nash Equilibria Are Competitive
In a Repeated
Economy*
MORDECAI
KURZ
Institute for Mathematical Studies in the Social Sciences,
Fourth Floor, Encina Hall, Stanford University, Stanford, California
94305
AND
SERGIU
Department
of Statistics,
HART
TelcAviv University,
Tel-Aviv 69978, Israel
Received November 20, 1981; revised June 22, 1981
Consider a finite exchange economy first as a static, 1 period, economy and then
as a repeated economy over T periods when the utility of each agent is the mean
utility over T. A family of strategic games is defined via a set of six general
properties the most distinct of which is the ability of agents to move commodities
forward in time. Now consider Pareto optimal allocations in the T period economy
which are also Nash equilibria in this family of strategic games. We prove that as T
becomes large this set converges to the set of competitive utility allocations in the
one period economy. The key idea is that a repetition of the economy when agents
can move commodities forward in the time acts as a convexification of the set of
individually feasible outcomes for player i holding all other strategies fixed. Journal
of Economic Literature Classification Numbers: 021, 022.
1. STATIC THEORY
DERIVED
FROM
DYNAMIC
CONSIDERATIONS
In "Yalue and Capital" [4, p. 115] Sir] ohn Hicks states the classical
distinction between statics and dynamics: Statics is timeless while dynamics
involves time. This sharp distinction was deemphasized by the extensive
developments of dynamic economic theory in the post-war period which,
among other things, sought to attain consistency between these two
approaches. Part of the process of establishing the compatibility of dynamics
Science Foundation
Grant SOC75'" This work was supported in part by the National
21820-AO J at the Institute
for Mathematical
Studies
in the Social Sciences,
Stanford
University, and by the Institute for Advanced Studies at thy Hebrew University, Jerusalem.
320
0022-0531/82/060320-27$02.00/0
Copyright
@ 1982 by Academic Press, Inc.
All rights
of reproduction
in any form
reserved.
PARETO
OPTIMAL
NASH
EQUILIBRIA
321
with statics entails the recognition that there are many static phenomena
which either vanish or change completely when placed in a dynamic context.
An example from economic theory which comes to mind and in which such
a situation arises is the difference in conclusions between temporary
equilibrium and full Walrasian equilibrium. Another example is the behavior
of an exchange economy with a storage technology but with or without
futures markets.
In the context of game theory this analysis revolved around the
conclusions drawn from a single game as opposed to the supergame~the
infinite repetition of the single game. It has long been recognized that the
transition from a game to the supergame enabled behavior to transit from
non-cooperative to cooperative mode. The celebrated "Prisoner's Dilemma"
of non-cooperative behavior is resolved in the supergame through the process
of the adoption of essentially a cooperative equilibrium strategy by all
players (see Luce and Raiffa [10, pp. 97-102]). Along the same line it has
long been known~as a "folk theorem" (e.g., see Hart [3 D--that for any
game the set of equilibria in the supergame coincides with the set of all
cooperative payoffs (in the single game) which are individually rational. This
interesting theorem shows that as we move from the "one-shot" to the
"repeated" game, the set of equilibria moves from the narrow set of
noncooperative outcomes to a set of cooperative outcomes, which is too
large to provide a definitive theory' of behavior.
In subsequent work Aumann [1] investigated the relationship between the
core of a game and the set of equilibria in the supergame. He shows that the
set of all strong equilibrium payoffs in the supergame is identical to the pcore of the single game. With~this same machinery, Kurz [7,8] examines the
theory of altruistic behavior. When considered in the context of a single
game no altruistic behavior is exhibited yet an extensive range of such
behavior is possible in the supergame. In a separate paper [9], Kurz studies
the process of inflation with the same analytical tools.
For some, these considerations may only be a reflection of the deeper principle that says that the foundation of every cooperative outcome is an
equilibrium of a non-cooperative game (e.g., Nash [11]). Yet we adopt the
view that it is the interaction of the non-cooperative set-up with the dynamic
structure of the economy which leads to the emergence of cooperative
behavior.
With these ideas as a background we explore in this paper the effect of
repeating an economy g, a finite or infinite number of times, on the outcome
of the implied game. The reader may note immediately the possible relation
to the Debreu-Scarf [2] replica economy. The difference is fundamental: in
the Debreu-Scarf theorem the core converges to the set of competitive
equilibria ,as the static economy is enlarged by replication. We, in this paper,
keep the size of the economy fixed but allow it to repeat over time, where the
322
KURZ
AND
HART
utility of an agent is the mean of his utilities in all periods. To underscore
this fundamental difference note that in the Debreu-Scarf replica economy
the convergence of the core to the competitive equilibrium occurs due to the
rising trading possibilities among the growing number of agents including
trades which involve identical agents. In our economy which repeats over
time agents do not have increased possibilities of trading with new agents but
have the option of reallocating their consumption bundles over time.
Whereas in the replication model, domination occurs with some of the
traders of the same type getting better off and the others worse off, in our
repeated model, this will not be acceptable unless the average utility over all
periods is increased.
Because of this fundamental difference between replication and repetition
over time, our Theorem 1 below will show that the (utility) core allocations
in the economy which repeats for T periods remains identically the same as
the core of the single period game and thus no convergence of the core of the
T period game (denoted CORE(g'T)) occurs. With this result in mind we
shift our attention to the set of points which are both Pareto optimal and
Nash equilibrium allocations in the T period repeated economy and show
that due to the repetition of the economy over time, it is this set which
converges to the set of competitive equilibria in the single period economy.
The key result of this paper can thus be stated as follows:
Consider a standard exchange economy which repeats for T many times and in
which agents can carry their commodity bundles forward in time. Then the set of
commodity or utility allocations, which are both Pareto optimal and Nash equilibria
in the T period economy converges, as T ~ CD, to the set of commodity or utility
competitive allocations in the single period economy.
The crux of the issue at hand is the assumption that agents can' carry their
commodity bundles forward in time and thus we shall be assuming the
possibility of storage. The reader may note that it is this unique and simple
assumption combined with the repetition of the economy over time that leads
to the convergence to the competitive. allocations. There are many recent
papers which showed that various one period games, give, as a solution, the
set of competitive allocations. What we show here is that the very simple
assumptions of the existence of storage possibilities and repetition over time
yield the competitive solution. To comment further on these results we
emphasize that we treat the repeated economy as a repeated game with a
"zero memory," thus establishing an independence over time in the strategies
which agents can adopt. The set of Pareto optimal allocations in the T
period supergame is known to be large. Moreover, we also know that the set
of Nash equilibria in the T period repeated game with zero memory is also
large. Yet we present here the surprising result that the intersection of these
two sets converges to the set of competitive allocations of the single, static,
economy.
PARETO OPTIMAL NASH EQUILIBRIA
323
2. THE ECONOMY
We start with a finite economy g' which is defined byg' =={(Xh, Uh' Wh)'
h = 1,2,..., H}, where
Xh = the consumption set of agenth,
Uh = the utility function of agent h defined over
consumption bundles chin X h ,
W h = the endowment
vector of agent h,
and we shall make the following assumptions:
=
ASSUMPTION 1.
Xh
ASSUMPTION 2.
Uh are monotonic for all h.
ASSUMPTION 3.
Uh are continuous for all h.
ASSUMPTION 4.
Uh are strictly concavefor
ASSUMPTION5.
Each individually
C=
Uh(X)
(CI
IR~
for all h.
all h.
rational Pareto optimal allocation
, Cz,..., CIl) satisfies, for all h,
(i)
Ch~ 0,
(ii)
there is a unique supporting hyperplane to the set {x E IR~
> Uh(Ch)}
I
at Ch'
Assumptions 1-5 are not unusual in this type of analysis; conditions that
would imply them could be given. For example, Assumption 5 will be
satisfied if for all h, Uh(') is differentiable in the interior of IR/+, it has infinite
derivatives on the boundary of IR~, and Wh'* O.
We denote by cOthe aggregate supply of commodities in g', thus (recall
Assumption 5(i»
Il
cO=
L
Wh
~
O.
(2.1)
h=1
Now define (c" cz,..., cH) to be a consumption allocation if r.:= I ch ~ W.
Corresponding to every consumption allocation we have the utility allocation
(U1(C1),uz(cz),..., UH(CH».With these we use the notation
COREG(g') = the set of all g' core commodity allocations,
CORE(g')
~
the set of all g' core utility allocations,
WG(g') = the set of all g' competitive commodity allocations,
W(g') = the set of all g' competitive utility allocations.
324
KURZ AND HART
We turn now to the economy which is repeated T times and which we
denote by g1'. In this economy we have again H agents with index
h = 1,2,..., H, and the consumption set of each agent is (rR/+r. Being a
repeated economy each agent has the repeated endowment vector
(T times).
(W", w" ,..., w,,)
For
any
C~lE R
~I-
sequence
cJ: = (cL cL..., cD of agent h with
of consumptions
all t, the utility level of h is defined by
1'1'
1;'
U,,(c,,)=-.i
T (=1J
(
u,,(c,,).
(2.2)
As we noted in the previous section, the key assumption of this paper is
made by allowing agents to carry their commodity bundles forward in time
in order to rearrange their consumption plans. Thus, although [5'1'should be
viewed as a repetition of g over T periods, there is a basic intertemporal
structure of allocations in g1' which is induced by this assumption. This is
summarized in Assumption 6 which we state now. To do this we need to
introduce the basic notations of the T period economy:
W~l = total commodity bundle in the possession of h at the
end of period t (the amount in "storage"),
z~ = net trade of agent h at time [(positive
coordinates for purchases),
d, = consumption bundle of agent h at time t.
We shall use the following notations:
(
((
(
x = (XPX2""'X",
t
( 1 2
X=X,X,...,X,
(
(
)
)
(
)
( I 2
x"=x",x",...,x,,,
with x standing for anyone of W, z, c, and so on.
We can now introduce our Assumption 6:
ASSUMPTION 6. Intertemporal
Links.
The feasible allocations
consist 01 all (W1',ZT,C1')= {(W~,z~,C~)~~=I};=1 such that for all t
(i)
W~l = W~l-I + w" + Z~l- cL where we define W~=O,
(ii)
(iii)
c~,) 0,
W~l)O,
(iv)
~h=IZh=
\;'/1
(
0.
in g1'
PARETO
OPTIMAL
NASH
325
EQUILIB'RIA
The conditions (i}-(iii) can be viewed as conditions of individual
feasibility while (iv) relates to aggregate exchange feasibility. Since in all of
this paper we are concerned only with Pareto-optimality in gT we do not
make separate use of conditions (i}-(iii) and (iv) and for this reason they are
.
combined here together.'
We note that a consumption stream cT = (c1, c2,..., cT) is feasible in gT
(i.e., there are WT and ZT such that (WT, ZT, CT) satisfies Assumption 6) if
and only if
(
H
L L c~< tro
T= I h= I
(2.3)
foralll<t<T.
In the proofs below we use (2.3) rather than Assumption 6 directly.
This completes the formal description of gr. However, before proceeding
to our analysis we can show immediately that Assumption 6 does not change
the set of Pareto optimal allocations that were not in g T without
Assumption 6. To see this note that for any Pareto optimal
CT=(C1,C2,...,CT) in g'T, the. strict concavity Assumption4
ensures
(Lemma
1 below)
that c(
=c=
constant
vector for all t. But this means that
commodities are not moved forward in any Pareto optimal allocation in g'T
and no use is made of Assumption 6.
We now turn to CORE(g'T) and ask if Assumption 6 combined with the
repetition over time induces any convergence of this core. For this, we define
feasibility for every coalition S c {I,..., H} using Assumption 6 with
LhES z~ = 0 as (iv). With this definition we now have
THEOREM 1.
(i)
(ii)
Theorem
Let Assumptions
1-6 hold, then
c E COREG(g') if and only if cT = (c, c,..., c) E COREG(g'T),
CORE(g') = CORE(gT).
1 states that the core of
g'T
consists
precisely
of the stationary
allocations in g'T, each one being an identical repetition of a core allocation
in g'. We note that Theorem 1 is also true without Assumption 6 (e.g., when
feasibility is required for each period t separately, or commodities may be
moved both forward and. backward in time, and so on) and thus the
possibility of rearranging consumption by moving commodities forward in
time has no effect on the core of g'T.
In relation to the Debreu-Scarf [2] replica economy, Theorem 1 shows
that the T repetition of the economy does not alter the core utility and
commodity allocations. Thus the idea of attaining the convergence of
CORE(g'T) in the repetitive economy in an analogous way to the
convergence of the core in the replica economy, must be abandoned.
The objective of this paper is to show that when the simple assumption of
326
,
KURZ AND HART.
moving commodities forward in time (Assumption 6) is made, the set of all
Nash equilibria and Pareto optima in g'T converges to W(g'). But to do this
we must now turn to a description of our strategic considerations relative to
which one can define the concept of Nash equilibrium.
3. A FAMILY OF FINITE STRATEGIC GAMES"
In order to talk about Nash equilibrium in g'T we need to talk about a
strategic game. However, Theorems 2 and 3 which are the main object of
this analysis remain true for a broad class of strategic games as long as they
have certain properties to be discussed now. Thus we define F(g'T) to be a
class of all the strategic games in gT, with the following properties:
P.I
The set of players is{ 1, 2,..., H}.
PLAYERS.
The set of strategies of player h is Sr.
P.2 STRATEGIES.
From now on, we will denote by or elements of Sr,oT = (oi, oi,..., o~),
an
d
T
O(h)
=
(
1'
I'
T
'T
°1"'" O"-I'°h+I"'"
°H
).
P.3 COMMODITY OUTCOMES, Each H-tuple of strategies
commodity outcome (WI', ZT, CT) which satisfies
(i)
Assumption
(ii)
-Z~l -< Wh for all h all t (zero memory).
01' results, in a
6 (feasibility),
Moreover, for every (WT, ZT, cT) satisfying (i) and (ii), there exists an Htuple of strategies whose commodity outcome is precisely (WI', ZT, CT) (comprehensiveness
).
PA PAYOFFS.
For each H-tuple
of strategies
01', the payoff of player h
IS
-
1
I'
L u,,(c~),
T 1=1
where
(WI', ZT, CT) is the corresponding
commodity
outcome.
P.5 INDIVIDUALSTRATEGICOPTIONS. Let 01' be an H-tuple of strategies,
resulting in the commodity outcome (WI', ZT, cT). Then, for each h:
(a)
Forward Reallocation.
For each t
<
T and v E
[R/+
with v -< c~,
there exists a strategy ar E Sr of player h, such that the commodity outcome
.
~T"7"'T
(W
, z , c ) correspondmg
.
I' o"I'h) satisfies
to (0 (h)"
"
PARETO
OPTIMAL
NASH
327
EQUILIBRIA
iT=ZT,
"1'
A
A
Ck = Ck
l'
and
Wk=Wk,
T
and
Wh =~,
"T
Ch
= Ch
l'
C~+I = C~+I
+v
for all k =1=
h,
for all ! =t=t,
W~ = W~ +
and
"I
I
Ch=Ch-V
l'
and
W~+I
t + 1,
v,
= W~+I.
,Note that a repeated application of P.5(a) for varying t, enables the player
h to reallocate his consumption! forward in time, namely, to replace cr by
any cr that satisfies L~= I C;, ~ L~=I c;, for all t.
(b) Option to Refuse Trading. For each sequence tl, t2,..., t;/ of dates,
there exists a strategy ar of- player h, such that the commodity outcome
(WT, iT, CT) corresponding to (CJfhPon satisfies
"I
Zh
=0
if
--
otherwise.
Z'h
t=tjforsomej=
1,...,J,
We remark immediately that the possibility of refusing to trade at time t
does not violate the feasibility of trades at all other dates due to P.3(ii). Thus
one needs to think of P.3(ii) and P.5(b) together. It is clear that this is not an
inconsequential assumption. However,' it should be pointed out that in the
finite repetition case we use it only for stationary commodity outcomes
which do not require any intertemporal transfers of goods (see Proof of
Theorem 2 and Lemma 1). Thus, in this case a change in trade in some
period cannot affect the feasibility of trades in the other periods. In this
sense, the assumption we actually use is to be viewed as much weaker than
the combination of P.3(ii) and P.5(b).
P.6
RESPECT
FOR
PRIVATE
OWNERSHIP.
Let
(WT, ZT, cT)
r
be
a
commodity outcome. Then every player h has a strategy a such that
(i)
(WT
,
the commodity
ZT , CT ),
outcome resulting from aT = (a;, ar,..., aJ;) is
(ii) for every player k and every strategy or. the commodity outcome
(\\,1', iT, CT) corresponding to (a;k)! oD satisfies,for all t, eitherl i~ "I
Zk = 0 .
z~ or
"either... or ..." refers to each ( separately, i.e., for some ('S, zi = z i, and for the others,
,i~= O.
I
328
KURZ AND HART
Discussion
Although not explicitly stated the condition of "zero memory" is at the
basis of the above properties. Simply stated this condition means that no
strategy can be defined as a function of past events. It implies that players
cannot threaten to retaliate in the "future" as a result of "past" or "present"
deviations. We do not assume here that such retaliations are prohibited but
rather specify in P.5-P.6 those individual actions which cannot lead to
retaliations.
The idea of "zero memory" also appears in property P.3(ii). To see this
note that due to zero memory the trades at times t cannot rely upon the.
actual occurrence of exchange or storage activities prior to time t, and thus it
is only natural that as an outcome condition for time t no individual
exchange cart' exceed the available resources of an agent at that period (this
is condition P.3(ii)). This is also related to an implicit demand that each
player accumulates commodities for consumption only, and thus if he wishes
to obtain a commodity for later consumption he should obtain it early and
carry it by himself.
Conditions P.3, P.5(a) and P.5(b) must be understood together. On the
one hand, any commodity outcQme (WT, ZT,CT) can be realized by some
strategy aT. In this sense the strategy aT realizes the exchanges z~ in [fT. On
the other hand, player h may deviate from aT under two special situations:
(a) He is able to move commodities forward in time (for consumption
or storage purposes), provided he already owns these commodities
under the
.
outcome
(b)
of dates.
of aT,
He may refuse to exchange any commodity at all in a specified list
The idea of "respect for private ownership" is brought into the model so
that no player can force other players to yield to him any portion of their
endowment if they do not wish to do so. This suggests, for example, that no
player can use a threat strategy which' will tax away part of the endowment
of. other players. These icieas are expressed by the property that relat,ive to
any feasible trade outcome ZT the players can adopt a type of strategy which
we denote by aT that ensures that a deviation by any player h means one of
two things: either h trades according to ZT or he does not trade ~t all. Note
that we do not make this assumption for all aT---'but, rather, that to each
commodity outcome corresponds at least one H-tuple of strategies that
respect private ownership.
A comparative reference must now be made to a growing literature on the
strategic approach to Walras equilibrium (e.g., Hurwicz [5, 6], Schmeidler
113] and many others). In this literature the market economy is characterized by"a game in which the strategies of the players are messages and the
329
PARETO OPTIMAL NASH EQUILIBRIA
outcomes are allocations. It is then proved that the outcomes of Pareto
optimal Nash equilibria coincide with the Walrasian correspondence. The
problem, however, is that the family of mechanisms which generate the
above result is rather limited. For example, in Hurwicz [5] and Schmeidler
[13] the strategies of the players. include price messages so that given the
prices proposed by others a player can then select from a feasible budget set.
In recognizing this limitation Hurwicz [6] finds it necessary to impose the
condition that, holding the strategies of all other players fixed, the set of
feasible outcomes for any given player is convex. This obviously generalizes
the idea of price messages for trading purposes.
.
The fundamental idea of our paper is that the large family of games
defined here do not require the convexity of individual feasible outcome sets
since repetition of the economy enables us to derive this convexity as a
conclusion. We regard the convexification which results from repetition to be
an interesting result which could have important implications elsewhere.
An Example of a Game in r(g'T)
Define a strategy a
(i)
(ii)
r of player h to consist
of two parts:
A "proposed" sequence ~r of net trades with
A "proposed" sequence of consumptions 'Yr.
-,~ ~ Wh
all
t.
The commodity outcome is then defined as follows:
H
I
Zh
t
Ch
yl
L ,~= 0
= "'h
if
=0
otherwise,
=
t
Yh
if
W t-l
h
I
+Zh
+ WhPYh'
'""-
t
otherwise,
=0
Wt=
h=l
Wt-l
+Z~+Wh-C~,
It is easy to check that all the assumptions and properties specified above are
satisfied; proposals for trade materialize only when unanimous agreement
takes place and each trader has complete control both over his storage (thus
over his intertemporal transfers of commodities) as well as over his
consumption-as
long as they are feasible.
Having defined the family r(g'T) we shall now proceed. to use the
terminology "Nash equilibrium in g'T" to mean "Nash equilibrium in any
game Y in the family r(g' T) which has the five properties P.I-P.5." Although
the set of Nash equilibria of each particular game Y depends upon the
. specific definition of each such game, our results, concerning the commodity
and utility outcomes, hold for any game Y in the .Camily r(g'T).
330
KURZ
HART
in gT define the average consumptions
For each feasible allocation
(a J",af,..., a~) and average
AND
aT
UT = (Uf, uf,..., U~) as follows
utilities
{,
1
I'
=
t
ah=-kch'
T t=1
1 I'
Uf:(CT) =
.L Uh(C~).
T
t= I
Now denote by
PO T
-
the set of all average utilities UT = (Ui, uf,..., Un
associated with Pareto optimal outcomes
PO~
=
(WT, ZT, CT) in gr.
th~ set of all average <;onsumptions aT = (af, ar,..., a~)
"
associated with Pareto-optimal outcomes in gr.
NE T = the set of all average utilities UT associated with
Nash Equilibria in r(g'T).
NE~
= the
set of all average consumptions
aT associated
with
Nash Equilibria in r(g'T).
We can now state our first main result:
THEOREM 2.
then
Let Assumptions
1-6 and Properties P.I-P.6
be satisfied
(i) W(g) = limT->co [POT n NET],
(ii) WoCg') = limT->co[PO~n NE~],
(iii) for every Pareto-optimal outcome. (WI', ZT, CT),
C I =c
2
.='..=c.
I'
Theorem 2 is the first of our two main convergence theorems. The set
PO Tn NE T is the set of Pareto
optimal
utility allocations
in g
T
which are
also Nash equilibria in F( g T). The theorem suggests that the limit of this set
is the set of competitive utility allocations in g. Similarly, the set
PO ~ n NE~ is a set of mean commodity allocations in the T period
economy and this set converges to the set of commodity allocations Wo(g).
Theorem 2 formalizes our earlier claim that if we start with any noncooperative behavior in a single economy then the repetitive economy with
the property specified in Assumption 6 will undergo a transformation into an
economy with a more specialized behavior leading to an outcome defined by
the set of competitive equilibria in the one period game. Alternatively one
PARETO OPTIMAL
NASH
331
EQUILIBRIA
can interpret the result of Theorem 2 to mean that a competitive behavior in
the one period economy will be strongly reinforced by the repetitive nature of
the economy and the ability of agents to transfer commodity bundles
forward in time.
We now motivate Theorem 2 with the aid of the familiar Edgeworth box.
Consider an economy with individuals I and II. In Fig. 1, E is the
competitive equilibrium while A is a Pareto optimal allocation on the
contract curve. Since the indifference curves are strictly convex a Pareto
optimal allocation A which is not a competitive equilibrium has the property
that the line wA contains points like B = (xf, xfl) which from the point of
view of I, are strictly better than A (thus UI < uf)o Such points satisfy the
condition that for some 0 < A. < 1
xf
~
AWI + (1 - A) x1.
We shall show that the repetition of the economy provides individual I a
strategy to attain convex combinations like xf and thus reject xA as a
potential Nash Equilibrium in r(g'T). To see this suppose A.= 1/4 and it is
proposeq to trade w into xA for T periods. Table I shows how individual I
can alternate his decisions over time to particupate and trade into xA or not
participate and stay at WI and in so doing attain xf.
The table shows that at t = 0 the individual consumes nothing, but in all
subsequent periods he consumes exactly xf=(1/4)wl+(3/4)x1,
and this
he does by not trading every fourth period. As the length of T increases the
initial loss at t = 0 becomes smaller and disappears in the limit. Also if A. is
not a rational number the approximation becomes more accurate as T
N
IT
>fa
0
~
:E
0
u
I
COMMODITY 1
FIGURE 1
332
KURZ AND HART
TABLE I
=
Decision Table to Attain x:
(1/4)
WI
+ (3/4) x~
An alternative program
Time
0
1
2
3
4
5
6
Proposed
program of
consumption
X~
x~
x~
x~
x~
x~
x1
Decision
taken
Resources
A vailable in t
(endowment
+ trade)
Do not trade
WI
Storage
Consumption
0
I
WI
3 A
Trade
x~
;jWI + ;jXI
J
;jWI
Trade
x~
I
3 A
;jWI
+ ;jXI
2
;jWI
I A
+ ;jX,
2 A
+ ;jXI
Trade
x~
lw
4
I
.WI
+
Do not Trade
WI
1
;jWI
I
J
+ lxA
4 I
3 A
+ ;jX1
3 A
Trade
x1
.WI '+
Trade
x~
tw( + ~x~
;jXI
3 A
;jXI
WI
J
;jWI
I A
+ ;jXI
2
2 A
+ .X.
;jWI'
becomes large and A is approximated by the fraction of the total number of
periods in T during which the agent does not trade. It follows that for large
T the Pareto optimal allocations which cannot be improved upon with the
convexifying strategy are allocations like E-the competitive ones.
4. THE INFINITE REPETITION
Our object now is to extend the analysis to goo, the infinitely
economy. Assumptions 1-6 and P~operties P.I-P.6 are carried over,
change being that the feasible commodity outcomes are now infinite
using the notation x (rather than x 00) for (x!, X2 ,..., Xl,...), they
repeated
the QpJY
streams;
will be
denoted by (W, z, c).
Unfortunately, this extension is not a straightforward procedure since
there are two basic difficulties. The first relates to the problem of defining
"domination" and thus Pareto optimum in goo and Nash equilibrium in the
infinitely repeated games F(g'oo). The second relates to the problem of
differentiating among different exchange patterns all leading to essentially the
same outcome.
To clarify the first difficulty consider any feasible allocation (W, z, c) in
goo. Define the terms
'
T
ah
T
Uh(Ch)
1
= -T
=-
~
t
1 {~
T
(4.1 )
2.., Ch'
l=!
L
t=!
t
Uh(Ch
).
(4.2)
333
PARETO OPTIMAL NASH EQUILIBRIA
Now, if we wish to compare c with any other feasible consumption
allocation c we may have a difficulty due to the fact that the sequences
jUk(Ch)}r=l and {UnCh)}r=l may not converge. In order to define a
preference relation for the comparison of Ch and Ch' we introduce two
domination relations discussed by Aumann [1]. The two domination
relations called >-u (upper) and >-L (lower) are defined as follows.
Let a = jat I~ 1 and 13= {Pt} ~ .be two infinite sequences.
DEFINITION4.1 a. The sequence a is said to Upper-dominate the
sequence 13(denoted a >-u13) if there exists an infinite sequence of dates tj,
j = 1, 2, 3,... and e > 0 such that
j = 1, 2,3,....
at} > Pt} + e,
DEFINITION4.1b. The sequence a is said to Lower-dominate
sequence 13(denoted a >-L13)if there exists a To and e > 0 such that
at > Pt + (;
the
for all t > To'
With these two definitions of domination w~ have two induced concepts of
Pareto optimum, and two induced concepts of Nash equilibrium. Next, let us
,address the second problem mentioned above. To do this we 'start with a
simple example (see Table II).
Consider an economy with two agents called 1 and 2 and two
commodities. The two endowments are WI = (0, 2) and W2= (2, 0). Assume
that the allocation [(1, 1), (1,1)] is Pareto optimal in g. Now consider in
Table II two different commodity outcomes in goo. Clearly c~ is the same in
A as in B except that in Bagent 1 consumes (1, 1) in the first period while
TABLE II
Commodity outcome A
Commodity outcome B
Time
ell
z~
W~
e~
z~
W~
I
(0,0)
(0,0)
(0,2)
( I, I)
(I, -I)
(0,0)
2
(I, I)
(2, -2)
(1, 1)
(1, 1)
(1, -1)
(0,0)
2
(1, I)
(0,0)
(0,2)
(1, I)
(1, -1)
(0,0)
4
(1, 1)
(2, -2L
(1, 1)
(1, 1)
(1, -1)
(0,0)
5
(1, 1)
(0,0)
(0,2)
(1, 1)
(1, -1)
(0,0)
6
(1, I)
(2, -2)
(1, I)
(1, 1)
(1, -1)
(0,0)
334
KURZ AND HART
he consumes (0,0) in A. However, by our standard of comparison in goo,
the consumption sequences in A and B will be equivalent. Yet the exchange
pattern in A leaves the agent much less freedom than in B. This is so since in
A the agent needs to maintain the storage strategy and if he wanted to alter
c~ so as to consume (1, 1) in every second period and (0, 2) at the rest of the
time points-he cannot do so. In situation B the agent is free of the storage
restrictions and is free to alter his consumption pattern if he wished.
The critical difference between A and B is found in the useless storage
activity which is carried out in A and our aim is to insist that in a Pareto
optimal program, the agents do not carry out unneeded storage. Clearly, one
way of doing so is the introduction of a direct storage activity which would
consume resources but this will complicate the present model and will
obscure the main results. An alternative way of removing useless storage is
by introducing a fiction in the form of a vector q E IR~ of "theoretical"
storage cost per unit of commodity and per unit of time. This means that if
(W, z, c) is a commodity outcome in goo, then we shall define the q-utility
outcome up to time T to be
1
T
t
t
U/t(c/t,q)=-~
~ [u/t(c/t)-q,W/t.
T [=1
]
(4.3)
We denote the infinite sequence of utilities by
(4.4 )
U/t(c/t, q) = {uh'(c/t, q)}~=I'
Now, for any q we can compare c with c using the definition (4.4) and the
two domination relations. in Definitions 4.1.a and 4.1 b; This procedure will
lead to a concept of Pareto optim.um which will depend upon q~a result
which we cannot accept since q is viewed as fictitious storage cost intended
to remove all unneeded storage. Since we do not want the cast of storage to
become an essential part of the payoff we shall let q decrease to 0 and seek
the limiting behavior. The following will make these ideas precise.
DEFINITION 4.2. Let q ~ O. A commodity
lower Pareto optimal with respect to q if
(i)
there is no commodity
outcome
outcome
(W, z, c) in goo is
CW,z, c) such that
U,Jc/t, q) > U/t(c/t, q)
L
for all 11;
(ii)
for all 11,the sequence {U;;(c/t, q)}~=1 has a limit ("summability").
We remark immediately that the summability condition (ii) is not related
at all to Pareto optimality; we include it in the definition so as to belable to
define "payoffs" (see also Aumann [1 D.
PARETO
OPTIMAL
NAS~
335
EQUILIBRIA
,I
Let us denote
POL(q)=the set of all limits of average utilities (i.e.,
{UrcCh,q)}r=l-see
(4.3» for L-Pareto optimal
outcomes with respect to q.
POGL(q) = the set of all limits of average consumptions -(Le.,
{an r;!=I-see (4.1» for L-Pareto optimal outcomes
.
with respect to q.
In a similar manner one defines the notion of Upper Pareto optimality
with respect to q, and the corresponding sets POu(q) and POGu(q).
We now turn to the issues of Nash equilibrium. The class of strategic
games r(gOO) is assumed
to satisfy the properties P.I-P.6 as do the games in
.
T
.
reg ) (I.e., take T = 00).
DEFINITION4.3. Let q ~ O. An H-tuple of strategies 0 resulting in the
commodity outcome (W, z, c) is a lower Nash equilibrium with respect to q
if
(i) for all h, there is no strategy Oh of player h such that the
commodity outcome (W, z, c) corresponding to (O(hl' °h) satisfies
Uh(Ch, q) >- Uh(Ch' q),
L
(ii)
for all h, the sequence
{UrcCh' q)}~=1 has a limit ("summability").
We now denote
NEL(q) = the set of all limits of average utilities for L-Nash
equilibrium with respect to q
NEGL(q) = the set of all limits of average consumptions for LN ash equilibria with respect to q.
Again by replacing "lower" with "upper" we obtain the sets NEu(q) and
NEGu(q).
.
It should be pointed out that, as q decreases to 0, the sets POL(q) need not
have a limit; and the same is true for NEL(q) and for the other sets which we
defined above. Let us denote'by limq->oH the limit as q ~ 0 and q ~ O. Our
result is
THEOREM3. Let Assumptions
and r(gOO). Then
(i) W(g)
NEu(q)], .
=
1-'6 and Properties P.I-P.6
limq ->o++[POL(q)
n
NEL(q)]
=
hold for goo
limq->oH[POu(q)n
r
336
KURZ
AND HART
(ii) WG(g) = limq--.o++[POGL(q) n NEGL(q)] = limq--.o++[POGu(q)n
NEGU(q)],
(iii) for any q ~ 0 and every allocation (W, z, c) in goo which is L or
U-Pareto optimal wih respect to q, the sequence a = {aT} ~= 1 converges.
Theorem 3 provides the natural extension of Theorem 2 to the infinite
economy goo. It says that in goo the set of all Pareto optimal allocations
which are also Nash equilibrium in r(gOO) have the same character as the
competitive allocations in g. This means that the utility limits of all such c
are exactly the utility allocations in W(g) and the limit mean consumption
allocations aT are exactly WGUf).
5. A FINAL NOTE
As indicated in Section I the present paper is part of a general search of
non-cooperative mechanisms which tend to change their nature due to
repetition. We think that the extent of such phenomena is more widespread
than is generally recognized. Our aim is to explore the factors which
generate the motivation for the convergence toward- a more cooperative mode
of behavior. In the present paper we find that the ability of agents to. carry
their endowments forward in time combined with the repetition of the
economy are adequate factors to generate a tendency toward competitive
equilibria.
6. PROOFS IN g
LEMMA 1.
T
-.,
Let cT E PO(g'T). Then cT is stationary, i.e., there is c such
that cl = Cfor aliI
t T. Furthermore, c is Pareto optimal in g', and there
< <
are no transfers of commodities between periods.
Proof
Define
ah
I {.
L,
T 1=
I
=-
1
for all h.
Ch
The stationary consumption
aT = {(a a2 ,..., aH)} i= 1 is feasible
l'
forwarding of commodities is needed), since
HIT
I
h=l
H
ah
=-
I I
TI=lh=l
(and
no
H
c~
<I
h=l
Who
(6.1)
PARETO
OPTIMAL
NASH
337
EQUILIBRIA
By Assumption 4, each Uhis strictly concave and since c T is Pareto optimal,
we must have c~ = ah for all t and all h. If b Pareto dominates a in g', then
bT = (b, b,..., b) Pareto dominates aT in g'T; therefore a is Pareto optimal in
g'. By monotonicity (Assumption 2) Lh ah = Lh Wh' hence no transfers of
goods are made between periods.
Proof of Theorem 1. Since every point in CORE(g'T) is also Pareto
optimal in g' T, it follows from Lemma 1 that it is stationary in utilities and
commodities. The theorem is now immediate.
LEMMA2. Let x be an allocation in g. Then x is competitive if and only
if x is individually rational and Pareto optimal in g' and, for all m = 1, 2,...,
m-l
Uh
(
1
mm
Xh + -
Wh
)
for all h.
-< Uh(Xh)
'-
Proof. If (p, x) is a competitive equilibrium
individually rational an~ Pareto optimal, and
p
,
.
m-l
( m
xh+
1
)
m Wh-<
p
(6.2)
in g', then x is clearly
.W h,
implying (6.2).
Conversely, if x is Pareto optimal, let p E IRI separate Y h = {y E IRIt
Uh(y) Uh(Xh)} from Xh' for all h; such a p clearly exists, and by
Assumption 2, p O.
From (6.2) and the concavity ofuh' we obtain Yhn{axh+(l-a)wh
0-<
>
I
>
I
a ~ 1}= 0. We can thus separate these two sets by a vector p k
E IRI.
However, since Ph also separates Xh and Yh, and since by Assumption 5(ii)
such a separating hyperplane is unique, it follows that Ph may be taken
without loss of generality to be Pk = p. But then P . Wh -<p . y for all y E Yh,
which implies P . Wh -<P . Xh' This holds for all h; together with Lh Wh =
Lh Xh' and P 0, we get p . Wh = P . Xh for all h, completing our proof.
>
Remark.
We actually proved that in the statement of Lemma 2 we could
have concluded that uh(axh + (1 - a) Wh) ~ Uh(Xh) for all 0 ~ a ~ 1.
Proof of Theorem 2.
We prove the theorem in two steps:
Step 1. limT-->oo[POTn NET] c W(g'),
Step 2. POT n NET ~ W(g').
Step 1. limT~oo[POT n NET] c W(g').
Let aT be an H-tuple
of
strategies with a commodity outcome (WT, ZT, cT) such that cT is Pareto
optimal in g'T and aT is a Nash equilibrium in some y E F(g'T). By Lemma 1
c T is stationary (Le., ct = c for all t), no commodities are transferred forward
338
KURZ
'AND HART
and e is Pareto optimal in fJ'. By P.5(b) and P.5(a); player h may'refuse to
trade at all dates, and consume his endowment Wh in each period. Since 01' is
a Nash equilibrium we conclude that e is an .individually rational Pareto
.
optimal allocation in g -hence we can use Lemma 2.
Now let player h use P.5(b) as follows: let m be a positive integer, and
consider the following new strategy of h: in each of the dates t - 1, m + 1,
2m + 1,..., km + 1,..., he does not trade (i.e., i~ = 0), and in the others
i~l= z~. His consumption will be Ch= 0 and
.
c~
= «m - 1)lm) Ch+ (1/m)wh
for all t > 1. We claim that this consumption plan is feasible for h (by use of
P.5(a»); indeed, consider the first m + 1 periods~ At t = 1, h has w", wVich is
consumed in 'the next m periods (at the rate of (11m) w" each); for
t = 2,..., m, only «m - 1)lm) e" is consumed, the remaining (11m) c" being
stored in order to be able to consume « m- 1)1m) e" in period m + 1 (when
there is no trade again). Therefore h can get [«m - 1)lm) Ch] + (11m) w" in
each of the periods 2,..., m + 1 by using
w" from period 1 and e" from
periods 2,..., m. Furthermore, at the start of the next cycle, at t = m + 2, h
has again w" (stored from t = m + 1), so that the whole process may be
repeated. (See Table III for an example.) The feasibility follows' from this
.
construction.
The utility to h of {c~}i= 1 is
~-U,,(o)
+ T;
1
1
u" (m:
c,,+ ~ w,,),
Since 01' is a Nash equilibrium, we must have
~
u,,(O)+ T;
1
1
u" (m:
c" + ~ W,,) < u,,(e,,).
(6.3 )
TABLE III
Example of the Use of Property P,5(a): m = 3
Total wealth
at! - 1
(W~- 1)
Net trade
at at
(z~)
W~-l + w"
0
0
w"
I
2
W"
2
1
+ }C"
3
}w"
4
2,
}lW ,,- t- }c"
5
w"
C" -' W"
c" - W"
0
c"-w,,
2
4
+ z~" - c~)
t
+ z~
C"
0
w" + c"
}W"
W~
(= Wt-I + w"
4
+ jC"
2
}w" + }c"
1
w"
2
jW" + jC"
I
jW"
1
+
2
}c"
2
}w" + }c"
2
1
1
2
jW" + jC"
}w" + }c"
w"
339
PARETO OPTIMAL NASH EQUILIBRIA
As T -+ 00, we get the inequality (6.2). The same construction may be done
for all m = 1,2,..., and for all h. Lemma 2 therefore implies that c E wdtW)
and (UI(CI)' U2(C2)"'" UH(CH» E W(tw).
Step 2. W(tw) c (POT n NET). Let C E Wo<g) and consider
the
stationary commodity outcome (WT, ZT, cT) given by c~ = Ch' z~ = Ch - Wh
and W~ = O. It clearly satisfies P.3(i) and (ii) and by comprehensiveness
there is aT resulting in it. Moreover, by P.6, we may choose aT so as to
respect private ownership for all h. Assume now that some player h deviates
and by P.6 his trades will necessatily be either Ch- Wh or O.
By concavity of Uh the average utility of h will be' at most
TI
uhTwh
(
T - TI
Ch,
)
T
+
where TI is the number of periods in which h's trade was O. By Lemma 2
(see remark at the end of the proof) C E WG(tw) implies that
TI
Uh
(T
Wh
+
T - TI
T
Ch
)
~ Uh(Ch)'
,thus aT is a Nash Equilibrium. As for Pareto optimality in tWT it follows
directly from the Pareto optimality of any Walrasian allocation in g and the
stationarity of cT.
It is interesting to question the need for the assumption of strict concavity
(Assumption 4). It was used in an essential way in Lemma 2 above and the
example to be presented now will show that our results will not hold without
it.
EXAMPLE. Let 1=2, H = 2, WI = (1,0), W2 =
utility function for h = 1,2 that is Uh(X,y) = (Vi +
but not strictly concave utility function as it can be
from the origin the function is linear. Now consider
(0,1),
and a common
y'y)2. This is a concave
seen that along any ray
the following plan:
For l = 1,2,..., T - 1: no trade and no consumption. For l = T:c: = «T/3), (T/3n,
ci = «2T/3), (2T/3)) and the corresponding trade at an exchange rate of 2/3 of
Commodity 1 for 1/3 of Commodity 2.
This plan is Pareto optimal in
tWT
since (lIT) Li=
I uI(cD
= uI(1/3, 1/3)
and (lIT) Li= I U2(C~) = u2(2/3, 2/3). The allocation
is also a Nash
equilibrium in gT since none of the agents can improve due to the fact that
the only available alternative is not to trade at all in period T giving each an
average lltility of Uh(Wh)' The key observation to make is that the mean
utilities
UI (1/3, 1/3) and u2(2/3, 2/3) are not a competitive equilibrium in g
340
KURZ
AND
HART
since the unique competitive price is p* = (1/2, 1/2) and for agent 2 the
point (2/3, 2/3) is outside his budget set
Bip*)
= {xlp* . (0, I)}.
7. PROOFS IN goo
LEMMA 3. Let q ~ 0 and let (W, z, c) be L-Pareto optimal with respect
to q. Then there is a E POG(g) such that, for all h,
(i)
I1m ah
l'
'
1'->00
= ah .
and for all c5 > 0,
(ii)
(iii)
(iv)
Proof.
lim ~#{t<Tlllc~-ahll<c5}=l,
1'->00 T
lim
~ # {t < T III W~
1'->00 T
II
< c5} = 1,
I
rim ~#{t<Tlllwh+z~-ahll<c5}=1.
1'->00 T
Denote by v h the expression
l'
Vh
= 1'->00
lim ~
T
L
(Uh(C~) - q
.W~).
(7.1)
1= 1
Since the sequence {a1'}r= 1 is bounded (for each h, by a and cO)let a be a
limit point of this sequence, The concavity of Uh implies (recall (4.1» that
l'
l'
uh(a,J
~
The continuity
1~
T
L ,
l'
Uh(Ch)
1=1
11'--'
~T
.
I
Wh
].
1=1
of Uh then implies
uh(ah) >- v h
Consider
.1
L , [Uh(Ch) - q
now the stationary
allocation
for all h.
(7.2)
(W, i, c) given by c~ = a h + (1/H)
L~=l (wh-ah) and i~I=C~-Wh'
This is clearly feasible with W~=O for
all h and all t, Since L~; = 1 (coh - all) >- a (by feasibility) it followsfrom
monotonicity that the payoff to Chis greater than uh(ah)' Since (W, z, c) is LPareto optimal with respect to q, (7.2) implies that in fact
I/{
Uh
(
ah
,
+ HL
h=l
(evh - all)
= u~(ah) = Vh,
)
341
PARETO OPTIMAL NASH EQUILIBRIA
hence it also follows that L~=l Wh = L~=l ah' Furthermore,
a EPOG(g),
otherwise, if b dominates it, the' stationary allocation associated with b will
L-dominate (W, z, c), hence it will also dominate (W, z, c) (by 7.2).
Let a' be another limit poiht of {aT}~=l' As above, uh(ah)=vh and
a' E POG(g); the strict concavity of all Uh implies a = a' (consider
(1/2) a + (1/2) a'). Hence the sequence aT must actually converge to a,
proving (i).
Next, fix h and let ..t be a super-gradient of Uh at ah (cf. Rockafellar [12,
Theorem 23.4, p. 217], and keeping in mind Assumption 5(i».
Define
lex) = uh(ah) - Uh(X)- ..t(ah- x),
for all x E fR~. Thenf(x) ~ 0 for all x, andf(x) = 0 only for x = ah (because
of the strict concavity of Uh)' Furthermore, f is a convex function, therefore
f(ax + (1 - a) ah) ~ alex) + (1 - a)f(ah) = alex) ~f(x)
for all 0 ~ a ~ 1 and x E IR1+.Hence f does not decrease as x gets "farther
away" from ah' and we get
inf{f(x)
x
II
I
The latter infimum
x - ah II
~ c5} = inf{f(x)
x
I II
x
is attained
since f is continuous
~
. Wht ]
-
ahll
and
= c5}.
c5
> O.
infimum by p.
Consider now
uh(ah) -
1
T
t
L.., [Uh(Ch)- q
t= 1
= ~ ~I [u.(a.) - u.(c~) -.t.
t
ah)
+ A . (ah 1
~T
~
+ q.
1
(T
(a. - c~)1
~
t
Wh
t~l
)
1
q.
/;;If(ch) +A' (ah -ah) +
(T
t
T
~A' (ah-dh)+p.
~
t-21
t
Wh
)
~#{t~'Tlllc~-ahll~c5}
l
+ c5. l(j(1min (q)j' T #
.
t
!
< TI
t
1~1
(W~)j ~ bl.
~
Denote this
342
As
KURZ
T
--7
the
00
left-hand
AND
side
HART
of the
above
inequality
converges
to
(v" - v,,) = 0 and since A(a" - aD --70 it follows that
p-
1
#{t ~ T
T
III c~ - alII!
> J}
l
I
+ J 1<,
m~n
(q)J'
-#
) <,l
T
.
However, since p
> 0,
t~ T
I
L
(W~)J > J
1=
1
!
~ O.
!
J> 0 and minI <,J<,l(q)J> 0, conclusions (ii) an.d (Hi)
of the lemma follow immediately.
To prove (iv) note that
(w" + zD - a" = (c~- a,,) +
W~+
1
-
W~
and this shows that (iv) follows directly from (H) and (Hi).
Proof of Theorem 3. The definitions of lower and upper domination
imply that for all q, POL(q)::) POu(q) andNEL(q)::) NEu(q), and thus
[POL(q) n NEL(q)] ::) [POu(q) n NEu(q)].
We recall that, for a collection {A(q) }q~O of sets,
lim sup A(q) =
q-tO++
lim inf A(q)
q-tO++
=
n U
A(q),
Un
A(q),
q~O
O<t,q<,q
q~O O<t,q<,q
and the limit exists if and only if lim sup = lim inf.
Our proof will therefore consist of the following two steps:
Step 1.
Step 2.
lim SUPq-to++[POL(q) 0 NEL(q)] C W(tw).
Step 1.
Urn SUPq-to++[POL(q)n NEL(q)] c W(tw)~
W(tw) c [POu(q)
.
n NEu(q)], for all q ~ O.
Let (J be an H-tuple of strategies with (W, z, c) as, commodity outcome
such that for every q ~ 0 there is 0 <{q ~ q with respect to which (W, z, c) is
L-Pareto optimal and L-Nash equilibrium. By Lemma 3, a = limT-tco aT is in
PO d g). This allocation is also individually rational since h can always
select the strategy which yields w" as a stationary consumption and no trade.
We have to show that it is a Walrasian equilibrium.
Thus assume a E WG(Jr). By Lemma 2 there are hand m such that
m
-
1
1
u" ( m a" + -;;; wIt ) > u~(a,,).
343
PARETO OPTIMAL NASH EQUILIBRIA
Define
e=
1
4""
m- 1
1
- u,,(a,,) > O.
]
[ u" ( m a" + m w" )
(7.3)
By the continuity of u" (Assumption 3), let b > 0 be such that
x -
m- 1
( m
ah +
1
m
<b
Wh
)
implies
u.(x)
- u.
I
(m:
1
u. + ~ w.)
< t.
(7.4)
I
Next, we call a date t "good" if it satisfies
(7.5)
II(W"+z~) - ahll<b,
and "bad" otherwise. Let {tjU~ I be the sequeJ1.ce of "good" dates (it is
infinite by (iv) in Lemma 3).
We now apply properties P.5(a) and P.5(b) of the strategic game to obtain
a strategy a" of player h according to which he refuses to trade on dates
tkm+I' k = 0, 1, 2,..., and then move commodities forward so that the
commodity outcome (W, z, c) resulting from (o("P a,,) satisfies the following:
for t = tkm+ I , k = 0, 1, 2,...,
"t
z"
=0
-zt
.
otherwise,
"
"t
C,,=Wh+Zh
for t "bad" date, -
t
for t = t I
=0
=
m -1
1
(w+z~)+-w"m
m
f
Z~(k-I)m+i
=~
m ( 1=2
).
+ Wh
for t = tj' j '* km + 1, k = 0, 1, 2,...,
for t
-
t km + I' k
-
1, 2,....
The idea here is similar to the one used in the finite case (see Table III).
On "bad" dates agent h continues to trade z~ as before and consumes
(w" + z~). Now, not counting "bad" dates, consider only the "good" ones
t I' tz, t3"'" The agent divides these dates into cycles of length m and at dates
tkm+ l' k = 0, 1, 2,..., he does not trade. He starts ofT the consumption and
storage program by consuming 0 at tI and- storing w" at that time. Following
this initial,.step, for each cycle of length m (after the initial date t 1) the agent
.
344
KURZ AND HART
1 dates only (m - 1)Im of the net
quantity (wit + z~J and 11m of the initial endowment wit out of storage. In
the mth date there is no trade and the amount which is in storage at the start
of the period is devoted to consumption. The cycle starts again by putting
into storage (at the end of the mth data) the endowment wit of the period.
We will show that there is To and q ~ 0 such that for all T> To and all
0 ~ q <: q.
consumes
during each of the first m
1
, -
,
T
-
,
T
~
I
~ [UIt(CIt)
I
L,
- qWIt]
> ult(alt) + e,
1= 1
thus proving that (J cannot be an L-Nash equilibrium for arbitrarily small q,
contradicting our assumption.
Indeed, W~l= wit for all t = tkm + l' k = 0, 1, 2,..., where "bad" dates do not
increase storage. Hence W~ is bounded by some ME IR~ (for example
M = mw). Hence
T
~
Next, let t=tj
8~
-
T
[uh (c~)
L
-
q W~]
/=1'
1
(k=
(cD
(7.6)
q . M.
-
a, +
~
1 (k=O, 1,2,...); then (7.5) implies
ill,)
In::.
=
1,2,...),
1 (ill,
+ z~)-
1
In
::.
a,
I
<
For t=tkm+I
U It
1=1
andj=tkm+
C ,:
>~ L
I
m-1
m
(j
< (j.
we have, again by (7.5) applied
to all t(k-l)m+;
for i = 2,..., m, k = 1, 2,...,
C~II
m-l
alt +~Wh
( m
m
)
<:~ £, IIz~(k-l)m+i+wh-ahll
m ;=2
II
<
m-l
m
'
(j
< (j.
Therefore, by (7.4) and the monotonicity of Uh (Assumption 2),
1
T
T
L Uh(C~)
1=1
1
> -T
+
#{t <: T I =
I
~,
# {t
II or I "bad"}
Uh(O)
< Tit" good"} [uh
C:
1
ah +
1
Wh) - e ,
]
PARETO OPTIMAL NASH EQUILIBRIA
By Lemma 3(iv), (lfT)#{t<Tlt"good"}
therefore there is To (large enough) such that
-T1;'~
At
Uh (Ch)
> Uh (
t=1
m- 1
m
345
converges
1
ah +-Wh
m
)
to 1 as T--+oo,
- 28
for all T> To' Let ij be such that ij. M < e, then, by (7.6) and (7.3), we
finally have
-
1
T
~
~
[Uh(Ch)At
t=1
for all T
> To
Step 2.
~
q
t
. Wh ] > Uh (
and all 0 4i;q
.
m- 1
m
ah
1
+ -f11.Wh ) -
38
> Uh (ah ) + e,
< ii, completingour proof.
W(g) c [POu(q)
n
NEu(q)]
for allq ~ O.
Let CE WG(g), and consider the stationary commodity outcome (W, z, c)
with c~ = Ch, z~ = Ch- Wh' W~ = O. We can use precisely the same
arguments as in the Proof of Step 2 in Theorem 2, to show that it is U-Pareto
optimal and U-Nash equilibrium with respect to any q ~ O. Actually, we
prove a stronger result, namely, that it cannot be dominated even in one date
T (i.e., for no T, the average payofT(s) may be higher). Also, since there is no
'storage cost in (W, z, c), q does not matter.
REFE REN CE S
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Studies 40 (1959), 287-324.
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Rev. 4 (1963), 235-246.
.
3. S. HART, Lecture notes on game theory, mimeo, Institute for Mathematical Studies in the
Social Sciences, Stanford University, 1979.
4. J. R. HICKS,"Value and Capital," Oxford Univ. Press (Clarendon), London, 1939.
5. L HURWICZ,Outcome functions yielding Walrasian and Lindahl al1ocations at Nash
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6. L. HuRwlcz, On allocations attainable through Nash equilibria, J. Eeon. Theory 21,
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346
KURZ AND HART
1J. J. F. NASH, Two-person cooperative games, Econometrica 21 (1953), 128-140.
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"Convex Analysis," Princeton University Press, Princeton, N. J.,
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