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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.

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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 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