Competition versus cooperation in economics

Competition versus Cooperation in Economics Different Behavior Leads to Equivalent Results in Large Economics*

Alan Kirman

Economists are concerned with the way in which resources are allocated. This may not be obvious to the uninitiated as he witnesses the almost religious war- fare between the proponents of various conflicting economic "theories." In fact, there is an enormous gulf between the highly simplified models within which results are proved and the complex world to which they are applied.

How does the economic theorist proceed? Having specified the characteristics of an economy he then goes on to define situations which are "equilibria" and to examine their properties. In general economists' models are essentially static, equilibria are not considered as resting points of any adjustment process, and indeed little is said, except on an intuitive level, about how equilibria are obtained. The reader may be disappointed with the almost caricatural nature of the models discussed here. It is possible to complicate them considerably thereby making them apparently more realistic and certainly less naive. Such complica- tion changes nothing in the fundamental nature of our models however, and the reader who is shocked by their limitations wou ld be equally shocked by the more sophisticated versions. It would just take him longer to find out! Thus the mathematician who turns to economic theory out of concern for the problems of the real world is frequently disappointed.

After this precautionary preamble I can now explain the nature of the nevertheless rather remarkable result which is the subject of this article. Two apparently wholly unrelated allocation mechanisms yield in large economies the same result. One of these mechanisms, the traditional price or "competitive" market mechanism, is based on individualistic behaviour by agents wi thout communica t ion be tween them, whilst the other, the "core", is based on the idea of cooperation and coalition formation.

We shall only look at exchange economies here, that is at economies where the goods to be allocated exist already and where there is no production. The sim- plest example is the agricultural economy in which production has already taken place and where farmers now gather at the market place to exchange their prod- ucts. In the competitive framework prices would be posted for each good. These prices serve to give a value to the "bundle" (i.e., n-tuple) of goods pos- sessed by each individual; this we call his "income". Each individual then choose or "demands" the best bundle (according to his preferences) available to him. This is the best bundle of those which have a value, at the given prices, less than this income. Now if the prices are such that the total of all demands is equal to the total of all the initial bundles or resources then we have a "'competitive equilibrium". I will make all this precise in a momen t, but in passing it should be said that economists have devoted considerable energy to proving that such equilibria exist, under various assumptions, and an excellent survey is that of Debreu (1982).

Alan Kirman

* My intellectual deb t to m y fr iend Werne r H i l de nb ra nd is obvious. No b lame for this p resen ta t ion shou ld be a t t ached to h i m however and the reader requi r ing a detai led account of the p rob lem presen ted here shou ld refer to his excellent survey , H i ldenb rand (1982).

26 THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2 �9 1986 Springer-Verlag New York

Let's turn our attention now to the "core" and see how different it is from the competitive solution. If we think of the agricultural market I ment ioned above then imagine that all the goods available are put in the centre and that some allocation of these goods to individuals is proposed. Now any individual or coalition of individuals may object to this allocation. Their objection must be founded on the fact that they could redistribute their initial resources amongst themselves, in such a way as to make all of them better off than they would be with the proposed allocation. To consider a rather silly example, imagine a school picnic to which all the children bring sandwiches. If each child ate his own sandwiches all sorts of opportunities to improve the happiness of the group might be lost. So some child proposes a redistr ibution of the sandwiches. Suppose that John prefers a ham sandwich to a cheese sandwich to a pat6 sandwich and starts with a pat6 sandwich whereas Jane has exactly the opposite preferences but starts with a ham sandwich. Now suppose that in the proposed allocation both John and Jane are to receive a cheese sandwich. Clearly, neither of them alone has an objection since their initial resources are worse, from their point of view, than the proposed allocation. However together if they simply exchange their initial sandwiches they can achieve something which they both prefer to the suggested allocation. Hence they have an objection or, to use the standard terms, they can "improve upon" the proposed allocation.

We say that the core of an economy is the set of allocations which cannot be improved upon by any coalition.

Suppose that in our picnic some child, having s tudied a little economics, proposes to announce prices for different types of sandwich and then adjusts those prices till an equilibrium is achieved. Each child announces his choice of sandwiches given the value of his own at each price system called. It does not seem obvious that the solution achieved in this way would bear any relation to the allocations which make up the "'core".

In fact, to remove any suspense, I shall give the basic results the last two of which are rather surprising. Firstly any competitive allocation is in the core, secondly, in very large economies the allocations in the core are competitive, thirdly as economies become large, core allocations are "'nearly competitive". Thus the connection between the two solutions is very strong and furthermore not only do we have an ec]uivalence theorem in the limit, but also asymptotic results.

I shall now present the formal model and sketch the way in which these results are obtained.

,t~E~t37.,t/Ru77oN OF 7~/E ~R~vbuJ/Cl4E-'~'

element of R ~. We consider a set A of consumers. The preferences of each individual, denoted ~ ("preferred or indifferent"), are elements of U~mo the set of all complete preorders on R+ e satisfying the following conditions:

(i) continuity Rc+Xlx > y} and {xly > x} are closed sets for all y E

(ii) strict convexity x, y E R~+, x > y implies {oLx + (1 - o0y } > y for all oL, 0 < o~ < 1;

(iii) monotonicity x, y E Rr x > y (i.e. x i > Yi for some i and x i/> Yi for all i) implies x ~ y.

These assumptions are technical and with the excep- tion of the first can be weakened or even dropped so just three remarks are in order. The first condition un- derlines the fact that we are working in a world where goods are infinitely divisible. The second says essentially that people prefer mixtures and the third that people always prefer more of any good.

In addition to preferences people will have initial resources, i.e., a bundle of initial goods x in R+ e with which to trade. This gives us:

An exchange economy is a mapping Z : A ---> ~ s~ o x Re+

where the preferences of individual a are denoted >~a and his initial endowments e(a).

Given this extremely simple economy we can now consider the allocations of goods in the economy and in particular those which are generated by the two mechanisms I have mentioned.

The Model: An Exchange Economy

There are f goods and a bundle of goods x is an

An allocation f of goods for the economy ~ is a mapping f:A--+Rt+. An allocation is feasible if Ea~_A f(a) <<- Y'a~_A e(a).

THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2, 1986 27

Figure 1

Competitive allocations

In this context each agent will choose the best bundle available to him in his "budget set".

The budget set of agent a is given by fMa,p) = {xlp " x p " e(a)}

where the prices are always "norma l i zed" so that ~,~=lPi = 1. This normalization has no consequences in our simple exchange economy.

The demand of agent a is given by ~b(a,p) = {x in ~ (a,p) I x ~ y for all y in ~(a, p)}.

6 is, with our simplifying assumptions, a continuous function of p.

A competitive or Walrasian equilibrium for the economy is given by a price system p and a feasible allocation f

such that f(a) = rb(a,p) for all a in A.

The allocation f is referred to as a competitive allocation and the set of all competitive allocations of the economy as W(~) in honour of Walras, a pioneer mathematical economist.

Before looking at the other solution let us have a look at an ingenious geometrical device developed by Edgeworth, an English economist, and known as the Edgeworth box. It illustrates a two person two good exchange economy. The idea is simple, if we look at figure I. There the total resources of the two individuals a and b are given by e = e(a) + e(b). This fixes the "origin" for consumer b. Consumer a views the world from his origin O~ and the "equivalence classes" or " indif ference curves" of his preferences may be traced, and are given by the solid curves in figure I, the further towards the north-east the greater the satisfaction for a. Conversely b starts from origin 0 b and his satisfaction increases as he moves south-west, and

his indifference curves are represented by dot ted lines. The point e in the box represents the initial endowments of the two consumers, e(a) seen from O a and e(b) seen from 0 b. Any point (xl,x2) in the box represents a feasible allocation for the two consumers, with f(a) = (xl,x2), F(b) = e - ( X l , X 2 ) . Now for a given price system p we can draw the budget sets for the two individuals by simply tracing the line {xlp �9 x = p �9 e(a)} which passes through e. The curve below this line represents the budget set for a and above it, that for b. Individual a chooses the best point for himself in the budget set ~(a,p) as does individual b in ~(b,p). If the two points coincide we have an equilibrium, shown as 00 in figure I. There are no really plausible assumptions which will guarantee the uniqueness of such an equilibrium but this is not important here.

Before turning to the other solution, the "'core", one observation is in order. All that I have said above de- pends on a simple premise, that once prices are given, then agents calculate their demand and equilibrium will be achieved if "'total demand" equals "total supply". However, in the picnic example, a clever child would soon see that since the equilibrium prices de- pend on the individual demands, he can influence the final outcome. So long as his influence is not negli- gible, and this is clearly the case if there is a finite number of consumers, he has an incentive to cheat, for example by misrepresenting his demand in order to guide the economy to a state which he would prefer to a competitive equilibrium. But since this possibility exists for every individual the outcome is unclear. So, it is clear that "price taking" behaviour is only reason- able if there are many individuals in the economy and, in fact, is only completely logical in the case where there is an infinite number of consumers. Economists who are perfectly happy to accept the infinite divisi- bility of goods as an idealisation have however balked

28 THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2, 1986

at the idea of accepting, even as an idealisation, an economy with an infinite number of individuals.

Aumann (1964) introduced such an economy and we can define, following Hildenbrand (1982).

An Atomless Exchange Economy Z is a measurable mapping of an atomless measure space (A, ~ , v) with v(A) = 1 intO~mo x R~+ with finite mean endowments; i.e., fedv < ~. An allocation for ~ is an integrable function f: A ~ Re+ and is feasible i f fA fdV = fA edv.

Before the reader abandons in the face of this rather nasty abstraction, let me repeat that it is the only for- mulation that is logically consistent with the traditional idea of "perfect competition", i.e., of price taking behaviour.

The observant reader will have noticed the disap- pearance of the qualification of preferences as strictly convex. This is one of the gains of assuming an infinite number of consumers: competitive equilibria exist without any assumption of convexity.

In what follows, the or-algebra ~ will have a very natural interpretation.

Core-Allocations

For the moment, let us go back to the finite exchange economy and consider the other solution. First the formal definitions. Considering a coalition as a non- empty subset of A,

A coalition S of agents in A improves upon allocation f if there is an allocation g with g(a) >a f(a) for all a in S and ~ ~ g(a) = ~a ~ e(a)

and

The Core of an Economy ~ denoted C (E) is the set of all allocations that no coalition can improve upon.

Returning to the Edgeworth box it is clear that the allocations which are below the indifference curve of individual a passing through e in figure II, will be "improved upon" by a alone since his initial endowments will give him more satisfaction. In the same w a y the area above the corresponding indifference curve through e for b can be el iminated by b alone. This leaves the "lens" shaded in figure II. Nothing in the lens can be improved upon by any individual but the two individuals together can improve upon h for example by using m. However allocations such as f and g cannot be improved upon at all. The set of points preferred by a to g has no intersection with those preferred by b. This is also true of f. The allocations which cannot be improved upon by the coalition of all agents together have a long tradition in economics and are called Pareto optima. Thus in our box the core allocations are the Pareto optima between f and g.

0 b

2

,,\

\ "

\, X a !

Figure 2

A glance at the box in the example shows one thing clearly. A competitive allocation is in the core. In fact, the proof of this is so easy that I can spell it out here.

PROPOSITION The set of competitive allocations of an economy, W(•) is contained in the Core of the economy C(~).

PROOF Suppose that there is an allocation f with f E W(~') but f ~ C(~). Then there is a coalition S and an allocation g with (i) g(a) >~a F(a) for all a in S and (ii) XaE s g(a) = Eae s e(a). Since f is a Walrasian equilibrium we know that for the corresponding prices p,

p . g(a) > p . e(a)

otherwise f(a) would not be demanded. Hence

P" Y~a~ e(a) < p . Y ~ s g(a)

a contradiction with (ii).

This simply says that if one has a competitive allocation, then anything preferred to that allocation by an individual has a higher value than his income a t those prices. But since income is determined by resources the only way in which an individual or coalition could make themselves better off would be to have more resources which is ruled out by the definition of the core.

Core and Competitive Equilibria: Results

Before looking at the main and striking result, it is worth first thinking for a moment about the nature of the core as opposed to the competitive solution. The core involves the verification by all possible coalitions that they cannot do better than a given allocation. Ex- cept in very small groups this sort of organizational problem would be very difficult to handle. Thus the core would really seem to make sense only in the context of a small number of agents, whilst the Walrasian

THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2, 1986 29

equilibrium is only really acceptable for a large group. In our book (1976), Hildenbrand and I suggested that the Core, like Corelli, was suitable for small orchestras, while the Walrasian equilibrium, like Wagner, needed a large orchestra; but that were one to score Corelli for a very large orches t ra , it m igh t s o u n d more like Wagner . This caused such a f f ron t to some of our music-loving colleagues that I hesitate to repeat it here. But whether one accepts the analogy or not, what we will see now is that there is not only similarity, but identity of our two themes in large economies.

Going back to the atomless exchange economy the core is n o w de f ined precisely as before, s imply re- placing sums with integrals and not ing that o4, the (r- algebra of subsets of A, should be considered as the set of all coalitions. We then have:

THEOREM (Aumann (1964)) If s (A, ~ , v ) ~ ~mo x Re+ is an atomless exchange economy with .fedv ~ O, then

W(~) = C(E).

This result says that in an economy in which perfect competition makes complete sense, there are no other core allocations than the competitive ones. Thus even though one migh t s o m e h o w prefer the cooperat ive basis of the core, in large economies where such a solution would be difficult to implement the simple individualistic competit ion mechanism will achieve the same result.

The first reaction to this result from an economist is to ask whether this is simply a peculiarity of economies " in the l imit" , or do there exist asympto t ic results wh ich say tha t in some sense core a l locat ions are " n e a r " to compet i t ive allocations in large but finite economies. Another way of formulating this is to say that in large economies, if we have a core allocation, we can always find a price system such that only a " few" agents would find their bundle "really worse" than what they would obtain in the competitive allocation corresponding to that price system.

To measure the "'difference" between the core and the set of Walras allocations in a given economy s we give the following definition

Let 8 (C(~), W(~)) be the smallest number 8 such that for every f in C(E) there is an f' in W(~) with

~f(a) - /'(a)l ~ 8.

I could finish on an elegant note if it were true that for any increasing sequence of economies (~n) we have

~(C(G), W(G)) ~ 0.

However, it is not. But such a result does hold for a ra ther special class of economics , " rep l ica economies". Thus precisely the result above holds if we take a given finite exchange economy and replicate its members, their endowmen t s and preferences. Thus if

we replicate an economy enough the difference between core allocations and Walras allocations disap- pear. Indeed we can even, for a reason that will be immediately obvious, say that the core "shr inks" to the set of competitive allocations. A little more for- mally

If ~: A --~ C pm~ x Re+ is an exchange economy then we denote by

En: A x (1 . . . . n) ~ C~mo X Re+ ~ o

its n replica where e(a,i) = e(a,j) and >(a,i) = ~a,j), l ~ i , j ~ n .

The first remark to make is that in this context in any core allocation, identical agents receive identical bundles. That is

If f E C(~n) then f(a,i) = f(a,j) for all a in A and for l ~ i , j ~ n .

This property is extremely useful since it means that we can describe any core allocation in terms of the or iginal e c o n o m y E. Since each agen t of the same " t y p e " receives the same b u n d l e we on ly have to specify that bundle for one agent of each type and this will allow one an easy representat ion of what is going o n .

Thus the basic result is

THEOREM If ~: A --~ ~mo x Re+ is an exchange economy with Xa~Ae(a) ~ 0 anS~~ n) the associated series of replica economies then 8 (C(2n), W(~n) ) ---~ 0 as n ~ oo.

This result is a version of that which was beautifully and simply proved by Debreu and Scarf (1963) but I will do no more than give a geometric indication of h o w certain core al locat ions are e l imina ted as the economy becomes larger.

Making use of the "'equal treatment" property and going back to our two man box we observe first of all that any core allocation for the nth replica can still be represented as a point in the original box. Thus in figure III, let us look at the allocation f which is in the core of this two man economy. Now consider the twofold replica of this economy, that is where a and b have "identical twins" each with the same preferences and e n d o w m e n t s as the i r b ro thers . N o w we have four people in the economy called (a,1), (a,2), (b,1) and (b,2). Is the allocation f where fla,1) = fla,2) = f(a) and f(b,1) = fib,2) = fib) still in the core? Now form a coalition of two type a individuals and one of type b and propose the allocation g

g(a,i) = lhff(a,i) + e(a,i)) i = 1,2 g(b,1) = fib, l).

Clearly by strong convexity and as is obvious from figure III,


Figure 3

g(a,i) >(a,/) f(a,i) i = 1,2

and

g(b,1) ~(b,1)f(b,1) Is this allocation feasible for the three? Well, recalling that in the original economy f was a feasible allocation, then we have

f(a,1) + f(b,1) = f(a) + f(b) = e(a) + e(b) = e(a,1) + e(b,1)

So it is now clear that

g(a,1) + g(a,2) + g(b,1)

= 2[f(a'l) +e(a' l) +f(a'2) +e(a'2) 1 2 + f(b,1)

= f(a,1) + f(b,1) + e(a,2)

= e(a,1) + e(a,2) + e(b,1)

since/(a,1) = f(a,2) = f(a) and e(a,1) = e(a,2) = e(a).

So the allocation is feasible. This is not quite enough wi th our de f in i t ion to s h o w tha t this coali t ion improves upon f since agent (b,1) is only indifferent between his allocation g and the allocation f but, in fact, by continuity, we would take an e of one good from an agent a and give it to b leaving both a and b still better off than they were in f. So f is no longer in the core of the twofold replica of E. Indeed it is easy to show in this case that anyth ing "below" a competitive allocation will be blocked by a coalition of n agents of type a and n - 1 agents of type b for n large enough and symmetrically for the allocations "above".

This geometric idea was already seen by Edgeworth (1881), an Oxford economist, and the modern litera- ture on the subject stems from Shubik's (1959) linking of Edgeworth 's idea to the modern game theoretic notion of the "core of a game".

This result is pictured in figure IV. As I ment ioned earlier the situation is much more

Figure 4

complicated once one enters the domain of non replica economies. Here the power of measure theoretic tools becomes really apparent.

To give just an idea of how one can proceed think for a momen t of what it means for a sequence of economies with an increasing number of agents to "converge". A simple notion is to consider the "'distribution of agents characteristics" P'n for the economy ~'n which we define as

1 #~nn 1 (B) for every B C ~mo X R~+ ~n(B)- #An s c o

Supposing that we have a sequence of economies (Sn) with #A n --) % whose distributions ~n converge to ~, then in fact we can associate with (8n) a "limit

THE MATHEMATICAL INTELLIGENCER VOL. 8, N(3.2, 1986 31

economy" ~ which is an atomless exchange economy wi th d i s t r ibu t ion of character is t ics t~. Taking a sequence ~n) E C(~n) we show how we can prove that ~ ( C ( ~ n ) , W ( E n ) ) ~ 0.

Now there are three steps to the proof. One shows that

1) There is a convergent subsequence of (fn) which I shall still call (fn)" Call the limit f;

2) f is a competitive or Walrasian allocation for the economy E~;

3) There are Walras or competitive allocations f~ for ~'n such that f~ ~ f.

I have s idestepped many problems here and for full details of how, and under what assumptions general results hold the reader should refer to the excellent survey by H i ldenb rand (1982). Step 3 of the above sketch provides a major pitfall. To ensure that it works we need that the set of equilibrium prices M(~ n) converges to the set of equilibrium prices M(~ ~). For this we need to impose more structure, in particular, differential structure on the economies we consider. In particular what we need for step 3 to work is that the limit distribution of characteristics should be "regular". For a detailed account of economies wi th a differential structure see Debreu (1976) and Dieker (1982).

I have certainly given more than sufficient detail for the reader who was only interested to know w h y in large picnics one might as well use prices for sandwiches ra ther t han initiate complicated negot ia t ions and coalition formation.

Conclus ion

I will conclude with three observations. Firstly, one might ask " h o w fast" the distance between the core and the set of Walrasian equilibria converges to zero. The answer for replica economies is given by Debreu, unde r s o m e w h a t more restrictive a s sumpt ions than those used here, and is essentially that the distance goes to zero as 1In.

Secondly, I remarked that the core might not be a very desirable no t ion in large economies since the number of coalitions who had to consider all the pos- sibilities available to them is very large. A nice result of Mas Collel (1978) shows that in fact " n o n competitive" allocations in large enough economies can be improved upon by more or less half of all the possible coalitions. This of course d iminishes somewha t the force of the objection.

Thirdly, one might not like the idea that all coalitions are possible. Suppose that coalitions can only form between people who " k n o w each other". Suppose furthermore that "knowing one another" is probabilistic. In this case the core is stochastic, but even if the prob- ability that individuals know each other goes to zero

as the number of individuals becomes large we can show tha t " n o n compet i t ive" allocations are a lmost certainly no longer in the core of large enough economies. See Kirman, Oddou and Weber (1984) for details.

To s u m up then , in the context of ra ther s imple economies, we are now able to prove a very general version of Edgeworth 's 19th-century results that co- operative behaviour in the large will give the same result as the informationally less demanding competitive price mechanism.

Lest the u n w a r y shou ld construe such results as giving some extra ethical justification of the market mechanism let me just say that what we have found is only a rather stronger version of the idea that the market mechanism is, in a certain sense, "efficient" and nothing more. For example should the distribution of initial resources be, in some sense "unfai r" , there is n o t h i n g about the core or the compet i t ive mechanism which would make the final outcome any fairer. But this is another story . . . .

References

Debreu, G., (1975) "The Rate of Convergence of the Core of an Economy", Journal of Mathematical Economics, 2: 1-18.

Debreu G. (1976) "Regular Differentiable Economics". Amer- ican Economic Review, 66, pp 280-287.

Debreu, G., (1982) "The Existence of Equilibria", in K. J. Arrow and M. D. Intriligator (eds) Handbook of Mathe- matical Economics, vol II, Amsterdam, North Holland, Chap. 15.

Debreu, G., and H. Scarf, (1973) "A Limit Theorem on the Core of an Economy" International Economics Revue, 4: 235-246.

Dierker E. (1982) "Regular Economics" in Handbook of Mathe- matical Economies. Ed. K. Arrow and M. Intriligator. Vol. 2, pp 795-830. North Holland Publishing Co. Am- sterdam.

Edgeworth, F. Y., (1881) "Mathematical Physics", London, Paul Kegan.

Hildenbrand, W., (1982) "Core of an Economy" in K. J. Arrow and M. D. Intriligator (eds) Handbook of Mathe- matical Economics, Amsterdam, North Holland, Chap. 18.

Hildenbrand W. and A. P. Kirman (1976) "Introduction tO Equilibrium Analysis", North Holland Publishing Co. Amsterdam.

Kirman, A. P., C. Oddou, and S. Weber (1984) "Stochastic Communications and Coalition Formation" Southern European Economics Discussion Paper, Bilboa, forth- coming in Econometrica.

Mas Collel, A. (1978) "A Note on the Core Equivalence The- orem: How Many Blocking Conditions are there?" Journal of Mathematical Economics, 54: 207-215.

Shubik, M., (1959) "Edgeworth Market Games", in: R. D. Luce and A. W. Tucker (eds) Contributions to the theory of Games IV., Annals of Mathematical Studies vol. 40, Princeton University Press, 267-278.

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Competition versus cooperation in economics