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# A Social Equilibrium Existence Theorem

Communicated by J. von Neumann, August 1, 1952

In a wide class of social systems each agent has a range of actions among which he selects one. His choice is not, however, entirely free and the actions of all the other agents determine the subset to which his selection is restricted. Once the action of every agent is given, the outcome of the social activity is known. The preferences of each agent yield *his* complete ordering of the outcomes and each one of them tries by choosing his action in his restricting subset to bring about the best outcome according to his own preferences. The existence theorem presented here gives general conditions under which there is for such a social system an equilibrium, i.e., a situation where the action of every agent belongs to his restricting subset and no agent has incentive to choose another action.

This theorem has been used by Arrow and Debreu^{2} to prove the existence of an equilibrium for a classical competitive economic system, it contains the existence of an equilibrium point for an *N*-person game (see Nash^{8} and Section 4) and, naturally, as a still more particular case the existence of a solution for a zero-sum two-person game (see von Neumann and Morgenstern, Ref. 11, Section 17.6).

In Section 1 the topological concepts to be used are defined. In Section 2 an abstract definition of equilibrium is presented with a proof of the theorem. In Section 3 saddle points are presented as particular cases of equilibrium points and in connection with the closely related MinMax operator. Section 4 concludes with a short historical survey of results about saddle points, fixed points for multi-valued transformations and equilibrium points.

Only subsets of finite Euclidean spaces will be considered here.

## 1. Topological Concepts

Two sets in *R*^{n} are said to be *homeomorphic* when it is possible to set up between them a one-to-one bicontinuous (*h* and *h*^{−1} continuous) correspondence *h* (called a homeomorphism).

A *convex cell C* in *R*^{n} is determined by *r* points *z*^{K}(*k* = 1, …, *r*); it is the set

The product of two convex cells *A* be generated by the *p* points *B* by the *q* points *C* the convex cell in *R*^{l+m} generated by the *pq* points (*x*^{i}, *y*^{j}). Obviously

A *geometric polyhedron* is the union of a finite number of convex cells in *R*^{n}. It is clearly closed.

The product of two geometric polyhedra *P*, *Q* is a geometric polyhedron. Let *A*_{i} (the *B*_{j}) are convex cells in *R*^{l} (in *R*^{m}). The relation

A *polyhedron* is a set in *R*^{n} homeomorphic to a geometric polyhedron (called geometric antecedent of the first one).

The product of two polyhedra is a polyhedron (since it is homeomorphic to the product of the two geometric antecedents).

Let *Z* of *R*^{n} is said to be *contractible*, or more precisely, deformable into a point *z*^{0} *ϵ Z*, if there exists a continuous function *H*(*t, z*) (called a deformation) taking *I × Z* into *Z* such that for all *z ϵ Z, H*(0, *z*) = *z* and *H*(1, *z*) = *z*^{0}.

The product of two sets *x*^{0}, *y*^{0}).

Finally the real function *completed real line* (which can naturally be defined directly^{5}).

## 2. Equilibrium Points

Let there be *ν* agents characterized by a subscript *ι* = 1, …, *ν*.

The *ι*th agent chooses an action *a*_{ι} in a set *ν*-tuple of actions (*a*_{1}, …, *a*_{ν}), denoted by *a*, is an element of *ι*th agent is a function *f*_{ι}(*a*) from

Denote further by *ν −* 1)-tuple (*a*_{1}, …, *a*_{ι−1}, *a*_{ι+1}, …,*a*_{ν}) and by *ι*th agent is restricted to a *non-empty*, *compact* set *ι*th agent chooses *a*_{ι} in *a*_{ι} on

This background makes the following formal definition intuitive:

*Definition a** *is an equilibrium point if for all* *and*

The *graph* of the function

Theorem. *For all ι* = *1*, *…*, *ν, let* *be a contractible polyhedron*, *a multi-valued function from* *to* *whose graph G*_{ι} *is closed, f*_{ι} *a continuous function from G*_{ι} *to the completed real line such that* *is continuous. If for every ι and* *the set* *is contractible, then there exists an equilibrium point.*

The proof uses as a lemma a particular case of the fixed point theorem of S. Eilenberg and D. Montgomery^{6} or of the even more general result of E. G. Begle.^{3}

Let *Z* be a set and *ϕ* a function associating with each *z ϵ Z* a subset *ϕ*(*z*) of *Z*. We have defined above the graph of *ϕ* as the subset of *ϕ* is said to be *semicontinuous* if its graph is closed. A *fixed point* of *ϕ* is a point *z** such that

Lemma. *Let Z be a contractible polyhedron and* *a semicontinuous multi-valued function such that for every z ϵ Z the set ϕ*(*z*) *is contractible. Then ϕ has a fixed point*. †

*ν* contractible polyhedra, is a contractible polyhedron (Section 1). Define on *ϕ* as follows:*ι* and *ϕ* is semicontinuous.

For this first define in

The equivalent definition

The graph

Consider the subset of

The conclusion of the lemma is then that there exists

The requirement that *joint* requirement on the two functions *Remark* tries to overcome this.

The function *continuous* at *n*,

### Remark:

*If* *has a compact graph* *and is continuous at* *, if* *is a continuous function from* *to the completed real line, then* *is continuous at*

We drop subscripts *f* took its values in the real line (the isomorphism

*Using only the compactness of G and the continuity of f we first prove:* For any sequence *ϵ* > 0, there is an *N* such that *n* > *N* implies

For every *n*, choose *a*^{n} *ϵ*

By the continuity of *f*,

*Using in addition the continuity of* *at* *we prove:* For any sequence *N* such that *n* > *N* implies

Choose *n*, *f*, *N* such that *n* > *N* implies

## 3. Saddle Points and MinMax Operator

In this section *X* × *Y* to the completed real line. A *saddle point* of *f* is a point

Corollary. *Let X, Y be two contractible polyhedra, and* *a continuous function from X × Y to the completed real line. If for every* *is contractible and for every* *is contractible, then* *has a saddle point.*

This corollary contains as more and more particular cases the saddle point theorems of Kakutani,^{7} von Neumann (Ref. 9, p. 307, and 10), and von Neumann and Morgenstern (Ref. 11, Section 17.6).

The special interest of saddle points comes from their intimate relation with the MinMax operator.

*From now on X, Y are assumed to be compact and* *to be continuous.*

We know from the Remark that *is a continuous function of x* [resp., *y*]. The following results, already given in Ref. 11, Section 13, are proved here for completeness.

(*a*)

Let

If

(*b*) *The existence of a saddle point* *implies the equality*

From the definition 1 it follows that*a*) gives the result. It also gives

(*c*) *The equality* *implies the existence of a saddle point.*

Assume that the equality holds and take

(*d*) *the set of saddle points is either empty or equal to A × B.*

## 4. Historical Note

A function *Z* to the completed real line *quasi-convex* (resp., *quasi-concave*) if for any

Let

In his first study on the theory of games, J. von Neumann^{9} proved:

(I). *Let* *be a continuous real-valued function for* *and* *. If for every* *the function* *is quasi-convex, and if for every* *the function* *is quasi-concave, then f has a saddle point.*

In another paper on economics^{10} he later proved a closely related lemma which S. Kakutani^{7} restated in the more convenient form of the following (equivalent) fixed point theorem:

(II). *Let Z be a compact convex set in* *and* *a semicontinuous multi-valued function such that for every* *the set* *is non-empty and convex. Then* *has a fixed point.*

The convexity assumptions were, however, irrelevant and S. Eilenberg and D. Montgomery^{6} gave a fixed point theorem where convexity was replaced by acyclicity. Their result was further generalized by E. G. Begle.^{3}

These last two theorems deserve particular attention as valuable contributions to topology whose origin can be traced directly to economics.

The notion of an equilibrium point was first formalized by J. F. Nash^{8} in the following game context. There are *s** such that for all

## Acknowledgments

One of the main motivations for this article has been to lay the mathematical foundations for the paper by Arrow and Debreu;^{2} in this respect I am greatly indebted to K. J. Arrow. Acknowledgment is also due to staff members and guests of the Cowles Commission and very particularly to I. N. Herstein and J. Milnor. I owe to J. L. Koszul and D. Montgomery references 6 and 3. Finally I had the privilege of' consulting with S. MacLane and A. Weil on the contents of Ref. 6.

## Footnotes

↵* Based on two Cowles Commission Discussion Papers, Mathematics 412 (Nov. 1, 1951) and Economics 2032 (Feb. 11, 1952). This paper has been undertaken as part of the project on the theory of allocation of resources conducted by the Cowles Commission for Research in Economics under contract with The RAND Corporation. To be reprinted as Cowles Commission Paper, New Series, No. 64.

↵† The statement of E. G. Begle (Ref. 3, p. 546) is indeed much more general and the existence theorem can accordingly be generalized. Instead of a contractible polyhedron one might take for example an Absolute Retract (as defined in [Ref. 4, p. 222]) using the fact that the product of two A.R. is an A.R. [Ref. 1, p. 197]. For finite dimensions “Absolute Retract” is equivalent to “contractible and locally contractible (Ref. 4, pp. 235–236) compact metric space” [Ref. 4, p. 240].

## References

- ↵Aronszajn, N., and Borsuk, K., “Sur la somme et le produit combinatoire des rétractes absolus,”.
*Fundamenta Mathematicae*,**18**, 193–197 (1932) - ↵Arrow, K. J., and Debreu, G., “Existence of an Equilibrium for a Competitive Economy,”.
*Econometrica*, in press (1953) - ↵Begle, E. G., “A Fixed Point Theorem,”.
*Ann. Math*.,**51**, No. 3, 544–550 (May, 1950) - ↵Borsuk, K., “Über eine Klasse von lokal zusammenhängenden Räumen,”.
*Fundamenta Mathematicae*, Vol. 19 (1932), p. 220–242 - ↵Bourbaki, N.,.
*Eléments de Mathématique*, Première partie, Livre III, Chap. IV, §4, Hermann, Paris, 1942 - ↵Eilenberg, S., and Montgomery, D., “Fixed Point Theorems for Multi-valued Transformations,”.
*Am. J. Math*.,**68**, 214–222 (1946) - ↵Kakutani, S., “A Generalization of Brouwer’s Fixed Point Theorem,”.
*Duke Math. J*.,**8**, No. 3, 457–459 (September, 1941) - ↵Nash, John F., “Equilibrium Points in N-Person Games,” Proc. Natl. Acad. Sci..
**36**, 48–49 (1950) - ↵Neumann, J. von, “Zur Theorie der Gesellschaftsspiele,”.
*Math. Ann*.,**100**, 295–320 (1928) - ↵Neumann, J. von, “Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,”.
*Ergebnisse eines Mathematischen Kolloquiums*,**8**, 73–83 (1937), (translated in*Rev. Economic Studies, XIII*, No. 33, 1–9 (1945–46) - ↵Neumann, J. von, and Morgenstem, O.,.
*Theory of Games and Economic Behavior*, 2nd ed., Princeton University Press, Princeton, 1947 (1st ed., 1944)

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