REGIONAL SCIENCE ASSOCIATION:

PAPERS, X V i l I , V I E N N A CONGRESS, 1966

GAME THEORY, LOCATION THEORY AND INDUSTRIAL AGGLOMERATION*

by Walter Isard**

A c l a s s i c p r o b l e m in L o c a t i o n T h e o r y a n d R e g i o n a l Science is i n d u s t r i a l a g g l o m e r a t i o n . A l f r e d W e b e r first posed this p r o b l e m n e a t l y in 1909 in his m a j o r b o o k on t h e l o c a t i o n o f i n d u s t r y . 1 T h e r e he p o i n t e d l y d i s c u s s e d t h e prob l e m of a g g l o m e r a t i n g s e v e r a l p l a n t s p r o d u c i n g a h o m o g e n e o u s c o m m o d i t y . M o r e r e c e n t l y , t h e p r o b l e m h a s e m e r g e d in a b r o a d e r c o n t e x t , w h i c h I find p a r t i c u l a r l y i n t e r e s t i n g - - n a m e l y , t h e f o s t e r i n g in a c o n s i s t e n t a n d efficient m a n n e r of t h e d e v e l o p m e n t of a n i n d u s t r i a l a g g l o m e r a t i o n in e a c h of s e v e r a l r e g i o n s of a nation, or in e a c h of s e v e r a l n a t i o n s of a c o m m o n m a r k e t s y s t e m . T h e p r o b l e m in a c o m m o n m a r k e t f r a m e w o r k is e s p e c i a l l y f a s c i n a t i n g , a n d of course is in k e e p i n g w i t h t h e p a r t i c u l a r a t t e n t i o n w h i c h w e h a v e b e e n g i v i n g to c o m m o n m a r k e t issues in o u r r e c e n t E u r o p e a n R e g i o n a l Science C o n g r e s s e s . If I m a y , let m e b e g i n w i t h t h e s u c c i n c t s t a t e m e n t of this W e b e r i a n p r o b l e m c o n t a i n e d in m y b o o k on Location and Space-Economy. 2 C o n s i d e r i n g s e v e r a l u n i t s of p r o d u c t i o n p r o d u c i n g a n i d e n t i c a l c o m m o d i t y , W e b e r a s k s u n d e r w h a t c o n d i t i o n s a n d w h e r e will t h e y a g g l o m e r a t e . H e p r o v i d e s p r e c i s e a n s w e r s to t h e s e questions. S e v e r a l i n d i v i d u a l u n i t s of p r o d u c t i o n will a g g l o m e r a t e w h e n (in r e l a t i o n to a n y a s s u m e d u n i t of a g g l o m e r a t i o n ) : (1) t h e i r c r i t i c a l i s o d a p a n e s 8 i n t e r s e c t a n d (2) t o g e t h e r t h e y a t t a i n w i t h i n t h e c o m m o n s e g m e n t t h e r e q u i s i t e q u a n t i t y of p r o d u c t i o n . S u p p o s e t h r e e u n i t s of p r o d u c t i o n , PI, P2, a n d Ps, e a c h t r a n s p o r t - o r i e n t e d , a r e l o c a t e d as in F i g . 1. A r o u n d e a c h a r e d r a w n its locational figure a n d c r i t i c a l * The author is grateful to the National Science Foundation for financial support of his research. This paper draws heavily upon research jointly conducted with Tony E. Smith, to be presented elsewhere in much more comprehensive and systematic fashion. ** The author is associated with the Regional Science Research Institute and the Department of Regional Science, Wharton School, University of Pennsylvania, Philadelphia, U.S.A. Uber den Standort der Indust~ien, Tfibingen, 1909, English translation with introduction and notes by Carl J. Friedrich, Alfred Weber's Theory of the Location of Industries, Chicago, 1929. 2 Massachusetts Institute of Technology Press, Cambridge, Mass., U. S. A., 1956, pp. 176182. 8 In this connection the critical isodapane for any unit of production is that locus of points for each of which transport costs in assembling the raw materials and shipping the finished product exceed the corresponding transport costs associated with the optimal transport point by a constant amount. This amount is equal to the economies of agglomeration that would be realized by association with the assumed unit of agglomeration. See Weber for extensive discussion of the critical isodapane and of its dependence upon locational weight, transport rates, the function of economy of agglomeration, and other variables.

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REGIONAL SCIENCE ASSOCIATION: PAPERS, XVIII, EUROPEAN CONGRESS, 1966

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ISARD: GAME THEORY, INDUSTRIAL AGGLOMERATION

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isodapane. The critical isodapanes do not intersect. Agglomeration is infeasible. In contrast stands the situation depicted by Fig. 2 where these same three units are assumed initially to lie closer to one another. Here, their critical isodapanes, the heavy dashed circles, do intersect. (For the present, ignore the dotted circles.) A la Weber, agglomeration will take place at a site within the common segment which is shaded. Weber's determination of the center of agglomeration is as precise as his statement of conditions under which agglomeration will occur. The center of agglomeration "will be located at that one of the several possible points of agglomeration which has the lowest transportation costs in relation to the total agglomerated output. ''4 This point is derived by means of a locational figure and analysis of the equilibrium of forces in much the same way as is the optimal transport point for any given unit of production. However, in the derivation of this point, Weber permits the use of new sources of raw material supplies (replacement deposits) for each unit of production. Weber gives a precise answer also to the question of the size of the unit of agglomeration to which each unit of production will be attracted. Each unit of production will select that unit of agglomeration whose center lies most distant from the relevant critical isodapane of the given unit of production. Weber's analysis is not unsophisticated. He does consider for each unit of production a function of economy of agglomeration which varies with the size of agglomeration. He admits exceptions to his conditions under which agglomeration will be precipitated. 5 He emphasizes labor locations as centers of agglomerations, where both cheap labor and agglomeration economies are obtainable, and introduces various realities into his analysis. Nonetheless it must be said that Weber's schema has limited application, especially in understanding the forces which determine the site at which agglomeration obtains in actuality. Imagine an entrepreneur who controls three units of production and who confronts the location problem, de novo. Considering the locational polygon of raw material sources and markets relevant for each unit and assuming that economies of scale are not operative, he could locate each unit at its optimal transport point. Or, he could locate the three units adjacent to each other at a center of agglomeration, thereby achieving localization economies but only by incurring larger transportation costs. This is one type of situation to which Weber's schema has most application. In this type of situation, each unit of production may be visualized as substituting transport outlays for production outlays of one sort or another when it shifts to the center of agglomeration. However, as Engl~inder and Palander have rightly indicated in their sharp criticism of Weber's agglomeration theory, this type of situation is not widely characteristic of reality. Societal development is an historical process. At any given point of time there exists an inherited physical structural framework. 4 Friedrich, op. cir., p. 138. 5 For example, the critical isodapane of a given unit of production may not quite reach the common segment formed by the intersection of the critical isodapanes of other units. Nevertheless, if the given unit's procuction is necessary for the group to attain the requisite total of production and if other units would enjoy sizable economies from agglomeration, the given unit of production can be induced to shift to the potential center of agglomeration by some form of subsidy or side payment.

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REGIONAL SCIENCE ASSOCIATION: PAPERS, XVIII, EUROPEAN CONGRESS, 1966

Plants have already been erected and are producing. To relocate these plants i n v o l v e s o p p o r t u n i t y costs, since one w o u l d f o r e g o the use of f a c i l i t i e s forced into obsolescence. C r i t i c s of W e b e r h a v e t h e r e f o r e e m p h a s i z e d t h e a d v a n t a g e s of e x i s t i n g p r o d u c t i o n p o i n t s as c e n t e r s of a g g l o m e r a t i o n , w h e t h e r t h e y reflect l a b o r o r a n y o t h e r f o r m of o r i e n t a t i o n . A s n e w u n i t s of p r o d u c t i o n c o m e into existence, t h e y w i l l t e n d to g a i n localization e c o n o m i e s b y a g g l o m e r a t i n g a r o u n d e s t a b l i s h e d p r o d u c t i o n points. M o r e i m p o r t a n t , e v e n if t h e o p p o r t u n i t y c o s t s of r e l o c a t i o n could be i g n o r e d a n d p l a n t s w e r e c o m p l e t e l y m o b i l e , t h e p r o b l e m is not as s i m p l e as W e b e r d e p i c t e d . I n s h i f t i n g to a c e n t e r of a g g l o m e r a t i o n , it is to t h e a d v a n t a g e of e a c h u n i t of p r o d u c t i o n to d e v i a t e as little as possible f r o m its o p t i m a l t r a n s p o r t site. A t t h e s a m e t i m e , t h e m a n a g e r s of t h e s e u n i t s of p r o d u c t i o n differ in b a r g a i n i n g a b i l i t y . T h e r e f o r e , it is to b e e x p e c t e d t h a t t h e c e n t e r of a g g l o m e r a t i o n will n o t b e a t t h e over-all m i n i m u m t r a n s p o r t c o s t p o i n t of a n e w over-all loc a t i o n a l polygon; r a t h e r , it will t e n d to lie w i t h i n t h e c o m m o n s e g m e n t c l o s e r to t h e firms w i t h g r e a t e r b a r g a i n i n g a b i l i t y . I t could e v e n lie a t a p o i n t outside t h e c o m m o n s e g m e n t if a n a p p r o p r i a t e s e t of side p a y m e n t s w e r e m a d e to firms w h o could n o t o t h e r w i s e be i n d u c e d to a g g l o m e r a t e . A n d , if costs of r e l o c a t i o n a r e r e i n t r o d u c e d into t h e p r o b l e m , t h e c e n t e r of a g g l o m e r a t i o n could lie a t t h e site of a n a l r e a d y e x i s t i n g p r o d u c t i o n point. Since this w o u l d e l i m i n a t e one g r o u p of r e l o c a t i o n costs, in m a n y s i t u a t i o n s e a c h u n i t of p r o d u c t i o n could be m a d e b e t t e r off t h r o u g h a n a p p r o p r i a t e s e t of side p a y m e n t s t h a n if all w e r e to s h i f t to W e b e r ' s over-all t r a n s p o r t o p t i m a l point. C l e a r l y , g a m e t h e o r y s t r i k e s a t t h e h e a r t of this l a t t e r t y p e of s i t u a t i o n . T h e s e v e r a l p a r t i c i p a n t s a r e t h e s e v e r a l u n i t s of p r o d u c t i o n . W h e t h e r t h e y be n e w u n i t s w i t h w h o m no r e l o c a t i o n costs a r e a s s o c i a t e d o r e x i s t i n g u n i t s c o n f r o n t e d w i t h r e l o c a t i o n costs, t h e y i n t e r a c t e n g a g i n g in v a r i o u s f o r m s of collusive action. T h e b a r g a i n i n g w h i c h e n s u e s is c o m p l i c a t e d not o n l y b e c a u s e of the i n n u m e r a b l e coalitions w h i c h a r e possible b u t also b e c a u s e of the d i f f e r e n t scales of a g g l o m e r a t i o n w h i c h a r e p o t e n t i a l l y feasible for e a c h u n i t of p r o d u c t i o n . 6'7 6 To spell out somewhat more the way in which game theory pertains to this phase of agglomeration theory, imagine there are three units of production (parties) placed as in Fig. 2. Their critical isodapanes intersect with respect to two sizes of agglomeration. (We already oversimplify the problem by considering only two sizes.) The critical isodapanes relevant for the smaller unit of agglomeration are the dotted circles; those relevant for the larger unit are the dashed circles. Any two parties could agglomerate to form the smaller unit of agglomeration. The third party would consequently gain nothing. It is therefore to his advantage to encourage the formation of the larger unit of agglomeration in which he could participate and from which he could reap gain. Leaving aside the determination of which party is the third party, we encounter the problem of identifying types of collusive actions which might develop. Whom will the third party approach to form a coalition ? To make an effective approach he must offer a gain to the cooperating (second)party which will be greater than what the latter obtains in the smaller unit of agglomeration. The third party may offer a side payment. Or he may propose to agglomerate at a site closer to the second party's initial location (optimal transport point); this proposal may, or may not, be contingent upon the participation of the remaining (first) party. Or the second party may be strong enough to force agglomeration at his own optimal transport poin% provided the first and third parties reap gain either directly or indirectly through side pay(cont'd next page)

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A g a i n s t this b a c k g r o u n d of W e b e r i a n t h e o r y and criticism, let m e proceed to the h e a r t of this paper. I m a g i n e three regions, A, B a n d C, each d e s i r i n g to p r o m o t e a n i n d u s t r i a l a g g l o m e r a t i o n . A m o n g o t h e r activities, w e m a y i m a g i n e t h a t each proposes a steel p l a n t for its a g g l o m e r a t i o n . H o w e v e r , i t is clear t h a t o w i n g to large scale a n d localization economies it w o u l d be m u c h m o r e desirable f r o m a n overall s t a n d p o i n t for the p r o d u c t i o n of steel i n g o t a n d other basic steel p r o d u c t s to be c o n c e n t r a t e d at one site. B u t w h i c h site ? Each r e g i o n w o u l d like the steel i n d u s t r y a t its i n d u s t r i a l district, a n d each is n o t w i l l i n g to yield to the other. Too f r e q u e n t l y , the r e g i o n s d o n ' t agree on a c o m p r o m i s e proposal and location, j u s t as m a j o r p o w e r s c a n ' t a g r e e on a full n u c l e a r test b a n . T h a t is, too f r e q u e n t l y the o u t c o m e c o r r e s p o n d s to a deadlock or s t a l e m a t e position w h e r e each r e g i o n is o p e r a t i n g a small, g r o s s l y inefficient steel mill a t l a r g e financial loss. A s a r e g i o n a l s c i e n t i s t a n d c e n t r a l p l a n n e r , I a m a w a r e of the likelihood t h a t this h i g h l y u n d e s i r a b l e s t a l e m a t e will m a t e r i a l i z e . I a m m o t i v a t e d to i n d u c e the r e g i o n s s o m e h o w to b r e a k the s t a l e m a t e a n d m o v e to a n i n t e g r a t e d a n d efficient i n v e s t m e n t a n d p r o d u c t i o n p l a n w h i c h will lead to large increases i n the Gross P r o d u c t s of the r e g i o n s - - i n c r e a s e s w h i c h m a y be considered " f a i r and e q u i t a b l e " b y all. 6 (cont'd) merits. However, the first party cannot be presumed to be an inactive participant. His power, like the power of any of the other two parties, rests in the fact that without his cooperation the additional gains of the larger unit of agglomeration are not possible. He too has bargaining power and can be presumed to exercise it. Costs of relocation complicate the problem still more by altering the probabilities of diverse moves. They significantly affect the range of collusive action. Furthermore, the problem as presented is not a constant-sum game. As Weber demonstrates, there is a center, the over-all optimal transport point, at which agglomeration can proceed with a minimum addition to the sum of the transportation costs of all parties. Any deviation from this point reduces the "surplus" or "net gain" to be apportioned among the participants. In certain situations it may therefore be useful to introduce a fourth participant, a dummy, in order to convert the problem into a constant-sum or zero-sum game. This entails further complexities, as well as does any variation from the symmetrical situation presented, such as with respect to initial geograpMc positions, size of output of each unit of production, ability to relocate as measured by opportunity costs, and so forth. 7 From an entirely different standpoint, however, Weber's agglomeration theory may be justly defended. Suppose a new area is to be opened for development by a governmental planning authority. Technological and other factors dictate, for any given commodity, the range of feasible scales for the units of production. Should these units be agglomerated to realize localization economies, or should they be spatially disconnected in order to reduce transportation costs ? From this social welfare approach, irrationalities and differences among managers in bargaining ability do not enter the problem. Nor do inherited physical structures. The localization economies achievable at Weber's over-all transport optimal point (and not at any other point) must be compared with the additional transport outlays occasioned by agglomeration at this point. Moreover, this social welfare approach implicit in Weber, though not generally realistic, provides a useful guidepost; in certain contexts it can indicate directions in which existing structure should be transformed in order to approach optimum resource utiIization. Hence, from these standpoints, too, the Weberian agglomeration theory is relevant, and likewise the substitutional locational framework within which it fits.

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REGIONALSCIENCE ASSOCIATION: PAPERS, XVIII, EUROPEAN CONGRESS, 1966

I make this point clear to the several regional planning authorities, and also make crystal clear that I shall propose that each regional planning authority makes moves, which any other regional planning authority has full power to veto at any time. In order m obtain the proper psychological mood for cooperation, I reiterate: A t any time in the scheme which I propose each regional planning authority has unrestricted power to exercise a veto, and, without cost, to bring everyone back to the initial deadlock point i f it so desires. I am not much of a psychologist. Also, I am aware that psychologists like most social scientists know very little about the complex motivation that may be behind actions chosen by regional planning authorities. However, as a central planner I must make suggestions which seem good to me, even though I cannot scientifically prove that they are optimal. So without a scientific proof, allow me to put forth the following alternatives to the regional planning authorities of A, B and C. Alternative 1. From m y studies of each region's plan for a steel mill, I judge that each region will lose $1 MM per year for the situation where each produces steel when compared with the situation where the production of all steel is concentrated at an intermediate site D. (I assume D is a port, and in all other respects qualifies as a prime site for large-scale steel production.) 8 Therefore, the first alternative to keep in mind is the attainment of a compromise joint action where an integrated steel works is erected and operated at D, and where each region has investment funds equal to the capitalized value of an annual income of $1 MM available for the development of steel fabrication and other industrial facilities. Alternative 2. From m y studies, I judge that if a single, large-scale steel works were erected and operated at any one of the three sites proposed by A, B and C respectively (with no steel production at the other two), the loss of $1 MM annually for each region would be avoided, and in addition there would be on other costs annual savings of $ 50,000. 9 Thus, there would be available investment funds equal to the capitalized value of $ 3.5 MM annual income that might be allocated to the two regions not producing steel. Such funds might be available for development of steel fabrication and other industry in these two regions. Alternative 3. Recognizing that there are indirect as well as direct gains from the development of major industrial facilities such as steel--gains not only in employment and income, but also in desirable economic activities sparked by steel production-and recognizing that sophisticated regional planning authorities may think in these terms, other investment funds may be applied to supplement the investment funds equal to the 8 Of course, as a regional scientist and central planner, I would need to fully develop the information base to support my statements. 9 I assume that the delivered price of steel would be set at the same level in all three regions.

ISARD: GAME THEORY, INDUSTRIAL AGGLOMERATION

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capitalized value of $ 3.5 MM annual income--specifically to three industrial complexes: (1) a steel-steel fabricating complex for one region, (2) an oil refining-petrochemicals complex in one of the two regions not engaging in steel production, and (3) a university-research and development-space-age industry complex in the other. Already, I have set down three alternatives, each of which requires considerable deliberation and analysis. To avoid making the problem too difficult and confusing, I suggest no further alternatives. Now, how to induce the regions to make a series of proposals, the end result of which will be one of these alternatives (or some equivalent) ? Of course, if one alternative appeals to all three in a manner which involves a feasible joint action--for example, where A wants steel, B chemicals and C research and development--then the toughest part of m y assignment is handled. However, suppose not. I suggest one of several procedures falling in a class of ineremax procedures. This procedure is an alternating leader-follower procedure. (It was first outlined in the Cracow Peace Research Conference last year in connection with the disarmament problem. 1~ [ employ it here again, rather than another procedure such as split-the-difference, because it is easier to develop in non-mathematical form. The full development of the incremax procedures is contained in another f o r d coming manuscript with T o n y E. Smith.) I ask each of the three regions to agree to the following rules: Rule (1). In making proposals designed to move away from the deadlock point, the (0, 0) point, no region shall put forth a proposal which will make any other region worse off. To illustrate this rule we must first digress and consider the structure of the hypothetical case of Fig. 3 with only two participants A and B. Let the origin represent the deadlock point, the (0, 0) point, for each of the two regions. Let us imagine that we can measure A ' s action (in terms of planned increases or decreases in steel production, petrochemicals, and research and development, transfer of funds to other regions for development, etc.) along the horizontal; and B's actions along the vertical, u Let us consider A ' s set of iso-Gross Product curves (the solid curves of Fig. 3) where each iso-Gross Product indicates the various combinations of actions of A and B which would achieve the same Gross Product for A. For example, the solid curve labeled $ 2 MM corresponds to the locus of joint actions yielding $ 2 MM annual income. B also has a set of iso-Gross Product curves which are the dashed curves. The first rule therefore states that on the first move when A and B are at the (0, 0) point, and where A and B receive annual incomes of $ 2 MM and $2.5 MM respectively, A must not propose any joint action representable by a point lying above the dashed $2.5 MM curve, for that joint action would 10 W. Isard and Tony E. Smith, "A Practical Application of Game Theoretical Approaches to Arms Reduction," Papers, Peace Research Society (International), Vol. IV, 1966. Also, see W. Isard and Tony E. Smith, "On the Resolution of Conflicts Among Regions of a System," Papers, Regional Science Association, Vol. XVII, 1966. 11 Since these actions involve many independent components, it is impossible to represent them along a single dimension. Thus Figure 3 cannot have any realistic counterpart. It is only suggestive.

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REGIONAL SCIENCE ASSOCIATION: PAPERS, XVIII, EUROPEAN CONGRESS, 1966 $1.5MM ~

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/ FIGURE 3 m a k e B worse off. And the rule also states that B m u s t not propose any joint action which lies below the undashed $ 2 MM curve, for such an action would make A worse off. T h e dotted area between these two curves m a y then be defined as the mutual improvement set. Rule (2). Each region m a y move as slowly as it desires toward a compromise solution. Each region is allowed to be as conservative as it wishes to be in taking steps effecting the cooperation. Rule (2) recognizes that one or more regions m a y consider the possibility of "excessively large" changes as an undesirable feature of an i m p r o v e m e n t procedure. Specifically, in our two-region situation, even though large changes in actions by each region are estimated to lead to significant increases in annual income of each region, any given region m a y anticipate that these large changes increase the risk of being unwittingly " o u t m a n e u v e r e d " by the other region and increase the possibility of making big mistakes rather than small ones. Moreover, the regions m a y place much greater confidence in their estimates of the effects of small changes than in their estimates for large changes. T h u s rule (2) states that on each move each region has the power to limit to any degree the amount of change in steel capacity and other action components to be proposed by any region. Put otherwise, rule (2) states that on any move no region can propose a joint action which does not lie in a square box the length of whose side is the smallest m a x i m u m allowable change that is considered acceptable by any region for that move. Together, rules (1) and (2) require that any joint action proposed by a region lie in the intersection of the box and the football--the densely dotted area of Figure 3. Rule (3). T h e regions shall randomly determine (say, by drawing lots) the order in which each shall serve as leader and the others as followers in the

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sequence of moves. Rule (4). The regions shall continue to make moves, alternating the roles of leader and follower in the order determined under rule (3), until they all are proposing the same joint action, that is until no one can make any further improvement given the first three rules. In short, the regions continue to make proposals untilL they reach (or are insignificantly distant from) some point S on the efficiency :frontier, depicted by arc EE t on Figure 3. If in the application of these rules, region A wins on the drawing and is the leader on the first move, A will propose the joint action A ' of Figure 3. With A p as the new reference point, a new improvement set is defined as well as a new box in accord with rule (2). B serves as leader on the second move and proposes that joint action in the intersection of the new improvement set and the new box which is optimal for B. B's proposal than serves as reference point for the third move on which A once again is leader. And so forth. Returning to our three-region case, 12 it is difficult for me to specify for any situation what joint action any region ought to propose on any move, or is likely to propose. Nor can I with any degree of confidence specify what the final solution might be. However I do have a certain degree of confidence that some cooperative solution can be achieved if the regions are properly informed of the possibilities. It might be that on the first move, the region which is designated by the random lottery to be leader proposes that (1) it expand the capacity of the steel mill in its plan and (2) that the two others contract their capacities with grants of investment funds so as to keep them on their same iso-gross product curves. But it might very well be that on this first move region A, the leader, proposes that one of the other two regions expand steel capacity while A and the third region contract. If on the first move A proposes steel expansion in its region, it might well be that C, who is randomly determined to be the second to serve as leader, m a y well suggest petrochemical expansion in its region. This expansion would be based upon (1)investment funds already assigned to C by A on the first move, (2) funds released to C by the contraction of its steel plant proposed both by A on the first move and by C itself on the second move, and (3) funds released by A and B because both these regions will prosper from C's petrochemical expansion. Region B, the leader on the third move, m a y have found one or both of the above proposals undesirable and m a y have already vetoed them. But if A and C have been careful, B m a y not be motivated to exercise the veto. And now it is B's turn to lead and take advantage of the opportunity to increase its Gross Product. B m a y propose taking an initial step in the construction within its domain of a university-research and development-electronics-space-age-industry complex, or a food-processing complex. B has abundant opportunity to amass capital for these purposes on account of (1) investment funds released by the already-proposed reductions in the capacity of its planned steel facilities, (2) funds which B can ask A to transfer to B's account because B m a y propose that A be assigned his contracted steel capacity, thus enabling A to remain at the lz I wish to reiterate that while Figure 3 illustrates in essence the leader-follower procedure, it fails, because of the limitations of diagrammatic analysis, to correspond to our particular problem in many respects.

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REGIONAL SCIENCE ASSOCIATION: PAPERS, XVIII, EUROPEAN CONGRESS, 1966

same welfare level after effecting the large transfer of investment funds to B, (3) funds transferred to B by C because part of B's proposal involves further petrochemical expansion in C, thus enabling C to remain as well-off after the transfer of these funds to B, (4) funds which m a y be allocated by both A and C to B because A and C will be better off from B's venture into research and space-age electronic industries (the growth of such industries would foster further industrial development in A and C). So the process continues. We m a y imagine that sooner or later, in an integrated fashion, an efficient complex emerges in each region's plan. Clearly, we cannot say that the region which turns out to be the leader on the first move will be the region to have the steel capacity. For there are also major advantages (as well as disadvantages) from the development of petrochemical and research complexes. And it m a y v e r y well turn out that a large-scale steel plant i s planned for an intermediate location because the region which is leader on the first move, while wanting a petrochemical complex, might be hesitant to see a steel complex develop in either B or C. Hence A elects to start off steel development at the intermediate location. Generally put, it seems to me that the regions m a y be said to face alternatives, each of which leads to a cooperative solution yielding each a per capita income (and level of economic development) greater than would exist if no cooperative solution were effected. M y time is up. Let me conclude by raising the question: Is all this discussion basic to agglomeration theory ? T h e answer is Yes. One m a y of course contend that agglomeration, if achieved, is achieved only because participants have been persuaded to adopt certain procedures. Such persuasion represents an art of administration or arbitration. It m a y be contended that there is no science, let alone regional science involved. Therefore no agglomeration analysis and theory is involved. T o this reasoning, I would object. W h a t I have done, of course, is to illustrate the workings of a procedure. But if you will examine the a r g u m e n t carefully, you will observe m a n y implicit assumptions. For example, I have assumed regions primarily motivated to maximize Gross Product. I have implicitly assumed certain properties of the outcome space which m a y be associated with the action spaces of the several regions. I have implicitly assumed certain attitudes which each region has with respect to other regions. And so forth. (All these assumptions are spelled out in a technical manuscript b y T o n y Smith and myself to be published shortly.) Now as a scientist, I am interested in asking: under w h a t set of assumptions in interdependent decision or game situations can I project that the behaving participants will agglomerate, or take a set of decisions reflected in regional plans so that each region contains a major industrial agglomeration and so that together these agglomerations comprise an efficient integrated national or common m a r k e t industrialization plan? T o sum up, Weberian analysis still contributes relevant analytic elements to w h a t a modern-day agglomeration theory must contain. However, the g a m e element m u s t also be a basic component of such theory. It is essential that we understand fully both the conflict and cooperative elements present in the interdependent decision situation associated with any given agglomeration problem. And we also m u s t have in the background theory which states under w h a t assumptions on action spaces, outcome spaces, preference functions, objective and

ISARD: GAME THEORY, INDUSTRIAL AGGLOMERATION guiding principles can lead (or do not lead) we may try to set up their own self-interest

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we project that "rational" actors will take actions which to agglomeration. Then, of course, as regional planners these conditions so that motivated individuals will out of achieve the agglomeration goals we have in mind.