CORRESPONDENCE 103

and the esthetic value of the hypothesis, e.g., its elegance and building in biology, while the property of living in water is moresimplicity. Many of the classical paradoxes with regard to induc- important for the purpose of classification of industries.tion can be resolved if we realize that the crucial factor determin- The conclusion is that classification is a value-dependent tasking the acceptability of a hypothesis is not confirmation but the and pattern recognition is mechanically possible only if we smug-extra-evidential extra-logical considerations [10]. gle into the machine the scale of importance of predicates.

This takes the process of induction clearly out of the reach Alternatively, we can introduce into the machine the scale ofof purely mechanical data processing. We can produce a sem- distance or similarity between objects. This seems to be anblance of inductive inference with a computer by formulating innocuous set of auxiliary data, but in reality we are therebyesthetic criteria in machine-legible sentences, but we have to telling the machine our value judgment, which is of an entirelypoint out first that these rules cannot be produced by a machine, extra-logical nature. The human mind has an innate scale ofand second that the criteria of esthetic values cannot be properly importance of predicates closely related to the sensory organs.formulated in terms of abstract symbols. (See the end of Section This scale of importance seems to have been developed duringIII.) We may, therefore, conclude that all machine imitation of the process of evolution in such a way as to help maintain andinduction has either to omit some important factor of induction expand life [12], [14]. The machine, having no organized life-or to smuggle in man-made factors under some disguise. sustaining inner value structure, cannot form classes of objects

or classify objects according to paradigms. In the humian mind,VII. PArrERN RECOGNMON on the other hand, paradigmatic symbols, being concrete objectsPattern recognition can mean various things, but a typical and having their place in the inner value structure, can generate

task of pattern recognition may be defined as follows. We are classes as cloud seeds can generate a large cloud around them.first shown a certain limited number of samples (paradigms) of In summary, we may say that artificial intelligence and humaneach of the given number of classes, and then we are shown an intelligence have an unbridgeable gap not only in procedure, butobject of unknown class and are required to place it in one of also in functional effect. If one wants, however, at least to narrowthe classes. It is obviously an inductive process, hence it is not the gap, the first thing one could try may be to develop a genuinea logical process or mechanical sorting that the computer can associative memory-that really deserves the name-to imitateperform without human aid. However, it is very instructive to some of the features of paradigmatic symbols.study more closely the nature of the problem. See [11 ] for therelationship between pattern recognition and induction. REFERENCES

Pattern recognition is a task of classification, but the classes [1] S. Watanabe, "Paradigmatic symbol," in Proc. 20th Int. Congr.Psychology, 1972.are defined neither by their intention nor by their extension. [2] , "Civilization and science-man and machine," Ann. Japan Ass.They are indicated only by paradigms. If the computer is ever Philos. Sci., vol. 1, no. 4, pp. 216-234, 1959.

-[31 , "Comments on key issues,", in Dimensions of Mind. Newto become capable of doing this job, it has to derive the class- York: New York Univ. Press, 1960, pp. 143-147.definingproperties from the paradigms, so that the task becomes [41 M. Polanyi, Personal Knowledge. London: Routledge and Kegandefining ~~~~~~~~~~~~~~~~~Paul,1958.one of intentional sorting, which the computer is capable of [5] s. Watanabe, "Les' l6ments humains arationnels dans la connaissancedoing. But the derivation of class-defining properties from scientifique," Arch. Philos., vol. 34, pp. 609-622, 1971.doing. Buttneaervaton f cassdenlngproertes rom [61 A. Sainuel, "Some studies in machine learning using the game ofparadigms is not a machine-feasible operation without human checkers," IBM J. Res. Develop., vol. 11, no. 6, pp. 601-617, 1967.171 H. Wang, "Toward mechanical mathematics," IBM J. Res. Develop.,aid. This can be seen by the following consideration. vol. 4, no. 1, pp. 2-22, 1960.According to the theorem of the Ugly Duckling, any pair [8] S. Watanabe, "Can cognitive process be totally mechanized?" inProc. Int. Conf. Philosophical Problems in Psychology, 1968. Hono-of nonidentical objects share an equal number of predicates as lulu, Hawaii: Univ. Press Hawaii, 1974.

any other pair of nonidentical objects, insofar as the number of [9] R. Carnap, Logical Foundation of Probability. Chicago, Ill.: Univ.Chicago Press, 1950.predicates is finite [10], [12]. That is to say, from a logical [10] S. Watanabe, Knowing and Guessing. New York: Wiley, 1969.point of view there is no such thing as a natural kind. In the case [11] ~, "Pattern recognition as an inductive process," in Methodologiesof Pattern Recognition. New York: Academic Press, pp. 521-534,of pattern recognition, the new arrival shares the same number 1969.ofpredicates with any paradigm of any class. This shows that [12] '"Une explication mathematique du classement d'objets," inof predilcates Withl any paradigm of any class. This shows that Information and Prediction in Science. New York: Academic Press,

pattern recognition is a logically indeterminate problem. The pp. 39-76, 1965.class-defining properties are generalizations of certain of [13] H. L. Gelernter, in Proc. 1st UNESCO Int. Conf. Information Pro-class-defining properties are generalizations of certain of the cessing, 1959; also in Computer and Thought. New York: McGraw-

properties shared by the paradigms of the class. Which of the [14] Hill 1963.[1]W. V. Quine, ontological Relativity and Other Essays. New York:properties should be used for generalization is not logically Columbia Univ. Press, pp. 114ff., 1969.

defined. If it were logically determinable, then pattern recognitionwould have a definite answer in violation of the theorem of theUgly Duckling.

This conclusion is somewhat disturbing because our empiricalknowledge is based on natural kinds of objects. The source of A Model for "'The Tragedy of the Commons"the trouble lies in the fact that we were just counting the number JAY M ANDERSof predicates in the foregoing, treating them as if they were all . ND ONequally important. The fact is that some predicates are more Asrc- omldnmcmdlwihrpeet h supinimportant than some others. Objects are similar if they share a in Hadi' pae,"h rgd fteComn,xiishscn

large~~ ~~ ~nubro-motn rdcts clusions explicitly. The calculus serves as an adequate language forImportant In what scale? We have to conclude that a predicate thle model, but its conclusions are exhibited graphically with the aid of ais important if it leads to a classification that is useful for some simulation lang,,age and computation. Technological solutions to thepurpose. From a logical point of view, a whale can be put "tragedy" necessarily fail; coercive solutions admit a range of pos-together in the same box with a fish or with an elephant. How- sibilities. Limitation of the use of the "commons" may prove moreever, for the purpose of building an elegant zoological theory, itis better to put it together with the elephant, and for classifyingindustries it is better to put it together with the fish. The property Manuscript received July 19. 1973.chaactrizngmammals is important for the puxrpose of theory BTnheMauthorpis Iwith the Department of Chemistry, Bryn Mawr College,

104 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1974

TABLE I models illustrate the time dependence of these relationships asCAPITAL AND COMMONS well.

Problem Commons Capital Prduct In fact, one simple explicit language for the expression ofOvergrazing the village Grass Sheep Wool dynamic models is the calculus. The model then appears as a

green system of differential equations. For example, we could writePolluting water Water Factory Gadgets dC = COveruse of wilderness Wilderness Men Recreation dt Tr

- MMn(aKC)

where C represents commons, Tr, is the regeneration time constantPROFIT for the commons, and K represents capital. Here we assume that

+ / + the regeneration of commons follows first-order kinetics (a rateINVESTMENT OUTPUT REGENERATION proportional to the remaining commons) and that the commons

N+ // a t j+ is depleted at a rate proportional to working capital, but no moreCAPITAL COMMONS than the existing commons.I t t\ 1 In the logistic growth curve often used by ecologists; however,DEPRECIATION USAGE the regeneration rate of a species tends toward zero as the popu-

Fig. 1. Cause-and-effect relationships between capital and commons lation of the species approaches the carrying capacity of thein "The Tragedy of the Commons." Arrow indicates cause-and-effect environment for that species. Accordingly, we represent therelationship, with tail at cause and arrowhead at effect. Sign at arrow-head indicates sense of relationship; for example, regeneration increases regeneration tme constant for the commons not as a constant,commons but usage depletes commons. but as a monotonically increasing function of commons'

Ir =To)Cp0 (2)

successful than taxation for the use of the "commons." The model lays Co- Cboth assumptions and conclusions open to critical examination, and so thatserves as a useful teaching tool. dC (04 Co mi

INTRODUCON dt cIOGarrett Hardin's seminal paper, "The Tragedy of the Com- The rate of change of capital depends on investment and

mons," [1 ] describes in general terms the problems surrounding depreciationthe depletion of common-property resources. He then applies dK= (a-b)Khis analysis to the population problem in particular. The model dtproposed here offers .no results not already anticipated by where a and b represent investment and depreciation, respective-Hardin; it does, however, exhibit Hardin's assumptions and con- ly. As long as a> b, K grows; ifK grows enough, the commonsclusions in a formal way, laying each open to examination and usage rate may become total, that is, mmn (aK,C) = C. Thencriticism.

Let us consider the "tragedy" in general terms. Men use dC 0.6C_ 0.4C2common-property resources such as space, water, air, etc. In a dt COsystem where these resources are free, the gain to the resourceuser is not reduced by a resource cost. Normally, therefore, the whch is necessarily negative, leading ultimately to no commons.user will be induced to use yet more of the "commons." What is the effect of regulation? Suppose that a limit of annual

Let us consider this economic-ecological system (tuly an commons use is imposed and that that limit is some fraction ofecosystem) as being made up of two interwoven parts: capital the original commons, say, 0CO. Then dC/dt = 0 establishes aand commons. "Commons" will represent any renewable condition forminimum stablecommonscommon-property resource, and "capital" will represent any -0.4 ± VO.16 - 1.6,8means of production of goods. Table I shows some examples. C=-0.Fig. 1 shows the cause-and-effect relationships between the 0.8elements of the system. Capital is regulated by investment and Clearly 1? . 0.1. That is, the commons will always decrease todepreciation, and commons by natural reWeneration and usage. zero-there will be a "tragedy"-unless a firm limit of no moreOne can deduce the behavior of the system by examining the than 10 percent of the commons used annually is imposed. Forfeedback loops (closed circuits of causal relationships) which this, the most generous limit, C(at equilibrium) = C0/2, or halfcharacterize the system. The stability of the commons-capital the commons will remain. This conclusion, representing thesystem hangs on the balance between the positive driving forces ultimate or equilibrium state of the system, is independent ofofinvestment and regeneration and the negative restraining forces the speed by which that equilibrium is reached (the constantof depreciation and usage. 0.4-=O in (2)).

In this correspondence we shall develop a model for "The Consider an alternative solution: the taxation of industry at aTragedy of the Commons" and formally examine Hardin's rate proportional to used commons. This solution admnits, byassumptions and conclusions. analysis analogous to that just undertaken, of a range of possi-

bilities for equilibria of both commons and capital. One canDvlJAvIC ODLS:TH CIXL* choose from a range extending from little capital and much

A model is nothing more than statements about cause and comons to much capital and little commons.effect. Models can be carred in one's mind, but mental modelsoften suffer from incompleteness and inaccuracy. Formal models 1I h rga culyue,ti qainapasa .[0exhibit cause-and-effect relationships explicitly, and dynamic (C - C + 1)] to avoid a singularnty.

CORRESPONDENCE 105

It is, therefore, clear that coercive solutions are required to savethe commons. Merely technological solutions, changing therelationship between capital and commons or the regenerationrate of the commons (the constants a in (1) or ro in (2)) are of noavail.

DYNAIc MODELS: SYSTEM DYNAMICSThe cause-and-effect structure shown in Fig. I can easily be

translated into a computer program using the techniques ofIT system dynamics [2] and the DYNAMO computer language [31.

The result is the flow diagram of Fig. 2.2CIR CRa \ JThe standard simulation of this model is shown in Fig. 3(a).The behavior of the system, as Hardin suggests, is characterized

!lCAP . COM -~' by the depletion of the commons accompanied by a rise and fall, .CCR F\ (overshoot and collapse) of capital.

COR ~ ~ _CUR~ LJp~i s1>-Hardin suggests also that this is a problem for which there is

"no technical solution." To test this, we simulate the same model,(QCtF- 4but with a lower commons-capital ratio; that is, less commons

-&dR is used per unit of capital to ensure the same output. One couldFig. 2. DYNAMO flow diagram for model. also simulate the effect of raising the regeneration rate of the

commons through technological means; the result is quali-tatively similar to that shown in Fig. 3(b). Our result indicatesthat capital rises higher and commons falls less rapidly before the"remorseless working of things"3 precipitates the same tragedy.What solutions to the tragedy are available? Hardin points

CAP I Tonly to coercive solutions: "mutual coercion mutually agreedupon." Within the context of the present model, we examine two

COMMONSX - possible coercive solutions: a) limiting the use of the commons;a b) charging for (taxing, internalizing external diseconomies) the

use of the commons. Figs. 3(c) and 3(d) show simulations of the/m \model under the conditions of implementing these coercive

solutions. In each case, a balance between commons and capitalis reached. In the language of control and feedback, a negative

aAp \ \ feedback loop including capital and commons is establishedwhich brings the system into a steady-state condition.

b__________________ The limit imposed in Fig. 3(c) is the most generous limit,10 percent of the commons per year. The choice of taxes levied,shown in Fig. 3(d), is arbitrary.

CONCLUSIONS

AP --Er-The formal dynamic model proposed here, whether expressedin the calculus or in a simulation language, offers no results notalready anticipated by Hardin; it does, however, exhibit Hardin's

cassumptions formally, laying them open to examination, criti-cism, and interpretation.

ACKNOWLEDGMENTThe author acknowledges valuable contributions from the

CAP students and faculty in Bryn Mawr's program in the Growth andStructure of Cities, from students in The Dynamics of Environ-

d mental Systems, for whom this essay was initially prepared, andTIME from Prof. G. Porter of the University of Pennsylvania, who

graciously reviewed an earlier draft.Fig. 3. Simulations of model for "The Tragedy of the Commons."(a) Standard simulation. (b) Simulation with reduced commons-capital Rratio, indicating that there are no technical solutions to "tragedy." EFERENCES

possible coercive solution to "tragedy." (d) Simulation with inter- 12]43-1a2d48,Dec. 13,rl1968.yo h omnSScec,vl 6pnahlzation of common-property resource costs, indicating another [2] J. W. Forrester, Princeiples of Systems. Cambridge, Mass.: Wright-

Industrial Dynamics. Cambridge, Mass.: M.I.T. Press, 1961.[31 A. L- Pugh, Dynamo II User's Manual. Cambridge, Mass.: M.I.T.

Press, 1970.

2 A technical appendix to this paper, containing DYNAMO fl*owdiagram, equations, and parameter changes used in the simulations shownin Fig. 3, can be obtained by writing directly to the author.

3 A. N. Whitehead, quoted by Hardin in [1].