Sociocybernetics pp 105-127 | Cite as

# Are societies Turing machines? Some implications of the cyclical majority problem, an NP complete problem, for cybernetic models of social systems

## Abstract

The paradigms of cybernetic models of social systems are frequently finite automata. Such models, implicitly or explicitly, require a decision procedure for multi-attribute value aggregation. This decision procedure requires either a hierarchical ordering of sub- and supra-systems, or is reduced to the cyclical majority problem. Many well-known applications of cybernetic concepts to the analysis of social systems have been aimed at constructing formal models of society and using these models to examine processes which could be described by finite automata^{1}. The outstanding work of Buckley^{2}, Parsons^{3} and Etzioni^{4} in sociology; of Deutsch^{5} and Easton^{6} in political science; and of Forrester^{7}, D. H. Meadows, D. L. Meadows, Randers and Behrens^{8}, and Mesarovic and Pestel^{9} in social systems simulation exemplify this approach. Simulations of political processes in Shaffer^{10}, Pool^{11}, and Klausner^{12} typify cybernetic models expressed as implicit theory embodied in computer programs (as noted by Browning)^{13}, as well as showing the behavior of the explicit processes which the programs model.

## Keywords

Turing Machine Criterion Function Decision Structure American Political Science Review Impossibility Theorem## Preview

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## Notes

- 1.For a model exhibiting an explicit attempt to construct a finite automata see G. H. Kramer, ‘An Impossibility Result Concerning the Theory of Decision Making’ in J. Bernard (ed.)
*Mathematical Applications in Political Science*Dallas, Texas: So. Meth. U. Press; 1966.Google Scholar - 2.W. Buckley,
*Sociology and Modem Systems Theory*. Englewood Cliffs, NJ: Prentice-Hall, 1967, and*Modern Systems Research for the Behavioral Scientist*Chicago: Aldine, 1968.Google Scholar - 3.T. Parsons,
*The Social System*. Glencoe, Ill: The Free Press; 1951.Google Scholar - 4.A. Etzioni,
*The Active Society: A Theory of Societal and Political Processes*. New York: The Free Press, 1968. See also Breed, W.,*The Self-Guiding Society*. New York: The Free Press, 1971.Google Scholar - 5.K. W. Deutsch,
*The Nerves of Government*. New York: The Free Press, 1963.Google Scholar - 6.D. Easton,
*A Systems Analysis of Political Life*. New York: Wiley, 1965.Google Scholar - 7.J. W. Forrester, ‘Understanding the Counter Intuitive Behavior of Social Systems’ in J. Beishon and G. Peters (eds.),
*Systems Behavior*. London: Harper and Row, 223–240, 1976, J. W. Forrester,*Urban Dynamics*. Cambridge, Mass.: The M.I.T. Press; 1971, J. W. Forrester,*World Dynamics*. Cambridge, Mass.: Wright-Allen Press, 1971.Google Scholar - 8.D. H. Meadows, D. L. Meadows, J. Randers and W. W. Behrens III,
*The Limits to Growth: A Report for the Club of Rome on the Predicament of Mankind*. New York: Signet, 1975.Google Scholar - 9.M. Mesarovic and E. Pestel,
*Mankind at the Turning Point: The Second Report to the Club of Rome*. New York: Signet, 1976.Google Scholar - 10.W. R. Shaffer,
*Computer Simulations of Voting Behavior*, New York: Oxford University Press, 1972.Google Scholar - 11.I. de S. Pool, R. P. Abelson, and S. L. Popkin,
*Candidates, Issues and Strategies: A Computer Simulation of the 1960 and 1964 Elections*, Cambridge, Mass.: The M.I.T. Press, 1965.Google Scholar - 12.S. Z. Klauner, (ed.),
*The Study of Total Societies*, New York: Doubleday, 1967.Google Scholar - 13.R. P. Browning, ‘Computer Programs as Theories of Political Processes.’
*Journal of Politics*24, 562–582, 1962.CrossRefGoogle Scholar - 14.E. Laszlo,
*Introduction to Systems Philosophy: Toward a New Paradigm of Contemporary Thought*, New York: Harper and Row, 1972, and his*The Systems View of the World*, New York: Braziller, 1972.Google Scholar - 15.E. Laszlo,
*Introduction to Systems Philosophy*, 98–117 (see Note 14).Google Scholar - 16.H. A. Simon,
*The Shape of Automation for Men and Management*, Harper and Row, p. 102, 99–100, 1965.Google Scholar - 17.H. H. Pattee,
*Hierarchy Theory, the Challenge of Complex Systems*, New York: Braziller, p. xi, 1973.Google Scholar - 18.H. A. Simon,
*The Shape of Automation*, p. 70–71 (see Note 16).Google Scholar - 19.
*Ibid*, p. 72.Google Scholar - 20.See E. Yourdon and L. L. Constantine,
*Structured Design*, New York: Yourdon, 1976; also O. J. Dahl, E. W. Dijkstra, and C. A. R. Hoare,*Structured Programming*, New York: Academic Press; 1972, especially Part III, ‘Hierarchical Program Structures’ by O. J. Dahl and C. A. R. Hoare.Google Scholar - 21.H. A. Simon,
*The Shape of Automation*, p. 101 (see Note 16).Google Scholar - 22.See A. C. Shaw,
*The Logical Design of Operating Systems*, Englewood Cliffs, NJ: Prentice-Hall, 1974, especially Chap. 8, ‘The Deadlock Problem’.Google Scholar - 23.See E. W. Dijkstra,
*A Discipline of Programming*, Englewood Cliffs, NJ: Prentice-Hall, 1976, and ‘Guarded Commands, Nondeterminacy and Formal Derivation of Programs’ in R. T. Yeh (ed.),*Current Trends in Programming Methodology*, Vol. I:*Software Specification and Design*, Englewood Cliffs, NJ: Prentice-Hall, 1977.Google Scholar - 24.See E. Yourdon and L. L. Constantine,
*Structured Design*, esp. Chap. 18, ‘Homologous and Incremental Structures’ (Note 20) and O. J. Dahl, E. W. Dijkstra, and C. A. R. Hoare,*Structured Programming*, ‘Hierarchical Program Structures’ by O. J. Dahl and C. A. R. Hoare (Note 20). Simon also refers to similar program structures as ‘productions’ in H. H. Pattee (ed.)*Hierarchy Theory*, ‘The Organization of Complex Systems’, p. 19 (see Note 17).Google Scholar - 25.See J. Martin,
*Computer Data-Base Organization*, Englewood Cliffs, NJ: Prentice Hall, 1975, especially Chap. 20, ‘Chains and Ring Structures’.Google Scholar - 26.K. Arrow,
*Social Choice and Individual Values*, New Haven: Yale University Press, 2nd edition, 1963.Google Scholar - 27.W. H. Riker, ‘Voting and the Summation of Preferences: An Interpretive Bibliographical Review of Selected Developments during the Last Decade.’
*American Political Science Review*55 (December): 900–911, 1961.Google Scholar - 28.G. Th. Guilbaud, ‘Theories of the General Interest, and the Logical Problem of Aggregations’ in P. F. Lazarsfeld and N. W. Henry (eds.)
*Readings in Mathematical Social Science*. Cambridge, Mass.: The M.I.T. Press, 262–308, 1966.Google Scholar - 29.R. G. Niemi, and H. F. Weisberg, ‘A Mathematical Solution for the Probability of the Paradox of Voting.’
*Behavioral Science*13, Nr. 4, (July): 317–323, 1968.CrossRefGoogle Scholar - 30.M. B. Garman and M. I. Kamien, ‘The Paradox of Voting: Probability Calculations.’
*Behavioral Science*13, Nr. 4 (July): 306–316, 1968.CrossRefGoogle Scholar - 31.F. S. Roberts,
*Discrete Mathematical Models*, Englewood Cliffs, NJ: Prentice-Hall, 1976, especially Chap. 7, ‘Group Decision Making’.Google Scholar - 32.C. D. Campbell and G. Tullock, ‘The Paradox of Voting — A Possible Method of Calculation.’
*American Political Science Review*60, 684–685, 1966.CrossRefGoogle Scholar - 33.D. Klahr, ‘A Computer Simulation of the Paradox of Voting.’
*American Political Science Review*60, Nr. 2 (June): 384–390, 1966.CrossRefGoogle Scholar - 34.J. E. Pomeranz and R. L. Weill, Jr., ‘The Cyclical Majority Problem.’
*Communications of the ACM*13, Nr. 4 (April): 251–255, 1970.CrossRefGoogle Scholar - 35.A. Kirman and D. Sondermann, ‘Arrow’s Theorem, Many Agents and Invisible Dictators.’
*Journal of Economic Theory*5, 267–277, 1972.CrossRefGoogle Scholar - 36.B. Hansson, ‘The existence of Group Preference Functions,’
*Public Choice*XXVIII, (Winter): 89–98, 1976.CrossRefGoogle Scholar - 37.P. C. Fishburn, ‘Arrow’s Impossibility Theorem: Concise Proof and Infinite Voters.’
*Journal of Economic Theory*2, 103–106, 1970.CrossRefGoogle Scholar - 38.B. Hansson, ‘The existence of Group Preference Functions.’
*Public Choice*, pp. 89–98 (see Note 36).Google Scholar - 39.J. E. Pomeranz and R. L. Weill Jr., ‘The Cyclical Majority Problem.’
*Communications of the ACM*. pp. 251–255 (see Note 34).Google Scholar - 40.R. M. Karp, ‘Reducibility Among Combinatorial Problems’ in R. E. Miller and J. W. Thatcher, (eds.),
*Complexity of Computer Computations*, New York: Plenum Press, 85–103, 1972.CrossRefGoogle Scholar - 41.R. G. Niemi and H. F. Weisberg, ‘A Mathematical Solution for the Probability of the Paradox of Voting.’
*Behavioral Science*13, Nr. 4, (July): 320, 1968. R. G. Niemi and H. F. Weisberg,*Probability Models of Collective Decision*, Columbus, Ohio: Merrill, 1972, especially Part 3, ‘The Paradox of Voting,’ pp. 181–272.CrossRefGoogle Scholar - 42.M. B. Garman and M. I. Kamien, ‘The Paradox of Voting: Probability Calculations.’
*Behavioral Science*13, Nr. 4 (July): 313, 1968.CrossRefGoogle Scholar - 43.P. C. Fishburn,
*The Theory of Social Choice*, Princeton, NJ: Princeton University Press, 1977, and his ‘Arrow’s Impossibility Theorem: Concise Proof and Infinite Voters.’*Journal of Economic Theory*2, 103–106, 1970.Google Scholar - 44.B. Hansson, ‘The Existence of Group Preference Functions.’
*Public Choice*, p. 97 (see Note 36).Google Scholar - 45.See P. K. Pattanaik,
*Voting and Collective Choice*, Cambridge, England: Cambridge University Press, 1971.Google Scholar - 46.K. Arrow,
*Social Choice and Individual Values*(see Note 26). Compare especially Theorem 2, p. 59, and Theorem 3 and its corollary, p. 63.Google Scholar - 47.For a formal treatment of Turing machines, see M. Minsky,
*Computation: Finite and Infinite Machines*, Englewood Cliffs, NJ: Prentice-Hall, 1967. For a discussion of NP completeness and complexity hierarchies, see A. V. Aho, J. E. Hopcraft and J. D. Ullman,*The Design and Analysis of Computer Algorithms*, Reading, Mass.: Addison-Wesley, 1974, also M. A. Arbib,*Theories of Abstract Automata*, Englewood Cliffs, NJ: Prentice-Hall, 1969.Google Scholar - 48.B. Weide, ‘A Survey of Analysis Techniques for Discrete Algorithms,’ ACM
*Computing Surveys*9, Nr. 4, (December): 291–314, 1977.CrossRefGoogle Scholar - 49.
*Ibid*, p. 306.Google Scholar - 50.
*Ibid*, p. 307.Google Scholar - 51.
*Ibid*. Weide also notes: ‘A language*L*_{1}is “polynomially reducible” to*L*_{2}if there is a deterministic polynomial-time algorithm which transforms a string*x*into a string*f(x)*such that*x*is in*L*_{1}iff*f(x)*is in*L*_{2}.’Google Scholar - 52.R. M. Karp, ‘Reducibility Among Combinatorial Problems,’
*Complexity of Computer Computations*(see Note 40), pp. 85–103.Google Scholar - 53.B. Weide, ‘A Survey of Analysis Techniques for Discrete Algorithms,’ ACM
*Computing Surveys*(see Note 48), p. 307.Google Scholar - 54.T. C. T. Kotiah and D. I. Steinberg, ‘Occurrences of Cycling and Other Phenomena Arising in a Class of Linear Programming Models.’
*Communications of the*ACM 20, Nr. 2 (February): 107–112, 1977.CrossRefGoogle Scholar - 55.R. L. Keeney and H. Raiffa,
*Decisions with Multiple Objectives: Preferences and Value Tradeoffs*, New York: Wiley, 1976, especially Chap. 10, ‘Aggregation of Individual Preferences,’ and Section 10.6.1, ‘The Supra Decision Maker Model.’Google Scholar - 56.See P. K. Pattanaik,
*Voting and Collective Choice*(Note 45). Also Y. Murakami,*Logic and Social Choice*, New York: Dover, 1968.Google Scholar - 57.G. Tullock, ‘The General Irrelevance of the General Impossibility Theorem.’
*Quarterly Journal of Economics*81, 256–270, 1967.CrossRefGoogle Scholar - 58.S. S. Stevens, ‘Measurement, Psychophysics and Utility,’ in C. W. Churchman and P. Ratoush (eds.)
*Measurement: Definitions and Theories*, New York: Wiley, 18–63, 1959.Google Scholar - 59.M. B. Garman and M. I. Kamien, ‘The Paradox of Voting: Probability Calculations,’
*Behavioral Science*, 306–316 (Note 42).Google Scholar - 60.C. H. Coombs,
*A Theory of Data*, New York: Wiley, 1964.Google Scholar - 61.R. G. Niemi, ‘Majority Decision Making with Partial Unidimensionality,’
*American Political Science Review*63, Nr. 2 (June): 488–497, 1969.CrossRefGoogle Scholar - 62.
*Ibid*, pp. 493–494.Google Scholar - 63.
*Ibid*, p. 488. Niemi defines single-peakedness as follows: ‘A set of preference orderings is single-peaked if there is an ordering of the alternatives on the abscissa such that when utility or degree of preference is indicated by the ordinate, each preference ordering can be represented by a curve which changes its direction at most once, from up to down (i.e. has at most one peak).’Google Scholar - 64.R. Levins, ‘The Limits of Complexity’ in
*Hierarchy Theory*, p. 114, (see Note 17), also S. A. Kauffman, ‘Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets.’*Journal of Theoretical Biology*22, 437–467.Google Scholar - 65.
*Ibid*, pp. 114–115.Google Scholar - 66.
*Ibid*, p. 115.Google Scholar - 67.See A. Ando, F. M. Fisher and H. A. Simon,
*Essays on the Structure of Social Science Models*, Cambridge, Mass.: The M.I.T. Press, 1963, for an extensive discussion of ‘near-decomposability’.Google Scholar - 68.H. A. Simon, ‘The Organization of Complex Systems,’
*Hierarchy Theory*, pp. 15–16 (see Note 17).Google Scholar - 69.See L. R. Sayles, ‘Matrix Management, the Structure with a Future,’
*Organizational Dynamics*5, Nr. 2 (August): 2–10, 1976; P. R. Lawrence, H. F. Kolodny and S. M. Davis, ‘The Human Side of the Matrix.’*Organizational Dynamics*6, Nr. 1 (Summer): 43–61, 1977; J. R. Galbraith, ‘Matrix Organization Design.’*Business Horizons*(February): 21–40, 1971; C. Argyris, ‘Today’s Problems with Tomorrows Organizations,’*The Journal of Management Studies*(March): 84–101, 1973; and S. M. Davis and P. R. Lawrence,*Matrix*, New York: Macmillan, 1977.CrossRefGoogle Scholar - 70.R. N. Rosecrance,
*Action and Reaction in World Politics*, Boston: Little, Brown, p. 222, 1963.Google Scholar - 71.
*Ibid*, p. 306.Google Scholar - 72.W. R. Ashby,
*An Introduction to Cybernetics*, New York: Wiley, 1963, especially Chap. 13.Google Scholar - 73.See K. Appel, and W. Haken, ‘The Solution of the Four-Color Map Problem.’
*Scientific American*237, Nr. 4 (October): 108–121, 1977.CrossRefGoogle Scholar - 74.W. R. Ashby,
*An Introduction to Cybernetics*(see Note 72), 22–23, also Chaps. 5 and 7.Google Scholar - 75.See H. Dooyeweerd,
*In the Twilight of Western Thought: Studies in the Pretended Autonomy of Philosophical Thought*. Grand Rapids, Mich.: Baker, 1960, and his*Transcendental Problems in Philosophic Thought*. Grand Rapids, Mich.: Baker, 1953.Google Scholar - 76.Simon also cautions against a ‘Laplacian’ reductionism. See H. A. Simon, ‘The Organization of Complex Systems’ in
*Hierarchy Theory*, pp. 24–27 (see Note 17).Google Scholar