¹ Readers wishing to sample for themselves the mathematical “frontiers” of economics, psychology, and management science will find Arrow, Kenneth J., Karlin, Samuel, and Suppes, Patrick, eds., Mathematical Methods in the Social Sciences, 1959 (Stanford, Stanford University Press, 1960)Google Scholar, a convenient collection.

² Simon, Herbert A., Models of Man (New York, 1957), p. ix Google Scholar.

³ Rapoport, Anatol, “Various Meanings of ‘Theory’,” this Review, Vol. 52 (December, 1958), pp. 972–988 Google Scholar.

⁴ Simon, Herbert A. and Newell, Allen, “Models: Their Uses and Limitations,” in White, Leonard D., ed., The State of the Social Sciences (Chicago, University of Chicago Press, 1956)Google Scholar.

⁵ Festinger, Leon, “The Relevance of Mathematics to Controlled Experimentation in Sociology,” International Social Science Bulletin, Vol. 6 (1954), pp. 622–627 Google Scholar.

⁶ Simon, Herbert A., “Some Strategic Considerations in the Construction of Social Science Models,” in Lazarsfeld, Paul F., ed., Mathematical Thinking in the Social Sciences (Glencoe, 1954)Google Scholar.

⁷ Coombs, C. H., Raiffa, H., and Thrall, R. M., “Some Views on Mathematical Models and Measurement Theory,” in Thrall, R. M., Coombs, C. H., and Davis, R. L., eds., Decision Processes (New York, 1954)Google Scholar.

⁸ Lazarsfeld, Paul F., “Evidence and Inference in Social Research,” in Lerner, Daniel, ed., Evidence and Inference (Glencoe, 1959), pp. 132–137 Google Scholar.

⁹ Kaplan, Abraham, “Sociology Learns the Language of Mathematics,” in Newman, James R., ed., The World of Mathematics, Vol. 2 (New York, 1956)Google Scholar.

¹⁰ Firey, Walter, “Mathematics and Social Theory,” Social Forces, Vol. 29 (October, 1950), pp. 20–25 CrossRefGoogle Scholar.

¹¹ The major works of Zipf, Rashevsky, Menger, and Bales are not included in this bibliography. Those wishing to investigate their contributions at first hand can follow the writings cited in the above critiques. A brief report and extensive bibliography of the recent work of Dodd, his colleagues, and his students will be found in Dodd, Stuart Carter, “Formulas for Spreading Opinions,” Public Opinion Quarterly, Vol. 22 (Winter, 1958), pp. 537–554 CrossRefGoogle Scholar.

¹² Bernard, Jessie, “Mathematical Studies in the Sociology of Conflict,” in The Nature of Conflict (Paris, UNESCO, 1957), pp. 64–73 Google Scholar.

¹³ Deutsch, Karl W., “On Communication Models in the Social Sciences,” Public Opinion Quarterly, Vol. 16 (July, 1952), pp. 356–380 CrossRefGoogle Scholar.

¹⁴ Arrow, Kenneth, “Mathematical Models in the Social Sciences,” in Lerner, Daniel and Lasswell, Harold D., eds., The Policy Sciences (Stanford, Stanford University Press, 1951)Google Scholar.

¹⁵ Lévi-Strauss, Cl., “The Mathematics of Man,” International Social Science Bulletin, Vol. 6 (1954), pp. 581–590 Google Scholar.

¹⁶ Kemeny, John G., “Mathematics Without Numbers,” Daedalus, Vol. 88 (Fall, 1959), pp. 577–591 Google Scholar.

¹⁷ Ibid., p. 578.

¹⁸ Von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behavior (Princeton, Princeton University Press, 1944, 1947, 1953)Google Scholar.

¹⁹ Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York, 1957), p. 485 Google Scholar.

²⁰ Snyder, Richard, “Game Theory and the Analysis of Political Behavior,” in Bailey, Stephen K. et al., Research Frontiers in Politics and Government (Washington, D. C., The Brooking Institution, 1955)Google Scholar.

²¹ Rapoport, Anatol, “Critiques of Game Theory,” Behavioral Science, Vol. 4 (January, 1959), pp. 49–66 CrossRefGoogle Scholar.

²² Rapoport, Anatol, Fights, Games, and Debates (Ann Arbor, University of Michigan Press, 1960)Google Scholar.

²³ Shubik, Martin, ed., Readings in Game Theory and Political Behavior (Garden City, 1954)Google Scholar.

²⁴ Bernard, Jessie, “The Theory of Games of Strategy as a Modern Sociology of Conflict,” The American Journal of Sociology, Vol. 59 (March, 1954), pp. 411–424 CrossRefGoogle Scholar.

²⁵ Von Neumann and Morgenstern, op. cit.

²⁶ Luce and Raiffa, op. cit.

²⁷ Ibid., p. vii.

²⁸ See for instance Arrow, op. cit., Lévi-Strauss, op. cit., and Kaplan, op. cit.

²⁹ Braithwaite, R. B., Theory of Games as a Tool for the Moral Philosopher (Cambridge, Cambridge University Press, 1955), p. 55 Google Scholar.

³⁰ Gabor, Denis and Gabor, André, in Journal of the Royal Statistical Society, Series A, Vol. 117 Part I (1954), pp. 31–72 CrossRefGoogle Scholar.

³¹ Braithwaite, op. cit.

³² New York, 1957. Those interested in the polemical literature generated by this thoughtful book could begin with Rogers, W. Hayward, “Some Methodological Difficulties in Anthony Downs's An Economic Theory of Democracy,” this Review, Vol. 53 (June, 1959), pp. 483–485 Google Scholar. Downs, answers in “Dr. Rogers's Methodological Difficulties—A Reply to his Critical Note,” this Review, Vol. 53 (December, 1959), pp. 1094–1097 Google Scholar. A less rigorous follow-up to his original work is his “Why the Government Budget is too Small in a Democracy,” World Politics, Vol. 12 (July, 1960), pp. 541–563 CrossRefGoogle Scholar.

³³ New York, 1951.

³⁴ Those interested in the more technical aspects of Arrow's work will find May, Kenneth O., “A Set of Independent, Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica, Vol. 20 (October, 1952), pp. 680–684 CrossRefGoogle Scholar, useful on the problem of simple majority decision. Readers more interested in the normative and moral implications of Social Choice will find value in Little, I. D. M., “Social Choice and Individual Values,” Journal of Political Economy, Vol. 60 (October, 1952), pp. 422–32CrossRefGoogle Scholar.

³⁵ Cambridge, Cambridge University Press, 1958. The work of Black was first published as a series of journal articles in 1948 and 1949. His book incorporates and enlarges upon this early work. Readers wishing to consult the original articles will find them listed under “Acknowledgements” in The Theory of Committees and Elections.

³⁶ In Part II of The Theory, Black discusses in some detail the contributions of Borda, Condorcet, Leplace, Nanson, and Galton to the mathematical theory of elections. A translation and critique of Borda's most relevant work can be found in de Grazia, Alfred, “Mathematical Derivation of an Election System,” Isis, Vol. 44 (June, 1953), pp. 42–51 CrossRefGoogle Scholar.

³⁷ In the Appendix to The Theory, Dodgson's little known pamphlets are reprinted.

³⁸ Robert A. Dahl, (Chicago, University of Chicago Press, 1956). For one criticism of Dahl's work see Morgan, Douglas N., “A Postscript to Professor Dahl's ‘Preface‘,” this Review, Vol. 51 (December, 1957), pp. 1040–1052 Google Scholar. Dahl defends himself in “A Rejoinder,” ibid., pp. 1053–1061.

³⁹ This Review, Vol. 52 (June, 1958), pp. 349–366; see also his bibliographical essay “Voting and the Summation of Preferences,” in this issue of the Review, below, pp. 900–911.

⁴⁰ An excellent discussion of the work of Arrow, Black, and Dahl will be found in Luce and Raiffa, op. cit., ch. 14. This chapter also will direct the reader to other, more technical work in the general area of “Group Decision Making.”

⁴¹ Psychological Review, Vol. 63 (May, 1956), pp. 181–194 CrossRefGoogle Scholar.

⁴² In Cartwright, Dorwin, ed., Studies in Social Power (Ann Arbor, University of Michigan Press, 1959)Google Scholar.

⁴³ The original formulation differentiated power based on attraction, expertise, ability to reward, ability to coerce, and legitimation.

⁴⁴ See for instance the distinction between types of power which is so central to the argument of Neustadt, Richard in Presidential Power (New York, 1960)Google Scholar. It should be noted that the March-Simon-Dahl approach to the study of power, detailed below, also does not distinguish between “types” of power.

⁴⁵ Herbert Simon's article was originally published in the Journal of Politics in 1953, and has been reprinted in his Models of Man, cited above, (note 2). Other articles in Models of Man, particularly “Bandwagon and Underdog Effects of Election Predictions,” would be of interest to political scientists concerned with the “Democratic Process.”

⁴⁶ This Review, Vol. 49 (June, 1955), pp. 431–451.

⁴⁷ Journal of Politics, Vol. 19 (May, 1957), pp. 202–226 CrossRefGoogle Scholar. Both March and Simon use “influence” synonymously with “power.”

⁴⁸ Dahl, Robert A., in Behavioral Science, Vol. 2 (July, 1957), pp. 201–215 CrossRefGoogle Scholar.

⁴⁹ MacRae, Duncan Jr., and Price, Hugh D., “Scale Positions and ‘Power’ in the Senate,” Behavioral Science, Vol. 4 (July, 1959), pp. 212–218 CrossRefGoogle Scholar.

⁵⁰ L. S. Shapley and Martin Shubik, in this Review, Vol. 48 (September, 1954), pp. 787–792. Strictly speaking, there is one other mathematical approach to the study of power. This was stated by Penrose, L. S., in “The Elementary Statistics of Majority Voting,” Journal of the Royal Statistical Society, Vol. 109, Part I (1946), pp. 53–57 CrossRefGoogle Scholar. On page 54 Penrose says that “In general, the power of the individual voter can be measured by the amount by which his chance of being on the winning side exceeds one half.” The author's argument is not very productive of new insights and seems to have died on the vine as far as political science is concerned.

⁵¹ The artificiality of our bibliographical distinctions here becomes extremely obvious. The Shapley-Shubik approach to the study of power could equally well be classified under “The Democratic Process.”

⁵² Luce, R. Duncan and Rogow, Arnold A., “A Game Theoretic Analysis of Congressional Power Distributions for a Stable Two-party System,” Behavioral Science, Vol. 1 (April, 1956), pp. 83–95, at p. 87CrossRefGoogle Scholar.

⁵³ As might be expected, an incisive critique of Shapley-Shubik and Luce and Rogow will be found in Luce and Raiffa, op. cit., pp. 253–259.

⁵⁴ Riker, William H. and Schaps, Ronald, “Disharmony in Federal Government,” Behavioral Science, Vol. 2 (October, 1957). pp. 276–290 CrossRefGoogle Scholar.

⁵⁵ Pennock, J. Roland replies to the Riker-Schap, analysis in “Federal and Unitary Government—Disharmony and Frustration,” Behavioral Science Vol. 4 (April, 1959), pp. 147–157 CrossRefGoogle Scholar. Interestingly enough, the author does not concern himself with the power index at all, but rather offers a counterargument in terms of an analysis of national and local voting patterns. Thus, in three brief intellectual generations a problem occasioned by the theory of n-person games has evolved into a contribution to the extensive, non-mathematical literature on federalism.

⁵⁶ Riker, William H., in Behavioral Science, Vol. 4 (April, 1959), pp. 120–131 CrossRefGoogle Scholar.

⁵⁷ Riker, William H., “A Method for Determining the Significance of Roll Calls in Voting Bodies,” in Wahlke, John C. and Eulau, Heinz, eds., Legislative Behavior (Glencoe, 1959)Google Scholar, is a much needed addition to the tools available to investigators of legislative behavior, although the essay does not bear directly on the problem of inter-party mobility.

⁵⁸ Recently, two volumes of his work have been collected and edited in convenient form. See Richardson, Lewis F., Arms and Insecurity (Pittsburgh, The Boxwood Press, 1960)Google Scholar and Richardson, Lewis F., Statistics of Deadly Quarrels (Pittsburgh, The Boxwood Press, 1960)Google Scholar. Richardson was trained in the physical sciences, turned to the social sciences later in life, and died in 1953 at the age of 72.

⁵⁹ The key article here is Rapoport, Anatol, “Lewis F. Richardson's Mathematical Theory of War,” Journal of Conflict Resolution, Vol. 1 (September, 1957), pp. 249–299 CrossRefGoogle Scholar. This article is followed by a biography of Richardson and a bibliography of his original articles. A critique of Rapoport's evaluation may be found in Nilson, Sten S., “Political Equilibrium,” Journal of Conflict Resolution, Vol. 3 (December, 1959), pp. 383–390 CrossRefGoogle Scholar. Rapoport answers in “Remarks on ‘Political Equilibrium’ by Sten S. Nilson,” ibid., pp. 391–393.

⁶⁰ See note 22, above.

⁶¹ See Richardson, Statistics of Deadly Quarrels.

⁶² In Journal of Conflict Resolution, Vol. 3 (December, 1959), pp. 326–342 CrossRefGoogle Scholar. In order to appreciate Burns's position fully, the reader would do well to consult Ash, Maurice A., “An Analysis of Power, With Special Reference to International Politics,” World Politics, Vol. 3 (January, 1951), pp. 218–237 CrossRefGoogle Scholar. This article was partially responsible for stimulating the analyses in two other articles by Burns. See his “From Balance to Deterrence: A Theoretical Analysis,” World Politics, Vol. 9 (July, 1957), pp. 494–529 CrossRefGoogle Scholar and “The International Consequences of Expecting Surprise,” World Politics, Vol. 10 (July, 1958), pp. 512–536 CrossRefGoogle Scholar. These two articles, in turn, support the conceptual framework employed in “A Graphical Approach to Some Problems of the Arms Race.”

⁶³ Op. cit., p. 328.

⁶⁴ New York, 1957.

⁶⁵ In World Politics, Vol. 11 (October, 1958), pp. 20–43 CrossRefGoogle Scholar. Another important addition to the literature on game theory and deterrence is Snyder, Glenn H., “Deterrence and Power,” Journal of Conflict Resolution, IV (June 1960), pp. 163–178 CrossRefGoogle Scholar.

⁶⁶ Kaplan, , System and Process, p. 223 Google Scholar. The entire passage is emphasized in the original.

⁶⁷ Cambridge, Harvard University Press, 1960. Schelling originally presented his arguments in a series of essays published in various journals. Some of these essays have been rewritten and incorporated into The Strategy of Conflict. Readers wishing to consult the originals will find them listed on page vii of The Strategy.

⁶⁸ Ibid., p. 83.

⁶⁹ Readers wishing to continue reading in this area will find a helpful discussion and abundant bibliographical leads in Snyder, Richard C. and Robinson, James A., National and International Decision Making (New York, The Institute for International Order, 1961), pp. 118–126 Google Scholar.

⁷⁰ In Journal of Conflict Resolution, Vol. 5 (June, 1961), pp. 167–178 CrossRefGoogle Scholar. Those not familiar with the work of Harary, Cartwright and others on graph theory and balance theory will find a short bibliography at the end of Harary's article.

⁷¹ The work of Herbert Simon, bridging as it does the several fields, is a notable exception to this generalization.

⁷² In Management Science, Vol. 6 (July, 1960), pp. 455–474 CrossRefGoogle Scholar.

⁷³ In Harvard Business Review, Vol. 34 (November-December, 1956). pp. 115–123 Google Scholar.

⁷⁴ In Haire, Mason, ed., Modern Organization Theory (New York, 1959)Google Scholar.

⁷⁵ In Management Science, Vol. 5 (July, 1959), pp. 387–403 CrossRefGoogle Scholar.

⁷⁶ Also relevant to studies of both formal and informal organization is the extensive mathematical literature on small groups. This literature has been organized and reviewed by Coleman, James S., “The Mathematical Study of Small Groups,” in Solomon, Herbert, ed., Mathematical Thinking in the Measurement of Behavior (Glencoe, 1960)Google Scholar.

⁷⁷ In Behavioral Science, Vol. 6 (January, 1961), pp. 72–78 Google Scholar.

⁷⁸ In Behavioral Science, Vol. 6 (April, 1961) pp. 148–152 Google Scholar.

⁷⁹ Ibid., p. 148.

⁸⁰ In this Review, Vol. 51 (March, 1957), pp. 1–12.

⁸¹ In this Review, Vol. 52 (June, 1958), pp. 321–338.

⁸² Kort answers Fisher's criticisms in “Reply to Fisher's Mathematical Analysis of Supreme Court Decisions,” this Review, Vol. 52 (June, 1958), pp. 339–348 Google Scholar.

⁸³ Another more straightforward method of predicting judicial decisions is offered by Nagel, Stuart, “Using Simple Calculations to Predict Judicial Decisions,” The American Behavioral Scientist, Vol. 4 (December, 1960), pp. 24–28 CrossRefGoogle Scholar.

⁸⁴ See Kort, Fred, “The Quantitative Content Analysis of Judicial Opinions,” PROD, Vol. 3 (March, 1960), pp. 11–14 Google Scholar, for a brief discussion of a content analysis-factor analysis approach to the study of sets of Supreme Court cases.

⁸⁵ Glencoe, 1959.

⁸⁶ S. Sidney Ulmer has also used the Shapley-Shubik index for analyzing behavior in the Supreme Court. See his “The Analysis of Behavior Patterns on the United States Supreme Court,” Journal of Politics. Vol. 22 (November, 1960), pp. 629–653 CrossRefGoogle Scholar, particularly pp. 639–640.

⁸⁷ The difficulties of building, operationalizing, and testing mathematical models of political phenomena—particularly the difficulties engendered by lack of relevant data—have been discussed by Duncan MacRae Jr. and R. Duncan Luce. See MacRae's, Dimensions of Congressional Voting, University of California Publications in Sociology and Social Institutions, Vol. 1, No. 3 (Berkeley and Los Angeles, University of California Press, 1958), pp. 354–382 Google Scholar. See Luce's, “Analyzing the Social Process Underlying Group Voting Patterns,” in Burdick, Eugene and Brodbeck, Arthur J., eds., American Voting Behavior (Glencoe, 1959)Google Scholar.