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Optimizing group judgmental accuracy in the presence of interdependencies

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Abstract

Consider a group of people confronted with a dichotomous choice (for example, a yes or no decision). Assume that we can characterize each person by a probability, p i, of making the ‘better’ of the two choices open to the group, such that we define ‘better’ in terms of some linear ordering of the alternatives. If individual choices are independent, and if the a priori likelihood that either of the two choices is correct is one half, we show that the group decision procedure that maximizes the likelihood that the group will make the better of the two choices open to it is a weighted voting rule that assigns weights, w i, such that

$$w_i \propto \log \frac{{p_i }} {{1 - p_i }}.$$

We then examine the implications for optimal group choice of interdependencies among individual choices.

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REFERENCES

  • Arrow, K. (1962). Social choice and individual values. New York: Wiley.

    Google Scholar 

  • Badger, W. (1972). Political individualism, positional preferences, and optimal decision rules. Probability models of collective decision-making. In R. Niemi and H. Weisberg (Eds.), Columbus, Ohio: Merrill.

    Google Scholar 

  • Baker, K.M. (1975). Condorcet: From natural philosophy to social mathematics. Chicago: University of Chicago Press.

    Google Scholar 

  • Baker, K.M. (1967). Scientism, elitism, and liberalism: The case of Condorcet. Studies in Voltaire and the Eighteenth Century 55: 129–156.

    Google Scholar 

  • Banzhaf, J.F. III. (1966). Multi-member electoral districts — Do they violate the ‘one-man-one-vote’ principle? Yale Law Journal 75: 1309–1388.

    Article  Google Scholar 

  • Barry, B. (1969). The public interest. Proceedings of the Aristotelian Society, Supplementary Volume 38: 1–18.

    Google Scholar 

  • Berge, C. (1975). Introduction à la theorie des hypergraphes. Montreal: Montreal University Press.

    Google Scholar 

  • Black, D. (1958). The theory of committees and elections. London: Cambridge University Press.

    Google Scholar 

  • Borda, J.C. (1781). Memoire sur les Elections au Scrutin. Histoire de l'Academie Royale des Sciences. Translated with commentary, de Grazia, 1953.

  • Bordley, R.F. (1980). Aggregating individual probability assessments. Research publication, General Motors Research Laboratories, 30 May 1980, No. GMR-3321.

  • Brams, S.J. (1975). Game theory and politics. New York: Free Press.

    Google Scholar 

  • Buchanan, J., and Tullock, G. (1962). The calculus of consent. Ann Arbor: University of Michigan Press.

    Google Scholar 

  • Condorcet, N.C. de (1785). Essai sur l'application de l'analyse à la probabilité des decisions rendues à la pluralité des voix. Paris.

  • Curtis, R. (1972). Decision rules and collective values in constitutional choice. In R. Niemi and H. Weisberg (Eds.), Probability models of collective decision making. Columbus, Ohio: Merrill.

    Google Scholar 

  • Davis, J.H. (1973). Group decision and social interaction: A theory of social decision schemes. Psychological Review 80 (2) (March): 97–125.

    Article  Google Scholar 

  • Dreyer, J., and Schotter, A. (1978). Power relationships in the international monetary fund: The consequences of quota changes. Unpublished Manuscript.

  • Dubey, P., and Shapley, L.S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4 (2) (May): 99–131.

    Article  Google Scholar 

  • Duda, R.O., and Hart, P.E. (1973). Pattern classification and scene analysis. New York: Wiley.

    Google Scholar 

  • Farquharson, R. (1969). Theory of voting. New Haven, Conn.: Yale University Press.

    Google Scholar 

  • Feinberg, W.E. (1971). Teaching type I and type II errors: The judicial process. The American Statistician, June: 30–32.

  • Fischer, D., and Schotter, A. The paradox of redistribution in the allocation of voting weights. Public Choice. Forthcoming.

  • Gelfand, A., and Solomon, H. (1973). A study of Poisson's models for jury verdicts in criminal and civil trials. Journal of American Statistical Association 68 (June): 271–278.

    Article  Google Scholar 

  • Gelfand, A., and Solomon, H. (1974). Modelling jury verdicts in the American legal system. Journal of American Statistical Association 69 (March): 32–37.

    Article  Google Scholar 

  • Gelfand, A., and Solomon, H. (1975). Analyzing the decision-making process of the American jury. Journal of the American Statistical Association 70 (June): 305–310.

    Article  Google Scholar 

  • Gelfand, A., and Solomon, H. (1977). Comparing juries: An argument in favor of twelve-member juries. Jurimetrics 17: 292–313.

    Google Scholar 

  • Gelfand, A., and Solomon, H. (1977). An argument in favor of twelve-member juries. In S. Nagel (Ed.), Modelling the Criminal Justice System, Sage Criminal Justice System Annual 7: 205–220. Beverly Hills, Calif.: Sage Publications.

    Google Scholar 

  • Gillespie, C.C. (1972). Probability and politics: LaPlace, Condorcet, and Turgot. Proceedings of the American Philosophical Society 116 (1) (Feb.): 1–20.

    Google Scholar 

  • Grainger, G.G. de (1956). La mathematique sociale du Marquis de Condorcet. Paris: Presses Universitaires de France.

    Google Scholar 

  • Grofman, B. (1975). Mathematics and politics: Mathematical reasoning and optimal jury decision processes. Plus, Reply. In M. Black (Ed.), Problems of choice and decision: Proceedings of a colloquium held in Aspen Colorado, June 24–July 6, 1974, 331–337; 544–547. Cornell University Program on Science, Technology and Society and Aspen Institute for Humanistic Studies: Photo-offset.

  • Grofman, B. (1975). A comment on democratic theory: A preliminary mathematical model. Public Choice 21 (Spring): 100–103.

    Article  Google Scholar 

  • Grofman, B. (1978). Judgmental competence of individuals and groups in a dichotomous choice situation. Journal of Mathematical Sociology 6 (1): 47–60.

    Article  Google Scholar 

  • Grofman, B. (1979). The case for majority jury verdicts. Trial Magazine, December: 23–25, 29, 47–48.

    Google Scholar 

  • Grofman, B. (1980a). Preliminary models of jury decision-making. In G. Tullock (Ed.), Frontiers of Economics 3. Blacksburg, Va.: Center for Study of Public Choice, Virginia Polytechnic Institute and State University.

    Google Scholar 

  • Grofman, B. (1980b). Mathematical modeling of jury/juror decision making. In B.D. Sales (Ed.), Perspectives in law and psychology, Volume II: The jury judicial and trial processes, 305–351, Plenum.

  • Grofman, B. (1981). Predicting the outcomes of a Keynesian beauty context. Unpublished manuscript.

  • Grofman, B., and Owen, G. (Eds.), Information pooling and group decision making. Westport, Conn.: JAI Press. 1984 forthcoming.

  • Grofman, B., and Owen, G. (1983). Thirteen theorems in search of the truth. Theory and Decision 15: 261–278.

    Article  Google Scholar 

  • Grofman, B., Owen, G., and Feld, L. (1982). Average competence, variability in competence, and the accuracy of statistically pooled group decision making. Psychological Reports 50: 683–688.

    Article  Google Scholar 

  • Grofman, B., and Scarrow, H. (1979). Game theory and the U.S. Courts. In S. Brams, G. Schwödiauer and A. Schotter (Eds.), Applied game theory. Vienna: Springer-Verlag.

    Google Scholar 

  • Grofman, B., Feld, S., and Owen, G. Group size and the performance of a composite group majority on multiple item decision tasks. Organizational behavior and human performance. 1984 forthcoming.

  • Keynes, J.M. (1936). General theory of employment, interest, and money. New York: Harcourt, Brace, and World.

    Google Scholar 

  • Klevorick, A.K., and Rothschild, M. (1978). The jury decision process. Grant proposal to the National Science Foundation.

  • Klevorick, A.K., and Rothschild, M. (1979). A model of the jury decision process. Journal of Legal Studies, January: 141–164.

  • Lucas, W.F. (1974). Measuring power in weighted voting systems. In Case studies in applied mathematics, 42–106. Mathematics Association of America.

  • Luce, R.D. (1959). Individual choice behavior. New York: Wiley.

    Google Scholar 

  • Margolis, H. (1976). A note on incompetence. Public Choice 26: 119–129.

    Article  Google Scholar 

  • Miller, N.R. (1977). Graph-theoretical approaches to the theory of voting. American Journal of Political Science 21 (November).

  • Miller, N.R. (1980). A new solution set for tournaments and majority voting. American Journal of Political Science 24.

  • Minsky, M., and Papert, S. 1969, Perceptrons: An introduction to computational geometry. Cambridge: MIT Press.

    Google Scholar 

  • Mirkin, B. (1979). Group choice. New York: Halsted Press.

    Google Scholar 

  • Moore, E.E., and Shannon, C.E. (1956a). Reliable circuits using less reliable relays, I. Franklin Institute, Philadelphia Journal 262: 191–208.

    Article  Google Scholar 

  • Moore, E.E., and Shannon, C.E. (1956b). Reliable circuits using less reliable relays, II. Franklin Institute, Philadelphia Journal 262: 281–297.

    Article  Google Scholar 

  • Nagel, S.S., and Neef, M. (1975). Deductive modeling to determine an optimum jury size and fraction required to convict. Washington University Law Review, 933–978.

  • Nevison, C.H. (1979). Structural power and satisfaction in simple games. In S. Brams, G. Schwödiauer and A. Schotter (Eds.), Applied game theory. Vienna: Springer-Verlag.

    Google Scholar 

  • Niemi, R., and Weisberg, H. (1972). The effects of group size on collective decision making. In Probability models of collective decision-making. Columbus, Ohio: Merrill.

    Google Scholar 

  • Nitzan, S., and Paroush, J. (1980a). Discussion Paper #8017. Bar-Ilan University, Israel.

  • Nitzan, S., and Paroush, J. (1980b). Investment in human capital and social self-protection under uncertainty. International Economic Review 21: 547–558.

    Article  Google Scholar 

  • Nitzan, S., and Paroush, J. (1982). Optimal decision rules in uncertain dichotomous choice situations. International Economic Review 23 (2) (June): 289–297.

    Article  Google Scholar 

  • Nitzan, S., and Paroush, J. (1984). Are qualified majority rules special? Public Choice 42 (3): 257–272.

    Article  Google Scholar 

  • Owen, G. (1975). Evaluation of a presidential election game. American Political Science Review 49: 947–953.

    Article  Google Scholar 

  • Owen, G. (1983). ‘Fair’ indirect majority rules. Unpublished manuscript.

  • Owen, G., Grofman, B., and Feld, S. (1981). A theorem on the optimal distribution of competence within a group. Unpublished manuscript.

  • Penrod, S., and Hasite, R. (1979). Models of jury decision-making: A critical review. Psychological Bulletin 86: 462–492.

    Article  Google Scholar 

  • Philips, L.D. (1973). Bayesian statistics for social scientists. New York: Orwell.

    Google Scholar 

  • Pierce, W. (1961). Improving reliability of digital systems by redundancy and adaptation. Ph.D. Thesis in Electrical Engineering, Stanford University.

  • Pinkham, R.S., and Urken, A.B. (1981). Competence and the choice of a voting system. Unpublished manuscript.

  • Plott, C. (1976). Axiomatic social choice theory: An overview and interpretation. American Journal of Political Science 20 (August).

  • Poisson, S.D. (1837). Recherches sur la probabilité des jugement en matière criminal et en matière civile: Précédées des règles générales du calcul des probabilités. Paris: Bachelier, Imprimateur Libraire.

    Google Scholar 

  • Rae, D.W. (1969). Decision-rules and individual values in constitution choice. American Political Science Review 63 (March): 40–56.

    Article  Google Scholar 

  • Riker, W.H., and Shapley, L.S. (1968). Weighted voting: A mathematical analysis for instrumental judgment. In J.R. Pennock and J.W. Chapman (Eds.), Representation. New York: Atherton Press.

    Google Scholar 

  • Roberts, B. (1973). Extension of optimality criteria—Axiomatic approach to institutional choice. Journal of Political Economy 81 (2): 386–400.

    Article  Google Scholar 

  • Sattler, H. (1966). Effect of group variability on pooled group decisions. Psychological Reports 18 (3): 676–678.

    Article  Google Scholar 

  • Schofield, N. (1971). Ethical decision rules for uncertain voters. British Journal of Political Science 2: 193–207.

    Article  Google Scholar 

  • Schofield, N. (1972). Is majority rule special? In R. Niemi and H. Weisberg (Eds.), Probability Models of Collective Decision-Making. Columbus, Ohio: Merrill.

    Google Scholar 

  • Schotter, A. (1979). Voting weights as power proxies. In S. Brams, G. Schwödiauer and A. Schotter (Eds.), Applied game theory. Vienna: Physica-Verlag.

    Google Scholar 

  • Shapley, L.S. (1962). Simple games: An outline of the descriptive theory. Behavioral Science 7: 59–66.

    Article  Google Scholar 

  • Shapley, L.S., and Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review 48: 787–792.

    Article  Google Scholar 

  • Straffin, P.D. (1978). Probability models for power indices. In P.C. Ordeshook (Ed.), Game Theory and Political Science, 477–510. New York: New York University Press.

    Google Scholar 

  • Taylor, M. (1969). Proof of a theorem on majority rule. Behavioral Science 14 (May): 228–231.

    Article  Google Scholar 

  • Thom, G. (1979). Bringing the left back home. New Haven: Yale University Press.

    Google Scholar 

  • Walbert, T.D. (1971). Note: Effect of jury size on probability of conviction — An evaluation of Williams v. Florida. Case Western Reserve Law Review 22: 529–555.

    Google Scholar 

  • Young, H.P. (1974). An axiomatization of the Borda Rule. Journal of Economic Theory 9: 43–52.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Economics, University of California at Los Angeles, 90024, CA

    Lloyd Shapley

  2. School of Social Sciences, University of California at Irvine, 92717, Irvine, CA

    Bernard Grofman

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  1. Lloyd Shapley
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  2. Bernard Grofman
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Additional information

This research was supported by NSF Grant # SES 80-07915. We acknowledge the assistance of Professor Thomas Cover, Department of Statistics, Stanford University, in identifying references to earlier results related to the key theorem in this paper; the assistance of the staff of the Word Processing Center of the School of Social Sciences, University of California, Irvine, and of Sue Pursche and Laurel Eaton for typing, editing, and bibliographic search for this manuscript; and the helpful suggestions of three anonymous referees.

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Shapley, L., Grofman, B. Optimizing group judgmental accuracy in the presence of interdependencies. Public Choice 43, 329–343 (1984). https://doi.org/10.1007/BF00118940

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