Optimizing group judgmental accuracy in the presence of interdependencies

AbstractConsider 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, pi, 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, wi, 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.

[1]  P.-C.-F. Daunou,et al.  Mémoire sur les élections au scrutin , 1803 .

[2]  Siméon-Denis Poisson Recherches sur la probabilité des jugements en matière criminelle et en matiére civile, précédées des règles générales du calcul des probabilités , 1837 .

[3]  H. Barger The General Theory of Employment, Interest and Money , 1936, Nature.

[4]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[5]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[6]  Claude E. Shannon,et al.  Reliable Circuits Using Less Reliable Relays , 1956 .

[7]  D. Black The theory of committees and elections , 1959 .

[8]  G. Granger La mathématique sociale du marquis de Condorcet , 1958 .

[9]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[10]  W. H. Pierce,et al.  IMPROVING RELIABILITY OF DIGITAL SYSTEMS BY REDUNDANCY AND ADAPTION , 1961 .

[11]  G. Tullock,et al.  The calculus of consent : logical foundations of constitutional democracy , 1962 .

[12]  James M. Buchanan and Gordon Tullock The Calculus of Consent , 1962 .

[13]  L. Shapley Simple games: an outline of the descriptive theory. , 2007, Behavioral science.

[14]  Lloyd S. Shapley,et al.  WEIGHTED VOTING: A MATHEMATICAL ANALYSIS FOR INSTRUMENTAL JUDGMENTS, , 1966 .

[15]  Banzhaf,et al.  Multi-Member Electoral Districts-Do They Violate the "One Man, One Vote" Principle , 1966 .

[16]  Howard E. Sattler Effect of Group Variability on Pooled Group Decisions , 1966 .

[17]  Michael Taylor,et al.  Critique and comment: Proof of a theorem on majority rule , 1969 .

[18]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[19]  D. Rae,et al.  Decision-Rules and Individual Values in Constitutional Choice , 1969, American Political Science Review.

[20]  W. E. Feinberg Teaching the Type I and Type II Errors: The Judicial Process , 1971 .

[21]  Richard G. Niemi,et al.  Probability models of collective decision making , 1972 .

[22]  Norman Schofield,et al.  Ethical Decision Rules for Uncertain Voters , 1972, British Journal of Political Science.

[23]  Alan E. Gelfand,et al.  A Study of Poisson's Models for Jury Verdicts in Criminal and Civil Trials , 1973 .

[24]  J. H. Davis Group decision and social interaction: A theory of social decision schemes. , 1973 .

[25]  Blaine Roberts An Extension of Optimality Criteria: An Axiomatic Approach to Institutional Choice , 1973, Journal of Political Economy.

[26]  Lawrence D. Phillips,et al.  Bayesian Statistics for Social Scientists. , 1973 .

[27]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[28]  Claude Berge Introduction à la théorie des hypergraphes , 1973 .

[29]  Alan E. Gelfand,et al.  Modeling Jury Verdicts in the American Legal System , 1974 .

[30]  H. P. Young,et al.  An axiomatization of Borda's rule , 1974 .

[31]  Bernard Grofman,et al.  A comment on ‘democratic theory: A preliminary mathematical model.’ , 1975 .

[32]  Guillermo Owen,et al.  Evaluation of a Presidential Election Game , 1975, American Political Science Review.

[33]  Michael Hughes Bayesian Statistics for Social Scientists. , 1975 .

[34]  Stuart S. Nagel,et al.  Deductive Modeling to Determine an Optimum Jury Size and Fraction Required to Convict , 1975 .

[35]  Alan E. Gelfand,et al.  Analyzing the Decision-Making Process of the American Jury , 1975 .

[36]  Howard Margolis A note on incompetence , 1976 .

[37]  Charles R. Plott,et al.  Axiomatic Social Choice Theory: An Overview and Interpretation , 1976 .

[38]  K. Baker Condorcet. From natural philosophy to social mathematics , 1974, Medical History.

[39]  Nicholas R. Miller,et al.  Graph- Theoretical Approaches to the Theory of Voting* , 1977 .

[40]  P. Ordeshook Game theory and political science , 1978 .

[41]  Bernard Grofman,et al.  Judgmental competence of individuals and groups in a dichotomous choice situation: Is a majority of heads better than one? , 1978 .

[42]  Andrew Schotter,et al.  The inevitability of the “paradox of redistribution” in the allocation of voting weights , 1978 .

[43]  Reid Hastie,et al.  Models of jury decision making: A critical review. , 1979 .

[44]  H. Nevison,et al.  Structural Power and Satisfaction in Simple Games , 1979 .

[45]  A. Schotter,et al.  Applied Game Theory , 1979, Physica-Verlag HD.

[46]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

[47]  M. Rothschild,et al.  A Model of the Jury Decision Process , 1979, The Journal of Legal Studies.

[48]  D. W. Bunn,et al.  Group Choice , 1980 .

[49]  Nicholas R. Miller A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting , 1980 .

[50]  Andrew Schotter,et al.  Power Relationships in the International Monetary Fund: The Consequences of Quota Changes , 1980 .

[51]  Shmuel Nitzan,et al.  Investment in Human Capital and Social Self Protection under Uncertainty , 1980 .

[52]  Bernard Grofman,et al.  Mathematical Models of Juror and Jury Decision-Making , 1981 .

[53]  Shmuel Nitzan,et al.  Optimal Decision Rules in Uncertain Dichotomous Choice Situations , 1982 .

[54]  Average Competence, Variability in Individual Competence, and Accuracy of Statistically Pooled Group Decisions , 1982 .

[55]  G. Owen,et al.  Thirteen theorems in search of the truth , 1983 .

[56]  William F. Lucas,et al.  Measuring Power in Weighted Voting Systems , 1983 .

[57]  Shmuel Nitzan,et al.  Are qualified majority rules special? , 1984 .

[58]  G. Owen Fair Indirect Majority Rules. , 1984 .

[59]  Scott L. Feld,et al.  Group size and the performance of a composite group majority: Statistical truths and empirical results , 1984 .

[60]  B. Grofman Information Pooling and Group Decision Making , 1986 .