Abstract
Delphi, as a procedure for aggregating judgments under uncertainty, has suffered from the lack of an underlying theoretical framework, especially one that related group estimates to decision processes. Attempts to introduce group judgment into existing theories of decision have run into difficulties exemplified by the Arrow impossibility theorem for group preferences, and an analogous theorem by the author demonstrating the non-existence of a general method of aggregating probability estimates.
It is shown that consistent group preference functions can be formulated by the use of anchored scales, i.e., indivdual preference scales with fixed reference objects. No general resolution of the aggregation problem for probabilities appears feasible, but a justification for the use of group probability judgments can be made, based on a family of theorems to the effect that the accuracy of a group judgment is always greater than (or at worst equal to) the average accuracy of the individual judgments. Some empirical data, and some analytical results, indicate that these aggregation rules are more generally applicable, and more powerful than has been assumed in the past.
This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Office of Naval Research Under Contract No. N00014-69-A-0200-4056/452. Reproduction in whole or in part is permitted for any purposes of the United States Government.
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Notes and References
Vide, Howard Raiffa, Decision Analysis, Addison-Wesley, Reading, Mass., 1968, or
Ronald Howard, “The Foundations of Decision Analysis,” IEEE Transactions on Systems Science and Cybernetics, Vol. SSC-4, No. 3. September 1968, pp. 211–219.
This suggestion has been made by a number of contributors to decision analysis, including Harold Raiffa, and Ward Edwards. Cf. Ralph L. Keeney and Craig W. Kirkwood, “Group Decision Making Using Cardinal Social Welfare Functions,” Technical Report No. 83, Operations Research Center, MIT, Oct. 1973.
Arrow, Kenneth, Social Choice and Individual Values, John Wiley and Sons, New York, 1951.
Dalkey, N., “An Impossibility Theorem for Group Probability Functions,” The RAND Corporation, P-4862, June 1972.
This example is similar to one discussed in Raiffa, op. cit.
This topic is explored in the Introduction to Dalkey, N., et. al., Studies in the Quality of Life, D.C. Heath, Lexington, Mass., 1972.
Plott, Charles R. and Michael E. Levine, “On Using the Agenda to Influence Group Decisions: Theory, Experiments, and Applications,” Presented at the Interdisciplinary Colloquium, Western Management Science Institute, UCLA, Jan. 1974.
Dalkey, N. Group Decision Analysis, forthcoming, c.f., Keeney, op. cit., and L.S. Shapley and M. Shubik, “Game Theory in Economics—Chapter 4: Preferences and Utility,” The RAND Corp., R-904/4-NSF, Dec. 1974.
von Neumann, J., and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, N.J., 1947.
Nash, J.F., “The Bargaining Problem,” Econometrica, Vol. 18, No. 2, Apr. 1950.
Dalkey, N., “Toward a Theory of Group Estimation,” in Linstone, H., and M. Turoff, The Delphi Method: Techniques and Applications, Addison-Wesley, Reading Mass., 1975.
Savage, L.J., “Elicitation of Personal Probabilities and Expectations,” J. Amer. Stat. Assoc., Vol. 66, Dec. 1971, pp. 783–801.
Brown, Thomas, “Probabilistic Forecasts and Reproducing Scoring Systems,” The RAND Corp., RM-6299-AEPA, July 1970.
Shuford, Emir H., Jr., Albert Arthur, and H. Edward Massengill, “Admissible Probability Measurement Procedures,” Psychometrika, 31, June 1966, pp. 125–145.
Cf., Savage, op. cit.
Dalkey, in Linstone and Turoff, op. cit.
Savage, op. cit.
Raiffa, H., “Assessments of Probabilities,” unpublished, 1969.
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Judea Pearl has obtained cognate results working with a different decision matrix, namely E Not E A 0 x B y 0 and a different formulation of the distribution function, namely identical, independent distributions on x and y, so that g(x,y) = f(x) f(y). In some respects this is a more general formulation than the gambling model since x and y are arbitrary payoffs. On the other hand, the assumption of independence is a definite restriction. The form of the distribution f that generates a given scoring scheme is, of course, quite different from the distribution D(u) that generates the same scoring scheme for the gambling model. Vide, Pearl, Judea, “An Economic Basis for Certain Methods of Evaluating Probabilistic Forecasts,” UCLA-ENG-7561, School of Engineering and Applied Science, University of California at Los Angeles, July 1975.
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Dalkey, N.C. (1976). Group Decision Analysis. In: Zeleny, M. (eds) Multiple Criteria Decision Making Kyoto 1975. Lecture Notes in Economics and Mathematical Systems, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45486-8_3
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DOI: https://doi.org/10.1007/978-3-642-45486-8_3
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