ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR PROGRESS REPORT THE MATHEMATICS OF MEASUREMENT BY PARTIAL ORDERING By C. H..9B.. and. R. M.- THRALL Project M965 OFFICE OF NAVAL RESEARCH, U. S. NAVY DEPARTMENT CONTRACT NO. Nonr 374(00), NR 041-011 June 1, 1952

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ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN PROGRESS REPORT FOR CONTRACT Nonr:371+00 THE MATHEMATICS OF MEASUREMENT BY PARTIAL ORDERING A. REPORT OF C. H. COOMBS 1) A General Theory of Psychological Scaling: this monograph is in press and I have spent considerable time recently in editorial tasks such as proof reading. 2) On the Method of Single Stimuli: this is a further development of the above scaling theory in the context of the Method of Single Stimuli. Conventional methods of analyzing multiple alternative items by Guttman scalogram analysis can be shown to infore certain conditions over and above unidimentionality. A new method is developed which avoids these extra conditions. 3) Measurement of a Social Utility: within the context of the general theory of scaling above a social utility is developed under the condition of a single latent attribute, which gives each individual an equal vote but weights individual utilities according to the strength of the preference. This requires the assumption of the existence of a common unit of measurement but it does not need to be computed or estimated to provide a simply ordered social utility. I have given several talks on this at the Cowles Commission Seminar and Staff Meeting. B. REPORT OF R, M. THRALL I+) Utility theory. I have been engaged in investigation of certain aspects of the axiomatic foundations of utility theory. This is in part a continuation of work which I began at RAND in the summer of 1951.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The starting point is the set of axioms used by Von Neumann and Morgenstern, and the method is to study what remains when these axioms are weakened in various ways. Of course, each weakening of the axiom results in a theory which has broader scope than the original one. The first part of this work has been described in several RAND research memoranda: Hausner, Melvin An Embedding of a Mixture Space in a Vector Space RM-697, October 1951. Hausner, M., and J. G. Wendel An Embedding of a Utility Space in an Ordered Vector Space RM-698, 4 October 1951. Hausner, M., and J. G. Wendel Ordered Vector Spaces RM-716 5 November 1951. Thrall, R. Mo. and N. C. Dalkey A Generalization of Numerical Utilities - I RM-724, 16 November 191.7 Currently I am considering generalization of the axiom of substitutibility of indifferent events in probability mixtures. I have discussed this work at a Staff Meeting of the Cowles Commission. 5) Social organizations. Professor R. C. Angell of the University of Michigan, Department of Sociology, proposed the following problem. Consider two organizations A and B (e.g., Rotary and Masons) in a community. We wish to know which organization is "higher" in the connunity. Suppose that organization A is divided into three subgroups Al, A2, A3 of equal size where Al is is made up of the most active members (officers, committee members, etc.) of A; A2 is the middle strata; and A3 the bottom strata of A. Let B1, B2, B3 be a similar stratification of B into equal subclasses. Let Sij - Order Ain B /Order An B and let Rx 1I. ij I The matrix R gives information about the relative social "height" of the two organizations; one could use the sign of the function f(R)- Eij(i-j)S ij to measure which is higher. The order thus obtained would not be transitive but might serve as a basis for some more refined order relation.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The subdivision into three equal parts can readily be generalized to the case of n equal parts. The further generalization to the case of n unequal parts creates a new problem which I call the bias problem. For example, if A = B and Al 1 B1, AlU A23 B1 U B2 then we get E(i-j) 5 ij <0 which would give A higher than itself. Apparently one should first adjust the matrix R to a new matrix R* in such a way as to compensate for the unequal subdivisions and then use f (R*) for the order criterion. I have devised two methods for doing this each of which has the desirable property f(R*5 ~ 0 for A = B, but neither of which is entirely satisfactory in that they do not yield R* =-1 I n when A =B. Details of this study will be continued in a working paper to be written soon. C. GENERAL REPORT 6) Experimental program. The work on utility theory and other studies in game theory indicate the need for experimental work to test the extent of validity of various axioms. The Ford Foundation has made a grant to support a seminar in the coming summer to design experiments in this field. This seminar and the ensuing experimental program should be of great help in the theoretical studies covered by this ONR contract. Conversely, the theoretical studies will help give direction to the summer seminar, and we have been much occupied this spring semester in various preliminaries to the summer program. 7) Mr. Walter Feit has continued his study of the problem of mapping various partly ordered sets onto simply ordered sets, and of related problems. 8) We are continuing our study on a general model for measurement theory and now have a rough draft of the first part. 9) Distribution list. Enclosed is a distribution list for technical reports submitted for your approval. 10) The paper on the lattice theoretic analogue of the midpoint problem now available in reprint form. A counterexample to the conjective mentioned has been found and this has lead to a renewed attack on the question as to which orders of the midpoints of n ordered points are possible. The lattice problem gives an upper bound

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which is more known to be too large for n > 5. 11) Interdepartmental seminar. Summaries of recent meeting enclosed.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Office of Naval Research Distribution List for Project Number M 965 Office of Naval Research 1 Dr. Dorothy C. Adkins 1 Branch Office Department of Psychology 150 Causeway Street University of North Carolina Boston, Massachusetts Chapel Hill, North Carolina Office of Naval Research 1 Dr. K. J. Arrow 1 Branch Office Economics Department 844 North Rush Street Stanford University Chicago 11, Illinois Stanford, California Office of Naval Research 1 Dr. Harold P. Bechtoldt 1 Branch Office Department of Psychology 34+6 Broadway State University of Iowa New York 13, New York Iowa City, Iowa Office of Naval Research 1 Dr. E. F. Beckenbach 1 Branch Office University of California 1030 East Green Street Los Angeles 24, California Pasadena I, California Dr. H. F. Bohnenblust 1 Office of Naval Research 1 California Institute Branch Office of Technology 1000 Geary Street Pasadena 4, California San Francisco 9, California Dr. S. S. Cairns, Head 1 Office in Charge 2! Department of Mathematics Office of Naval Research University of Illinois Navy #100 Urbana, Illinois Fleet Post Office New York, New York Dr. John B. Carroll Peabody House Office of Naval Research 10 Kirkland Street Logistics Branch, Code 436 Cambridge 38, Massachusetts Navy Department, T3-Building Washington 25, D. C. Dr. C. H. Coombs Department of Psychology Acting Chief, Planning 1 University of Michigan Research Branch Ann Arbor, Michigan Office of Comptroller of the Army Dr. Herbert S. Conrad 1 Pentagon Building U. S. Office of Education Washington 25, D. C. Washington 25, D. C. Attn: Mr. Sidney Kaplan

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Dr. Arthur H. Copeland, Sr. 1 Dr. Harold 0. Gulliksen 1 Department of Mathematics Department of Psychology University of Michigan Eno Hall Ann Arbor, Michigan Princeton University Princeton, New Jersey Dr. Lee J. Cronbach 1 1007 South Wright Street Dr. Theophil H. Hildebrandt 1 Champaign, Illinois Department of Mathematics University of Michigan Dr. Edward E. Cureton 1 Ann Arbor, Michigan 1907 Lake Avenue Knoxville 16, Tennessee Mr. Charles J. Hitch 1 RAND Corporation Dr. John H. Curtiss 1 1500 Fourth Street Bureau of Standards Santa Monica, California Washington, D. C. Dr. Paul Horst Dr. George B. Dantzig 1 Department of Psychology Air Comptroller's Office University of Washington Headquarters, U, S. Army Seattle, Washington Air Force Washington 25, D. C. Dr. Harold Hotelling 1 Department of Mathematical Dr, Allen L. Edwards 1 Statistics Department of Psychology University of North Carolina University of Washington Chapel Hill, North Carolina Seattle 5, Washington Dr. Alston S. Householder 1 Dr. Leon Festinger 1 Oak Ridge National Laboratory Department of Psychology Oak Ridge, Tennessee University of Minnesota Minneapolis, Minnesota Dr. Leonid Hurwicz 1 University of Minnesota Dr. Merrill M. Flood 1 Minneapolis, Minnesota RAND Corporation 1500 Fourth Street Dr. E. Lowell Kelly 1 Santa Monica, California Department of Psychology University of Michigan Dr. David A. Grant 1 Ann Arbor, Michigan Department of Psychology University of Wisconsin Dr. John L. Kennedy 1 Madison 6, Wisconsin RANID Corporation 1500 Fourth Street Dr. J. P. Guilford 1 Santa Monica, California P. 0. Box 1134 Beverly Hills Dr. Tjalling C. Koopmans 1 California Cowles Commission University of Chicago Chicago 37, Illinois 6

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Dr. Paul F. Lazarsfeld 1 Naval Research Laboratory 9 Department. of Sociology Washington 25, D. C. Columbia University New York 27, New York Mr. M. L. Norden 1 Operations Research Office Dr. Sebastian B. Littauer 1 The John Hopkins University Industrail Engineering 6410 Connecticut Avenue Department Chavy Chase, Maryland Columbia University New York, New York Dr, Howard Raiffa 1 Department of Mathematics Logistics Research Project 2 Columbia University George Washington University New York, New York 707 22nd Street, N.W. Washington 7, D. C. Dr. George Y. Rainich Department of Mathematics Dr. Irving Lorge 1 University of Michigan Box 130 Ann Arbor, Michigan 525 West 120th Street New York 25 New York RAND Corporation Library 1 1500 Fourth Street Dr. W. G. Madow I Santa Monica, California Department of Mathematics University of Illinois Dr. Arnold E. Ross, Head 1 Urbana, Illinois Department of Mathematics University of Notre Dame Dr. Donald G. Marquis 1 Notre Dame, Indiana Department of Psychology University of Michigan Dr. Phillip J. Rulon 1 Ann Arbor, Michigan 13 Kirkland Street Cambridge 38, Massachusetts Dr. Jacob Marschak 1 Cowles Commission Dr. Paul A. Sammuelson 1 University of Chicago Department of Economics Chicago 37, Illinois Massachusetts Institute of Technology Dr. Quinn McNemar 1 Cambridge 39, Massachusetts Department of Psychology Stanford University Dr. L. J. Savage 1 Stanford, California University of Chicago Chicago 37, Illinois Dr, George A. Miller 1 Department of Psychology Dr. Samuel A. Stouffer 1 Massachusetts Institute Laboratory of Social of Technology Relations Cambridge 39, Massachusetts 303 Emerson Hall Harvard University Dr. Oskar Morgenstern I Cambridge 38, Massachusetts Department of Economics and Social Institutions Dr. Calvin W. Taylor 1 Princeton University Department of Psychology Princeton, New Jersey University of Utah Salt Lake City, Utah - ~~~7

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Dr. Robert L. Thorndike 1 Weapons System 1 Teachers College Evaluation Group Columbia University Pentagon Building 525 West 120th Street Washington 25, D. C. New York 27, New York Dr. Robert J. Wherry 1 Dr. R. M. Thrall I Department of Psychology Department of Mathematics Ohio State University University of Michigan Columbus 10, Ohio Ann Arbor, Michigan Dr. Raymond L. Wilder 1 Dr. L. L. Thurstone 1 Department of MathematicsUniversity of Chicago University of Michigan Chicago 37, Illinois Ann Arbor, Michigan Dr. C. B. Tompkins 1 Dr. Dael L. Wolfle 1 RAND Corporation Commission on Human Resources 1500 Fourth Street and Advanced Training Santa Monica, California 2101 Constitution Avenue, N.W. Washington 25, D. C. Dr. A. W. Tucker 1 Fine Hall, Box 708 Dr. Jacob Wolfowitz 1 Princeton University Department of Mathematics Princeton, New Jersey Cornell University Ithaca, New York Dr. Ledyard R. Tucker 1 Educational Testing Service Mr. Marshall K. Wood 1 20 Nassau Street Air Comptroller's Office Princeton, New Jersey Headquarters, U, S. Army Air Force Dr. Von Neumann 1 Washington 25, D. C. Institute for Advanced Study Princeton University Dr. Max A. Woodbury 1 Princeton, New Jersey Department of Mathematics Stanford University Dr. D. F. Votaw, Jr. 1 Stanford, California Department of Mathematics Yale University New Haven, Connecticut Dr. W. Allen Wallis 1 Committee on Statistics 207 Haskell Hall University of Chicago Chicago 37, Illinois Dr. J. L. Walsh 1 Department of Mathematics Harvard University Cambridge 38, Massachusetts 8

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