Professor R. Duncan Luce died on August 11, 2012. His website is being maintained here a service to the academic community. Professor Luce remained an active scholar right up to his death and his record here has been updated to reflect posthumously published work.

R. Duncan Luce1925-2012

The representational theory of measurement concerns the types of data that can be summarized in some numerical way. Much general theory has been developed ( exposited in Foundations of Measurement, vols. 1,2,3, 1971, 1989, 1990, with D.M. Krantz, P Suppes, & A. Tversky) and still is being actively explored. Although some of my efforts continue on general topics of measurement, I have mostly been applying since 1990 some of these ideas to individual decision making, where the numerical measures are called utility and subjective probability or weights, and to global psychophysics spanning full dynamic ranges of intensity such as loudness. Recently, with colleagues, I have examined the kinds of behavioral laws that link riskless and risky utility, in particular, are developing theories that address the issue of the utility of gambling. Accompanying the theoretical work is an empirical program of Michael Birnbaum in which these plausible behavioral properties are tested in computer-based, laboratory experiments. In the auditory domain, Ragnar Steingrimsson and I have extensively and successfully evaluated the psychophysical model including the forms for psychophysical and weighting functions.Currently we are working on the time-order error, and he is evaluating the general model in the brightness domain. Many tricky and interesting questions arise about how best to evaluate these properties.

Ph.D. Massachusetts Institute of Technology, 1950 (Mathematics); Editorial Board: British Journal of Mathematical Social Sciences and Statistical Psychology, Journal of Mathematical Psychology, Mathematical Social Sciences Psychological Review; Fellow: Society of Experimental Psychologists; Member: American Academy of Arts and Sciences, National Academy of Sciences, American Philosophical Society; American Psychological Association Distinguished Scientific Contribution Award, 1970; National Medal of Science, 2003. Honors.

Curriculum vitae Pre-1990 publications here



Publications since 1990


(1990) (- - -, Krantz, D. H., Suppes, P., & Tversky, A.) Foundations of Measurement, Vol. III, Academic Press
(1993) Sound & Hearing, Hillsdale, NJ: Erlbaum.
(2000) Utility of Gains and Losses: Measurement-Theoretic and Experimental Approaches., Mahwah, NJ: Lawrence Erlbaum Association. See Errata

Edited Books

(1995) (- - -, D’Zmura, M., Hoffman, D, Iverson, G., & Romney, A.K., Eds.) Geometric representations of Perceptual Phenomena: Papers in honor of Tarow Indow on his 70th birthday. Hillsdale, NJ: Lawrence Erlbaum Associates.

Book Reviews

Scientific Papers


(a) “On the possible psychophysical laws” revisited: Remarks on cross-modal matching. Psychological Review, 97, 66-77.
(b) (Bostic, R., Herrnstein, R. J. & - - -) The effect on the preference-reversal phenomenon of using choice indifferences. Journal of Economic Behavior and Organization, 13, 193-212.
(c) Rational versus plausible accounting equivalences in preference judgments. Psychological Science, 1, 225-234; and in W. Edwards (Ed.) (1992) Utility Theories: Measurements, and Applications. Boston: Kluwer Academic Publishers. Pp 187-206.
(d) (Narens, L. & - - -). Three aspects of the effectiveness of mathematics in science. In R. Mickens (Ed.) Mathematics and Science. World Scientific Press. Pp. 122-135.


(a) Rank- and sign-dependent linear utility models for binary gambles. Journal of Economic Theory, 53, 75-100.
(b) (- - -, & Fishburn, P.C.) Rank- and sign-dependent linear utility models for finite first-order gambles. Journal of Risk and Uncertainty, 4, 29-59.
(c) What is a ratio in ratio scaling? In S.J. Bolanowski & G.A. Gescheider (Eds.) Ratio Scaling of Psychological Magnitudes: In Honor of the Memory of S.S. Stevens. Hillsdale, NJ: Erlbaum. Pp. 8-17.


(a) Where does subjective expected utility fail descriptively? Journal of Risk and Uncertainty, 5, 5-27.
(b) (- - -, & Narens, L.) Intrinsic Archimedeanness and the continuum. In C.W. Savage & P. Ehrlich (Eds.) Philosophical and Foundational Issues in Measurement Theory. Hillsdale, NJ: Erlbaum. Pp. 15-38.
(c) A theory of certainty equivalents for uncertain alternatives. Journal of Behavioral Decision Making, 5, 201-216.
(d) Singular points in generalized concatenation structures that otherwise are translation homogeneous. Mathematical Social Science, 24, 79-103.
(e) A path taken: Aspects of modern measurement theory. In A. F. Healy, S. Kosslyn, & R. Shiffrin (Eds.) From Learning Theory to Connectionist Theory: Essays in Honor of William K. Estes, Vol. 1. Hillsdale, NJ: Erlbaum. Pp. 45-64.
(f) (Hunt, E., & - - -) SOAR as a world view, not a theory. Behavior and Brain Science, 15, 447-448.


(a) (- - -, Mellers, B., & Chang, S.-J.) Is choice the correct primitive? On using certainty equivalents and reference levels to predict choices among gambles. Journal of Risk and Uncertainty, 6, 115-143.
(b) (- - -, & Narens, L.) Comments on the "non-revolution" in the representational theory of measurement. Psychological Science, 4, 127-130.
(c) Let's not promulgate either Fechner's erroneous algorithm or his unidimensional approach. Behavior and Brain Sciences, 16, 155-156.
(d) Reliability is neither to be expected nor desired in peer review. Behavioral and Brain Sciences, 14, 399-400.


(a) Thurstone and sensory scaling: then and now. Psychological Review, 107, 271-277.
(b) (- - -, & von Winterfeldt, D. What common ground exists for descriptive, prescriptive, and normative utility theories. Management Science, 40, 263-279.
(c) (- - -, & Narens, L.) Fifteen problems in the representational theory of measurement. In P. Humphreys (Ed.), Patrick Suppes: Scientific Philosopher, Vol. 2: Philosophy of Physics, Theory Structure, Measurement Theory, Philosophy of Language, and Logic. Dordrecht: Kluwer Academic Publishers. Pp. 219-245. Comments by Patric Suppes.
(d) (Cho, Y., - - -, & von Winterfeldt, D.) Tests of assumptions about the joint receipt of gambles in rank- and sign-dependent utility theory. Journal of Experimental Psychology: Human Perception and Performance, 20, 931-943.
(e) (Chung, N.-K., von Winterfeldt, D., & - - -) An experimental test of event commutativity in rank-dependent utility theory. Psychological Science, 5, 394-400.


(a) Joint receipt and certainty equivalents of gambles. Journal of Mathematical Psychology, 39, 73-81.
(b) (Fishburn, P.C., & - - -) Joint receipt and Thaler's hedonic editing rule. Mathematical Social Sciences, 29, 33-79.
(c) Four tensions concerning mathematical modeling in psychology. Annual Reviews of Psychology, 46, 1-26.
(d) (- - -, & Fishburn, P.C.) A note on deriving rank-dependent utility using additive joint receipts. Journal of Risk and Uncetainty, 11, 5-16
(e) (Cho, Y., & - - -) Tests of hypotheses about certainty equivalents and joint receipt of gambles. Organizational Behavior and Human Decision Processes, 64, 229-248.


(a) Commentary on aspects of Lola Lopes paper. Organizational Behavior and Human Decision Processes, 65. 190-193.
(b) The ongoing dialogue between empirical science and measurement theory. Journal of Mathematical Psychology, 40, 78-98.
(c) (Aczel, J., - - -, & Maksa, Gy.) Solutions to three functional equations arising from different ways of measuring utility. Journal of Mathematical Analysis and Applications, 204, 451-471.
(d) When four distinct ways to measure utility are the same. Journal of Mathematical Psychology, 40, 297-317.


(a) (von Winterfeldt, D., Chung, N.-K., - - -, & Cho, Y.) Tests of consequence monotonicity in decision making under uncertainty. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 406-426.

Several unresolved conceptual problems of mathematical psychology. Journal of Mathematical Psychology, 41, 79-87.

(c) Associative joint receipts. Mathematical Social Sciences, 34, 51-74.
(d) The past seven years: 1998-95. In A.A.J. Marley (Ed.) Choice, Decision, and Measurement: Essays in Honor of R. Duncan Luce. Mahwah, NJ: Lawrence Erlbaum Associates. Pp. 3-14.
(d) Quantification and symmetry: Commentary on Michell, Quantitative science and the definition of measurement in psychology. British Journal of Psychology, 88,395-398.


(a) (Iverson, G., & - - -) The representational measurement approach to psychophysical and judgment problems. In M.H. Birnbaum (Ed.) Measurement, Judgment, and Decision Making. San Diego: Academic Press. Pp. 1-79.
(b) Coalescing, event commutativity, and theories of utility. Journal of Risk and Uncertainty, 16, 87-114. Errata (1999, p. 99).
(c) (Pales, Z., Volkman, P., & ---) Heyers-Ulam stability of functional equations with a square-symmetric operation. Proceedings of the National Academy of Sciences, 95, 12772-12775.


(a) On the interplay of riskless and risky utility. In J. Shanteau, B.A. Mellers, & D. Schum (Eds). Decision Science and Technology: Reflections on the Contributions of Ward Edwards. Norwell, MA: Kluwer Academic Publishers. Pp. 9-26.
(b) Binary gambles of a gain and a loss: an under studied domain. In G. Herden, N. Koche, C. Seidl, & W. Trockel (Eds.) Mathematical Utility Theory. Wien, New Yrok: Springer. Journal of economics, Supplement 8, pp.181-202.
(c) Personel reflections on an unintentional behavioral scientist. Aequationes Mathematiae, 58, 1-13.
(d) Where is mathematical modeling in psychology headed? Theory & Psychology, 9, 723-737.
(e) (Chechile & ---) Reanalysis of the Chechile-Cooke experiment: Correcting for mismatched gambles. Journal of Risk & Uncertainty, 18, 321-325.


(a) (- - - &, Marley, A.A.J.) Separable and additive representations of binary gambles of gains. Mathematical Social Sciences, 40, 237-356.
(b) (- - - & Marley, A.A.J.) On elements of chance. Theory and Decision, 49, 97-126.
(c) (Aczél, J., Falmage, J.-C., & - - -) Functional equations in the behavioral sciences. Mathematica Japonica, 52, 469-512.


(a) Conditions equivalent to unit representations of ordered relational structures. Journal of Mathematical Psychology, 45, 81-98.
(b) Reduction invariance and Prelec’s weighting functions. Journal of Mathematical Psychology, 45, 167-179.
(c) (Sneddon, R., & - - -) Empirical comparisons of bilinear and non-bilinear utility theories. Organizational Behavior and Human Decision Processes, 84, 71-94.
(d) (Marley, A.A.J., & - - -) Ranked weighted utilities and qualitative convolution. Journal of Risk and Uncertainty, 23, 135-163.
(e) Zimmer, K., - - -, & Ellermeier, W. Testing a new theory of psychophysical scaling: Temporal loudness integration. In E. Sommerfeld, R. Kompass & T. Lachmann (Eds.) Fechner Day 2001. Proceedings of the seventeenth annual meeting of the International Society for Psychophysics. Lengerich: Pabst. Pp. 683-688.


(a) (Marley, A.A.J., & - - -). A simple axiomatization of binary rank-dependent expected utility of gains (losses). Journal of Mathematical Psychology, 46, 40-55.
(b) (- - - & Suppes, P.) Representational measurement theory. In (H.Pashler & J.Wixted, Eds.) Stevens’ Handbook of Experimental Psychology, 3rd Edition, Vol 4. New York: Wiley, Pp. 1-41.
(c) (Aczél, J., & - - -). Two functional equations preserving functional forms. Proceedings of the National Academy of Sciences, 99, 5212-5216.
(d) (Ng, C. T., - - -, Aczél, J.) Functional characterizations of basic properties of utility represesntations. Monatshefte für Mathematik, 135, 305-319.
(e) A psychophysical theory of intensity proportions, joint presentations, and matches. Psychological Review, 109, 520-532.
(f) (Cho, Y.-H., & - - -,Truong, L.) Duplex decomposition and general segregation of lotteries of a gain and a loss: An empirical evaluation. Organization Behavior and Human Decision Processes, 89, 1176-1193.
(g) (Aczél, J. & - - -) Functional equations in the behavioral and social sciences. In N. J. Smelser & P. B.Baltes (Eds.) International Encyclopedia of the Social and Behavioral Sciences, Vol. 9. Oxford: Elsevier. Pp. 5828-5833.


(a) (Marchant, T., & - - -) Technical note on the joint receipt of quantities of a single good. Journal of Mathematical Psychology, 47, 66-74.
(b) (Aczél, J.,- - -, Ng, C. T.) Functional equations arising in a theory of rank dependence and homogeneous joint receipts. Journal of Mathematical Psychology, 47, 171-183.
(c) (Aczél, J.,- - -,& Marley, A.A. J.) A functional equation arising from simultaneous utility representations. Results in Mathematics, 43, 193-197.
(d) Rationality in choice under certainty and uncertainty. In S. Schneider & J. Shanteau (Eds). Emerging Perspectives in Judgment and Decision Making . Cambridge , England : Cambridge University Press. Pp. 64-83.
(e) Whatever happened to information theory in psychology? Review of General Psychology, 7, 183-188.
(f) Increasing increment generalizations of rank-dependent theories. Theory and Decision, 55, 87-146.


(a) Symmetric and asymmetric matching of joint presentations. Psychological Review, 111, 446-454. Errata.
(b) (- - - & Steingrimsson, R.) A model of ratio production and estimation and some behavioral predictions. In Berglund, B (Ed.) Fechner Day 2003. Proceedings of the 19th Annual Meeting of the International Society for Psychophysics. Lengerirch, Germany: Pabst. Pp. 157-162.


(a) (- - - & Marley, A. A. J.) Ranked additive utility representations of gambles: Old and new axiomatizations. Journal of Risk and Uncertainty, 30, 21-62
(b) Measurement analogies: Comparisons of behavioral and physical measures. Psychometrika, 70, 227-251.

Marley, A.A. J. & - - -) Independence properties vis-à-vis several utility representations. Theory and Decision, 58, 77-143.

(d) (- - -, & Steingrimsson) Scientific psychology: Science versus easy accessibility? Observer, 18, 13.
(e) (Steingrimsson, R., & - - -) Evaluating a model of global psychophysical judgments I: Behavioral properties of summations and productions. Journal of Mathematical Psychology, 49, 290-306. Errata.
(f) (Steingrimsson, R., & - - -) Evaluating a model of global psychophysical judgments II: Behavioral properties linking summations and productions. Journal of Mathematical Psychology, 49, 308-319. Errata.
(g) Editorial. Journal of Mathematical Psychology, 49, 430-431.


(a) (- - - & Steingrimsson, R.) Global psychophysical judgments of intensity: Summary of a theory and experiments. In H. Colonious & E. Dzharfov (Eds.) Measurement and Representations of Sensations. Mahwah, NJ: Erlbaum. Pp. 89-129. Errata.
(b) (Steingrimsson, R., & - - -) Empirical evaluation of a model of global psychophysical judgments III: A form for the psychophysical and perceptual filtering. Journal of Mathematical Psychology, 50, 15-29. Errata.


(a) (Steingrimsson, R., & - - -) Empirical evaluation of a model of global psychophysical judgments IV: A form for the weighting function. Journal of Mathematical Psychology, 51, 29-44. Errata.
(b) (Aczél, J., & - - -) A behavioral condition for Prelec's weighting function on the positive line without assuming W(1) = 1. Journal of Mathematical Psychology, 51, 126-129.


(a) (Ng, C. T., - - -, & Marley, A. A. J. ). On the utility of gambling: Extending the Approach of Meginniss. Aequationes Mathematicae, 76, 281-204.

( - - -, Ng, C. T., Marley, A. A. J., & Aczél, J). Utility of gambling I: Entropy-modified linear weighted utility. Economic Theory , 36, 1–33. Errata.

(c) ( - - -, Ng, C. T., Marley, A. A. J., & Aczél, J). Utility of gambling II: Risk, Paradoxes, and Data. Economic Theory , 36, 165–187. Errata.
(d) (Marley, A. J., - - -, & Kocsis, I.). A Solution to a Problem Raised in Luce & Marley (2005). Journal of Mathematical Psychology, 52, 64–68
(e) Purity, resistance, and innocence in utility theory. Theory and Decision, 64, 109-118.
(f) Luce's choice axiom. Scholarpedia, 3(12):8077

Correction to Luce (2004). Psychological Review, 115, 601.

(h) (- - - & Steingrimsson, R.) Note on a changed empirical inference in several Steingrimsson and Luce articles due to C. T. Ng's correction of an error in Luce (2004). Journal of Mathematical Psychology, 52, 263-264.
(i) (- - -, & Narens, L.) Measurement, theory of (PDF: Pre-print). In Blume, L. & Durlauf, S. N. (Eds.) The New Palgrave Dictionary of Economics. Second Edition. Palgrave Macmillan. Online 2010.
(j) (Narens, L., & - - -) Meaningfulness and invariance (PDF: Pre-print). In Blume, L. & Durlauf, S. N. (Eds.)The New Palgrave Dictionary of Economics. Second Edition. Palgrave Macmillan. Online 2010.


(a) (- - -, Marley, A.A. J., & Ng, C.T.). Entropy-related measures of the utility of gambling. In Brams, S., Gehrlein, W., & Roberts, F. (Eds) The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishbur, Springer. Pp. 5-25.
(b) (Ng, C. T.,- - -, & Marley, A.A. J.). Utility of gambling when events are valued: An application of inset entropy. Theory and Decision, 67, 23-63. Errata.

(Ng, C. T.,- - -, & Marley, A.A. J.). Utility of gambling under p-additive joint receipt and segregation or duplex decomposition. Journal of Mathematical Psychology, 55, 273-286.

(d) A Functional Equation Proof of the Distributive-Triples Theorem. Aequationes Mathematicae, 78, 321-328.



Interpersonal comparisons of utility for 2 of 3 types of people. Theory and Decision, 68, 5-24.

(b) (Aronson, E., - - -, Steele, C., Suppes, P.). Garner Lindzey: Obituary. Proceedings of the American Philosophical Society, 154, 104-107.
(c) Behavioral Assumptions for a Class of Utility Theories: A Program of Experiments. Journal of Risk and Uncertainty, 41, 19-27.
(d) (- - -, & Steingrimsson, R, & Narens, L.). Are Psychophysical Scales of Intensities the Same or Different When Stimuli Vary on Other Dimensions? Theory with Experiments Varying Loudness and Pitch. Psychological Review,117, 1247--1258.


(a) (- - -, & Steingrimsson, R). Theory And Tests Of the Conjoint Commutativity Axion for Additive Conjoint Measurement. Journal of Mathematical Psychology, 55, 379-385.
(b) (- - -). Inherent Individual Differences in Utility. Frontiers in Psychology, 2, 297.


(a) (- - -). Predictions About Bisymmetry and Cross-Modal Matches From Global Theories of Subjective Intensities. Psychological Review, 119, 373-387.
(b) (- - -). Torgerson's Conjecture and Luce's Magnitude Production Representation Imply an Empirically False Property. Journal of Mathematical Psychology, 56, 176-178.

(Steingrimsson, R., - - -, & Narens. L.).  Brightness of Different Hues is a Single Psychophysical Ratio Scale of Intensity. American Journal of Psychology, 125, 321-333.


(Steingimsson, R., & - - -). Predictions From a Model of Global Psychophysics About Differences Between Perceptual and Physical Matches. Attention, Perception, & Psychophysics, 74, 1668–1680.



(a) (- - -). Analogs In Luce's Global Psychophysical Theory Stevens' Psychophysical Regression Effect, American Journal of Psychology, 126, 45-52. With a preface by Ehtibar Dzhafarov.

(- - -). The Incompleteness Of Holder's Theorem During Most of The 20th Century. PDF-PrePrint. To appear in a Festschrift on the occasion of Suppe's 90th birthday.

Last updated May 2013