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<\/a><\/p>\nDuncan Luce passed away earlier this year. William Batchelder has written (for the Society for Mathematical Psychology) the following biography of Duncan Luce’s intellectual contributions:<\/p>\n
FACETS OF DUNCAN LUCE\u2019S RESEARCH CAREER
\nDuncan Luce (1925-2012) was one of the pioneers in establishing mathematical psychology as a field of study. It is arguable that without Duncan\u2019s leadership, intellect, and scholarly skills, the field would not have existed. He was the main force behind the publication of the two readings volumes and the three handbook volumes in mathematical psychology edited by Luce, Robert Bush, and Eugene Galanter from 1963-65. These five books together served to define our field. Later, he played a crucial role in starting our flagship journal, the Journal of Mathematical Psychology, as well as our Society. While everyone in the Society for Mathematical Psychology knows about Duncan and some aspects of his work, there are facets of his work that many members may be unaware of. The main purpose of this note is to celebrate Duncan\u2019s work by focusing on a few of these facets.
\nDuncan received his PhD in Mathematics at M.I.T. in 1950. His thesis was titled \u201cOn Semigroups,\u201d an area in abstract algebra. His earliest work in the social and behavioral sciences was in the area now called social networks. In his very first paper, published in 1949 with A. D. Perry, he mathematically defined the concept of clique and exploited the idea of representing graphs in matrix form, where matrix manipulations could be used to reveal the structure in a graph. This approach to analyzing graphic structures has become standard in the field of computer science, an area that hardly existed at the time of Duncan\u2019s early work.
\nDuncan\u2019s first academic position was as co-director of a network laboratory at M.I.T. from 1950-53, and his first tenure track position was as an Assistant Professor of Mathematical Statistics and Sociology at Columbia University (1954-57). It was in this period that Duncan acquired his lifelong interest in decision theory, and his 1957 book with Howard Raiffa, Games and Decisions: Introduction and Critical Survey, has become a classic in the economic sciences.
\nPerhaps the earliest hint of his future interest in psychology is seen in a paper with L. S. Christie in 1956 titled \u201cDecision structure and time relations in simple choice behavior.\u201d This paper foreshadowed the publication of Duncan\u2019s well-received book, Response Times, published 30 years later. Apart from this 1956 paper almost all his early work was in the areas of social structure and decision theory, areas well outside of psychology at that time.
\nIn 1959 Duncan published a very influential book, Individual Choice Behavior: A Theoretical Analysis. It is in this small book with a red cover that Luce\u2019s choice axiom is proposed. One consequence of the choice axiom is the so-called ratio rule, namely if several possible choice alternatives have positive valued strengths, then the probability of any one of them being selected is its strength divided by the sum of the strengths of all the other available alternatives. The ratio rule has been employed by a number of cognitive modelers in moving from latent representations of response strengths into actual manifest responses. It is perhaps ironic that the ratio rule is only a small consequence of Luce\u2019s choice axiom, which also includes the case where one or more of the choice alternatives has zero strength in certain contexts. In fact, his book explores the consequences of the choice axiom in many areas including paired-comparison scaling, Fechnerian scaling, signal detection, utility theory, and learning theory.
\nThe style of the 1959 book, like almost all of Duncan\u2019s work, is to proceed rationally with definitions, axioms, theorems, and proofs. The primary goal is to make progress by discovering the consequences of simple, non-trivial assumptions, rather than, say, inventing complex hypothetical structures with the primary goal of fitting data. Nevertheless, in all cases the aim of Duncan\u2019s work is to discover the testable consequences of one\u2019s assumptions. Duncan did not pose the choice axiom as the \u2018correct theory\u2019 of choice behavior. Instead, it was intentionally posed as an elegantly simple theory with many surprising consequences. Then when some choice phenomena is found not to satisfy the consequences of the choice axiom, it is often clear exactly what aspects of the axioms are in need of elaboration. Indeed, later work by others on choice theory has worked in the important concepts of item similarity, context effects, and time to respond that were intentionally missing from the choice axiom.
\nIn 1959, Duncan published another classic work, \u201cOn the possible psychophysical laws.\u201d In it he shows that given only knowledge of the scale type (ratio, interval, etc.) of an independent and dependent variable, one can determine the possible functional forms relating the two variables that are invariant under permissible scale transformations. That paper was not without controversy, because in many scientific laws there are dimension-absorbing constants that free up possible functional forms and thus delimit the applicability of Duncan\u2019s results. Nevertheless, the 1959 paper was seminal in directing his interest to dimension analysis, an area associated at that time with mathematical physics. This interest in dimensional analysis was one of the strands that lead to Duncan\u2019s long-term interest in the foundations of measurement discussed later.
\nIn addition to the axiomatic\/theorem approach, the 1959 paper on psychophysical scales reveals another hallmark of Duncan\u2019s approach to formal theory. The idea is that a theory formulated by empirically motivated axioms can give rise to functional equations whose solution provides the possible functional relationships between theoretical and behavioral variables. The solutions to these equations can suggest experiments that have the potential to falsify the theory, and if falsified one can look at specific axioms for what went wrong and how to fix it. In Duncan\u2019s later efforts to exploit this approach to theory construction, others assisted him, including the mathematician J\u00e1nos Acz\u00e9l, perhaps the World\u2019s most respected solver of functional equations.
\nFoundations of measurement became a central topic of Duncan\u2019s research from the middle 1960s up to the publication of the second and third volumes of the Foundations of Measurement in 1990, with Patrick Suppes, David Krantz, and Amos Tversky. In addition to the co-authors of the three foundational volumes, other mathematically savvy colleagues such as Louis Narens, Jean-Claude Falmagne, and Tony Marley joined him in this monumental effort. The thrust of the work was more directed to philosophy of science rather than to psychology. For this reason, the approach was mostly concerned with finding proper axiomatic formulations in idealized, error free settings rather than in more realistic settings involving measurement error. Despite the lack of concern with measurement error, some of the axiomatic work in foundations has been very influential in psychology such as Duncan\u2019s 1964 paper with the statistician John Tukey on conjoint measurement. This paper has guided experimental psychologists to more carefully regard the hypothesis of an interaction between experimental variables.
\nMuch of Duncan\u2019s more empirical work was in the areas of psychophysics, with a special interest in acoustics. This work started in the middle 1960s, and much of it was carried out with his close association with David M. Green. Green\u2019s active acoustics laboratory and Duncan\u2019s mathematical ideas gave rise to some influential theoretical papers such as his 1972 paper with Green, \u201c A neural timing theory for response times and the psychophysics of intensity.\u201d Somewhat uncharacteristically for Duncan, he published an undergraduate text, \u201cSound and Hearing,\u201d in 1993, based on a course he developed at Harvard. I am certain many of us can understand how difficult it must have been for Duncan to find things to teach at the undergraduate level in an American university.
\nIn his last ten years, Duncan published over fifty papers, and in these papers all the themes discussed above were represented many times over. Much of this work was coauthored with others mentioned earlier, and some of it was greatly assisted on the empirical side by his research association with Ragnar Steingrimsson.<\/p>\n
William H. Batchelder
\nOn behalf of The Society for Mathematical Psychology<\/p>\n