The Society for Mathematical Psychology promotes the advancement and communication of research in mathematical psychology and related disciplines. Mathematical psychology is broadly defined to include work of a theoretical character that uses mathematical methods, formal logic, or computer simulation. The Society journal is the Journal of Mathematical Psychology.
Facets of Duncan Luce’s Research Career
Duncan Luce (1925-2012) was one of the pioneers in establishing mathematical psychology as a field of study. It is arguable that without Duncan’s 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’s work by focusing on a few of these facets.
Duncan received his PhD in Mathematics at M.I.T. in 1950. His thesis was titled “On Semigroups,” 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’s early work.
Duncan’s 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.
Perhaps the earliest hint of his future interest in psychology is seen in a paper with L. S. Christie in 1956 titled “Decision structure and time relations in simple choice behavior.” This paper foreshadowed the publication of Duncan’s 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.
In 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’s 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’s 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.
The style of the 1959 book, like almost all of Duncan’s 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’s work is to discover the testable consequences of one’s assumptions. Duncan did not pose the choice axiom as the ‘correct theory’ 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.
In 1959, Duncan published another classic work, “On the possible psychophysical laws.” 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’s 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’s long-term interest in the foundations of measurement discussed later.
In addition to the axiomatic/theorem approach, the 1959 paper on psychophysical scales reveals another hallmark of Duncan’s 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’s later efforts to exploit this approach to theory construction, others assisted him, including the mathematician János Aczél, perhaps the World’s most respected solver of functional equations.
Foundations of measurement became a central topic of Duncan’s 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’s 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.
Much of Duncan’s 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’s active acoustics laboratory and Duncan’s mathematical ideas gave rise to some influential theoretical papers such as his 1972 paper with Green, “ A neural timing theory for response times and the psychophysics of intensity.” Somewhat uncharacteristically for Duncan, he published an undergraduate text, “Sound and Hearing,” 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.
In 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.
William H. Batchelder
On behalf of The Society for Mathematical Psychology