THE 1998-99 MARSCHAK COLLOQUIUM AT UCLA



JANUARY 29 , 1999 -- 1:00 to 3:00 P.M. AT THE ANDERSON SCHOOL IN C-301

JANOS ACZEL, Visiting the Institute for Mathematical Behavioral Science, University of California, Irvine and Professor of Mathematics, Emeritus, University of Waterloo, Waterloo (Canada)  AND

R. DUNCAN LUCE, Distinguished Research Professor of CognitiveSciences and Research Professor of Economics, University of California, Irvine

"UTILITY OF  JOINT RECEIPT OF UNCERTAIN ALTERNATIVES AND FUNCTIONAL EQUATIONS"

FIELD/SUBFIELD: Psychology/Cognitive Sciences, Measurement and
Utility Theory

(Cosponsored by the Cognitive Science Group)

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ABSTRACT

In this second lecture on utility theory and functional equations, we add to the primitives of uncertain alternatives and preference the concept of status quo, gains and losses, and a binary operation of joint receipt. For gains and losses separately, assume the rank-dependent representation for binary gambles. A rational property, with some empirical support, called segregation is assumed to link uncertain alternatives to joint receipt. A simple functional equation argument shows that joint receipt has an additive representation V and that U is either proportional to V or they are exponentially linked, and in the latter case U over joint receipt has a simple polynomial form. Conversely, if U has that polynomial form, (U,W) is separable, and segregation holds, then (U,W) forms a rank-dependent representation over uncertain alternatives. We understand axiomatically what gives rise to the polynomial form, namely, that joint receipt has an additive representation, and separately we understand what underlies a separable form. But we have no assurance that the same U is involved. A necessary behavioral condition for that to be true is called joint-receipt decomposition. The converse of showing that this condition together with the two separate axiomatizations yields a common U that has both the polynomial form over joint receipt and the separable form over gambles entails solving a difficult functional equation. The method of solution is outlined. It is then noted that this property also forces a quite general representation to devolve to the rank-dependent one.

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JANOS ACZEL AND R. DUNCAN LUCE: BIOGRAPHIES

Janos Aczel has been with the University of Waterloo (Canada) since 1965 as Professor and (since 1969) Distinguished Professor of Mathematics (and directed its Centre for Information Theory and Qualitative Economics); since 1993 Distinguished Professor Emeritus. Previous academic positions were at the University of Cologne (Germany), the Kossuth University of Debrecen, the Technical University of Miskolc and the University of Szeged (Hungary). He had shorter time visiting appointments at 20 universities and two research institutes in North America, Europe, Australia and Africa. His Ph.D. is in mathematical analysis from the University of Budapest.

He is Fellow of the Royal Society of Canada (Academy of Science) and Foreign Member of the Hungarian Academy of Sciences. He received the J. R. Cajal Medal from the Council of Scientific Research (Madrid) and honorary degrees from the University of Karlsruhe (in Economics), the University of Graz, and the Silesian University, Katowice (in Mathematics).

His research, which is both theoretical and applied, includes functional equations and applications, in rough chronological order to nomography, webs, geometrical objects, probability theory, information measures, index numbers, group decision making, aggregation, production functions, laws of science, theory of measurement and utility theory. He is the author or co-author of 8 books, editor of two books, author or co-author of over 250 scientific papers, and editor of several journals, including Aequationes Mathematicae, which he founded. He would like to hope that another book by him and a coauthor, on mathematics and economics is nearing completion. 

R. Duncan Luce is Distinguished Research Professor of Cognitive Sciences and Research Professor of Economics at the University of California, Irvine, where he has been since 1988. Until last July he directed UCI's Institute for Mathematical Behavioral Sciences. Previous academic positions were at Harvard, UCI, the Institute for Advanced Study (Princeton), the University of Pennsylvania, and Columbia University. His Ph.D. is in applied mathematics from MIT.
   
His honors include membership in the American Academy of Arts and Sciences, the American Philosophical Society, the National Academy of Sciences, and the Society for Experimental Psychologists. He has also received a Distinguished Scientific Contributions Award from the American Psychological Association and UCI's Distinguished Faculty Lectureship Award for Research.
  
His research, which is both theoretical and empirical, has spanned several topics including, in
rough chronological order, game theory, probabilistic choice theory, psychophysics, response times, theories of additive and non-additive measurement, and utility theory. He is the author or co-author of 7 books, editor or co-editor of 11 books, and author or co-author of over 200 scientific papers. Another monograph, including among many other things the topics of these lectures, is nearing completion.

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