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Stochastic models of choice and reaction time: new developments
Mathematical
Social Sciences
23 (1992) l-3
North-Holland
Stochastic models of choice and reaction New developments
time:
A.A. J. Marley Department of Psychology, Quebec H3A IBI, Canada
McGill
University,
1205 Avenue
Docteur
Penfield,
Montreal,
The Irvine Research Unit in Mathematical Behavioral Sciences at the University of California Irvine (R.D. Lute, Director) is the premier research and teaching group in mathematical behavioral sciences, which takes as part of its mandate the support of summer research workshops at Irvine. For the month of August 1990, it sponsored the Workshop on Measurement and the Mathematical Representations of Empirical Structures. The topics of this workshop complemented those of the recently published final volumes of Foundations of Measurement (Suppes, Krantz, Lute, and Tversky, 1989, Vol. II; and Lute, Krantz, Suppes, and Tversky, 1990, Vol. III). Specifically, there were four sub-conferences on: Stochastic Models of Choice and Reaction Time (A.A.J. Marley, organizer); Color Theory (Carol Cicerone, organizer); Decision Theory (Chew Soo Hong and R. Duncan Lute, organizers); and Knowledge Spaces and Geometrical Representations (Jean-Claude Falmagne and Jean-Paul Doignon, organizers). The present three issue special volume of Mathematical Social Sciences contains articles based on work presented at, and further developed during and after, the sub-conference on Stochastic Models of Choice and Reaction Time; I believe it is planned that future issues of the journal will contain some or all of the material from the other three sub-conferences. Although the number of contributors and the time constraints of the meeting allowed for less research interaction at the subconference than might have been desirable, it is clear from the relations between the various papers (discussed below) that interactions before, during and after this meeting led to significant collaborative work and new results on several important problems. The papers in the volume’s first issue emphasize characterization problems that arise in the study of stochastic choice models. The first three papers (Marley, Heyer and Niederte, Clark) present recent characterizations of stochastic choice models with emphasis on the random utility concept. Marley presents a general summary of his own recent results on such characterizations, and also states a series of open problems. Heyer and Niederee develop an approach to probabilistic measurement that is motivated by a generalization of the random utility concept, and Clark ap0165-4896/92/$05.00
0
1992-El
sevier Science Publishers
B.V. All rights
reserved
2
A.A. J. Marley
/ Stochastic
plies ideas related to the latter concept
models
of choice and reaction
to probabilistic
time
social choice. The next three
papers (Fishburn, Suck, Stern) focus on binary choice. Fishburn and Suck each consider the special case of the random utility problem where we wish to characterize all systems of binary choice probabilities on a finite set that are induced by probability distributions over the family of linear orders of the set. Fishburn brings together recent results on this problem from two previously rather distinct literatures - one from probability, preference, and psychology, the other from linear programming and linear orders. Suck uses results from the latter literature to advance our understanding of the binary choice problem. When you read these articles and the work that they summarize, you will realize that there has recently been a burst of activity on the binary choice problem - stimulated to a significant extent, I believe, by interactions initiated in the planning and execution of the subconference. Stern’s paper brings us back to empirical reality by questioning whether we can ever empirically discriminate between standard linear paired comparison (binary choice) models. The final paper in the first issue (Resnick and Roy) uses quite sophisticated mathematical techniques in extending the ‘horse race’ random utility model (for choice probabilities and reaction times) to the case of infinite choice sets. All of the above papers (except Stern) are purely theoretical. The papers in the second and third issues are in addition seriously concerned with the empirical adequacy of the results that they present. The lead paper in the volume’s second issue (Iverson and Sheu) continues the characterization theme of the first issue and presents fascinating new characterizations of Thurstonian models using the properties of partially stable distributions. The papers by Ashby, Lee and Balakrishan and by Smith both begin with the recognition that Lute’s biased choice model has been the probabihstic ‘model of choice’ for describing identification data. However, no one has ever developed a completely satisfactory ‘process’ interpretation of that model nor developed an understanding of when one would expect it to fail. These two papers include new models that sometimes produce better fits than the biased choice model and they also present interpretations of why the various models do or do not fit identification data. In the final two papers of the volume’s second issue, Ennis and Mullen study the advantages gained when the variability of physical stimuli is acknowledged and modelled, and Bockenholt focuses on extensions of binary choice models to cases where the objects of choice are multidimensional. The volume’s third issue opens with Busemeyer and Townsend’s paper which makes important new contributions to the study of stochastic dynamic models of decision making, including the derivation of practical solutions for certain predictions regarding choice probabilities and distributions of response times. These results substantially generalize previous results on stochastic dynamic models. It appears that a major reason for Busemeyer and Townsend’s success in obtaining tractable results is that they assume spatially non-homogeneous but temporally homogeneous processes. Temporally non-homogeneous models tend to be more
A.A. J. Marley
/ Stochastic
models
of choice and reacrion time
3
computationally intractable. For instance, the next paper by Heath develops such a temporally non-homogeneous diffusion model for two-choice decision making that can be applied to situations where the stimuli (and hence the decision processes) are non-stationary. Each of the final two remaining papers (Candel, Schweickert) makes a significant contribution to some aspect of the general stochastic choice problem. Candel combines stochastic Coombsian (or ideal based) unfolding with feature representations of objects of choice to obtain an interesting new class of stochastic choice models. Schweickert develops techniques to test whether reaction times in an experiment could be produced by factors influencing processes in a critical path network, and if such a network exists, to discover the arrangement of processes and bounds on their durations. The diversity of papers in these special issues, ranging from the highly theoretical to the quite empirical, demonstrates the wide range of issues currently considered in the area of stochastic models of choice and reaction time. I believe these contributions are among the best in the area and that they will stimulate further exciting theoretical and empirical work.
Acknowledgments On behalf of the contributors, I wish to thank the Irvine Research Unit and especially R. Duncan Lute and Louis Narens for their financial and other support of the sub-conference. We also all received extensive help from Nancy Pinkerton, then of the Irvine Research Unit, in planning our participation; in addition, I am grateful for her help both before and after the meeting. Finally, and importantly, I thank Susan Gregus at McGill for carefully and cheerfully carrying out the many tasks involved in my part of the planning of the meeting and in the extensive followup editorial process for the special issues.
References R.D. Lute, D.H. Krantz,
P. Suppes and A. Tversky,
tions, Axomatization, P. Suppes, D.H. Krantz,
and Invariance (Academic Press, San Diego, 1990). R.D. Lute and A. Tversky, Foundations of Measurement,
cal, Threshold,
and Probabilistic
Representations
Foundations
(Academic
of Measurement,
Press,
San Diego,
Vol. III, RepresentaVol. II, Geometri1989).
Social Sciences
23 (1992) l-3
North-Holland
Stochastic models of choice and reaction New developments
time:
A.A. J. Marley Department of Psychology, Quebec H3A IBI, Canada
McGill
University,
1205 Avenue
Docteur
Penfield,
Montreal,
The Irvine Research Unit in Mathematical Behavioral Sciences at the University of California Irvine (R.D. Lute, Director) is the premier research and teaching group in mathematical behavioral sciences, which takes as part of its mandate the support of summer research workshops at Irvine. For the month of August 1990, it sponsored the Workshop on Measurement and the Mathematical Representations of Empirical Structures. The topics of this workshop complemented those of the recently published final volumes of Foundations of Measurement (Suppes, Krantz, Lute, and Tversky, 1989, Vol. II; and Lute, Krantz, Suppes, and Tversky, 1990, Vol. III). Specifically, there were four sub-conferences on: Stochastic Models of Choice and Reaction Time (A.A.J. Marley, organizer); Color Theory (Carol Cicerone, organizer); Decision Theory (Chew Soo Hong and R. Duncan Lute, organizers); and Knowledge Spaces and Geometrical Representations (Jean-Claude Falmagne and Jean-Paul Doignon, organizers). The present three issue special volume of Mathematical Social Sciences contains articles based on work presented at, and further developed during and after, the sub-conference on Stochastic Models of Choice and Reaction Time; I believe it is planned that future issues of the journal will contain some or all of the material from the other three sub-conferences. Although the number of contributors and the time constraints of the meeting allowed for less research interaction at the subconference than might have been desirable, it is clear from the relations between the various papers (discussed below) that interactions before, during and after this meeting led to significant collaborative work and new results on several important problems. The papers in the volume’s first issue emphasize characterization problems that arise in the study of stochastic choice models. The first three papers (Marley, Heyer and Niederte, Clark) present recent characterizations of stochastic choice models with emphasis on the random utility concept. Marley presents a general summary of his own recent results on such characterizations, and also states a series of open problems. Heyer and Niederee develop an approach to probabilistic measurement that is motivated by a generalization of the random utility concept, and Clark ap0165-4896/92/$05.00
0
1992-El
sevier Science Publishers
B.V. All rights
reserved
2
A.A. J. Marley
/ Stochastic
plies ideas related to the latter concept
models
of choice and reaction
to probabilistic
time
social choice. The next three
papers (Fishburn, Suck, Stern) focus on binary choice. Fishburn and Suck each consider the special case of the random utility problem where we wish to characterize all systems of binary choice probabilities on a finite set that are induced by probability distributions over the family of linear orders of the set. Fishburn brings together recent results on this problem from two previously rather distinct literatures - one from probability, preference, and psychology, the other from linear programming and linear orders. Suck uses results from the latter literature to advance our understanding of the binary choice problem. When you read these articles and the work that they summarize, you will realize that there has recently been a burst of activity on the binary choice problem - stimulated to a significant extent, I believe, by interactions initiated in the planning and execution of the subconference. Stern’s paper brings us back to empirical reality by questioning whether we can ever empirically discriminate between standard linear paired comparison (binary choice) models. The final paper in the first issue (Resnick and Roy) uses quite sophisticated mathematical techniques in extending the ‘horse race’ random utility model (for choice probabilities and reaction times) to the case of infinite choice sets. All of the above papers (except Stern) are purely theoretical. The papers in the second and third issues are in addition seriously concerned with the empirical adequacy of the results that they present. The lead paper in the volume’s second issue (Iverson and Sheu) continues the characterization theme of the first issue and presents fascinating new characterizations of Thurstonian models using the properties of partially stable distributions. The papers by Ashby, Lee and Balakrishan and by Smith both begin with the recognition that Lute’s biased choice model has been the probabihstic ‘model of choice’ for describing identification data. However, no one has ever developed a completely satisfactory ‘process’ interpretation of that model nor developed an understanding of when one would expect it to fail. These two papers include new models that sometimes produce better fits than the biased choice model and they also present interpretations of why the various models do or do not fit identification data. In the final two papers of the volume’s second issue, Ennis and Mullen study the advantages gained when the variability of physical stimuli is acknowledged and modelled, and Bockenholt focuses on extensions of binary choice models to cases where the objects of choice are multidimensional. The volume’s third issue opens with Busemeyer and Townsend’s paper which makes important new contributions to the study of stochastic dynamic models of decision making, including the derivation of practical solutions for certain predictions regarding choice probabilities and distributions of response times. These results substantially generalize previous results on stochastic dynamic models. It appears that a major reason for Busemeyer and Townsend’s success in obtaining tractable results is that they assume spatially non-homogeneous but temporally homogeneous processes. Temporally non-homogeneous models tend to be more
A.A. J. Marley
/ Stochastic
models
of choice and reacrion time
3
computationally intractable. For instance, the next paper by Heath develops such a temporally non-homogeneous diffusion model for two-choice decision making that can be applied to situations where the stimuli (and hence the decision processes) are non-stationary. Each of the final two remaining papers (Candel, Schweickert) makes a significant contribution to some aspect of the general stochastic choice problem. Candel combines stochastic Coombsian (or ideal based) unfolding with feature representations of objects of choice to obtain an interesting new class of stochastic choice models. Schweickert develops techniques to test whether reaction times in an experiment could be produced by factors influencing processes in a critical path network, and if such a network exists, to discover the arrangement of processes and bounds on their durations. The diversity of papers in these special issues, ranging from the highly theoretical to the quite empirical, demonstrates the wide range of issues currently considered in the area of stochastic models of choice and reaction time. I believe these contributions are among the best in the area and that they will stimulate further exciting theoretical and empirical work.
Acknowledgments On behalf of the contributors, I wish to thank the Irvine Research Unit and especially R. Duncan Lute and Louis Narens for their financial and other support of the sub-conference. We also all received extensive help from Nancy Pinkerton, then of the Irvine Research Unit, in planning our participation; in addition, I am grateful for her help both before and after the meeting. Finally, and importantly, I thank Susan Gregus at McGill for carefully and cheerfully carrying out the many tasks involved in my part of the planning of the meeting and in the extensive followup editorial process for the special issues.
References R.D. Lute, D.H. Krantz,
P. Suppes and A. Tversky,
tions, Axomatization, P. Suppes, D.H. Krantz,
and Invariance (Academic Press, San Diego, 1990). R.D. Lute and A. Tversky, Foundations of Measurement,
cal, Threshold,
and Probabilistic
Representations
Foundations
(Academic
of Measurement,
Press,
San Diego,
Vol. III, RepresentaVol. II, Geometri1989).