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Some thoughts on “A stochastic model for individual choice behaviour”
THEMA 9 A. -- Symposium M A T H E M A T I S C H E M O D E L L E IN DER PSYCHOLOGIE MODI~LES MATHI~MATIQUES EN PSYCHOLOGIE M A T H E M A T I C A L MODELS 1N PSYCHOLOGY
Organisator-Priisident:
H. O. GULL1KSON(Princeton, N. J.. USA ~
Referenten: R, J. AUDLEY, H. O, GULLIKSON,M. B. JONFS, ~W, WITI'E Diskussion: A. R. JONCKItEERE, W. S. TORGERSON
SOME T H O U G H T S ON "A STOCHASTIC MODEL F O R I N D I V I D U A L CHOICE BEHAVIOUR'" R. J. AUDLEY London In a paper published recently (1), I outlined a stochastic model for individual choice behaviour. The model was very simple, but it did seem to share several properties with observed behaviour in choice situations. It therefore seemed worth considering as an initial working hypothesis which could be used m guide experimental work and theoretical speculation. But further perusal of the relevant psychological literature caused me to wonder why the model approximated any characteristics of choice behaviour at alI, even in those situations which I so happily selected for consideration. However, I now think that there are some reasons why the model may still serve as a working hypothesis. In this paper today, after outlining the intentions of the present research and briefly describing the model, I shall give some attention to alternative interpretations of it, which lead to rather different lines of theoretical development and seem applicable to somewhat different experimental situations. It should at once be pointed out, that the present work is not immedi36o
MATHEMATISCHE MODELLE IN DER PSYCHOLOGIE
361
ately directed towards providing a conceptual scheme relating variations in stimulus conditions to the ensuing variations in obse',ved responses, which is perhaps the classical approach of the psychologist. Rather, at the present stage in speculation, the intention is to outline as possible description of behaviour at a choice point with specific emphasis upon the relations between the various types of measure which can be observed in hmnan and animal choice behaviour. Thus I am concerned with extent to which it is possible to explore, both conceptually and experimentally, the nature of the processes involved in a cboice between alternatives in situations where it can be assumed that neither the subject or the environment is changing in any way. This, I agrce, is an ideal situation not likely to be met in experimental reality. But perhaps apprc, ximations to this ideal can be found and in any case thf: ideal situation serves as a suitable starting place for speculation. ~'n addition it must be emphasized, although I shall not dwell on the po'nt, that the present work is directed towards a study of the behaviov.r of individual subjects. The literature both experimental and theoretical even on this limited aspect of choice behaviour is vast and unorganized and I shall not attempt to relate the present work to earlier formulations. In contemporary research, there are some similarities to the axiomatic treatment presented b) Duncan Luce (3). There are, however, differences both o[ emphasis and content, the discussion of which would require at least another paper, and perhaps a great deal of argument. There are much greater similarities to the work of Gordon Bower (2) and his paper has been of great value in my reappraisal of the present choice model. To simplify exposition, I shall confme attention to situations where knowledge of the correctness of the selection of a given alternative either is not known until the selection is made, as for example in a discrimination task, or where such knowledge does not enter, as for example in preference studies. This avoids the more complicating circumstances of say, disjunctive reaction time studies, where the subject can match his choice with a signal before completing the overt response. The model involves two assumptions. The first assumption is that for given stimulus and organismic conditions, there is associated with each posxible choice response a single parameter. This parameter determines the probabilivy that in a small interval of time (t, t + 2xt) there will occur an "#nplicit" response of the kind with which the parameter is associated. This in essence, supposes each response is being emitted in implicit form at a characteristic mean rate. If there were ~or example,
362
l'lt• MA 9
two alternative responses, A and B, then there will be implicit r~sponses say, a and b, with associated rates of responding say, ,1 and /3. No specific meaning is given to the term "implicit response" which is left conveniently open to alternative interpretations. The second assumption is that an overt etloiee response occurs when a run o] K implicit responses o] a given kit~l appears, this run being uninterrupted by occurrences o[ implicit responses o[ other kinds. For example, if K = 2, then the events supposed to be going on during the choice between overt responses A and B might be as follows: Start of 1 trial
Time -~ a
a b
~
overt response
b
My original conception of the model was as a kind of behavioural synapse, with the uninterrupted sequence of implicit responses mimicking temporal and possibly spatial summation. From this very simple model it is easy to derive for each alternative response, firstly the probability of overt occurrence and secondly the distribution of appropriate response times to be expected over a sequence of simiIar trials. Furthermore, if the implicit responses are identified with some kind of orientating or preparatory response, one can obtain the distribution of VTE's, that is the number of vacillations between the alternatives before an overt choice is made. Again, there are distinct distributions associated with the behaviour preceding the choice of a given alternative. I also suggested in the earlier paper that the sequence of implicit responses preceding an overt choice might serve as a basis for the analysis of a subject's judgement of confidence in his selected response. For example, it would seem likely that perfect confidence in a judgement might arise in trials where implicit responses of only one kind emerged and iqt seems almost tautnlogous to suggest that doubt might reflect a considerable amount of vacillat~on between the various implicit alternatives before an overt response occ'ars. Although this is a very elementary model it does specify the nature of the distribution of several raspon~;e variables and provide predictions concerning the relations between them. There arc two kinds of predictions available. The first is concerned with results obtained under near-identical stimulus conditions. Even under these relatively restricted circumstances it is possible to test various aspects of the model. For example, in a r
MATHEMAT1SCHE MODELLE IN DER PSYCHOLOGIE
363
choice situation, knowing the mean time taken for all choices, and the probability of a given choice, it is possible to predict the mean difference in time between the two available choices. Or again, if a hypothesis concerning the basis of confidence judgements is adopted, for example that outlined above, then it is only necessary to know the probability of a given response in order to predict the relative occurrence os judgements of various degrees of confidence, and also the accoracy of the subject within each confidence category. Furthermore, the predictions are not limited to average values, but extend to all statistical properties of the distributions involved. The second kind of prediction which can be made concerns tile way in which the various response measures vary with each other if stinmlus conditions are changed. Here for example, if an individual subject is tested over a series of discrimination problems, there is a direct relationship between the probability of a correct response and the expected number of VTE's which will precede firstly correct, and secondly incorrect responses. This type of prediction, I think, to be of considerable interest in so far as under some circumstances it should be correct not only for a given subject over a set of different discriminations but also for the results of a number of subjects in one problem. This is possible because the san~e parameters are involved in the hypothetical descriptions of both response measures. Qualitatively these predictions seemed to be as good as one could hope of an initial hypothesis. For example, dominant responses in a situation are expected on the average to have a quicker.response time and confident responses are expected to be quicker and more accurate than others, expectations which find support in the literature and for which 1 also have confirmatory data. Consideration of the way in which the response measures vary with each other as stimulus conditions are changed s the predicted relationships in qualitative agreement with experimental findings. Now I do not wish to appear immodest in my clalrus for the model, fidly realising that the justification of the model must depend upon more detailed quantitative investigations. Furthermore, the model as it stands is certainly too general to be likely to stand up to such quantitative tests in specific situations. But as I have been stressing throughout this paper, ou the basis of qualitative evidence, it does seem to be a suitable scheme from which to develop appropriate models for various experimental situations. Closer examination of the literature on choice behavio~r, however, suggested that the eharacterisation of a two choice situation as a compe-
364
THrMA 9
tJtion between two alternative responses must be too gross an oversimplification. It seems much more likely that the model ;is it stands, when interpreted as a behavioural synapse, is more appropriate for situations where there is only one actual choice available and the decision concerns whether to make this choice or reject it: for example, an animal in the single-door jumping stand situation of Schlosberg and Solomon (4), where the choice is between jumping or not jumping in the presc ::c of a given stimulus card. This, of course, presupposes an adequate experimental analysis of what is meant by ~he response of not jumping, when this is treated as a definite response and not just as the absence of any activity. Perhaps, ,and I have no great confidence in the suggestion, tile unmodified model may be applicable to the psychophysical method of single stimuli, where a subject is presented with a series o f say, weights, and is asked to categorisc each one as it appears as light or heavy. On this first interpretation of the model it seems unlikely that is should ever satisfactorily describe the results obt*dned in a choice between two posidvc alternatives, as in the constant method or T-.maze. Much of the experimental work of the '30s and "40s indicates that something like a double approach-avoidance model is required. And I conclude from this work that the two choice situations should perhaps be characterised in something like the following way. choice of A O
N
clloice of B O
/
final response
preparatory orientation
start of tria!
Where the circles indicate successive stages in choice behaviour and the arrows indicate possible transitions between these. Now it is interesting to note that me model 1 have described today, can be viewed as an elementary version of this more elaborate schanre. For suppose we consider ",.he ease arising when k = 2, then we can produce the following diagram.
MATHEMATISCHE MODELLE IN DER PSYCHOLOG1E O
365
O
O .._
O
o where p = ~ - ~
and q = ~--~+fl
aud ~ and /r are the hypothetical rates of responding with the implicit alternatives a and b. This interpretation of the model makes it very similar to that proposed by Gordon Bower for the description of V T E s at a choice point. A personal communication from him on this point has had an obvious influence on my speculations and I am pleased to have this opportunity to acknowledge this. This second kind of interpretation of the model suggests that it may be profitable to make more detailed observations of both human and animN choice behaviour with a view to determining the complete sequence of activities involved in a single act of choice. Thus, much 0f what is hypothetical, as in the choice model I have proposed, may become unnecessary because ~ e comptition of responses may be already sufficiently explicit if the choisc situation is observed with sufficient ingenuity. The possibilities of doing this are clearly indicated by some earlier work, for example, that of Miller, Brown and others. More recently this has been attempted by Bower in the case of the choice behavior of rats, although not for individual subjects, which I consider to be crucial for this kind of study. The essential difference between this second interpretation and the first is in treating a given choice act as a pattern of imerlocking simpler acts of choice. It is conceivable that both interpretations of the model m a y be involved in a satisfactory description of choice behaviour. On the one hand, a description of the pal:tern of activities involved in the selection of one response from a set of alternatives, and on the other, a description
366
r nE:',ta 9
of the competition involved in moving from one clement of tile pattern to another. In conclusion. 1 must apologise for a lack of empirical props to accompany nay speculations, but 1 tbink that several possible Imes of experimental investigation, some of which are under way, are suggested by the present work. I also feel vep,.' stron)ly that this kind of analysis of choice must necessarily be invoh'ed in any tbco O' of behaviour which attempts quantitative predictions, file basic unit of observation ill psy chology is surely a single act of choice by an individual subject and it is towards an urldcrstandlng of this unit that the present work is directed.
REFERFNCES l. AtJDLE~, ", R. J., A SIo~hasl[c Model for Individual Choice geha~iour. PsychoI. Rev., 1960, 67, 1--15. 2. BOWER,G. H., Choice Point Beha~,iOtll. Chap. 6 in Stu(t'i~,r itl M~lthl.~tl~itical Learning Theory. R. R. Bush rind W. K. Estcs (Edrs) Stanford Univ. Press. 1959. 3. LL'CE. R. D.. lndilulual Choice Behaviour. A Theoretical Anabsis. New York: Wiley, 1959. al. SCHLOSBERO, H. and SOLONION, R. L., Latency of Response in a Choice Discrimination. J. exp. PsychoL, 1943, 33, 22-39.
I N T E R C U L T U R A L A T T I T U D E COMPARISONS HAROLD GULLIKSEN Pr~ncelon, N.J. (USA)
Psychologists and sociologists have studied the area of preferences and values. Thurstone (1959), for e• obtained preference ratings for different nationalities and ratings of tbe seriousness of various crimes. These ratings were obtained from different groups of judges, in order to determine the similarities and differences of these groups. However, the concept of a single prestige scale applying to all the persons in the group has begun to trouble some investigators. It is not very satisfactory, to use predefined subgroups, such as Urban-Rural, EducatedUneducated. Weahhy-Poor, in an attempt to predefine some subgroups that will have a uniform opinion. It is, however, possible to ask directly if there are different schooIs of thought with respect to these hierarchies. Do some people view things one way, and some another? In the present approach, the structure is. not obtained by comparing various arbitrarily
Organisator-Priisident:
H. O. GULL1KSON(Princeton, N. J.. USA ~
Referenten: R, J. AUDLEY, H. O, GULLIKSON,M. B. JONFS, ~W, WITI'E Diskussion: A. R. JONCKItEERE, W. S. TORGERSON
SOME T H O U G H T S ON "A STOCHASTIC MODEL F O R I N D I V I D U A L CHOICE BEHAVIOUR'" R. J. AUDLEY London In a paper published recently (1), I outlined a stochastic model for individual choice behaviour. The model was very simple, but it did seem to share several properties with observed behaviour in choice situations. It therefore seemed worth considering as an initial working hypothesis which could be used m guide experimental work and theoretical speculation. But further perusal of the relevant psychological literature caused me to wonder why the model approximated any characteristics of choice behaviour at alI, even in those situations which I so happily selected for consideration. However, I now think that there are some reasons why the model may still serve as a working hypothesis. In this paper today, after outlining the intentions of the present research and briefly describing the model, I shall give some attention to alternative interpretations of it, which lead to rather different lines of theoretical development and seem applicable to somewhat different experimental situations. It should at once be pointed out, that the present work is not immedi36o
MATHEMATISCHE MODELLE IN DER PSYCHOLOGIE
361
ately directed towards providing a conceptual scheme relating variations in stimulus conditions to the ensuing variations in obse',ved responses, which is perhaps the classical approach of the psychologist. Rather, at the present stage in speculation, the intention is to outline as possible description of behaviour at a choice point with specific emphasis upon the relations between the various types of measure which can be observed in hmnan and animal choice behaviour. Thus I am concerned with extent to which it is possible to explore, both conceptually and experimentally, the nature of the processes involved in a cboice between alternatives in situations where it can be assumed that neither the subject or the environment is changing in any way. This, I agrce, is an ideal situation not likely to be met in experimental reality. But perhaps apprc, ximations to this ideal can be found and in any case thf: ideal situation serves as a suitable starting place for speculation. ~'n addition it must be emphasized, although I shall not dwell on the po'nt, that the present work is directed towards a study of the behaviov.r of individual subjects. The literature both experimental and theoretical even on this limited aspect of choice behaviour is vast and unorganized and I shall not attempt to relate the present work to earlier formulations. In contemporary research, there are some similarities to the axiomatic treatment presented b) Duncan Luce (3). There are, however, differences both o[ emphasis and content, the discussion of which would require at least another paper, and perhaps a great deal of argument. There are much greater similarities to the work of Gordon Bower (2) and his paper has been of great value in my reappraisal of the present choice model. To simplify exposition, I shall confme attention to situations where knowledge of the correctness of the selection of a given alternative either is not known until the selection is made, as for example in a discrimination task, or where such knowledge does not enter, as for example in preference studies. This avoids the more complicating circumstances of say, disjunctive reaction time studies, where the subject can match his choice with a signal before completing the overt response. The model involves two assumptions. The first assumption is that for given stimulus and organismic conditions, there is associated with each posxible choice response a single parameter. This parameter determines the probabilivy that in a small interval of time (t, t + 2xt) there will occur an "#nplicit" response of the kind with which the parameter is associated. This in essence, supposes each response is being emitted in implicit form at a characteristic mean rate. If there were ~or example,
362
l'lt• MA 9
two alternative responses, A and B, then there will be implicit r~sponses say, a and b, with associated rates of responding say, ,1 and /3. No specific meaning is given to the term "implicit response" which is left conveniently open to alternative interpretations. The second assumption is that an overt etloiee response occurs when a run o] K implicit responses o] a given kit~l appears, this run being uninterrupted by occurrences o[ implicit responses o[ other kinds. For example, if K = 2, then the events supposed to be going on during the choice between overt responses A and B might be as follows: Start of 1 trial
Time -~ a
a b
~
overt response
b
My original conception of the model was as a kind of behavioural synapse, with the uninterrupted sequence of implicit responses mimicking temporal and possibly spatial summation. From this very simple model it is easy to derive for each alternative response, firstly the probability of overt occurrence and secondly the distribution of appropriate response times to be expected over a sequence of simiIar trials. Furthermore, if the implicit responses are identified with some kind of orientating or preparatory response, one can obtain the distribution of VTE's, that is the number of vacillations between the alternatives before an overt choice is made. Again, there are distinct distributions associated with the behaviour preceding the choice of a given alternative. I also suggested in the earlier paper that the sequence of implicit responses preceding an overt choice might serve as a basis for the analysis of a subject's judgement of confidence in his selected response. For example, it would seem likely that perfect confidence in a judgement might arise in trials where implicit responses of only one kind emerged and iqt seems almost tautnlogous to suggest that doubt might reflect a considerable amount of vacillat~on between the various implicit alternatives before an overt response occ'ars. Although this is a very elementary model it does specify the nature of the distribution of several raspon~;e variables and provide predictions concerning the relations between them. There arc two kinds of predictions available. The first is concerned with results obtained under near-identical stimulus conditions. Even under these relatively restricted circumstances it is possible to test various aspects of the model. For example, in a r
MATHEMAT1SCHE MODELLE IN DER PSYCHOLOGIE
363
choice situation, knowing the mean time taken for all choices, and the probability of a given choice, it is possible to predict the mean difference in time between the two available choices. Or again, if a hypothesis concerning the basis of confidence judgements is adopted, for example that outlined above, then it is only necessary to know the probability of a given response in order to predict the relative occurrence os judgements of various degrees of confidence, and also the accoracy of the subject within each confidence category. Furthermore, the predictions are not limited to average values, but extend to all statistical properties of the distributions involved. The second kind of prediction which can be made concerns tile way in which the various response measures vary with each other if stinmlus conditions are changed. Here for example, if an individual subject is tested over a series of discrimination problems, there is a direct relationship between the probability of a correct response and the expected number of VTE's which will precede firstly correct, and secondly incorrect responses. This type of prediction, I think, to be of considerable interest in so far as under some circumstances it should be correct not only for a given subject over a set of different discriminations but also for the results of a number of subjects in one problem. This is possible because the san~e parameters are involved in the hypothetical descriptions of both response measures. Qualitatively these predictions seemed to be as good as one could hope of an initial hypothesis. For example, dominant responses in a situation are expected on the average to have a quicker.response time and confident responses are expected to be quicker and more accurate than others, expectations which find support in the literature and for which 1 also have confirmatory data. Consideration of the way in which the response measures vary with each other as stimulus conditions are changed s the predicted relationships in qualitative agreement with experimental findings. Now I do not wish to appear immodest in my clalrus for the model, fidly realising that the justification of the model must depend upon more detailed quantitative investigations. Furthermore, the model as it stands is certainly too general to be likely to stand up to such quantitative tests in specific situations. But as I have been stressing throughout this paper, ou the basis of qualitative evidence, it does seem to be a suitable scheme from which to develop appropriate models for various experimental situations. Closer examination of the literature on choice behavio~r, however, suggested that the eharacterisation of a two choice situation as a compe-
364
THrMA 9
tJtion between two alternative responses must be too gross an oversimplification. It seems much more likely that the model ;is it stands, when interpreted as a behavioural synapse, is more appropriate for situations where there is only one actual choice available and the decision concerns whether to make this choice or reject it: for example, an animal in the single-door jumping stand situation of Schlosberg and Solomon (4), where the choice is between jumping or not jumping in the presc ::c of a given stimulus card. This, of course, presupposes an adequate experimental analysis of what is meant by ~he response of not jumping, when this is treated as a definite response and not just as the absence of any activity. Perhaps, ,and I have no great confidence in the suggestion, tile unmodified model may be applicable to the psychophysical method of single stimuli, where a subject is presented with a series o f say, weights, and is asked to categorisc each one as it appears as light or heavy. On this first interpretation of the model it seems unlikely that is should ever satisfactorily describe the results obt*dned in a choice between two posidvc alternatives, as in the constant method or T-.maze. Much of the experimental work of the '30s and "40s indicates that something like a double approach-avoidance model is required. And I conclude from this work that the two choice situations should perhaps be characterised in something like the following way. choice of A O
N
clloice of B O
/
final response
preparatory orientation
start of tria!
Where the circles indicate successive stages in choice behaviour and the arrows indicate possible transitions between these. Now it is interesting to note that me model 1 have described today, can be viewed as an elementary version of this more elaborate schanre. For suppose we consider ",.he ease arising when k = 2, then we can produce the following diagram.
MATHEMATISCHE MODELLE IN DER PSYCHOLOG1E O
365
O
O .._
O
o where p = ~ - ~
and q = ~--~+fl
aud ~ and /r are the hypothetical rates of responding with the implicit alternatives a and b. This interpretation of the model makes it very similar to that proposed by Gordon Bower for the description of V T E s at a choice point. A personal communication from him on this point has had an obvious influence on my speculations and I am pleased to have this opportunity to acknowledge this. This second kind of interpretation of the model suggests that it may be profitable to make more detailed observations of both human and animN choice behaviour with a view to determining the complete sequence of activities involved in a single act of choice. Thus, much 0f what is hypothetical, as in the choice model I have proposed, may become unnecessary because ~ e comptition of responses may be already sufficiently explicit if the choisc situation is observed with sufficient ingenuity. The possibilities of doing this are clearly indicated by some earlier work, for example, that of Miller, Brown and others. More recently this has been attempted by Bower in the case of the choice behavior of rats, although not for individual subjects, which I consider to be crucial for this kind of study. The essential difference between this second interpretation and the first is in treating a given choice act as a pattern of imerlocking simpler acts of choice. It is conceivable that both interpretations of the model m a y be involved in a satisfactory description of choice behaviour. On the one hand, a description of the pal:tern of activities involved in the selection of one response from a set of alternatives, and on the other, a description
366
r nE:',ta 9
of the competition involved in moving from one clement of tile pattern to another. In conclusion. 1 must apologise for a lack of empirical props to accompany nay speculations, but 1 tbink that several possible Imes of experimental investigation, some of which are under way, are suggested by the present work. I also feel vep,.' stron)ly that this kind of analysis of choice must necessarily be invoh'ed in any tbco O' of behaviour which attempts quantitative predictions, file basic unit of observation ill psy chology is surely a single act of choice by an individual subject and it is towards an urldcrstandlng of this unit that the present work is directed.
REFERFNCES l. AtJDLE~, ", R. J., A SIo~hasl[c Model for Individual Choice geha~iour. PsychoI. Rev., 1960, 67, 1--15. 2. BOWER,G. H., Choice Point Beha~,iOtll. Chap. 6 in Stu(t'i~,r itl M~lthl.~tl~itical Learning Theory. R. R. Bush rind W. K. Estcs (Edrs) Stanford Univ. Press. 1959. 3. LL'CE. R. D.. lndilulual Choice Behaviour. A Theoretical Anabsis. New York: Wiley, 1959. al. SCHLOSBERO, H. and SOLONION, R. L., Latency of Response in a Choice Discrimination. J. exp. PsychoL, 1943, 33, 22-39.
I N T E R C U L T U R A L A T T I T U D E COMPARISONS HAROLD GULLIKSEN Pr~ncelon, N.J. (USA)
Psychologists and sociologists have studied the area of preferences and values. Thurstone (1959), for e• obtained preference ratings for different nationalities and ratings of tbe seriousness of various crimes. These ratings were obtained from different groups of judges, in order to determine the similarities and differences of these groups. However, the concept of a single prestige scale applying to all the persons in the group has begun to trouble some investigators. It is not very satisfactory, to use predefined subgroups, such as Urban-Rural, EducatedUneducated. Weahhy-Poor, in an attempt to predefine some subgroups that will have a uniform opinion. It is, however, possible to ask directly if there are different schooIs of thought with respect to these hierarchies. Do some people view things one way, and some another? In the present approach, the structure is. not obtained by comparing various arbitrarily