Search behavior with and without optional stopping

ORGANIZATIONAL BEHAVIOR AND H U M A N PERFORMANCE 7~

1-17 (1972)

Search Behavior with and without Optional Stopping 1 AMNON RAPOPORT, ROBERT W. LISSITZ, 2 AND HUNTER A. McALLISTER

The University o] North Carolina, Chapel Hill Forty-six college students participated individually in a discrete search problem in which they attempted to find an object hidden in one of four distinguishable locations. They were told the prior detection probability, the conditional miss probability, and the per trial cost of search for each loeation. Ss in Group I were required to search until either the object was found or their resources were exhausted, whereas Ss in Group II could terminate the search at any trial, paying a penalty if they stopped searching before finding the object, but receiving a reward upon finding it. To a first approximation the mean data of both groups can be described by the optimal policy, prescribing minimization of expected loss and no differences between the two groups. Most of the Ss deviated from this policy slightly in the direction of maximizing detection probability and a few Ss deviated more seriously in the direction of minimizing per trial costs. These deviations, however, resulted in only a moderate mean proportional increase in search costs. Attempts to account for trial-by-triM search behavior were unsuccessful. INTRODUCTION T w o p a r a d i g m s for s e q u e n t i a l decision m a k i n g , b e t t e r k n o w n as " o p t i o n a l s t o p p i n g " p a r a d i g m s , h a v e p a r t i c u l a r l y a t t r a c t e d t h e a t t e n t i o n of p s y c h o l o g i s t s i n t e r e s t e d in h u m a n decision b e h a v i o r . T h e less w e l l - k n o w n p a r a d i g m , f r e q u e n t l y used to m o d e l s e q u e n t i a l search processes such as house b u y i n g (see, e.g., R a p o p o r t & T v e r s k y , 1966, 1970), is concerned w i t h a decision m a k e r ( D M ) , who can s a m p l e c o s t l y o b s e r v a t i o n s xl, xz, . . . , x,~, one a t a t i m e , f r o m a single d i s t r i b u t i o n f u n c t i o n F ( x ) . A f t e r p u r c h a s i n g t h e n t h o b s e r v a t i o n D M h a s to decide w h e t h e r to c o n t i n u e s a m p l i n g o b s e r v a t i o n s or to stop. I n t h e f o r m e r ease, he p a y s a c e r t a i n cost, c~+~, a n d t a k e s t h e (n @ 1 ) t h o b s e r v a t i o n , w h e r e a s in t h e l a t t e r ease t h e decision process is t e r m i n a t e d a n d D M receives a n e t p a y This research was supported by a gTant No. M-10006 from the National Institute of Mental Health to the University of North Carolina. The authors wish to thank Thomas S. Wallsten for a critical reading of the manuscript. Now at the University of Georgia, Athens, Georgia. While at the University of North Carolina Dr. Lissi~z was supported by P.tI.S. traineeship MU-08258 from the NIMN, Public Health Service. 1 © 1972 by Academic Press, Inc.

2

RAPOPORT~ LISSITZ, AND 1VICALLISTER

off f , ( x i , Z2, . • • , x ~ ) . The DM's objective is to find some stopping rule, stated in terms of a single decision boundary, that maximizes his expected gain. The more popular paradigm, employed in studies of information purchasing and signal detection (see, e.g., Edwards, 196.5 ; Rapoport & Burkheimer, in press), assumes a well-defined data generating process known to be in either state $1 or So with prior (subjective) probabilities p and 1 -- p, respectively. On each stage of the sequential process an observation is made by DM, the result of which is a random variable x which has a known distribution function F ~ ( x ) if S, is the true state ( u = 0,1). Observations are discrete, independent of one another, and costly; their number may or may not be bounded. After taking the nth observation DM has to decide whether to continue sampling or to make a terminal decision D, that S, is the true state (i = 0,1). In the former ease he pays a certain cost c.÷~,~ (if S~ is the true state) and takes the (n q - 1 ) t h observation, whereas in the latter ease the process is terminated and DM pays the cost c~. The DM's objective is to find a stopping rule, stated in terms of t w o decision boundaries, that minimizes the expected loss of the process. Despite the differences both paradigms provide DM with only o n e c h o i c e on each stage, namely, to continue or to stop purchasing information. Both assume a single source of information which is not under DM's control and are concerned only with the problem of stopping, not of information acquisition from several sources. Many search problems, tactical decision situations, and equipment checking problems include multiple information sources that may independently land sequentially provide DM with relevant, costly, but fallible information. Each information source typically has a different probability of providing valid information and a unique cost. The DM's problem at each stage of the sequential process is to decide not only whether to continue purchasing information but also, if continuing, which source to consult and what type of information to obtain. There have been only a few experimental studies concerned with multiple-source optional stopping process. Kanariek, ttuntington, and Petersen (1969), in a study simulating certain tactical decision making situations, offered three binary information sources to their Ss, each source having a certain known probability of providing valid information and a certain cost for consulting it. On each stage of the process the S could either make a terminal decision D~ that S~ is the true state and pay a (positive or negative):,eost ci~, depending on the decision D~ and the state S, (i,u = 0,1), or he could purchase information from one of the three sources. In another experiment Rapoport (1969) investigated a

SEARCH BEHAVIOR

multiple information source search problem in which the Ss were not allowed to stop. This problem, which resembles several real-life search processes, has served as the starting point for the present study. The regular search problem studied by t~apoporb (1969) is concerned with a single target object that cannot move, located in one of r possible locations (r --> 2). The r locations may be searched one at a time, and it is assumed that the outcomes of the search are independent, conditional only on the inspection procedure used and the location of the object. The search terminates when the object is found. It is assumed that D M possesses a prior probability, p~, that the object is in location i

( i = 1, . . . , r , pi>O,

~ p ~ = 1), i=1

as well as a fixed conditional miss probability, m~ (0 < mi < 1), that if the object is in location i it will not be detected on a single look there, m~ is assumed to be independent of the stage number or the history of the search. Associated with each location i is a cost, c~, for a single look there. It is assumed that the DM's obieetive is to find a decision policy that minimizes the expected total cost of the search. The mathematical aspects of the regular search problem have been studied by Black (1965), Chu (1966), Matula (1964), and others. The optimal policy which minimizes the expected total cost of the searching process has a very simple structure (DeGroot, 1970). ]~or any sequential search procedure let P~. denote the probability that the object will be found for the first time, thus ending the search, during the jth search of location i (j = 1, 2 , . . . ) . Clearly, P ~ > P~,j÷I. Then, regardless of the number of times other locations have been searched before the jth search of location i is made, Pi~ is given by P~. = pi(1 - m~)m~~-1.

(1)

The optimal policy, ~r°, specifies that if all values of the ratio P~j/ci in the two-dimensional j by i matrix are arranged in order of decreasing magnitude, then this ordering is the optimal sequence in which the searches should be made. For example, if the particular ratio P~j/c~ is the nth largest value in the optimal ordering, then the jth search of location i should be made at the nth ~stage of l~he searching process. The optimal policy at each stage is "to search in the location for which there is the highest probability per units search cost of finding the object in the next search [DeGroot, 1970, p. 427]." If at any stage two or more values of the ratio P~j/c~ are equal, these values may be ordered arbitrarily among themselves. Suppose that the search costs are equal, i.e., c~ = c for all i. Then'it

RAPOPORT~ LISSITZ~ AND MCALLISTER

can be shown that the policy ~r° will, in addition to minimizing the expected total cost of the searching process, minimize the expected number of searches needed to find the object. Under these conditions the optimal policy is to order all probabilities P~j, determined by (1), in a decreasing sequence and at the nth stage of the process to search in the location indicated by the nth term in the sequence. The regular search problem does not allow DM to stop searching unless, of course, the obiect is found. In practice, however, almost all search problems involve a terminal reward, R, for finding the target, a terminal penalty, B, for not finding it, and, in addition, allow DM to terminate the search whenever he wishes. We shall refer to these problems 'as optional stopping search problems. Clearly the introduction of terminal reward and penalty interacts with the search cost and thus affects the search policy. The more searches made by DM the heavier the search costs, the greater the probability of receiving the reward, R, and the smaller ~he probability of paying the penalty, B. The greater R or B relative to the search cost, c~, the more times should DM search hoping that the reward or penalty may overcome the search cost. In some cases when R or B is very small relative to c~, it may even be desirable for DM not to search at all but simply to pay the penalty B (Chu, 1966). The optional stopping search problem has been studied by Ross (1968) and Chu (1966). The latter has derived the optimal search policy minimizing overall expected loss--a linear combination of expected search cost, expected penalty, and expected reward. Let L(P~,rr) denote the expected loss for performing the first n searches with some search policy =, P~(~) denote the (cumulative) detection probability of finding the object during the first n searches with search policy 7r,

k=l

(where P(k,~) is the kth term in the sequence of probabilities P~ ordered by policy ~), and 0(P~,~) denote the expected search cost for performing the first n searches with the search policy ~r. The expected loss depends on the search policy and the corresponding search terminating point. It can be expressed as

L(P~,Tr) = B[1 - P~(~)] - RP~(~r) ÷ C(P,~,~). (2) The optimal search policy which minimizes L(P~,~) with respect to P~ and ~ is to search with policy v °, the optimal policy for the regular search problem, and terminate the search when L(P,,~ °) attains its minimum value. A description of the decision rules for terminating ~r° is unnecessary for the purposes of the present experiment but may be found in Chu (1966).

SEARCH ]BEHAVIOR

5

The primary purpose of this study was to investigate and compare human search behavior in the regular and optional stopping search problems. If the optimal terminal searching stage in the optional stopping search problem is denoted by t, then according to 7r°, there should be no differences in search behavior between the two problems in stages 1, . . . , t. Policy ~r° should be employed in both problems until either the object is found or Ss exhaust their resources for purchasing information. Since both the experimental setup and population of the Ss were changed, a secondary purpose of this study was to replicate the results of one of the three groups in the regular search condition reported by Rapoport (1969). METHOD

Subjects A total of 46 mate students from the University of North Carolina at Chapel Hill participated in the experiment. The first 23 were assigned to Group I and the last 23 to Group II. The Ss were run individually for one session which lasted approximately 80 min for Group I and approximately 65 rain for Group II.

Apparatus The apparatus consisted of r - 4 opaque fish bowls, 4.5 in. high and 5 in. in diameter at the top. These were placed on a platform 14.5 in. above the table at which the experimenter (E) and S sat. The bowls were placed side by side so that all were visible to the S and easily reached. They were labeled from A to D and placed on the platform so that A was always on the S's left, next to it was B, then C, and finally D was on the right. Each bowl i, i = I, 2, 3, 4, contained M~ white marbles of uniform size, shape, and weight. In addition, there was one black marble of the same size, shape, and weight, which was to be the object of the search. Each bowl was marked with its letter, the number of marbles it contained, and the cost, c~, to the S for one search of that bowl. E sat across from the S, mixed the marbles every few trials, and recorded the S's choices and the cumulative search cost.

Procedure for Group [ The Ss for Group I were brought into the experimental room individually, seated, and handed a set of instructions for the regular search problem. After reading these, they were given an opportunity to examine the experimental materials and to ask questions. Table 1 shows the information provided to the Ss as well as the values of the conditional miss probabilities, m~, and the prior probabilities, p~. The Ss were told that initially

RAPOPORT: LISSITZ, AND MCALLISTER

TABLE 1 PAR&METERS FOR THE TWO SEARCH

PROBL~3MS

Bowl

Mi m~ c~ p~

A

B

C

D

4 .750 4 .25

7 .857 3 .25

13 .923 2 .25

19 .947 1 .25

Note: R = 30, B = 50 for Group II. they had 100 points, that each point was worth 1 cent, and t h a t the game would end when either the 100 points were exhausted by the .search costs or the object was found. The S's task was to find the black marble which was substituted by the E for a marble in one of the four bowls. The substitution was accomplished by shaking a small box containing four letters (A, B, C, and D ) , the E blindly selecting one of the letters and substituting the black marble in the bowl with this letter, (i.e., pA = PB = p'c ---- p• = .25). This process was done behind a screen and was therefore blind ~o the S, although the procedure was thoroughly explained to him. The S was then instructed to proceed with the search, attempting to maximize his gain on each game. The search was conducted by the S at his own rate by sampling one marble with replacement on each stage of the process. The S proceeded to do as instructed, and the E recorded his searches and made available the cumulative search cost after each search. The Ss were not told how m a n y problems (search games) they would play until the end of the eighth at which time they were told they were finished and the appropriate amount of money was paid to them. The amount of money was equal to the total number of points remaining from 100 for each of the eight games.

Procedure for Group II The procedure for Group II, the optional stopping search group, differed from Group I in only two ways. The first change had to do with search termination. The Ss were penalized B = 50 points for not finding the black marble on a given problem and given an extra bonus of R = 30 points for finding it. T h e y were allowed to stop on any stage prior to finding the object by paying the penalty but keeping the unused points left ~rom the initial 100. The second change from Group I was an increase in the number of problems to a total of 12, so t h a t the mean gain would be approximately equal for both groups. This goal was achieved; the ob-

SEARCH BEHAVIOR

7

served mean and standard deviation of gain were $3.11 and $.81 for Group I and $3.06 and $.80 for Group II, the difference between the groups being nonsignificant at the .05 level. RESULTS

Comparison o/Groups I, II, and RT The first comparison between Groups I, II, and Rapoport's (1969) Group II, which will be referred to as Group RT, is in terms of the proportions of searches in each bowl made by each S. A one-way analysis of variance with repeated measures performed on the proportions of searches in each bowl summed over problems for each ~ separately, yielded nonsignificant results at the .05 level. Despite the differences in apparatus and nationality of Ss between Groups I and RT, the group search beharlot, as measured globally by the proportion of searches in each location, was highly similar. A psychologically more interesting finding is that the knowledge that they can terminate the search at any moment does not affect the Ss' search behavior as measured by the proportion of searches in each location. The mean proportions are presented in the last row of each of the three matrices in Table 2. To further analyze the results several search policies, previously introduced by Rapoport (1969), will be considered. The first is the optimal search policy, vo, which dictates to search on each trial that location for which the probability of finding the object per unit search cost is maximum. Table 3 presents the sequence of optimal searches, ~o, for n--I , . . . , 50, as well as the corresponding detection probability, p,(~o), and the corresponding detection probability divided by the cost of search,

Q~(~o) - ~ p(~,~o)/~, k=l

where c~ is the search cost paid on trial k,/c = 1 , . . . , n. The table shows that the first search should be made in Bowl A, the second in D, and so on, until the object is found. If ~r° is employed, the object will be found in eight trials with probability .2043, in 27 trials with probability .50,1], etc. A second policy to be considered, denoted by ~ , is one that minimizes the expected number of searches needed to: f~nd the object (or equivalently minimizes the expecte d total cost of the search, if it is assumed that c~ is constant for all i). "The detection probabilities corresponding to v l P,,(Trl), are presented in the last column Of Table 3 (the sequence of searches, ~r~, for n ---- 1 , . . . , 50, is presented in the first column of Table 2 in Rapoport, 1969). A third policy, denoted by ~r~, is one that minimizes the search cost per brial, which in the presen t case means searching Bowl

8

RAPOPORT~ LISSITZ; AND MCALLISTER TABLE 2 STEPWISE OPTIMAL SEARCHES SUMMED OVEI~ ALL PROBLEMS AND ALL SS

FOR EACH GROUP Group I; observed decision

Predicted search (by d°)

A B C D Sum

A

B

C

D

Sum

202 (.041) 33 (.007) 5 (.001) 639 (.131) 879 (.180)

214 (.044) 39 (.008) 29 (.006) 923 (.189) 1205 (.247)

190 (.039) 21 (.004) 51 (.010) 1047 (.215) !309 (.268)

295 (.060) 129 (.026) 40 (.008) 1023 (.210) 1487 (.305)

901 (.185) 222 (.045) 125 (.026) 3632 (.744) 4880 (1.000)

Group R; observed decision

Predicted search (by 6°)

A B C D Sum

A

B

C

1)

Sum

181 (. 046) 25 (. 006) 11 (. 003) 452 (. 114) 669 (. 169)

247 (.062) 52 (.013) 26 (.007) 651 (,164) 976 (.246)

343 (.087) 103 (.026) 12 (.003) 717 (.181) 1175 (.296)

343 (.087) 143 (.036) i0 (.003) 647 (.163) 1143 (.288)

1114 (.281) 323 (.082) 59 (.015) 2467 (.623) 3963 (i.000)

Group II; observed decision

Predicted search (by 6°)

A B C

D Sum

A

B

C

D

Sum

295 (. 062) 38 (. 008) 16 (. 003) 633 (. 133) 982 (. 206)

269 (.057) 66 (.014) 45 (.009) 742 (.156) 1122 (.236)

247 (.052) 44 (.009) 30 (.006) 861 (.181) 1182 (.248)

372 (.078) 245 (.051) 31 (.007) 825 (.173) 1473 (.310)

1183 (.249) 393 (.083) 122 (.026) 3061 (.643) 4759 (1,000)

SEARCH BEHAVIOR

TABLE 3 T h E OPTIMAL SEARCH POLICY FOR GROUPS I AND I I n

7:.o

p..@o)

Q~OrO) p~(~rl)

n

lr°

P~(~r°)

Q.(Ir°)

P,, (Tr~)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A D D B D A D D B D C D D C A B D C D D C B D C D

.0625 .0757 .0881 1238 1356 1825 1937 2043 2349 .2450 .2642 ,2737 .2827 .3005 .3356 .3619 .3704 .3868 .3949 .4025 .4177 .4402 .4474 ,4614 .4683

.0156 .0288 .0412 .0532 .0650 ,0767

.0625 .1094 ,1451 .1802 ,2109 .2372

26 27 28 29 30 31

A D C B D C

.4946 .5011 .5140 .5333 ,5395 .5514

.2485 .2550 .2615 .2679 .2741 ,2800

.5496 .5606 ,5712 ,5816 .5918 .6018

.0879

.2635

32

D

.5572

.2859

.6113

,0985 1087 1187 1283 1378 1469 1557 1645 1733 1818 .1900 .1981 ,2057 .2133 ,2208 .2281 .2350 .2419

.2860 .3057 .3250 .3442 .3620 .3785 .3949 .4100 .4246 .4390 .4530 .4661 .4790 .4915 .5036 .5155 .5273 .5385

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

D B C D C D A B D C D C D B D C D A

,5628 .5793 .5903 .5955 .6056 .6106 .6304 .6446 .6493 .6586 .6631 .6717 .6760 ,6881 .6921 ,7001 .7039 ,7187

.2914 .2969 .3024 .3077 .3127 ,3177 ,3266 .3274 .3321 .3367 .3412 .3455 .3498 .3538 .3578 .3618 .3656 .3693

.6207 .6297 .6386 .6473 .6558 .6641 .6722 .6802 .6879 .6955 .7029 .7101 .7170 .7238 ,7303 .7369 .7431 .7494

D on each t r i a l (with a cost of one u n i t ) . T h e v a l u e s of P . 0r 2) are easily c o m p u t a b l e a n d n o t presented here. T h e m a j o r d i s a d v a n t a g e of ~r°, ~r1, 772 as possible models for e v a l u a t i n g a n d describing t r i a l - b y - t r i a l i n d i v i d u a l search b e h a v i o r is t h a t t h e y do n o t allow for errors. Once a n S has m a d e a n incorrect search r e l a t i v e to a p a r t i c u l a r search policy, the c o n d i t i o n a l sequence from t h a t t r i a l on does n o t correspond to the original sequence. A model is required t h a t , w h e n provided with a n objective criterion a s s u m e d to be optimized b y the S a n d with the search h i s t o r y of the S up to a n d i n c l u d i n g t r i a l n, will prescribe the search for t r i a l n -1- 1, w i t h o u t a s s u m i n g o p t i m a l i t y of the preceding n searches. W h e n tile criterion is the m i n i m i z a t i o n of the expected t o t a l cost of the search, we shall denote the stepwise o p t i m a l model b y 8°. C l e a r l y , 3 ° should be c o m p u t e d s e p a r a t e l y for each S a n d each problem. T o test the policy ~o .stepwise o p t i m a l searches were c o m p u t e d separ a t e l y for each S a n d each p r o b l e m a n d t h e n s u m m e d over all Ss a n d p r o b l e m s w i t h i n each of the three groups. T h e r e s u l t s are p r e s e n t e d in

10

RAPOPORT, LISSITZ, AND MCALLISTER

Table 2 in three 4 X 4 matrices, where the columns of each matrix give the total number of searches in each bowl and the rows give the stepwise predicted frequencies. Because the total number of choices made are different for each group, the proportions of observed and predicted searches are given in parentheses. As can be .seen, the policy 8o is inadequate for describing the trial-by-trial search behavior of the Ss in all three groups. Stepwise optimal behavior, as predicted by 8°, would be characterized by a set of large main diagonal values. Inspection of the main diagonal shows that 3° accounts for only 26.9'% of the choices of Group I, 22.5% of the choices of Group RT, and 25.5% of the choices of Group II. When the stepwise predicted choices are considered, the 16 proportions in each of the three matrices seem to be highly correlated, providing further evidence for the similarity among the three groups. Ordering the 16 cells in each matrix in terms of tile magnitude of the proportions and calculting Spearman rank-order correlation.s confirms this observation. The correlation between Groups I and RT was .879 and that between Groups I and I I was .915, both correlations being highly significant (p < .01). In an attempt to construct a descriptive model for search behavior in the regular search problem, Rapoport fitted, unsuccessfully, a Markov chain model to the observed .searches of Group RT. The same attempt made in the present study led to similar results, namely, the order of the required Markov chain was greater than two. It was felt that the corresponding loss of parsimony made attempts to construct r-order (r--> 3) Markov chain models undesirable. The one-step observed transition proportions summed over problems and Ss in each group are shown in Table 4. Despite the rejection of a first-order Markov chain model, the results reported in Table 4 show interesting consistencies. The average behavior is a weak tendency to search in the same location repeatedly, or when changing, to choose an adjacent location. The mean run length for each bowl was computed for each group. The means for Group I were 2.13, 2.24, 2.40, and 3.43 for Bowls A, B, C, and I), respectively. The corresponding means for Group II were 2.23, 2.34, 2.39, and 3.58. None of the four differences between the means of the two groups was significant at the .05 level, and in both groups the mean run length increased a.s the per trial cost decreased. A comparison of the three matrices in Table 4 shows them to be highly ~imilar; Ordering the transition proporti:ons by their magnitude and calculating rank-order correlations yielded a value of .980 for the comparison of Groups I and I~T and a value of .982 for the comparison between Groups I and II. £ one-way analysis of variance with repeated measures performed on the individual one-stage transition proportions (summed over all problems for a given S) yielded nonsignificant results'at the .05 level, showing again no differences between the three groups.

11

SEARCH BEHAV1OR

TABLE 4 ONE-STAGE TRANSITION PROPORTIONS FOR EACH GROUP

Trial n A

B

C

D

.104 .233 .597 .123

.079 .074 .213 .728

.056 .153 .758 .096

: .057 .056 .123 .807

.110 .194 .599 ,124

.095 .080 .201 .740

Group I Trial n - 1

A B C D

.552 .127 .060 .067

Trial n - 1

A B C D

.678 .080 .033 .041

Trial n - 1

A B C D

.579 .126 .064 .064

.265 .565 .130 .081 Group RT .210 .712 .086 .057 Group II .216 .600 .136 .072

T h e S e a r c h P o l i c i e s ~o ~1, a n d ~r2

T.he failure of the M a r k o v chain model of order one or two, of the stepwise policy ~o, and of other models not discussed here, as well as a careful inspection of individual data, rule out the possibility of providing a simple and accurate characterization of trial-by-trial individual search behavior. Postexperimental interviews with individual Ss indicated no consistent patterns of search. I n searching for the object almost all Ss reported paying varying degrees of attention to both the cost of search and the detection probability, realizing that a bowl with a higher detection probability per trial had associated with it a higher search cost per trial. A few SS attempted the search pattern A, B, C, D, A, B, C, D , . . . , whereas a few others searched Bowl A for a few trials then shifted to Bowl B for a few more trials and so on, but none of them employed the same search policy for the entire session. Only one S in Group I I adopted the policy of simply paying the initial penalty with no searches, and he did this just once. But even if tria]-by-trial search behavior cannot be presently accounted for, the question remains as to how well did the Ss, who had been specifically instructed and presumably motivated to maximize gain, perform relative to the optimal policy ~r°. Recalling that, as the Ss clearly reported, bosh detection probability and cost of search per trial affected their search behavior, it seems reasonable to compare the group data to

12

RAPOPORT~

LISSITZ~

AND

~CALLISTER

the policies ~o, ~ , and 7r2. As indicated above, 7ra assumes maximization of detection probability regardless of cost of search, whereas 7r2 assumes minimization of cost of search per trial entirely disregarding the detection probabilities. Both ~r1 and 7r~ can serve then as bases for the evaluation of the empirical results, since they represent two ends of a continuum with T° falling between them. T h e .analysis m a y be performed alternatively in terms of P . ( ~ ) or d(P~,~). Figure 1 portrays the predicted detection probabilities p~(~o), P~(TF), and P.(~r2), for n ~- 1, . . . , 50. I t can be shown t h a t limP~(~r °) = l i m P . ( ~ F ) = 1, as n--~ co, but t h a t limP~@r ~) -----p~ = .25 as n--~ o~. I n addition, Fig. 1 depicts the observed mean detection probability for Group I, denoted b y P ~ ( I ) , for n = 1, . . . , 38. P , ( I ) is based upon the first 38 searches summed over all problems and Ss in Group I. I t was computed b y taking the n u m b e r of problems in which t,he obiect was found on or before trial n and dividing this n u m b e r by the total n u m b e r o~ problems, which is 23 X 8 -~ 184 for Group I. The analysis was stopped at trial 38 since this was the last trial on which all Ss either had found the obieet or had resources to continue searching. The results of this analysis are similar to the earlier R a p o p o r t study in suggesting that, on the average, Ss place slightly more emp.hasis on detecting the obieet t h a n they do upon the per trial costs relative to 7r°, and t h a t the group data follow the predictions of ~r° fairly closely. We shall return to this point in the analysis of individual data below. A related but somewhat different perspective can be gained by evaluating the results of Group I in terms of the mean cumulative cost. Figure 2 portrays the relation between the expected search cost, C (P.,Tr), and the I

I

,3

I

I

I-

l

I

L

/

,2

2 Pn{~ )

-

I

I

0

0

5

IO

15

20

25 Trial

30

35

40

45

50

Fro. 1. Predicted and observed detection probability vs trials for policies 7ro, ~r1, 7r~, and for Group I.

13

SEARCH BEHAVIOR

70

I

1

I

I

1

I

I

~

/

I

c~

_

C

f,s

i

(Pn

m 2(2

0 i (

Fzo. 2. Expected for G r o u p I .

I

I

.I

.2

I

I

I

.5 .4 .5 Detection Probability

~

.6

1

.7

I

.8

search cos(: vs detection probability for policies o, 7r~, 7r~, and

detection probability, P . (~r), for the three policies 7 c, ~r1, and u2. Note that in 5his figure ~o lies below the other two policies. Indeed there is no policy which lies below 7% The figure also shows the mean cumulative cost of search of Group I, C(Pn,I), plotted against the mean detection probability, P~(I). The figure shows that for P , > .1 the proportional increase in expected cost relative to 7r°, [C(Pn,I) - - C ( P , , T r °) ]/C'(Pn,I), which provides a reasonable measure of efficiency, is never larger than 26%, and that it tends to decrease as P~ increases. An apparent inconsistency between Figures 1 and 2 should be clarified. As indicated above, most of the Ss seem to be basing their decisions on both maximizing the detection probability and minimizing the per trial cost with more emphasis on the former relative to 7r°. As we shall show below in an analysis of individual rather than group search behavior, a few ,8s, though, put their emphasis heavily upon per trial costs. In Figure 1, averaging t.he majority of Ss who employed a mixed 7r° and ~r* policy and a smaller number of Ss with a mixed 2 and 7~° policy resulted in a curve pulled downward close to 7~°. In Fig. 2, since ~2 is above ~r1 and ~o the effect of the averaging process is to pull the curve upward away from

~.o.

In an effort to analyze the results portrayed in Figs. 1 and 2 in terms of individual rather than group data relative to 7r°, a statistic a . was defined as the predicted detection probability for Tc, P . (=~), minus the observed detection probability for a given S, a given problem, and a given trial. To compute the observed detection probability for each trial, we assumed knowledge of the parameters pl, ml, c~, as well as Bayes formula;

14

RAPOP0,RT~ LISSITZ~ AND 2~ICALLISTER

i.e., we made the rather drastic assumption t h a t the subjective probabilities were identical to p~ and m+, that utility was linear in c+, and that the search information was processed optimally in updating the subjectve probabilities. If, for a moment, we let the mean detection probability for Group I, P , ( I ) , in Fig. 1 indicate the observed detection probaNlity for a single subject and a single problem, than as = P , ( ~ r °) - - P ~ ( I ) for n = 1, . . . , 38. The data of interest are the mean ~,~, denoted b y a, averaged over all trials and all problems for each S in Groups I and II and the corresponding standard deviation (SD). Since, if n > 1, ~ is not grossly biased b y the number of searches made and since the same optimal policy is dictated for both groups, a is a reasonable measure for comparing Groups I and I I to each other and for comparing individual search behavior in both groups to 7r°. Table 5 presents the ~ values and the corresponding standard deviations for individual Ss in Groups I and II. Most of the Ss in both groups achieved negative Ms, indicating a bias toward maximizing detection TABLE 5 MEANS AND STANDARD DEVIATIONS OF o~n

Group I

Group

II

Subject

a

SD

a

SD

1

081 012 022 013 020 018 043 061 - 020 -- 025 - - . 022 • 067 --. 025

.121 .022 .018 .045 .017 .021 .008 .070 .020 .011 .018 .094 .009 .051 .013 .013 .012 .014 .037 .005 .029 .015 .041

--.041 .067 .030 --.010 .045

.015 .112 .042 .016 .030 .018 .021 •014 .025 .010 .029 .012 .021 .006 .022 .020 .028 .028 .019 .032 .016 .029 .021

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

•

.019 024

--.

047 - - . 037 - - . 040 • 009 --. 050 --.

--.030

- - . 042 008

--.

--.007

--.032 --.048 --.009 --.018 --.023 --.051 .005 --.050 --.012 --.019 --.030 --.044 --.033 .054 --.017 --.016

--.022

SEARCH

BEHAVIOR

15

probability vs minimizing per trial search costs relative to ~r°. The analysis shows that the five Ss in each group who biased their search behavior in the direction of minimizing per trial costs were more extreme in their behavior than were the Ss who exhibited the opposite bias. In other words, with few exceptions, the positive a's are the largest in absolute value. Comparing Groups I and II, it is seen that the difference between the mean ~ values of the two groups is almost nonexistent. The mean a for Group I is --.011 (SD -----.031) and for Group II is --.012 (SD = .025).

Terminal Searching Trial In determining the values of B and R for Group II, an attempt was made to compromise two goals. The first was to have the optimal terminal searching point as far away in a problem as possible to encourage the Ss in Group II to make many searches, as many as Group I, and hence allow a meaningful comparison between the search behavior of the two groups. The second goal was to keep B and R sufficiently small so that the stopping option could be seriously considered and frequently exercised by the Ss. The resulting expected loss function, L (P~,Tr°), as well as the functions B [ 1 -- P~ (Tr°) ], RP~ (~r°), and C (P~,Tr°) are portrayed in Fig. 3 as a function of the detection probability P~ (Tr°). The striking thing about L (p~,~o) is its slope; L(P~,~r °) is almost parallel to the z-axis in the region 0 < P~(Tr°) < .9, slightly decreasing (with a few very minor reversals which cannot be observed in the figure) as p~(w0) increases. Figure 3 shows clearly t h a t the actual trial terminating the search hardly affects the magnitude of the expected cost. 7° I ~° /

~

,

,

,

,

,

, E /2 ~(~°'~)~/

.I

.2

.3

I .4

I .5

I .6

I .7

50 20

00

.8

.9

Detection Probability Fie. 3. Optimal expected loss vs detect,ion probability for Group I I .

16

RAPOPORT~ LISSITZ; AND MCALLISTER

This insensitive optimal stopping policy was reflected in the trial on which the Ss in Group II chose to stop searching. The search was terminated without finding the object (and the penalty B was paid) on 184 out of 276, problems. The frequency of observed stops plotted against n forms a relatively .flat curve skewed toward the upper end with a range from no trials (stop without any searching) to the 39th trial and a modal trial number of 14 (10 out of 134 stops). DISCUSSION The results reported above support the hypothesis that the availability of a stopping option with the introduction of a penalty for terminating an unsuccessful search and a reward for finding the object have no effect on search behavior. The analyses performed on the proportion of searches in each location, on the individual one-stage transition proportions, and on the statistic ~ yielded nonsignificant differences between the groups in each ease. Moreover, analyses of additional sequential statistics such as number of runs (i.e., consecutive searches in the same bowl) and mean run length, calculated separately for each S but not reported above, also yielded nonsignificant differences between Groups I and II. Analyses of individual data show, relative to 7r°, small deviations (biases) towards maximization of detection probability (policy 7r1) for most Ss and somewhat larger deviations toward minimization of per trial costs (policy 7r2) for about 20% of the Ss. These deviations result in a mild mean proportional increase in search cost whie.h is never larger than 26% for Group I and typically smaller than that for Group II. Independently of these deviations, relative to ~° which generally prescribes runs of length one and only infrequently runs of length two (see Table 3), there is an observed tendency to repeat searching the same location for several trials, a tendency which increases 'as the per bowl cost decreases, and then to move to an adjacent location. This later finding should be interpreted cautiously, however. The possibility of an artifact cannot be ruled out since both this study and l%apoport's (1969) placed and labeled the four bowls in exactly the .same way. It is instructive to compare the present study to that of Kanariek et aI. (1969). Despite the many differences between the two studies, the Ss in the Kanariek et al, study could terminate information acquisition on any trial and could choose any of several sources of information on every trial with the actual sampling cost depending upon the chosen source. Also in both studies, (with the exception of one experimental condition in the Kanariek et aI. study), the greater the diagnosticity (m~ in our study) of the source, the greater the cost assessed for consulting (or searching) that source. Results reported by Kanariek et al. show that as in the pres-

SEARCI-I B E H A V I O R

17

ent study, Ss did not consult the information sources in an optimal fashion. As in our study, the Ss' performance was clearly influenced b y the costs for consulting information sources, but, unlike our study, a bias towards "conservatism" was observed in which Ss showed a reluctance to expend their resources for consulting the more diagnostic information source. I t m a y be recalled t h a t the Ss in our study exhiM~ed the opposite bias. The discrepancy between the two studies m a y be accounted for. I n the K a n a r i c k et at. study the optimal model prescribed almost exclusively, for all the experimental conditions, the consultation of the most diagnostic (and most costly) source of information, i.e., Source A. Hence, unlike our study, any discrepancy from optimality in the K a n a r i e k et at. study was necessarily towards w h a t they called "conservatism." As K a n a r i c k et aI. have realized, "To evaluate whether this reluctance to obtain the best information in a multi-source situation is unique to the conditions of this experiment or whether it is a more general finding, an experimental design is required where the optimal behavior requires consulting Sources B and C as well as Source A [p. 382]." This is precisely what was done in the present study. REFERENCES BLACK, W. L. Discrete sequential search. In]ormation and Control, 1965, 8, 159-162. C ~ , W. Wi Optimal adaptive search. Report SEL-66-085 (TR No. 6252-1), Stanford Electronics Laboratories, Stanford, California, September 1966. DEGRoo% M. It. Optimal statistical decizions. New York: McGraw-Hill, 1970. EDWARDS,W. Optimal strategies for seeking information: Models for statistics, choice reaction times and human information processing. Journal oJ Mathematical Psychology, 1965, 2, 312-329. KANARm~, A. F., tI~¢TINGTON,J. M., & PETERSE~, R. C. Multi-source information acquisition with optional stopping. Human Factors, 1969, 11, 379-385. MAT~LA, D. A. A periodic optimal search. The American Mathematical Monthly, 1964, 71, 15-21. RAPOPO~T, A. Effects of observation cost on sequential search behavior. Perception and Psychophysics, 1969, 4, 234-240. RAPOeORT,A. & BT:RXHEIM~R,G. J. Models for deferred decision making. Journal o] Mathematical Psychology. In press. RAPOPORT,A. & TVERSXY,A. Cost and accessibility of offers as determinants of optional stopping. Psychonomic Science, 1966, 4, 145-146. RAPOPORT,A. & Tv~s~v, A. Choice behavior in an optional stopping task. Organizational Behavior and Human Per]ormance, 1970, 5, 105-120. Ross, S. M. A problem in optimal search and stop. Operations Research, 1969, 17, 984-992. RECEIVED:

October 27, 1970