Search for rules for search

Journal of Eco:lomic Behavior and Organization 3 (1982)65-e,1. North-Holland

S E A R C H F O R R U L E S FOR S E A R C H J o h n D. H E Y * University of York, York YOI 5DD, :5 K Received June 1981, final version received July 1982

This l~aper presents the findings of some prelinfinary 'laboratory' investigati~, as into 'actual" search behavior. Speeitieally we looked at situations in which searchers' initial in~mretic,v -,bout the distributions was negligible. It seemed likely that simple 'rules of thumb" wo~tC '~; used in such ,'~tuations. Our findings confirmed this view: we ":dzn..t~ed five such rules "~,i~ic~ betw~n them "accounted' for a high proportion of observed behaviour. Moreover, these ~les, ,lthough not 'optimal') appear to be reasonably good and may be fairly robust; joint~, taey may consti :late a bettt;r explanation (and predictor) of actual search behaviour than do the carrently populur supposedly optimal rules.

1, Introduction and overview T h e p u r p o s e of this p a p e r is to present the fiadings of some p r e l i m i n a r y l a b o r a t o r y - t y p e investigations into 'actual' search behaviour. P r i m a r i l y m o t i v a t e d by a g r o w i n g c o n v i c t i o n that the increasingly sophisticated models of o p t i m a l search b e h a v i o u r a p p e a r i n g in the e c o n o m i c s literature are a decreasingly a c c u r a t e reflection of reality, o u r investigations were designed to discover the actual rules that individuals use w h e n carrying out search activity. O f p a r t i c u l a r interest to us were the rules used in situations whe,:e initially searchers h a d essentially no k n o w l e d g e of the distribution over w h i c h they were searching. C a s u a l empiricism a n d i n t r o s p e c t i o n suggested to us t h a t simple 'rules of t h u m b ' w o u l d be e m p l o y e d in such situations [sec H e y (1981)]. O u r investigations were ~esigned to see whether, in fact, this was the ease, a n d is so, to d e t e r m i n e the folm of the roles of t h u m b :Lctually employed. T h e findings r e p o r t e d [ e r e are clearly preliminary, in the sense tlaat our investigations suggested l.~ us a ,.vhole range of other possibilities which *1 am greatly indebted to Stephen Baker, Athan Balfc,ussias, Stella Balf~ussias. Peter Barker. Richard Barnett. Robert Beedl'am, Eduard Berenguer, Nell Blake, Da~,i~ G~wland, t~rla~: Hillier, John Hutton, Pa't Kemwn. Men, Korchid. i'eter Lambert, Guillermo Lopez, Anne Ludbrook, Edwaid Lynk, Alan Neilson, Don Poskitt, John Rae, Martin Reynold~,~, Robert Risohx~, Harold Seddighi, Peter Solar, Eiieen St:tcliffe, Andrew Tremayne, Stephen Trotter,

Anthony Weekes, Jayne Wilae, Robert Woodlield and Simon Woodward for taking part in m? experime~lt. I am also greatly indebted to Professors Richard Day and Sidney Winter and two anonymous referees for helpful comments on an earlier draft ~f this pape:. ()16~' "~.& ~. g " ,~/{*~ . :00/'0

lf){~¢~ 75 ~' I OQ~ .... ,. "¢dl'~ .......... ~o. N o r t h - } [ o l l a a d

would be of interest to explore, but nevertheless they are instructive and of considerable interest in their own tight. In parficdar, they suggest that some very .simple roles of thumb m 'acoount' for a vet3~ Mgh proportion of ol~eerved behaviour. Moreover, a p r e ] ~ ~ examination of these rules i~[~ates that, although they are of necesdty not o p ~ a l , they are ~ t ~ a n ; ~ mpo. ......~ ~t property in the r~etonably good .~i~,~my ~ f ~"~,~r l y obus ,.x0vancly unco~ain environm~t in whi,I~h our ir,vestigations are ~t. Our invcstiga|[ons are net ~ f i r s t su~:~laboratory,tyl~ invtstigations into s~u,~.h behavior: Prcvious.stu:.~.~=s..in~t~:~ ~ out by psychologists in the Lit, '60~ [Rapoport ~-adTvc~=ky ( i ~ i and 1970~ and Kahan, Rapopot~ and Jon¢,,(l~7)] a u d a ~ m o m , ~ ~ o n e b y ~ n o m i s t s , Schotter and B~-aunste~ (1981). However, the.' studies differ quite mexk¢dly from ouxs m that ~.ey: w c r e ~ i n l y . c o n ~ m c d with..whe~, observed search behaviour was broadly .consistent w i t h . ~ p ~ c f i o n s of optimal sea~h theory;, m contras:rg o ~ m ~ ~ t i o n s were d ~ e d R, delve: deeper, and to r ~.

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The l~per is organised ~s follows. I n ~ next section we describe our ¢xpcrim~tal proctgtur¢. In s~fion 3 we describe ~ ' v a r i o u s 'rules of thumb" we ideaRifiod, and in section, assess tkc implicatiom 0?these rules (at both an iindividuad and a market level). Finally, section 5 offers some comments in cor~lusion.

We era'tied out two sets of investigations: the first set very much a pilot study; tl~ second set a more detailed pJ.'o~3¢oI analysis. The ba~c framework of both sets was essentially the sa~e, ~ was the pro'pose: to investigate the 'actaaF behaviour~of s e a r c ~ sea~hittg :br the lowc:st price of some go~t, in a si~fion whct¢ their prior ]mowkxlgo of the price distribution was essemfiall~y non.ex~."~nL. Volwateers for the experiment w=re ~ven detailed written instructions in advana~ The actual e x ~ e ~ n t was carried out with the volunteer alone in a room with a VDU terminal ( c o ~ e d i to the UniVersity's main computer), n~nning a simple interaL~v.: program. ']'he subject was asked to envisage a ~t~ation m which he or she had decided to buy some item (a consumer durable, for cJr~unple), cad had decided on a spe~fic make and model. Moreover, the subject had also made up his or her mind as to the maximum amount that thaw Were willing to spend on this item. This maximum amount, w l ~ h we denoted by £x,. £s a f a m ~ concept ~n d0mand theoD, particularly fi~lr~ b l e goods: it is the t,~m/tp r ~ . ~ i ~i~t~fd:~O~i~~~ ~tkata very 1 ~ . (formally an infinite WtLmber) Ofsho~ sold tiffs | t e ~ but at ~ s i b l y diffe~g prices, initially, the, se~cher was #yen no klea of what prisms were av~ab!e, but could get any

J.D. Hey, Search for rules for search

67

number of price quotes (from different shops) by paying a cost Le per qt,tote. Full recall was available (that is, '.he searcher could return to any shop at any ..,~tage and buy the i~em at the price quoted). Moreover, subjects were told that there gas nothing special about the order in which quote,~ were obtained. To enable subjects to familiarise themselves with the search problem, they were atlowed as many 'trial runs' ~s they req~ired; it was emphasised that t h e e trial runs were simply to demonstrate the mechanics of the experiment and were otherwise uninformative. Very few subjects aad more than 2 such ~ a l rims. After these trial runs, the ~archer moved on to the 'teal thing'. ~m the pilot study, this consisted of a total of 9 search problems. Tiffs preyed to be too many; in the main study, just 4 search problems were prei.entedi. In all cases, but unknown to the subjects, the underlying price' distribution was normal. The values of the mean ~ ) a n d ~aandard deviation (~) ~or the four cases in the 'real thing' of the main study are given in tables 1 to 4; again these were unknown to the subjects. They were chopin to make the exl~riment 'realistic' in the light of the empirical price distributions reported in Pratt, Wise and Z~khauser (1979) The volunteers were told that £x, ~.he maximum amount they were willing to spend on the item, was to cover both purchase and search. ~ In view of this, we necessarily had to give them the option of giving ur, searching without buying (if, for example, their lowest price so far enc,~untered plus their accrued search costs exceeded £x, and they saw no point in continuing), la the pilot study, we rather poorly specified this option; as a consequence some confusion may have resulted. However, in the main ,.tudy, we tightened up the speofication, so that searchers were allowed to give up without buying only ff their accrued search costs plus their milfimum price quote exceeded £x. (In fact, only 2 o~" the 124 cases ended in the searcher exercising this option.) Conversely, searchers were not allowed to buy ff their lowest quote plus the accrued search costs at that stage exceeded £x. Each s~.rch problem (whether in the 'trial runs' or the 'real thing') began with the sear¢t~or being told x and c, and being given an initial pri~ quote. After each quote) they wore asked whe~:her they wish to buy (at the ~owest price so far available, if i~asible), to continue searching, or ~o give up without buying (if permitted by the rules). After eacL new quote was generated, subjects ~ere reminded of x and c, and the price quotes so fc,r obtained (in chronological order); ~nd the amount (:hey would spenc~ in total (purchase plus ~arch) ff purchase ~ook place at the lowest price so far obtaiped. LThis a~umption is slightly at odds with the assumption usually employed in consualer scitreh theory. In this it is normally assm,a ~ that ourchaso will take place irrespective of the prco paid - - as Ion3 .ts it i,~ sufiieienfly Ic,v. This implies '.oth a ~ro/~come-elasticity a~ld a zero prit~,--elasticity of demand (in the relevant ~'ange); a rather embarra~ing combination. Our assumption, although still restrictive, i.~ more plausible.

Table I Raw d,,ta of Case 1.

Consistent

with

m|~$) i'b

0 O) O)

C E A*

J~,¢8

A ~73~'-:I~.~ 19135

(7) (8)

(9) (Io~ (il)

:7..~85

~41

A-

.

185.91 .:183.-70

c

(E)

~13::~"t75.32

C E A*

.274:91"

~.05 ~to

C- E

~:: 281X~:- 2 ~ I :

175~0

.248,(X)

229.19. ~}1.49

24i67 (l) A B A

~..~ 207.19. 2").25.10 22e.4~

- ( i 2 ) -~

(13)

237,55:. 23,5.12 2.13280 17:L82 234.(}8 226.17 252.13 21,7-63 197.47 185.97 224.58

-(ttq

.235a$- 246,50~,-189i~/~

(1"0

22&Of _-21,SA9 ;..-2~14..

(18)

191.30

i76.42 19921 241L~ ..... ~ : 7 6 2t7..90 198.55

236.25

(2O)

-174.';3

207.65

~2~L~

24665

229.95

189.70 B A* B A C A A B C A

148~3

(22) 244:18 19~76 -

(25) (26) (2"0 (28) (29) 00) 01)

237105

~-: -

24~17 211.66 222~47 197.22 219.85 1.76.59 204.99 24~t.58 219A4 2!3.71 205.74 .

Z.~9.38

.,-L~gY~.~: ~ . 5 3 .... 246.9 243.57 192,14 236.50 ~ L 4 8 21~ 215.34 183,T7 2]I.~, _

D E D E

B C C

D E E

C B

E D

. ~

~(!)-~,~th on b o ~ of ~

D

(c) E

221.12 221.29

D

~

,

-mixt~ al B ~

C (or of D and E),(letterin brae.ketindicatescase c o ~ t ~ rule~ ~¢itlwr Aor B or C'. 27 (out ,ff 3!) (I borderline), (b) number A Or'D or E~ 27 (out of 31) (1 borderline), (¢) subjects consistent

~ a ) ~ ~ t ctmsistent w i t h : ~ with none of rules (nor mixture): 9, ] 4 aud 29.

T h e p r o g r a m kept a reeord tff the subject's responses: to be specific, the number of trial r u m used, and the sequences of price quo'tes generated in t ~ h o f t ~ ~. searc!h problems in tb.e 'real thing'. In addition: ihe artswers to t w o questior, s .d~signed: ~to elicit some. information abowz the searcher's a ,tirade to ~ - m ~ m ~ ~ = : - -:~ - .....~ . . . . . in t ~ ~jo~u~ly~:~n0~ ~ 6 n ~ i n f 0 ~ a t i o n was gathered. This did not cremate any problems when it came to a~essing whether the aetua! search

:. O. Hey, Search fo~ rules for search

69

Table 2 Raw , a m of Case 2. Parameters: x = 100, c = 2.0,/z = 86.0, o =: 10.0. Consistent with rule(s)~.b

S no.

(I) (2) (3) (4) (5) (6) (7) (8) (9) (I0) (11) (12) (13) 114) (15) (16) (17) (18) (19) (20)

(21] (22) (23} (24) (,"5) (26) (27) (28) (29) (30) (31)

Price quotes obtained in chronological ord~-r 84,25 108.34 88.64 94.35 87.41 84.57 82.33 79.37 68.63 101.92 88.25 ~Z63 93.97 94.30 100.98 106.32 106.71 72.53 87.41 79.18 79.09 92.41 86.71 116.08 94.25 67.45 84.82 64.82 86.07 85.90 90.18

80.74 71.37 71.74 91.76 97.96 70.20 76.18 101.40 103.38 11)6.10 103.63 99.00 93.69 82.05 74.79 85.51 69.20 90.86 92.52 72.87 90.91 87.84 80.05 67.07 91.08 85.62 78.50 82.33 96.86 82.29 85.93 74.71 86.51 90.98 71.14 85.02 88 18 6 8 . 8 , ~ 94.60 68.5t 98.17 80.08 80.18 91.80 84.82 91.13 76.11 88.58 89.31 88.71 91.52 89.48

111.78 101.82

8 A* A* B g C B 13 C ("

1t30.65 67.11

10:'.95 84.42

71.05

85.74

~ 1.02

D

D D E D D E E

j •

A* 1~ B ,.. A

86.34 90.97 70.14

D D E

A ~

89.57 73.2.: 85.08

B A C

83.76

8 2 .~

D

E

92.75 A* (I, C D B B C A

E D D E E

"(I) consistent with mixture of B and C (or of D and E). t(a) number consistent wdh either A or B or C: 27 (out of 31), (bl nul:ber consistent with either A ~',r D or E: 28 (out of 31), (c) sabjects consistem ~mth nor,e of rules tnor mixture): 11 an~ 23.

behaviour was consist :r~ with optimal search b e h a v i o ~ : r - which ~s an assessment of vital interest to memb, rs of the as if' schoo.l of methodolcgy. We simply calculated lhe optima! reservation trice, and checked whethe: the subject stopped searching with the first quote below this reservation price. ~n :he pilot study, 48% of the cases ec~;formed with this; in the m~.in st~dy 42?o. Howew;r, when it came ~o investigating, wb?t the other 52~, .(or -~8°J~.o,d i i and indeed whether the 48~/o (o~- 42~o) were ,wu~ally following aa optima~

Table 3 Raw da~a of Ca.~ 3. Parameters: x= 250, c = 5,0, p= 215.0, a=25,0.

S 1~10.

quotes o b t a i ~ l in ch:o~mlogical order

(I) (2)

(3) (4)

(6) (7)

~A7 2t5.~3 27.7.6.~ 20~e,9

223.55

214,71

2~5.46 I.~5~85 176.29

23%34

253,66

I98.83

231.18 204.63 203,88

~9o:7~ 178.24 237 35 17~77

{8) (9) (10)

206.42 209.~ 235,12 223:)7 197.09 234,i15

[II)

(12)

A•

240.25

206.13 210.27

208.84

224a3 233.51

205,45 227.32

(19)

245.88 t88.5~t 196.02 200.62 22o.ii 20133 199.34 203.15 ~:,8.97 222.9O i"~a37 189.35 225.62 209.54

C B 255.31 '225.83 207.87 212..49 187;48 256.95

-

(2o)

(22]'.. (23~ {24,~

(25),

2~4.o~

(29)

207.86 217.87 i81.25 :~.29.46 :214.09 Z~.5.75 24&15 23O34 229.01

(30]

200.59

(260

(2~ [28)

AI

A*

(t) (1)

A*

177.85 203.77 289.36 204.65 226.73 217.05

206.0t 214.38

181.56 226.34

209.23

241,88

~I) ~

E D

A* A* A*

2i3.04 228.62

(21)

D 1/

B D A*

263A5 241.57

(~5~ (17)

B C

A*

(14)

06)

181.88 209.59

226.66 222.04

211.'/9

23~; "2 248.e6

201.22

225:05

,wJud I

(18)

C¢ n,sistent with ~ulo(sP'b

A A* A C B A B A A* B

D E D D D D

with m ~ r e orb ~nd C(or o l d and E). :

~!a)~ ~o~ist.ent , ~ : m t h e r A or B o r (~ 25(out of 31), ,b)number con~ with either A o r D or E: 25 'e',:. o . . 1). (c) subjects comistent ~ t h nm~e of rules (nor mixture}: 6, 8, 12 and 2~.

s,~ategy, we found the lack of ~d~fit~ohal evider~ce a ~rious problem 2 Accordingly, ~ major ivmovafi~ ~n ~© main study was~the use of a tapev-.corder, ~l:oSul~lement the rawplf.ice~uote data, ~xough the recording of the s u ~ s "thinking aloud": ~ is a M ~h tuque famifiar to psychologists, ~' pilot s , ~ . ( ~ t q a ~ ~ ~ c k ~ ~m~ig~it~~ study)i~ an effect to ini~r from the daa f~ ~ ~ ~ ~ ~ t k e ~ ~ , ~ ~ ~ ~these a~aalyses, which Were partially s~cce~sfu|, am ~ in an ~ ¢kaRof tbh~ l~Im, a~.:~ablet~rom the author on r~quest.

J.O. Hey, Search for rules for search

71

Table ,1 Raw data of Case 4. Parameters: x = 100, c = 4.t,, # = 86.0, o = lif0 Consistent with rute(sp .~

5

no.

l'rice quotes obtained in chronological order

(1) (2) (3) (¢) (5) (6) (7) (8) (9) 00) (11) (12)

7go02 78.56 8~.00 91.I4 93.36 7,L36 93.23 7~.99 85.20 78.05 87.93 83.46

!5.66 79.90 74.57 74.57

ll,dl.~, ~L~r]

QI "~,1 J a.....,,.~

'71 7"1 u , i t . i ,...

(15) (16) (17) (18) (20) (21) (22} (23) (24) (25) (26) (27) (28) (29) (30)" (31)

73.39 75.99 85.46 91.67

88.40 81.76 86.34 79.79 94.89 77.83 97.30 89.96

86.31

83.72

82.89

87.05

83.75 95.36 83.58 72.32

87.89

85.09

88.19 84.C5 82.11 79.19 64.: I 87,, ~ 88.52 79.70 82,31 79.00 99.92

79.(~4 76.?0 72.36 78.69

C (1) A

77.0t~

B

A g4.41 gi.32 87.34 80.43 85.35 75.82 89.80

84.83

72590

06.89

96.83

93.70 A•

75.81 775)4 76.59

A A B 77.73

"; _0 B Ilk

B

D

(1) B

D

A* B B

A* C E B

70.~

D D

D

A* A D A* A* A*

"(1) coasisteat with mixture of B and C. b(a) number consistent with either A or B or C: 24 (out of 29), (t~) mlmber consistent with either A or D or E: 24 (out of 29), (c) subjects consi:;~ent with none of tales (nor mixture): 6, 11 and 15.

from whom v,e borrow the term prowcol analysis to describe o a r m a i n ~tudy. W e apprecia.te t h a t o u r use of thi,,, technique m a y h a v e been r a t h e r ,a~ve, a n d t h a t or,r e x p e " i m e n t s m a y ha,,e been i m p r o v e d by better briefir~g (the 'performan,,e' of o a r subjects in ~hiL':ing a l o u d varied sigmficantly); but, nevedheie,;s, tiffs i n n o v a t i o n a p p e a r e d to be well w o r t h w h i l e lr. particu!ar, we feel that it h e l ~ d to ensure tht~t subjects took the exercise seriously (o~, at least, m o r e seriously t h a n they otherwise m i g h t h a v e done). To this extent, i t p r o v i d t ~ s o m e c o u n t e r v a i l i n g force against the a r g u m e n t that s~.ch hboratorytype exgerimerAs are artificial, in that n o reai incentive is

72

.',:~, ltey., Search for ~

for search

provided to tk: subjects for accurate behaviour. In futttre work, we intend to exploit the obvious reward sttnJeture [by giving the subjects (a proportion of) the balance of x saved on one or more of the search pro~lems]; we hope that the restdts of the experiments we have already carried out will convince a grant-givfiag b(~ly of the worthwhileness of this line of inquiry. InJtialiy, we h ~ envisaged the tape-recordings as supplementing the pricequote d a t a - - in ~ e sense of guidin~ th~ empifieaI investigation of the latter. In the event, ~e found that the tape-recordings almost supplanted the data -in the sense ~hat ~veral simple rules were repeatedly mentioned by the subjects as the reasons for their decisions (both ex ante and ex post)' in view of this, we ~lecided not to relx~rt in this paper the results of more conventioaal c~pirieal analyses, but rather to report the rules that we identified as emexging from the protocol analysis. ".['his we do in the next ~ction.

At this p~hat, we refer to the tape-recordings. Although behaviour naturally varied frc.m subject to subject, some popular themes emerged. We list below the most commonly apparent rules, :identifying them by the letters A to E indttdve, We also give some examF!es of quotes from the taperecordings as supporting (though not necessarily ,.onclusive) evidence. 3

Rule A ('~servation" rule). "su~mciently" low.

Stop searching if a wice quote is received that is

If this rule is invoked, then the final quote is tl~e minimum of the quotes obtaJn~3. ~Unles~ of course, the reservation vah~.~ increases at some stage: there is s ~ evidence to suggest that subj~t:: may have used a lower reservatiot~ va~lue with respect to the first price quote than with respect to sulTceqm.m quotes.)The rule may well be invoked probabilistically; that is, there may be ,~ 'continuing function'

where p is the probability of search continuing, when tire latest quote (Xmin) is th,: m,:nk'aum of those received. Some instance:i cf this rule are (where ~tmbers in square brackets are price quotes received): (subject 19, case 3) '... [£22862, £200.52] .... well, this is considerably lower ... £40 in ~otal less than my maximum.., i think I'll buy ...'; (subject 2~, case 4) 'The maxin~um price z¢-~orae i~,~rthet exami~-s ~ given in an earlier draft of this paper. In addition, full ~hand~ritten~ transcrip~ of the t~ pc-recordings are availabie.

J.D. Hey, Search .for rules for search

73

is £ t 0 0 ... [£79.10] ... the price quote is :",79.10 and so ... seeing it's substantially below the limit, we'll buy i t . . . ' .

Rule A* ('optimal reserxatioa' rule). Stop .~earching if a price quote is received l:~ss than the optimal {witi~ respect to the true distribution) reservation value. Obviously, subjects did no t k o o w the true distribution, so at most their behaviour could appear 'as if' they were using Rule A * It may be useful to note at this stage that very few subjects in the protocol analysis stopped after the first search" raany remarked that they woulcl always have at least 2 searches (unless ;he first was exce "tiom,lly low). Illustration:~ include: (subject 25, case 2)~... now it would be rather silly to buy on tht basis of just one search ...'; (subject 26, case 2 ) ' . . . y a a never plump for the first one ...'; (subject 28, case 4) 'I would always search ... ~o as to obtain a second quote ...'. All the remaining rules inco~porat.• at least two ~ , . , , h ~ . Rules B and D i n v o ! w ~,t i~,~t 9, while C and F i:~.vo!ve at least 3.

Rule B ('one bounce' rule) Have at leasc 2 searches; stor ira pric~ quote is received larger than the previous quote. This rvl~ can be regarded a~ equivalent to 'playing winning s t r e a k f --'keep on going while you are gec:ing lower and lower quotes, stop as soon as your luck runs out'. Illustration~ of this rule include: tsubiect 15, case 2 ) ' . . . [£100.98, £67.09]... the first tir~e I found £100 and the second time t ~bund £67 ... so there is a big difference ... so I might find sorJe~h;e~g, ycu know, less than that ... so I'll keep e a searchinLg ... [£91.08] ... oh no, this is a higher price ... £91 ... no, no, ... I'll stop here ...'; (subjec~ 16~ case 2) '.,. [£106.32, £85.62] .,. I think I'lJ search once more ... [£78.:f;] ... ah~ price quote n u m b e r 3 i~ £78.50 .., ,Jhich again is ... about £7 ~,h.~aper ... if l bought at this pric: Fd be spenc ing £84.50... mm ... again, I feel.., it seem,: as though the pr:~ce is getting :heaper at each stage ... FII go on for one more, ~.nd see what this next px ce is ... if it goes up at all, I'll decide to b**v at the £78.50 .. [£86.34] ... ees. now the 4th price quote is £ 8 6 . ~ . . . . Estops]'. Although this appears a fairly c o m m o n rule 125°~ of the ob:erved ~;equences were consistent with his rule), it appears rather strange, pa, t~y in ~hat it ignores t h : magnitude f the search cost, but mainly in theft it endows the order of the price quotes with some significance. For exam,~le, a l;,erson using this rule would stop after receiving £23~3, £ t90, £210 in that ~rd~:', but would continue after receiving £230, £210, £i90 in that order. We fad *o see the rationale behino this phenomenon, although cthers hav-z trio?, to convince t~s of its rea:~onablenes3.

74

J,D. Hey, Search flTr rules for search

Rule C Ctwo bounce' rule), t~laoe at least 3 searcSe:': stop if" both the last quote an.4 the next to the last are larger than the second to the last This rule can be r e g a r d ~ as equivalent to playing winning sttea~:s and ili~,oring t l~ odd loser. Illustrations of this rt:tle include: (zubject 4, cam 1) '... [£'232.73, £185.98] ... there is quite a wide distribution to *hese prices ... Fm going to search t~ar a third price to see if that holds ... [£280.54] ... the third qu,:r~e is Mgher £280.54 .. so I don't feel I'm going to get a lower price quote than £185 ... the search cost is very low ... 1'11 search once more ... [£23Z91] ... agai~a, it's a higher prier .., £232.91 .. at this point I can buy far a total, spending of £195.98 ... so I'm going m b u y . . . ' ; (subject 25, case 3) ° . . [£21405~ £207,86] .~. so i'm doing a little bit better, but I think we should search further ... [£226.73] ... no, I don't like that quote ... 111 searcl~ once more and ~ e what happens to thal; ... [£209.23] ... O.K., I've ~pent some money on search procedures, but I now believe that £207.86 for t h e e ~ g t o f thi.~ i t e m i~q m fite ~ a n c l . . I .w i.l l .h u.vd "1 ~ - ~

': Thi.~ ~_a.~cmahlv ~r~mmnn 'd

rule (18% of the observed sequences were consistent with this rule) suffers fron~ the same two oddities as does Rule B. Rules D and E below re~,ove the first, but not the second, oddity. Rule D ('modified one l~ounce' rule). Hate at least 2 searches: stop if a price auote is received lffrger than the previous quote less the search cost. Th~s rule adjusts the quotes by the cost of obtaining them; to continue, the price must fall on each quote by at least the search cost. Note that a seque:ace of quotes which decrease in chronological order, and in which the gaps are all greater tkan ~:he search cos,, except for the final gap, is consistent with N~th this rule and Rule A (exaraples of fl:is are subject 9, case 4 and subject 24, ease 3). Illustrations of this rule include: (subject 9, case 4) '... [£8520, £80.43, £77.9.41] ... out the difference ... [is) .. smaller than the cost of searching ... we wi~l stop the search ...'; (subject 2~, case 4) '... [£79.70, £72.32, £70.81 ] ... searching doesn t seem to pay ... so I ~vfll btty ...'. Rule ~,r"('modified two t'ounce' rule). Have at least 3 search?s: stop if both the last quote exceeds t)ze second to It.st less twice the search cc ~t and the next to last e x ~ e & the second to last less ,~he search cost. As ~fitti Rule D, this rttle adjusts ~he quotes by the cost of obtaining them; otherwise it is the same as Role C. Again, as with Rule D, s,~me sequences can be consistent both w~fl~ ithis rule and with Rule A (e~amt,les of this are subject 8, case 4 and su~,:ieet 31, ease 2). Illustrations of this rule include (subject 1, case 1) '... [£24 $.49, £207.06, £222.84] ... I will sea;:ch once more . . [£2! '.29~ ... again the ,.catch has thrown up a price which is above the

J. l). Hey, Sea,wh for rides for search

75

minimum ... therefore, I think I will buy ... the new 7rice inclusive of searching is also greater ,hart the second best minimum ...'; (subject ~, case 4) '... [£78.99, £87.34, £75.°1] ... [1 will buy, because if search] I will have to add a,aother search [cost]...'. Rules D and E are resgectively consistent with 309/0 and 209'0 of all observed sequences. Full details are set out in tables 1 to 4. Some sequences (49/0 of the total) are consistent wilh a random mixture of B and C (and/or of D and E). Some sequences are not consistent with any of our rules: there are 12 (or 109/o) such instances. Examination of these sequences shows that they have one obvious feature in c o m m o n - they consi,.t of a relatively large number of searches (an average of 5.25 searches per p,~rson per case, compared with an overall average of 3.10. Moreover, in the majority of the~e cases the subject obtained a relatively low quote e,,rly on, which tempted them to continue for some .:onsiderable time. From the tape-recordings it appears that ex post they we,re annoyed at having ~one on search,;ng for 'tco long'. Other evidence tLat the tape-reeordiags reveal'~d was that the outcome of one run occasionally influenced tbc strategy used in the subsecluent runs. Even with as few a:; 4 cases, behaviour was sometimes on a quite different basis in the last case than i ~. was in the first. Table 5 reveals that subjects' behaviour ~vas consistent with different rule:; in di, t'erent c a ~ s (though there are a few :nstances of consistent behavicur!. Indeed, we suspect that a mixture of A, B and C, or a mixture of A, D ,-.nd E, is probably a better guide to b e h a v k u r than any rare strat:g 2" alo,v' Eml?irieal investigation should reveal the l~recise form ot the mixtt~re and tine form of the reservation rule used under A. We carried out som, preliminary investigations to determine the form of the latter,, using pro~.i ~, analysis; tLe results 4 were encouraging but inconclusive. Although our simple rales between them can account for a high. proportion of the observed sequences, we do not claim that they exhaust ai'. possibilities. Indeed, an examination of the price sequenc::s on their own might reveal many other possible explanations. However, our tape-recordings add strength to the validity of our rules as, at least, partial! explar.at~ons of observed behaviour.

4. Implications of the rules This section looks at the i~lplications in two die,rent areas lir~.~. ~.vi,,h respect to the, (ii~dividual) expel:ted outcom~ following2 the various ruz~ ~and mixtures of them); second, wJ,h respect to market level implicatioas. W~, b e # n with the former. 4Reported mai, eartier draft c.,f I!~is paper.

70

,i'.D. ~¢[ey,Search fbr rtdes fi,," seacch Table 5 Summary by subject? •

c

¢

s

1

2

3

4

s

1

2

"

4

(D (3~ (3) (4) (5) (6) (7) (8) (9) {I0) (11) (I2) (13)

CE A* ~ CE A

Ill) A* A* BD BD

BD CE A* BD A*

CE (1) AD BD A

(17) (18): (19) (20) (21)

CfE)

CE

(~)

BD A CE A A BD

CE A A* I~D A C~

A ~' A* A* A* A A*

CE A*

BD BD

~.*

A* AE

(23) (24)

CE A

A*

CE CE

A* CE BD

AD BD

(25) (26) (27) (28) (29) (30) (31)

(C)E BD CE CE

AD CE BD A BD AD A* BD

(1) A* (2) BD A* BD BD A* CE BID A* AD A* A* ~,*

(t) A liD t~

04) (15) (16)

BD A*

A* A* A* BD

RD

A* A* (1':, {I)

BD (2) A*

CE BD

(I) CE D BD BD CE AE

BD

"(t} consistent with mixture of B and C (and/or of D a m E), (2) subject gave up without buying (letter in brackets indicates ease on borderline of being consistent with rule).

In view of tt~ teehni, eal difficulty in deriving the stoppin3 distribution and the distribution of the number (;f searches following Rates B to E, we resorted to a numedcad simulation. This simulation was carried out with respect tc, 3 of the 4 eases in our protocol analysis (Case 3 was omitted ~use of its simila~4ty in this respect with Case 2); for ~ c h simulation 1500 repetitiom were use& (As this is a tdatively small number, some random variation may be coatained in our' simulation results). Tabi~s 6 to 8 contain the simulation results: ir each table, the first cobamn specifies the form of rule being followed [an entry 'A (200.0)' indicates a reservation rule with reservat£)n price 2(3&~ an asterisk indicates the optimal reservation price with respe~::t to tb: tn,'e distribution; an entry '0.6B, 0.4C' indicates a stochastic ~fixture ~,ith a 60% chance of B being ir~voked and a 40% chance of C being invokec!]; the second column indicate~ the egpected total cost (search plus p~archase) following the rule; the colurans headed °RPi' indicate ris[ prem/um~ defined as follows. Consider an individual with constant absdute risk aversion who would pay a risk-premium of £10i when co.,ffronted with a garable involviug a 50-50 chance of winning £0 or £100 (tF,as, regarding the gamble as having a certainty equivalent of £(50-10i). Then Rav'i is the risk-premium attached by that ~ndividual to the distribution of iota| c~s~s gear:rated by the spedfied rifle. (Thus the individual regards ~he certaia!v e q u i / a k a t as tac " co~t t;it~, . .. " - expected "a , ta! R Pi}, C]eariv the h{gher is t ~.hc raore risk-a.,:r,~,e is ~hc individua!.

J.D. Hey, Search for rules fer search

/7

Table 6 The outcome of various rules m Case 17

Rule

Expected total cost

A (230.(') & (217.5) `a, t205.0) &*{192.5) ,at (180.0) A {t67.5) A {155.0)

207.49 201.78 195 ~-.~ 191.98 199.56 208.75 459.61

i]; 0.6B, 0.4C 0.413, 0.6C C

RPI

RP2

RP3

RP4

6.18 158.35

2510 2.38 2,01 24/ 6.79 11.38 208.10

5.57 4.66 3.8" 4.51 10.79 ,7.06 239.64

13.10 11.48 9.37 9.86 19.15 26.75 270.16

202.93 202.23 20t.04 199.37

1.61 1.36 1.34 1.24

3.54 2.97 2.94 2.74

6.53 5.41 5.40 5.07

14.04 11.34 11.56 ! 0.89

1.27 1.04 0.89 1.14

3.50

D

L, VL.~'~

,u~,~ ¢~a

t• . r~'7 ~ !

j.t,~,Jn~

"1 KQ

6 • 7t ~v

x ~..,,,t.,

0.6D, 0.4E 0.4D, 0.6E E

202.71 201.93 200.61

1.50 1.48 1.30

3.28 3.27 2.90

5.~5 6.04 5.42

12.25 t3.09 12.35

A A A A

203.63 200.02 203.85 200.85

1.54 1.43 1.65 1.43

3.40 3.20 3.'68 3.21

6.28 6.01 6.96 6.0,.~

13.89 13.49 16.78 13.84

and and mad and

B C D E

I A R9

"In the last f o a l a probabi~istie version of A was u,,.,:~, of tlae fi,llowing form: P(,~top ~, :=t~= x) = P(Z > (x - 114)/54), wh err. Z is N(0. l).

There are a number o!" interesting features contained i,,, tables 6 to 8 First. and rather obviously, A* has the lowest total expected cost.',, and almost (out not quite) the lowest R/'i values. The reservation rule w~t.I a slightly higher reservation price (than :M optimal one) d ~ s slightly b e t e r in this resr~e,:t., and so does Rule E (tl',e 'tn',,dified two boric,." rule). If using a reservation value rule, it is deafly better to err on the side of a too high value than a too low value. [As we remarked in Hey (1981), it is better to search too little rather than too mueh.] As we have already noted, Rule E does reasonably well as far as the RP: are concerned; it also does qui,e well wiO respec: to total expected cost. Surprisingly, Rules B and D p, rform roughly the sam. hroughoet, ~:, did Rules C and E. When we first ,aw this result, we suspected v mi:,:ake ip. tI, e computer program, but on elf':king we found this vea~ n'.i t:~e case. Ur:dcr B more search is undertaken ~han under D {because D ai~plies a :;:ricwr criterion for eor~tinuing) ami consequently under B the pra:hase price wn6.,: to be lower. Ho, wever, tL~ beJlefil of a reduced purcha;e p~ice is almo.,,'. exactly o~s~t by the cos*.., of t~e extra searci~ activ~y. A .,im~ar argurrc~ explai~s t?~e ,~imilari~v ~,etween ~u~es C ::rid F. JEBO

D

J.D. Hey, Sea, :h f~r rules for search

78

Teble 7 Tlt~ outcome of v,.',~ous ~

Retie

Expected total ~ t

85~58 ,3M A (8~1) A*(81.1) A (76.1)

A (?i~a) A {66.1)

8"2.01 8081 83.2~ 96,5~ 150,~5

in ~

Z"

PdPl

RP2

RP3

RP4

0.27 0.21 0.19 0.22 0.60 2~93 20.17

0;6i 0,47 0.42 0~47 L24 ~ 5.64 33.36

1.13 0.89 0.79 0,86 2.10 8.72 44.74

2.51 1.98 1.80 ' t.8:1 3.87 13.62 58.18

0.26 0.23 0.23

1.06 0.94 0.92 0.83

2.28 2.04 2,00 !.80

:&..,

34.13 84,41 84.39 84A1

0.~

0.58 0.51 0.50 0.45

D 0.6D, 0AE 0.4D, 0.~D E

84.39 84.36 84.34 84.41

0,25 0.24 0.23 0.19

0.55 0.53 0,51 0.*t2

1.02 0.99 0.95 0.78

"LI9 2.16 2.08 1.67

AandB A a~d C A a~d D AandE

83.47 83.58 ~3.71 83.42

0.28 0.24 0.28 0.26

0.61 0,52 0.62 0.57

1.11 0.96 1.14 1.06

2.41 2.07 2.44 2.31

0.6B, 0.4C OAK 0.6C t't

q n the l a s t four, a prolmb~istic ver:.ion of A was used, of the followhtg form: P(stop[ x ~ - - x)-:P(Z>O:-64.96)/15.65), where Z is N(0,1). (These paramete~ were estimated from the observed data using probi~ analysis.)

As might have ~ ~ed, RUle B (or D) mlds to do better than Rule C (or E) when ~ co~:ts aJ~ high (Ca~ 4),:wMle the converse is true when search costs are low (Casv 1); mixtures do not 9~pear to be attractive, even in redUdng tlie ~ . F;mally, it appears beneficial to incorporate some (sto~astic) ~ f i o n rule, as the 'fin~ four rows indicate (though the decrease in the expected tota~i cost may be offset by an increase in the R P i - the very risk -avers ~ shotdd avoid the combination of A with one of the othcr raie~). However, cue over-riding impression remains: our 'rules of thumb' do reasovably well. In particular they appear to be reasonably robust ~ whicb ~s something that Cannot ~; claimed for a (badly chosen) reservation ruie However, the generality ~ :~heseproperties lemains to be investigated. 5 ~ h e desi~ of such an invcgdg~ltioti is not withmxt its proHems. 1¢eMly, one would like to know how we~ the vraious ~!es ,:lo on 'av~age; a:~d how robust th,:y are ,o changes in th~ underb~ng pnc~ d~s~budon, But tMs requires the spoSfic~t~on of a se'! of possible distributions, and ~.he results of ~he ~nvcsfi~tio~t migh'-~well be v,:y seI~sitive ~o the ~:pe~t~iioa .ef '.h.i: ee. ~)~ha~ constitmes a ._,~ . . . . . . . speci'icafiot~ ~s ~ot ob ,ou.,~.

J.O. Hey, Search jar rules for search

79

Table 8 rb.- outco,ne of var~ous rules in f'~,~e 4?

Expected Rule

total cost

RP I

RP2

~P3

RP4

A[10i .0) A (f,.0) A (91.0) A*(86.0) A (81.0) A (76.0) A (7:.0)

85.57 88.16 86.55 86.(14 87.38 95.79 127.11

0.31 ,1).26 0.24 0.26 0.51 1.92 i(,.43

0.69 0.57 0.54 0.57 1.08 3.77 18.18

1.27 1.07 ).99 ~.04 1.87 5.96 ,:5.51

2.76 2.37 2.16 2.20 3.58 9.6? 35.00

B 0.6B, 0.4C 0.4B, 0.6C C

89.16 90.84 91.61 93.27

027 0.26 0.27 0_26

0.59 ~,.56 0.59 O.55

10¢~. 1.03 108 ! .03

2.32; 2.19 2.3t~ 2.20

D 0.61), 0.4E 0.41), 0.6E E

89.36 90.38 90.92 92.04

0.26 0.25 0.24 0.21

0.57 0.55 0.52 0.46

~.05 1 02 096 0 84

2.32 L22 2.13 t.82

A A A A

87.12 87.83 87.54 87.57

0.33 0.35 0.32 0.33

0,.73 0.70 0.70 0.73

1.]4 1..~8 1.20 1.33

2.89 2.91 ~,g2 2.g4

and and and ant

B C D E

-

"In the last four, a probabilistic version of A was used, of the following form:/'(stop ] Xmt~= x) = P(Z > ( x - 82)/8)), w aere Z is N(0, ! |.

Let us now briefly explore the market level implications. Supr',ose that the p.d.f, of prices across firms in the market is f(x) (.his b~ing tt, e distributior which is being searched), and that the ratio of total searchers (buyers)t firms is N. Denote by g(x) the stopping price p d f . - - this, of course, bei',g determined by f ( . ) as well as by the search rule~ emp!oyed. The numbe, of customers an individual firm charging x gets ;,. on average, Ng(x)/f(xjcall this n(x). We are interested in the lbrm of the ~unc d,m n(-). If all searchers follow a reservation strategy witl" :he same res::vation value x*, then n(x)= N/F(x*) for x x*. Tk,s, as has been ,yell documente,a in the literature, has important implications as far as market equilibrium is concerned. If ah searchers t~llow a reservar~o~ strategy, but with diffeTing reservation xalues, nix) is a decreasing function cf x up ~o the largest r, sfrvafion value, after which it is zero. From ot~r simulations, some provi~mal evidence about the form of n~x) implied by e~r various rifles can be cbta led. The sharpest contrast is between R,,!es B an,:] C (and between D and E): under B, n(x) starts lower and falls more siowiv pFeL~-~r~s' a r c ~ o w e than under C Under B (or l)). there~bre, ~colTtDeutlve " ' Comparing ? with D im~d ~-with E) slows a tendency tO ,~x) under B t~

s~art higher and fall faster thzn under D, but this ,~iffenmce is not so marked as that between B and C (land between D and E). These features of n(x) appear to accord with. ~r~,dtion. The in'teresting qtaestion 6f their implications for market eq~:~tib~'iu~ remains to be explo~'ed. Two other market level implications s!aoui~' be noted. First, it Rules D or E are being used, an increase in s,~ar~.h ~:osts ~1I trend to reduce. ~eareh activity an~t hen~ shift the stopping dis tri~,uti~}ni tightwads (but lhere will ~ no sud~ effect if Rules B or C are being used). Second, if any of the Rule~ B to E a:~e being used, then an increase in the ~spersr~n of the j'(.) distribution will tend to increase search activity and hence shift the s~opping distribution leftwards. 5. ConcinsiaB As we re,~aarked earlier, our analysis, though instructive, has ~'aised more possibilities than it has investigated. Numerous further possible extensions spring to mind. The major problem, inevitable with this ~ype c,f '~aboratory' ex~riment is to ensure that the subjects' responsesare accurate. As a referee remarked, w~thout a financial reward ele~l~ ~ to performance on the search experiment itself, subjects n~ty be plo~ding th,:~r own, alternative, definitions of s u ~ . Thus, our 'rules of thumb' may reflect 'attempts to ~v,-in" some perceived "game" with the experimente~ rather than reflect ccnventionali search behaviour. We realise that this is potentially a problem, but feel thai: certain features of our CXl.~rirnental p]:oo~ure would work to lessen its impact: first, the way the experimt.'nt was organised, there was no obvious 'game to play' other than that of doing well on the search experiment :itself; second, virtually ~dl of tl~~. st~bjects were economists, and were thus wilting and able to play the game according to the rules; third, the presence of the tape-recorder made it diffieu:t for the subject to play at,other game without it being fairly obvious. (Thou[:h we appreciate that the need to i ts " ~'haviour.) Ideally, some financial 'think aloud" mzy in it~lf alter suhjec ~ncentive, lfi~ed ~o their performance, ~hould be incorporated. By the very t'ature of the experiment, there is a natural way t h a t thi,,~ could be done; as g e remarked earlier, we hope to implemen ~. this in the future. 6 The ~ape.reeording o f the 'thin!dug ,aloud' ¢¢as important; but we think that this could be improved v~ith careful .b~iefing. As far as the experiment itself was ConcernS, nurnorous 'variations on a theme' sp::ing to mind: differer~t kinds of distributions, more comp~,~,c~Lted problems, and so on. It would be interesting ~o explore how differ~:~,:t mbjef'ts respond to the same se| of price quotes, t(n our experiments, the r~ndorn number generator was initiated by the ftme of the experiment; thus, different ~ubjects got different sequences. We did thi2 ~o a,,; to have a sufficiently wide dispersion of .

.

.

.

~Though it s~toutd be not~,~l ti~at rt,~ardless experiments have a very respectable pedigr~; see, for e~:ampte~Kalmem an alad Trotsky (1979)~

J.D. r:le~,, Search for rule ~ ]'or search

81

experience; also to minimise any possible collusion between subjects3 It would also be interesting t," be able to manipulate the price quotes ;n th2 light of what the subjects were saying. (~'his would have to be car~"ally arrangec so that the subjects were aware neither of being listened ~o nor of their quotes being 'massaged'.j It would also be of considerable iwtere.,t to make the whole expeiiment 'more realistic' by giving a real-lifi~ exaznple, with real-life shops, and g~enuine quotes. The trouble with this is that all subjects would have to be carefully examined, prior to the experiment, to find out the extent of trteir prior knowledge of the search problem in question. In any ease, we were more interested in finding out how pe~ple behaved in the worid of the theory,*,i rather than in the real world: we ,ranted to investigate the kffamous 'as if' o n ~' its own ground. (For much tht same reasons, we de: ided against an investigation into actual see.reh l~,~haviour in the rea! wo~id interestir:g though this would be.) Although there are many other avenues to Ix; exploced, we feel that our results are of significance. In par,icular, we appear to have unearthed sonic 'rules of thumb' which people actually use, and wlfich are also reasonably sensible. Although people do not always behave optimally, at least they seem to behave reasonably. Hopefully, these findings might point the way forward for research in economics.

References Hey, J.D, 1981, Are optimal sea-ch rules reasonable? And vice versa? (Ar,d does it matter anyway?), Journal of Economic Behavior and Orgar~ization 2, 47 70. Kahan, J.P., A. Rapoport, and, L.V. Jones, 1967, Decision making in a seout~nual search ta=k. Perception and Psychophymcs 2, 374-376. Kahneman, D. and A. Tversky, 1979, Prospect theory: An analysis of de~sion undc.r ri~k, Econometrica 17, 263-291. Pratt, J.W., D.A. Wise and R. Zeckhauser, 1979, Price differences in almost con: petitive market.t, Quarterly Journal of Economics 93, 189-211. Rapoport, A. and A. Tversky, ltMo6, Cost and a ~ b i l i t y , ff offers as determinants of optioaal stopping, Psychonomic Science ~,, 145-146. Rapoport, A. and A. Tversky, 1970, Choico behaviour in an optimal stopping task, Organizatio~ml Behaviour and Human Performance 5, 105-120. Schotter, A. and Y.M. Braunstein, 1981, Econonfic search: An experimental study, Economic Inquiry 19, 1-25.