Monty Hall's Three Doors: Construction and Deconstruction of a Choice Anomaly Author(S): Daniel Friedman Source: the American Economic Review, Vol

American Economic Association

Monty Hall's Three Doors: Construction and Deconstruction of a Choice Anomaly Author(s): Daniel Friedman Source: The American Economic Review, Vol. 88, No. 4 (Sep., 1998), pp. 933-946 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/117012 Accessed: 12/05/2009 08:50

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http://www.jstor.org Monty Hall's Three Doors: Constructionand Decoristruction of a Choice Anornaly

By DANIEL FRIEDMAN*

Sometimes people make decisions that show "Let's Make A Deal" asked his final seem inconsistent with rational choice the- guest of the day to choose one of three doors ory. We have a "choice anomaly" when (or curtains). One door led to the "grand such decisions are systematic and well doc- prize" such as a new car and the other two umented. From a few isolated examples such doors led to "zonks" or worthless prizes such as the Maurice Allais (1953) paradox and as goats. After the guest chose a door, Monty the probability matching puzzle of William always opened one of the other two doors to K. Estes (1954), the set of anomalies ex- reveal a zonk and always offered the guest the panded dramatically in the last two decades, opportunity to switch her choice to the re- especially following the work of Daniel maining unopened door. The stylized fact is Kahneman and Amos Tversky (e.g., 1979). that very few guests accepted the opportunity By now the empirical literature offers doz- to switch. ens of interrelated anomalies documented in Nonswitching is anomalous because in the hundreds of articles and surveys (e.g., Colin game just described the probability of win- F. Camerer, 1995). ning is 1/3 for nonswitchers and 2/3 for The problem is not just theoretical. Econo- switchers, as explained in popular articles by mists' traditional analytical tools are based Steven Selvin (1975a, b), Nalebuff (1987), squarely on rational choice theory, so anom- Marilyn vos Savant (1990), and Leonard alies provoke debates about the practicalvalue Gillman (1992). Forgoing the incremental of these tools. The present paper offers some probability of 1/3 on more than $10,000 in new laboratoryevidence bearing on the anom- incremental prize value represents a serious alies debates. It focuses on a particularchoice loss to a nonswitching guest. Indeed, it might task, chosen (after a search) for its robustness seem that millions of dollars were left on the and persistence. table because the game ran in the United States daily from 1963 - 1968 and later L. Construction weekly until as late as 1986, with over 3,800 shows altogether (Alex McNeil, 1991 pp. As recounted by Barry Nalebuff (1987), 369-70). host Monty Hall of the once-popularTV game But the stylized fact is not quite what it seems. In reply to my recent questions by phone and e-mail, Monty Hlallsays that to the * best of his recollection, he rarely offered Department of Economics, University of California, the switch and cannot how Santa Cruz, CA 95064. The National Science Foundation guests option say supported the work under Grant No. SBR-9310347, and often it was accepted (see also John Tierney, Brian Sniegowski and Hugh Kelley provided research as- 1991). His longtime production assistant, sistance. I learned a lot from comments by seminar and Carol Andrews, asserts that Monty never of- workshop participantsat the 1995 Amsterdam Workshop fered the switch option to the final guest. in Experimental Economics, Cornell University, the Eco- nomic Science Association, the University of California- Nalebuff says he personally recalls watching Santa Cruz, and the University of Southern California, the show and seeing the switch option offered especially from Peter Diamond, Thomas Gilovich, Uri (as do I), but he cannot recall "whether Gneezy, Lori Kletzer, Graham Loomes, Robert Sugden, Monty offered this option all the time and and Donald Wittman. Two anonymous referees offered his the offer at all valuable suggestions for the final revision. I retain whether or not making was sole responsibility for any errors, misinterpretations,and connected to whether you picked the right eccentricities. door." 93-3 934 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

Recovered memories are unreliable ology, organic chemistry, and politics at the (Elizabeth F. Loftus, 1997) so one might wish University of California, Santa Cruz (UCSC) to extract evidence from tapes of Monty Hall' s and at Cabrillo Community College. Each sub- show. Unfortunately there are two problems. ject entered a quiet room and sat at a table First, the tapes are not available. Inquiries to opposite the conductor with no other subjects Andrews and to the Museums of Broadcasting present. After reading the instructions, each in New York and Los Angeles disclose avail- subject completed a series of ten trials (or able tapes for at most five shows, hardly an "periods"). In each trial, the subject initially adequate sample. Second, the game show en- picked one of three face-down cards. Then the vironment was poorly controlled. Some of the conductor turned over a nonprize card that the difficulties can be seen in the work of Andrew subject did not choose and offered her the op- Metrick (1995), who analyzed tapes from an- portunity to switch to the other face-down other popular TV game show, "Jeopardy." card. Finally both face-down cards were Metrick finds that people do not always best turned over, one of which was the prize card. respond in a simultaneous move game ("Final Each trial the subject earned 40 cents when her Jeopardy"). However, his "empirical best re- final choice turnedout to be the prize card and sponse" assumes that each player faces the 10 cents otherwise. The instructions(available overall empirical distribution of play. Actual on request) briefly and unambiguously ex- players may have well-founded beliefs that plained the procedure. their specific opponents differ from average, and they may have idiosyncratic preferences B. Results (e.g., looking timid while losing is worse than going for it but still losing). Hence, despite Table 1 summarizes the results. Subjects' careful work, Metrick's failure rate may be relevant decision each period is binary, either overstated. The three-door game poses addi- to switch (the rational choice) or to remain tional problems. Perhaps guests were victims with the card initially chosen (the irrational of Monty Hall's unique personality or their choice). UCSC Humanities (and Arts) stu- own ''room temperatureIQ's" (Hal Erickson, dents switched most often (in 34.4 percent of 1989 p. 230). Even worse, if Monty offered 250 trials) and UCSC Natural Science majors the option to switch only when the guest's ini- switched least often (in 23.3 percent of 270 tial choice was correct, then nonswitching trials). Subjects who took only lower- would be optimal after all. division statistics courses switched less often Existing academic literatureincludes at least than those with more statistics or none; ap- two studies inspired by Monty Hall's game parently here, as elsewhere, a little knowl- show, but none to my knowledge that actually edge is a dangerous thing! Female subjects tests the three-door choice task. Ruma Falk were slightly more rational (31.8 percent) ( 1992) analyzes conceptually some related than males (24.9 percent). Of the 104 sub- judgment tasks. In order to test regret theory, jects tested, only 6 switched more than half Thomas Gilovich et al. (1995) conduct labo- the time. (One of the 6 subjects was already ratory experiments that disguise an unwinna- familiar with the task, having read and re- ble task as the three-door task. membered vos Savant's [1990] column in Parade magazine; no other subject admitted A. Method to prior knowledge even at the exit interview after final payment.) The essential ingredients of the three-door Figure 1 shows the time trend. The switch task are an initial risky choice, the revelation rate started out extremely low (less than 10 of inforimationconcerning a rejected choice, percent in the first trial) and increased fairly and the opportunity to reconsider the initial steadily over the next several trials. But it stag- choice. My experiment uses these ingredients nates at about 40 percent after the sixth trial in a mannerparallel to the stylized game show. and actually declines in the last trial to about One hundred four subjects were recruited, 30 percent. The overall switch percentage is mainly from large introductory classes in bi- 28.7 percent. VOL. 88 NO. 4 FRIEDMAN: CHOICEANOMALIES 935

TABLE 1-CHOICES AND EARNINGS IN RUNI and Martin W. Weber (1992). Unfortunately, many of the leading theories still forbid selec- Switch tion of a stochastically dominated action and rate Mean so cannot accommodate the new data. Gener- Pool Nobs percent earnings alizations that can accommodate dominated School choices are unlikely to generate sharp enough UCSC-Lower predictions to be useful in applications. For division 420 26.7 $2.35 present purposes, perhaps the best way to pro- UCSC-Upper ceed is to review briefly some related division 340 32.0 $2.30 Other schools 280 27.1 $2.31 anomalies. First, the three-dooranomaly may be related Major to the "gambler's fallacy" or more generally, Natural Sciences 270 23.3 $2.32 the "illusion of control" (Camerer, 1995). A Social Sciences 360 30.6 $2.30 subject may believe that, despite our best at- Humanities/Arts 250 34.4 $2.37 Undeclared 160 23.8 $2.37 tempts to randomize and a careful explanation of our attempts, he can somehow intuit which Sex face-down card is the prize card and so he may Male 490 24.9 $2.34 regardhis initial choice as the most likely. Re- Female 550 31.8 $2.32 vealing the nonprize status of a card he already Previous statistics class believed less likely (especially if he thought it None 620 31.0 $2.38 second most likely) may then only confirm his Lower division 340 24.2 $2.32 initial belief. Such beliefs conceivably were Upper division 80 28.8 $1.98 rational in the TV game show, but clearly are Age irrational in the trials just described because 17-19 440 27.3 $2.32 35.0 percent of the choices to remain won the 29-23 440 31.3 $2.35 prize, while 70.3 percent of the choices to 24+ 160 24.4 $2.31 switch were winners. Second, the anomaly may be related to the "nonrational escalation of commitment" fea- tured in Max H. Bazerman (1990). Having C. Discussion made his initial choice, the subject is reluctant to change it; he sees persistence as a virtue and The three-door task is now a true anomaly. flip-flopping as a vice. Economists might re- The data are not hypothetical: more than 100 gard this as a cousin of the sunk cost fallacy: real people left lots of real money on the table the subject rationally should regard his initial in a controlled laboratory setting. The ob- decision as sunk and of no relevance to his served behavior is not "approximately" ra- decision between the last two cards, but he tional; most of the people most of the time may be psychologically unable to let bygones made the irrational choice when the rational be bygones. Such reluctance to switch has choice was just as convenient. And with ten been called the "endowment effect" in a trials, each laboratory subject had the oppor- slightly different context, e.g., Jack L. Knetsch tunity to become familiar with the task and to ( 1989). As noted before, reluctance to switch find the rational decision. might be rational in the TV game show if How might this anomaly be explained? A Monty offered the switch option more often straightforwardresponse to any choice anom- when the initial choice was correct, but it def- aly is to accommodate it by relaxing some of initely is irrationalin the experiment where the the standard axioms of subjective expected switch option is always offered. utility and Bayesian decision theory. Such re- Third, subjects often misapply Bayesian up- sponses to previous anomalies has produced a dating in various ways (Camerer, 1995) and vast theoretical literature on generalized ex- may have done so here. With a prior proba- pected utility; for an introduction see, for ex- bility of 1/3 for each card, a random infor- ample, Mark J. Machina (1987) or Camerer mation process (i.e., equally likely a priori to 936 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

100 -

80 -

-70 -

60 -

A 50 - 50. 40 -

230-

20 -

0 1 2 3 4 5 6 7 8 9 10 Period

FIGURE 1. SWITCHING PERCENTAGE IN RUNI

reveal any one of the cards) that happens to and a 67-percent switch rate for good Bayes- reveal a nonprize, nonchosen card will yield ians. Probability matching may partially ac- Bayesian posterior probabilitiesof 1/2 for each count for the rarity of subjects who always of the nonchosen cards. That is, in accord with switch. The underlying reasons for probability most people's intuition, Laplace's principle of matching are unclear (it seems to be regarded insufficient reason applies to the posterior as as innate in much of the psychological litera- well as to the prior probabilities. In this case ture) but Sidney Siegel (1961) demonstrates there would be no advantage (or disadvan- that its effects diminish when salient cash re- tage) to switching. However, the actual infor- wards become more intense. mation process is not random; the conductor One conclusion seems clear. The three-door must reveal a nonprize, nonchosen card, so the task now deserves a place among the leading Bayesian posterior probability remains at 1/3 choice anomalies. The laboratory evidence for the initially chosen card and moves to 2/3 strongly corroborates anecdotal evidence for the other card. Such a nonrandom process (e.g., Massimo Piattelli-Palmarini,1994).' In- is outside most people's experience, so they deed, I am not aware of any anomaly that has may be unaware of the advantage of switching produced stronger departuresfrom rationality to the card with the 2/3 win probability. in a controlled laboratory environment. The A fourth anomaly may be worth mention- ultimate reasons for its strength remain un- ing. Probability matching behavior, docu- clear, but a proximate reason is that the task mented in numerous studies such as Estes (1954), consists of not always choosing the most likely alternative (as rationality would dictate), but rather choosing each alternative 'This popular book on "mental tunnels," i.e., choice in proportionto its likelihood. Like the Bayes- anomalies, refers to the three-door task as a "supertunnel ian failurejust discussed, probabilitymatching ... the best and most striking instance" (p. 7). Despite the fact that he offers only anecdotal evidence, Piattelli- is not potentially a complete explanation of the Palmarini devotes his last chapter ("Grand Finale") to three-dooranomaly because it would predict a the three-door task and emphasizes the Bayesian update 50-percent switch rate for confused Bayesians failure. VOL. 88 NO. 4 FRIEDMAN: CHOICE ANOMALIES 937 ties together several classic anomalies, includ- 2. Trackrecord (Track) .-Each subject wrote ing the gambler's fallacy, commitment esca- down the results each period on a record sheet lation and the endowment effect, Bayesian that included her own Run2 cumulative cash updating failures, and probability matching. It eamings as well as the cumulative earnings for would appear difficult to accommodate the the strategy "always remain" and for the three-door anomaly into a useful generaliza- strategy "always switch." Usually (but not in- tion of expected utility theory. variably) the "always switch" strategy did best. II. Deconstruction 3. Written advice (Advice).--Before the first period of Run2, a subject in this treatmentwas The anomalous laboratory evidence con- handed a page with two paragraphsin random fronts economists with at least two hard ques- order. One paragraph recommended always tions. Why are people so irrational?When (if switching and explained succinctly why ever) can people become more rational in switching improves the odds. The other para- tasks of this sort? In an attemptto find answers graph, wntten in a similar style, recommended to these questions, I had each subject in the always remaining and tried to evoke the "es- experiment continue with a second part. calation" and "gambler's fallacy" justifica- tions. The paragraphs, reproduced in the A. Method Appendix, were not presented separately to avoid confounding demand effects with learn- After completing Runl, the ten trials de- ing through written advice. Questionnaires scribed in the previous section, each subject reveal that subjects initially found the para- was paid accumulated earnings (between graphs equally persuasive. $1.00 and $4.00) in cash and asked to com- 4. Comparative results (Compare).-The plete a second set of 12 or 15 trials, called broad results for the first 40 subjects were Run2. The baseline procedures for Run2 were compiled and summarized for some later sub- exactly as in RunI with the addition of a non- jects. After the sixth period in Run2, each sub- salient $3.00 cash payment. Most subjects in ject in this treatment received a four-line Run2 received one or more of the following statement pointing out that 62.3 percent of the alternative treatments, chosen to encourage switch choices won the prize versus 30.5 per- more effective learning. cent of the remain choices. The summary also 1. Intense incentives (referred to below as In- revealed that the majority of choices were to tense). -The prize card was worth +$1.00 remain. The conductor believes that all sub- each trial and the other two cards were worth jects in this condition found the summary -$0.50. The initial $3.00 cash payment was credible. put on the table at the beginning of Run2. Each The Run2 data comprise 1,407 observations trial $1.00 was added or $0.50 subtractedfrom of the binary variables choice and outcome, the pile; at the end of the run, the subject re- gathered from 103 subjects. The subject who ceived the final pile.2 Note that each period the had piior knowledge of the task is excludedfrom expected value for remaining was (1/3)$1 + the analysis to follow because his choices (al- (2/3) (-$0.50) = $0.00 and for switching was most all to switch) might be incorrectlyattrib- (2/3)$1 + (1/3) (-$0.50) = $0.50 with in- uted to the treatmentshe received (Intense and tense incentives, versus $0.20 and $0.30 re- Advice) ratherthan to his plior knowledge. The spectively for remaining and switching with data include the actualchoices of the seven sub- the baseline incentives. jects dismissed early because of bankruptcy,but of course exclude the choices they would have made in later periods had they not been dis- 2 Seven subjects experienced bankruptcy (i.e., ex- missed (about four periods per bankruptsub- hausted the cash pile before the last trial), most of them ject). As explained in the Appendix,the design in trial 10 or later, at which point they were dismissed and received no furtherpayment. A referee notes correctly that mixes the treatmentsin a balanced fashion to the Intense treatmentcombines higher stakes with possible permit unbiased estimation of the direct and asymmetric responses to gains and losses. pairwise impacts of the treatments. 938 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

10 -0

90 -

80 -

"E 70

6o6

t~ 50-

2 40-

303 -

20-

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

FIGURE 2. SWITCHING PERCENTAGE IN RUN2

B. Results The other three treatmentsincreased the fre- quency of rational behavior. The impact of Surprisingly,the switch rate in Run2 did not Track was not quite significant at the 0.1IO begin where it left off in Runt. Figure 2 shows level in either early or late periods separately, that the overall switch rate was less than 25 but was significant at the 0.05 level overall percent in the first period, but soon rose above according to the Fisher exact test. Advice had 40 percent and remained mostly in the 40-50- no significant effect in early periods (or over- percent range after period 4. It rose to 55 per- all) but was significant at the 10-percent level cent in period 10 and trailed off somewhat in in later periods; perhaps it took some time for the last few periods. subjects to figure out which part of the advice Table 2 summarizes the impact of the treat- jibed with experience. By design, Compare ments. Overall, 46.0 percent of the 1,407 can have no effect on periods 1-6, so it is no choices were to switch. The switch rate rose surprise that it is insignificant in early periods. from 39.8 percent in the earlier periods ( 1-7 ) It is highly significant in later periods and to 52.6 percent in the later periods (8-15). overall. The combination of Advice and Com- The increase is highly significant (at least the pare may be especially powerful. Switch rates 0.0001 level) according to Fisher's exact test in later periods remained above 50 percent for (e.g., William J. Conover, 1980). Intense in- the 20 subjects with this combination and ex- centives did not increase the switch rate, ceeded 70 percent in several periods. mainly because in early periods only 38 per- Table 3 presents a finer-grained analysis. cent of subjects with intense incentives chose The dependent variable all regressions is bi- to switch. (By luck of the draw, many of the nary choice, with value 1 if the subject subjects with intense incentives did better, or switched and 0 if the subject remained that pe- would have done better, by not switching in riod. The controls for nuisance variables in- the first two or three periods, and it took a clude a constant (here the same for all while for the impact of early experience with subjects, but in the Appendix allowed to vary intense incentives to die out.) across subjects) and the trend variable Period VOL. 88 NO. 4 FRIEDMAN: CHOICE ANOMALIES 939

TABLE 2-SWITCH RATES BY TREIATMENT

Percent Percent Percent Nobs All periods Periods 1-7 Periods 8-15 0 Overall 1,407 46.0 39.8 52.6 (p-value) (0.000)

1 Intense 791 43.9 37.9 51.3 Not Intense 616 48.7 42.7 54.0 (p-value) (0.968) (0.915) (0.782)

2 Track 647 48.0 41.9 54.8 No Track 760 43.4 36.9 49.8 (p-value) (0.048) (0.101) (0.113)

3 Advice 804 47.4 40.6 55.0 No Advice 603 44.1 38.7 49.5 (p-value) (0.122) (0.327) (0.087)

4 Compare 647 50.3 40.2 59.4 No Compare 760 42.4 39.5 46.0 (p-value) (0.001) (0.458) (0.000)

Notes: Numbers of observations (Nobs) are almost evenly divided between Periods 1-7 (overall 51.6 percent) and Periods 8-15 (overall 48.4 percent of Nobs). Reportedp-values are the probability that Fisher's exact test incorrectly rejects the null hypothesis of no effect in favor of the one-sided alternativehypothesis that the treatment(or period number in item 0) has a positive effect on switch rates. The four treatments (Intense, Track, Advice, and Compare) are defined in the text.

(the period number in Runi or in Run2). negative. The significant positive coefficient The explanatory variables include dummies estimates for Switchbonus and Period suggest for each of the four treatments and for the that subjects do learn fromi personal experi- interaction of interest, Advice * Compare. ence and from time on task. The coefficient Two other explanatoryvariables are suggested estimates for Switchwon and the interactions by various learning theories (e.g., Drew raise the possibility of weak recency effects Fudenberg and David Levine, 1998). Ficti- and subadditive impacts of the treatmentswith tious play (or reinforcement learning) sug- personal experience. gests the variable Switchbonus, defined as the Two caveats are in order. First, the signif- earnings from always switching minus earn- icance levels in both tables assume indepen- ings from always remaining, cumulated from dent observations, but in fact successive the first period in RunI to the most recent pe- observations from any given subject share a riod. Cournot behavior (or directional learn- common history and therefore are unlikely ing, e.g., ReinhardSelten and Joachim Buchta, to be truly independent. Second, the depen- 1995) suggests Switchwon, a dummy variable dent variable in the regressions has limited equal to 1 if and only if switching would have (in fact binary) range, so the linear model won the prize in the most recent period. For underlying the OLS estimates is inappropri- completeness, the table also considers the in- ate. Orley Ashenfelter et al (1992) copes teractions of the continuous learning variable with analogous econometric problems aris- with Advice and Compare. ing in pairwise bargaining experiments by The ordinaryleast-squares (OLS) estimates supplementing the OLS estimates with confirm the Table 2 findings on the four treat- probit estimates, with and without fixed pair ment variables, but do not confirm the conjec- effects. In the Appendix, I report probit es- ture that Advice and Compare are more timates with and without random individual powerful in tandem; the relevant interaction subject effects; the estimates confirm and ex- near the bottom of Table 3 is insignificant and tend the inferences drawn from Table 3. 940 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

TABLE 3-OLS REGRESSIONS FOR SWITCH RATE

Variable Run1 t-statistic Run2 t-statistic Constant 0.125 4.24 0.191 4.42 (0.029) (0.043)

Period 0.017 3.18 0.012 3.71 (0.005) (0.003)

Switchbonus (cumulative earnings for 0.116 4.85 0.030 3.95 switching - not switching) (0.024) (0.008)

Switchwon (=1 if switch won in 0.039 1.03 0.114 3.83 previous period) (0.038) (0.030)

Intense (sensitive earnings) - -0.089 -3.05 (0.029)

Track (write out cumulative earnings - - 0.104 2.63 for all strategies) (0.039)

Advice (read conflicting - 0.127 2.52 explanations) (0.050)

Compare (told other subjects' - 0.080 1.87 performance) (0.043)

Switchbonus* Track - -0.017 -1.63 (0.010)

Switchbonus*Advice - -0.014 -1.55 (0.009)

Advice* Compare - -0.045 -0.77 (0.059) Adjusted R2 0.061 - 0.053 NOBs 1,040 - 1,407 -

Notes: The dependent variable is 1 in periods the subject chose to switch and is zero in other periods. The table reports ordinaryleast-squares coefficient estimates (and standarderrors) and student t-statistics.

The significantly positive point estimates A2 the solution is T 57. I rely on this indi- for Switchbonus have the important implica- rect evidence because running 60 trials by tion that switching will usually be chosen hand seems infeasible, and even on the com- when sufficiently favorable evidence accu- puter there is the problem of credibly precom- inulates. Indeed, the probit estimates in the mitting the prize card position before the Appendix imply that subjects would choose subject decides whether to switch. Of course, switch more than 90 percent of the time after there is also the usual boredom problem. So- about T = 60 Run2 trials with intense incen- lutions to these problems are possible (e.g., tives. The calculationis as follows. Set 0.90 multisession by-hand experiments with se- N(z), or z 1.3, for the unit normal cumu- questered subjects) but not easy. lative distribution function N with argument z = const + coeff x Switchbonus. Recall that C. Discussion with intense incentives the expected value of Switchbonus is 0.50 x T, and solve for T. With const = -0.863 - 0.238 1.1 and Evidently people are capable of learning to coeff = 0.084 from the last column of Table overcome the three-door anomaly. Most of VOL. 88 NO. 4 FRIEDMAN: CHOICE ANOMALIES 941 the people made the rational choice most of Directions the time in the more favorable conditions. Put subjects in the laboratory environment. The rationality rate sometimes exceeded the Run them through the standard procedures, classic "probability matching" benchmarkof carefully excluding any opportunitiesfor them 2/3 for transparentrisky choices with the same to leam from their experience or from imitat- odds. Coefficient estimates imply that, given a ing more successful subjects. Tabulate and an- sufficiently favorable track record, subjects alyze the results statistically. Write up the would make the rational choice almost all the results using rhetoric appropriatefor your tar- time. get social science journal. This recipe can The three-door task is not uniquely easy serve for one or more publications and may to learn. To the contrary, the evidence in produce a new entry in the pantheon of choice Section I shows that it is exceptionally dif- anomalies. ficult. If people are not hardwired to behave irrationally here, then perhaps they are ca- I do not claim that previous authors have pable of learning to behave rationally in consciously used the recipe, but it may give other tasks. Indeed, it now seems reasonable some insight into even the best empirical to make the following. anomalies papers. Take Craig Fox and Assertion .-Every choice "anomaly" Tversky (1995), for example, the first anom- can be greatly diminished or entirely elimi- aly paper I encountered after writing down the nated in appropriately structured learning recipe. The underlying rule of thunmbis to environments. avoid decisions when ill informed, especially Gerd Gigerenzer (1991, and numerous later to avoid dealing with better-informed people studies) offers many supporting examples; I with divergent interests. Such rules definitely welcome readers' suggestions of possible are useful in most field settings and can be counterexamples to the assertion. derived rigorously from expected utility the- If the assertion is correct then there is no ory under appropriateassurnptions (e.g., Jack such thing as an anomaly in the traditional Hirshleifer and John G. Riley, 1992 pp. 204ff, sense of stable behavior that is inconsistent 307ff). Using variations on the famous Daniel with rationality. There are only pseudo- Ellsberg (1961) paradox, the laboratory en- anomalies describing transientbehavior before vironment evokes comparisons to better- the learning process has been completed. informed people and to tasks about which the The argument can be taken a step further. subject has better information. It is inappro- The contrapositive form of the assertion says priate to avoid decisions in the laboratoryen- that the absence of adequate learning oppor- vironmnentbecause better informatioinwill not tunities is a necessary condition for observing be forthcoming nor will the subject actually pseudo-anomalies. A moment's reflection on contract with better-informed people. Even sufficient conditions leads to the following. though the authors use mild financial incentives at one point, the lab environments all are RECIPE FOR PSEUDO-ANOMALIES one shot with no opportunities to learn from Ingredients own (or others') experience. Standard statis- 1. A common and useful rule of thumb, e.g., tical techniques demonstrate that subjects procrastinatewhen anticipating important avoid the ill-informed tasks when they are new information. compared to better-informed tasks or better- 2. A laboratory environment that evokes the informed people. The rhetoric is a bit subdued rule of thumb when it is inappropriate,e.g., but appropriatefor an economics journal; the a lab environment where new information authors say that they ".... provide evidence is never important. against the descriptive validity of expected 3. Subjects inexperienced in the laboratory utility theory" (p. 586) that may be consistent environment. with some generalizations of that theory. 4. Standardlaboratory procedures. Does the recipe work when followed con- 5. Standard statistical techniques and stan- sciously? The main problem is in finding rules dard rhetoric. of thumb that have not already been used. For 942 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

example, I discovered that the procrastination of the anomaly and the reasons for its persis- rule of thumb has already been thoroughly and tence. One promising angle is to try to distance imaginatively studied in Tversky and Eldar the subject from the initial choice among the Shafir ( 1992). In Section I of this paper I used three doors in order to reduce the strength of instead a combination of the less-exploited commitment and/or the illusion of control. rules, "don't flip-flop" and "insufficient rea- Relevant treatments might include initial son." Opportunitiesto use the recipe still re- choice by a random device or by a second sub- main for social scientists with imagination and ject or by the same subject at an earlier time. a knowledge of the literature. Another angle is to manipulate the subject's What are the theoretical implications? Op- initial choice by beginning with a more trans- tical illusions arising from misleading visual parent task such as the million-door problem cues are interesting but do not imply the need suggested in vos Savant (1990). to modify the theory of optics. Likewise, ir- A third angle is to put the three-doortask in rationalchoices arising from incomplete learn- a richer institutionalframework such as a mar- ing do not imply the need to modify standard ket in choice opportunities. A subject could choice theory. switch, remain, or sell the choice or buy addi- tional choices at a price set exogenously or in III. Conclusion a double-auction market. Economists might predict that the market price would quickly Some readers will rightfully object to the move to the maximal expected value ($0.50 facile tone of the preceding subsection. De- with intense incentives) minus a small risk pre- spite the use of strong treatments, many sub- mium, with only switchers buying and all non- jects continued to make lots of irrational switchers selling. Psychologists (along with choices. The three-dooranomaly declined sub- some other experimentalists and observers of stantially but did not actually disappear in 15 humannature) might predict a core of stubborn periods. nonswitching nonselling subjects, even when In my view, the real message to economists prices are above the confused Bayesian ex- is neither "ignore anomalies because they will pected value ($0.25 with intense incentives). eventually disappear" nor "rewrite choice Such experiments, in tandem with the senous theory to accommodate the evidence." Rather, modelling of learning processes within eco- economists should focus their energies on two nomic institutions, hold the promise of ending sorts of questions. First, which learning envi- the now-sterile debate on anomalies (see also ronments encourage or discourage specific L. Jonathan Cohen, 1981; Robin M. Hogarth kinds of anomalies? Second, which institu- and Melvin W. Reder, 1987; Lola Lopes, 1990; tions are sensitive to anomalous choice behav- Vernon L. Smith, 1991 ) and creating a produc- ior? Anomalies can be ignored when the tive new research agenda on rationality. economic situation of interest (for example, a simple competitive market) involves an insti- APPENDIX: RUN2 DETAILS tution insensitive to anomalous behavior or when the institution allows effective learning. Otherwise, economists should take anomalies Advice treatment text. -Please read the seriously, not by relaxing axioms of choice [following] two pieces of advice on how to theory, but ratherby modelling the specific in- earn money in this experiment. The pieces dis- stitutional sensitivity and/or the incomplete agree on what you should do; you must make learning. up your own mind. Explorationof the three-dooranomaly is far Advice S1: You have 1 chance in 3 of pick- from complete. It surely would be useful to ing the Prize card initially. If you did pick it develop the technology to run each subject for initially you will win the prize if you Remain. 30 or more trials and to fit individual learning You have 2 chances in 3 of not picking the models to the individual data. With or without Prize card initially. If you did not pick it ini- such technology, new treatmentscould isolate tially you will win the Prize if you Switch. So, the underlying reasons for the initial strength the overall chance of winning is 2 chances in VOL. 88 NO. 4 FRIEDMAN: CHOICE ANOMALIES 943

TABLE Al-NUMBER OF SUBJECTS IN ALTERNATIVE RUN2 TREATMENTS

Intense = I Intense = 0

Track = 1 Track = 0 Track = 1 Track = 0 Total Advice= 1 Compare= 1 10 0 0 10 20 Compare = 0 20 10 + 1* 10 0 41

Advice = 0 Compare= 1 0 11 11 1 23 Compare-0 10 0 0 10 20

TOTAL 40 22 21 21 104

Notes: The four binary treatmentvariables (Intense, Track,Advice, and Compare) are defined in the text. The eight cells with ten (or 11) subjects comprise a half-fractionalfactorial design that confounds Comparewith the three-way interaction of the other variables. The asterisk (*) identifies an individual subject excluded from subsequent analysis because of prior knowledge.

3 if you Switch and only 1 in 3 chances if you sion of the relation between logit, probit, and Remain. Therefore, you will do the best in the standard regression techniques and a discus- long run if you always Switch. sion of random and fixed effects. Advice Rl: The conductor may try to dis- The first two columns of Table A2 report tract you by offers to Switch, but these offers the results for Runt . Random effects (individ- are made for his/her own reasons and are not ual subject differences in the basic tendency necessarily in your interest. When only 2 cards to switch) do appearto be present, as indicated remain face down, you have at least 1 chance by the diagnostic statistic Rho (significant at in 2 of picking the Prize card and Switching better than the 0.1-percent level) and by the will not improve your chances. You should not improved log-likelihood reported at the bot- let yourself be distracted. Therefore, you tom of the second column. The Switchwon co- should always Remain with your initial choice efficient changes sign but is insignificant either and never Switch. way. The other coefficients (and their signifi- Design. -Table Al shows that 80 subjects cance levels) are quite consistent in both col- were run in conditions that comprise a half- umns. The significantly negative constant fractional factorial design in the four treat- indicates a strong tendency to remain rather ments with ten subjects per cell. Twenty other than switch, all else equal. The significantly subjects were run in the condition Track and positive Period coefficient indicates an up- Intense only; the remaining three subjects ward trend in the switch rate. Learning from were run in miscellaneous conditions. The accumulated experience has an even stronger fractional factorial design allows estimation of and more significant impact as indicated by the the direct impact of each treatment and the Switchbonus coefficient. two-way interactionsbut not the three-way in- The Run2 results in the remaining columns teractions; see, e.g., George Box et al., 1978). include the impact of the treatments (none of Probit estimation. -Table A2 reportsmax- which were used in Runi ) and tell a somewhat imum likelihood probit coefficient estimates more complex story. The full specification in using the Limdep 7.0 package on all data. third column indicates that the average bias Logit regressions, with and without fixed ef- against switching is less than in RunL but still fects, produced very similar but less complete highly significant. The -0.814 constant indi- results; the software could not handle the full cates a basic tendency to switch in about unbalanced panel in the logit regression cor- N(-0.814) = 21 percent of trials, where N responding to the last column in Table A2. See is the unit normal cumulative distribution William H. Greene (1990) for a full discus- function. Trend as measured by the Period 944 THE AMERICANECONOMIC REVIEW SEPTEMBER1998

TABLE A2-PROBIT ESTIMATES (AND P-VALUES) FOR SWITCH RATE

Variable Runl Runl Run2 Run2 Constant -1.090 -1.166 -0.814 -0.863 (0.000) (0.000) (0.000) (0.001)

Period 0.055 0.064 0.032 0.043 (0.001) (0.001) (0.000) (0.000)

Switchbonus 0.325 0.385 0.082 0.084 (0.000) (0.000) (0.000) (0.001)

Switchwon 0.106 -0.146 0.293 0.131 (0.344) (0.250) (0.000) (0.137)

Intense - - -0.234 -0.238 (0.002) (0.109)

Track - - 0.276 0.286 (0.009) (0.117)

Advice - - 0.337 0.380 (0.012) (0.253)

Compare - - 0.208 0.229 (0.069) (0.411)

Switchbonus* Track - - -0.045 -0.052 (0.099) (0.249)

Switchbonus* Advice - -0.038 -0.034 (0.113) (0.395)

Advice*Compare - - -0.116 -0.087 (0.458) (0.799) Random effects No Yes No Yes

Rho - 0.384 0.453 (0.000) (0.000) NOBs 1,040 1,040 1,407 1,407 -Log-likelihood 589.9 581.3 927.8 897.5

Notes: The dependent variable is 1 in periods the subject chose to swtich and is 0 in other periods. The table reports maximum likelihood probit coefficient estimates with and without random effects (and correspondingp-values). Rho is the standardHausman test statistic for the presence of random effects.

coefficient is weaker than in Runl but still five times more intense than in Runl. If sub- highly significant. The fictitious play variable jects respond to the strongerincentives but less Switchbonus also remains highly significant. than proportionately,then we would see a neg- The Coumot variable Switchwon has a signif- ative Intense coefficient and/or a smaller icantly positive coefficient here. Since Switch- Switchbonus coefficient in Run2, as indeed we bonus already captures the most relevant do.) Track and Advice have positive and sig- positive aspects of intense incentives, the significant coefficients. The Compare coefficient nificantly negative Intense coefficient should is marginally significantly positive (at the p = be thought of as measuring a residual impact. 6.9-percent level), but the a priori most im- (The incentives for most subjects in Run2 are portant interactions are less significant. VOL. 88 NO. 4 FRIEDMAN: CHOICE ANOMALIES 945

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