- Email: [email protected]
Let's make a deal
Economics Letters 61 (1998) 175–180
Let’s make a deal Scott E. Page* Department of Economics, University of Iowa, Iowa City, IA 52242, USA Received 3 March 1998; accepted 25 June 1998
Abstract This paper describes experiments of variants of ‘‘Let’s Make a Deal.’’ The experiments support three hypotheses: (1) people do better as the number of doors increases, (2) the performance improvement is gradual, (3) people cannot represent their logic using probability. 1998 Elsevier Science S.A. All rights reserved. Keywords: Experimental economics; Bounded rationality JEL classification: C91
1. Introduction In recent years, few mathematical problems have received as much exposure as the ‘‘Let’s Make a Deal’’ (LMAD) problem (Vos Savant, 1990; Tierney, 1991). Perhaps, awareness of the game show of the same name explains LMAD’s popularity, making it an intriguing slice of reality and not some abstract, recondite question. In this case, familiarity breeds not contempt but contemplation. Unfortunately though, the level of thought often falls short. Most people, statisticians and mathematicians, solve LMAD incorrectly. The failure of so many people to solve correctly the LMAD problem demonstrates a systematic violation of rationality and has led some to argue that economists ‘‘should look for alternatives to Bayes’ Rule’’ (Nalebuff, 1987). In this paper, I describe experiments on the LMAD problem with 3 doors, 10 doors, and 100 doors. By exaggerating the probabilities, I simplify the problem. My findings support three hypothesis: first, that as the number of doors increases, people get better at solving the problem; second, that this improvement is gradual; and third, that people who understand the problem with a larger number of doors cannot apply it in the three door case. The findings speak to the strength of the LMAD bias, but do not imply that people cannot learn to solve the LMAD problem. Evidence suggests they can (Friedman, 1997). Instead, the findings demonstrate the subtlety of the underlying logic and provide support for drawing a distinction between having ‘‘a feel for’’ underlying logic and understanding that logic well enough to apply it in less obvious instances. The remainder of this paper contains a description of the LMAD problem, a summary of *Tel.: 11 319 3351015; fax: 11 319 3351956; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00158-X
176
S.E. Page / Economics Letters 61 (1998) 175 – 180
experimental results, and comments on the relevance of these experimental findings to institutional and organizational design.
2. The let’s make a deal problem The LMAD problem is easy to describe. The LMAD Problem One of three doors has been randomly chosen to conceal a prize. Each hides the prize with equal probability. A contestant selects a door. After she selects, she will be shown what is behind a door randomly chosen with equal probability from among those doors that she did not choose that also do not conceal the prize. The contestant is then allowed to switch her original door for other unopened door. If the contestant knows all three stages of the game, should she switch? All three stages of the decision problem must be known to the contestant or else she may believe that the offer of a switch depends on her initial choice. As the problem is stated, the contestant should accept the offer to switch. The probabilistic argument as to why to switch can be readily found (see Gardner, 1959, for an early version), but I will offer a ‘‘flash of insight’’ solution that makes the correct decision transparent. Suppose that after selecting a door, the subject can trade her door for both of the remaining doors. She doubles her odds by accepting the trade, so clearly she would accept. If after the switch, she is then told that to save the trouble of opening both doors, one of her doors that is empty will be opened for her. This is equivalent to the offer made in LMAD. In experiments, most people think that switching is no better than staying put. This logical error occurs because people do not recognize the asymmetry of the test of likelihood. The originally chosen door could not have been opened in the second stage of the problem. The unchosen door could have been. Thus, the LMAD bias is a failure to recognize that the revelation of the incorrect door only provides information about the likelihood of the unchosen door. The LMAD bias should be distinguished from the bias of distributing ignorance equally across verbally defined categories (Dawes, 1988). Agents are not merely counting events and assigning equal probability to each of two events. They are mistakenly updating probabilities based upon information.
3. The experiments Exaggerating a bias often enables people to learn to overcome it. Only Sherlock Holmes recognized that the dog not barking was a valuable clue, but almost anyone would comment upon the silence of 101 dalmations. As Nalebuff (1987); Vos Savant (1990), and others have suggested, increasing the number of doors simplifies the LMAD problem. If the contestant chooses from among say, 100 doors, and is then shown that 98 of the 99 unchosen doors do not conceal the prize, then she is likely to switch to the remaining unchosen door if given the opportunity. The asymmetry of the likelihood test becomes obvious. Knowledge of Bayes’ rule would not be necessary to make the correct decision. In the experiments, I considered three cases: a 3 door, a 10 door, and a 100 door problem.1 For convenience, I refer to these as LMAD(3), LMAD(10), and LMAD(100). To minimize framing 1
Copies of the experiment sheets are available from the author.
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effects (Kahneman and Tversky, 1979), the second stage of the game was described as ‘‘n 2 2 incorrect doors have been randomly selected from among the n 2 1 unchosen doors’’. Saying that ‘‘the correct door must either be the original choice or the remaining unopened door’’ makes the unchosen door focal. I ran two sets of experiments to test three hypotheses. In the first experiment, subjects played either LMAD(3), LMAD(10), or LMAD(100). In the second experiment, some played LMAD(3) and others played both LMAD(3) and LMAD(100) simultaneously. The first hypothesis is that as the number of doors increases, more subjects would learn to switch. The second hypothesis is that the performance improvement would be gradual. The third hypothesis is that subjects would not be able to apply the logic learned with more doors to the three door case. The evidence supports all three hypotheses.
3.1. Experiment [1 The first experiment involved 50 subjects, all MBA students at Northwestern University. All had completed at least one semester of probability theory. Winners, those that had selected the correct door, received 5 dollars. Losers got nothing, even if they played the correct strategy. Subjects were given as much time as they wanted to complete the task. No subjects took longer than 15 minutes. The table below shows the proportion of subjects who switched in each of three experiments, the probability of switching and 95% confidence intervals.
[ Doors
3 10 100
[ switch 2 of 17 8 of 17 14 of 16
P(switch) 0.116 0.471 0.875
95% C.I. (0.02,0.33) (0.26,0.69) (0.66,0.98)
These data demonstrate that the problem becomes easier as the number of doors increases, the first hypothesis. Less than 12% of subjects make the correct decision in LMAD(3) and over 87% choose correctly in LMAD(100).2 The data also support the second hypothesis. The bias diminishes gradually as the number of doors increases. In LMAD(10), around 50% of the subjects switch to the unopened, unchosen door. Formal statistical comparisons of differences show that the 95% confidence intervals for LMAD(3) and LMAD(10) and for LMAD(10) and LMAD(100) overlap slightly. That the correct insight does not become apparent to everyone all at once suggests heterogeneity in the ability to grasp the logic of LMAD. The game sheets asked subjects to estimate the probability that their initial and final selections were correct. The former question provided a check that they understood the game and could do simple probability calculations. The latter tested whether anyone stayed with a selection that she thought had a lower probability because she ‘‘felt lucky’’ (none did), and to discern how many subjects could calculate correct probabilities of being correct using Bayes’ rule. The sheets show that everyone correctly calculated the initial probability of being correct. This was not true of the final probabilities. In LMAD(3), 6 of the 17 subjects correctly estimated the final probability that their initial choice was correct was 1 / 3, although only 4 chose to switch. Analysis of their answer sheets and subsequent discussions revealed that 4 of these subjects thought that the probability the other door was correct 2
Friedman (1997) also finds a probability of around 10% switches in the first round of his experiments.
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also equaled 1 / 3. These subjects had sub-additive probability distributions over outcomes, a common assumption in formal bounded rationality models (Camerer and Weber, 1992; Ghirardato, 1996). In LMAD(10), 5 of 16 subjects correctly calculated the probability that the other door was correct at 9 / 10. All of these subjects switched. One other subject felt the probability of the unchosen door was larger than 1 / 2, switched, but was not capable of explaining why. The other 11 subjects, all calculated that the probabilities were equal, but two switched anyway based on ‘‘intuition’’ and ‘‘gut 99 feelings’’. In LMAD(100), eight subjects calculated the probabilities correctly ( ] 100 for the unchosen door). All eight switched. The other eight subjects, calculated the probabilities of the two doors as equal, but six of eight switched. They too, felt that switching improved their odds of winning but lacked the ability to justify their actions. The fact that the subjects could not calculate the probabilities correctly or even make an incorrect calculation supporting the more likely outcome supports the third hypothesis. Their inability to verbalize and / or quantify their logic bodes ill for their chances of applying it elsewhere. For an insight to be transferable, we must have a language with which to store and retrieve it. The second experiment provides more tangible support for this hypothesis.
3.2. Experiment [2 The second set of experiments involved 46 MBA students who had just completed a 3 week study of probability theory and Bayes’ Rule.3 The subjects were randomly divided into two groups. Group 1 played LMAD(3). Winners were paid 5 dollars; losers were paid nothing. Group 2 played LMAD(3) and LMAD(100) simultaneously. They were told that they would be given two sheets, each with a decision problem. Some of them were given LMAD(3) on top, and others were given LMAD(100) on top. The subjects in Group 2 were told that if they got the correct answer to LMAD(3), they received 3 dollars. A correct answer in LMAD(100) paid 2 dollars. Incorrect answers paid nothing. As before, subjects were asked to calculate the probabilities that their choices were correct after both rounds. The findings are summarized in the table below. Group
[ Doors
[ switch
P(switch)
95% C.I.
1 2 3
3 10 100
3 of 24 4 of 22 18 of 22
0.125 0.182 0.818
(0.03,0.29) (0.06,0.37) (0.63,0.94)
Group 1 served as a control group. There is no statistical difference between Group 1 and Group 2 on LMAD(3). In addition, the probability of switching for Group 2 in LMAD(3) (18%) does not differ significantly from that of subjects in the first experiment (12%). Moreover, the probability of switching in LMAD(100) for subjects in Group 2 is also statistically indistinguishable from subjects in the first experiment. The fact that the percentage that correctly solved LMAD(3) does not differ statistically for subjects who did not simultaneously play LMAD(100) substantiates the hypothesis that ‘‘gut feelings’’ are not transferable from LMAD(100) to LMAD(3). A closer look at the data shows that the four subjects in 3
Unlike the first group of subjects, these students were taking a class I was teaching.
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Group 2 who switched in LMAD(3) also did so in LMAD(100). Unfortunately, all four incorrectly estimated the probability of being correct after switching in both LMAD(3) and LMAD(100). These subjects either recognized the similarity of the problems and took the same action in both, or they understood the logic behind LMAD. Twelve of the 18 subjects (66%) who switched in LMAD(100) calculated the probability of being correct after switching to be ]12 , 5 of the 18 subjects (28%) correctly estimated the probability to be 99 ] , and one subject, one who switched in both problems, could not even write down a probability 100 estimate. A puzzling feature of these data is that none of the five subjects who correctly estimated the probability of being correct in LMAD(100) after switching, were able to make the same calculation in LMAD(3). When questioned as to why they did not apply their mathematical logic derived from LMAD(100) to LMAD(3), one subject said ‘‘even though they were worded identically, they seemed different somehow’’.
4. Variations on a theme Problems similar to LMAD that confuse most people can be constructed easily, though they rarely receive such warm or intelligent responses, save card games, bridge specifically. Fun and games aside (and LMAD is fun) there remains a nagging question as to the relevance of these experiments. Can we learn anything from them? I think we can. First, notice that not only can we make LMAD less difficult by increasing both the number of initial doors and number of opened doors, we can also make it more difficult.4 Second, the tasks confronting people in economic and political environments also often outstrip human abilities. When constructing institutions, we should ask, can we shift incentives so that we (so to speak) ‘‘increase the number of doors and the number of doors revealed’’ and thereby turn LMAD(3) into LMAD(100)? At present, most analyses of mechanisms and institutions focus on the equilibria generated. We include incentive compatibility and individual rationality constraints, but with few exceptions (Mount and Reiter, 1990), we do not gauge the difficulty of calculations confronting economic agents. Equilibria that require super-human logic may have little practical relevance. Effective institutions should structure incentives so that people can use ‘‘gut feelings’’ as well as Bayes’ rule to make optimal decisions.
Acknowledgements The ideas put forth in this paper have been improved through conversations with many colleagues and friends. I would especially like to thank Max Bazerman, Jenna Bednar, Andreas Blume, Dan Friedman, Gordon Green, Michael Kirschenheiter, John Miller, Rebecca Morton, and Roger Myerson. 4
1 1 Consider the problem that for lack of a better name I’ll call LMAD( ] ). In LMAD( ] ) there are 100 cards randomly 100 100 arranged face down on a table. One of the cards has a gold star on its face. If the contestant picks that card, she receives $100. The contestant randomly picks a card. Among the 99 cards she did not choose, at least 98 do not have the gold star. The contestant is randomly shown one of these incorrect, unchosen cards and offered the opportunity to switch to one of the other 98 unchosen cards. If she knows all of the stages of the decision process, should she switch? Yes, just as in LMAD, her odds of winning increase if she switches. Yet, probably even fewer than 10% of subjects would switch when playing 1 LMAD( ] ). 100
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S.E. Page / Economics Letters 61 (1998) 175 – 180
References Camerer, C., Weber, M., 1992. Recent developments in modeling preferences: uncertainty and ambiguity. Journal of Risk and Uncertainty 5, 325–370. Dawes, R., 1988. Rational Choice in an Uncertain World. Harcourt Brace Jovanovich, Orlando FL. Friedman, D., 1997. Monte Hall’s three doors: construction and deconstruction of a choice anomaly. Manuscript, University of California Santa Cruz, Department of Economics. Gardner, M., 1959. Mathematical recreations. Scientific American, October, pp. 174–182. Ghirardato, P., 1996. Coping with ignorance: unforeseen contingencies and non-additive uncertainty. Forthcoming in Journal of Economic Theory. Kahneman, D., Tversky, A., 1979. Prospect theory an analysis of decision under risk. Econometrica 47, 263–291. Mount, K., Reiter, S., 1990. A model of computing with human agents. Center for Mathematical Studies in Management Science, discussion paper 890. Northwestern University. Nalebuff, B., 1987. Puzzles: choose a curtain, duel-ity, two point conversions, and more. Journal of Economic Perspectives 1 (1), 157–163. Tierney, J., 1991. Behind Monty Halls doors: puzzle debate and answer? New York Times, July 21, p. 1. Vos Savant, M., 1990. Ask Marilyn. Parade Magazine, September 8, 1990, December 2, 1990, February 17, 1991.
Let’s make a deal Scott E. Page* Department of Economics, University of Iowa, Iowa City, IA 52242, USA Received 3 March 1998; accepted 25 June 1998
Abstract This paper describes experiments of variants of ‘‘Let’s Make a Deal.’’ The experiments support three hypotheses: (1) people do better as the number of doors increases, (2) the performance improvement is gradual, (3) people cannot represent their logic using probability. 1998 Elsevier Science S.A. All rights reserved. Keywords: Experimental economics; Bounded rationality JEL classification: C91
1. Introduction In recent years, few mathematical problems have received as much exposure as the ‘‘Let’s Make a Deal’’ (LMAD) problem (Vos Savant, 1990; Tierney, 1991). Perhaps, awareness of the game show of the same name explains LMAD’s popularity, making it an intriguing slice of reality and not some abstract, recondite question. In this case, familiarity breeds not contempt but contemplation. Unfortunately though, the level of thought often falls short. Most people, statisticians and mathematicians, solve LMAD incorrectly. The failure of so many people to solve correctly the LMAD problem demonstrates a systematic violation of rationality and has led some to argue that economists ‘‘should look for alternatives to Bayes’ Rule’’ (Nalebuff, 1987). In this paper, I describe experiments on the LMAD problem with 3 doors, 10 doors, and 100 doors. By exaggerating the probabilities, I simplify the problem. My findings support three hypothesis: first, that as the number of doors increases, people get better at solving the problem; second, that this improvement is gradual; and third, that people who understand the problem with a larger number of doors cannot apply it in the three door case. The findings speak to the strength of the LMAD bias, but do not imply that people cannot learn to solve the LMAD problem. Evidence suggests they can (Friedman, 1997). Instead, the findings demonstrate the subtlety of the underlying logic and provide support for drawing a distinction between having ‘‘a feel for’’ underlying logic and understanding that logic well enough to apply it in less obvious instances. The remainder of this paper contains a description of the LMAD problem, a summary of *Tel.: 11 319 3351015; fax: 11 319 3351956; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00158-X
176
S.E. Page / Economics Letters 61 (1998) 175 – 180
experimental results, and comments on the relevance of these experimental findings to institutional and organizational design.
2. The let’s make a deal problem The LMAD problem is easy to describe. The LMAD Problem One of three doors has been randomly chosen to conceal a prize. Each hides the prize with equal probability. A contestant selects a door. After she selects, she will be shown what is behind a door randomly chosen with equal probability from among those doors that she did not choose that also do not conceal the prize. The contestant is then allowed to switch her original door for other unopened door. If the contestant knows all three stages of the game, should she switch? All three stages of the decision problem must be known to the contestant or else she may believe that the offer of a switch depends on her initial choice. As the problem is stated, the contestant should accept the offer to switch. The probabilistic argument as to why to switch can be readily found (see Gardner, 1959, for an early version), but I will offer a ‘‘flash of insight’’ solution that makes the correct decision transparent. Suppose that after selecting a door, the subject can trade her door for both of the remaining doors. She doubles her odds by accepting the trade, so clearly she would accept. If after the switch, she is then told that to save the trouble of opening both doors, one of her doors that is empty will be opened for her. This is equivalent to the offer made in LMAD. In experiments, most people think that switching is no better than staying put. This logical error occurs because people do not recognize the asymmetry of the test of likelihood. The originally chosen door could not have been opened in the second stage of the problem. The unchosen door could have been. Thus, the LMAD bias is a failure to recognize that the revelation of the incorrect door only provides information about the likelihood of the unchosen door. The LMAD bias should be distinguished from the bias of distributing ignorance equally across verbally defined categories (Dawes, 1988). Agents are not merely counting events and assigning equal probability to each of two events. They are mistakenly updating probabilities based upon information.
3. The experiments Exaggerating a bias often enables people to learn to overcome it. Only Sherlock Holmes recognized that the dog not barking was a valuable clue, but almost anyone would comment upon the silence of 101 dalmations. As Nalebuff (1987); Vos Savant (1990), and others have suggested, increasing the number of doors simplifies the LMAD problem. If the contestant chooses from among say, 100 doors, and is then shown that 98 of the 99 unchosen doors do not conceal the prize, then she is likely to switch to the remaining unchosen door if given the opportunity. The asymmetry of the likelihood test becomes obvious. Knowledge of Bayes’ rule would not be necessary to make the correct decision. In the experiments, I considered three cases: a 3 door, a 10 door, and a 100 door problem.1 For convenience, I refer to these as LMAD(3), LMAD(10), and LMAD(100). To minimize framing 1
Copies of the experiment sheets are available from the author.
S.E. Page / Economics Letters 61 (1998) 175 – 180
177
effects (Kahneman and Tversky, 1979), the second stage of the game was described as ‘‘n 2 2 incorrect doors have been randomly selected from among the n 2 1 unchosen doors’’. Saying that ‘‘the correct door must either be the original choice or the remaining unopened door’’ makes the unchosen door focal. I ran two sets of experiments to test three hypotheses. In the first experiment, subjects played either LMAD(3), LMAD(10), or LMAD(100). In the second experiment, some played LMAD(3) and others played both LMAD(3) and LMAD(100) simultaneously. The first hypothesis is that as the number of doors increases, more subjects would learn to switch. The second hypothesis is that the performance improvement would be gradual. The third hypothesis is that subjects would not be able to apply the logic learned with more doors to the three door case. The evidence supports all three hypotheses.
3.1. Experiment [1 The first experiment involved 50 subjects, all MBA students at Northwestern University. All had completed at least one semester of probability theory. Winners, those that had selected the correct door, received 5 dollars. Losers got nothing, even if they played the correct strategy. Subjects were given as much time as they wanted to complete the task. No subjects took longer than 15 minutes. The table below shows the proportion of subjects who switched in each of three experiments, the probability of switching and 95% confidence intervals.
[ Doors
3 10 100
[ switch 2 of 17 8 of 17 14 of 16
P(switch) 0.116 0.471 0.875
95% C.I. (0.02,0.33) (0.26,0.69) (0.66,0.98)
These data demonstrate that the problem becomes easier as the number of doors increases, the first hypothesis. Less than 12% of subjects make the correct decision in LMAD(3) and over 87% choose correctly in LMAD(100).2 The data also support the second hypothesis. The bias diminishes gradually as the number of doors increases. In LMAD(10), around 50% of the subjects switch to the unopened, unchosen door. Formal statistical comparisons of differences show that the 95% confidence intervals for LMAD(3) and LMAD(10) and for LMAD(10) and LMAD(100) overlap slightly. That the correct insight does not become apparent to everyone all at once suggests heterogeneity in the ability to grasp the logic of LMAD. The game sheets asked subjects to estimate the probability that their initial and final selections were correct. The former question provided a check that they understood the game and could do simple probability calculations. The latter tested whether anyone stayed with a selection that she thought had a lower probability because she ‘‘felt lucky’’ (none did), and to discern how many subjects could calculate correct probabilities of being correct using Bayes’ rule. The sheets show that everyone correctly calculated the initial probability of being correct. This was not true of the final probabilities. In LMAD(3), 6 of the 17 subjects correctly estimated the final probability that their initial choice was correct was 1 / 3, although only 4 chose to switch. Analysis of their answer sheets and subsequent discussions revealed that 4 of these subjects thought that the probability the other door was correct 2
Friedman (1997) also finds a probability of around 10% switches in the first round of his experiments.
S.E. Page / Economics Letters 61 (1998) 175 – 180
178
also equaled 1 / 3. These subjects had sub-additive probability distributions over outcomes, a common assumption in formal bounded rationality models (Camerer and Weber, 1992; Ghirardato, 1996). In LMAD(10), 5 of 16 subjects correctly calculated the probability that the other door was correct at 9 / 10. All of these subjects switched. One other subject felt the probability of the unchosen door was larger than 1 / 2, switched, but was not capable of explaining why. The other 11 subjects, all calculated that the probabilities were equal, but two switched anyway based on ‘‘intuition’’ and ‘‘gut 99 feelings’’. In LMAD(100), eight subjects calculated the probabilities correctly ( ] 100 for the unchosen door). All eight switched. The other eight subjects, calculated the probabilities of the two doors as equal, but six of eight switched. They too, felt that switching improved their odds of winning but lacked the ability to justify their actions. The fact that the subjects could not calculate the probabilities correctly or even make an incorrect calculation supporting the more likely outcome supports the third hypothesis. Their inability to verbalize and / or quantify their logic bodes ill for their chances of applying it elsewhere. For an insight to be transferable, we must have a language with which to store and retrieve it. The second experiment provides more tangible support for this hypothesis.
3.2. Experiment [2 The second set of experiments involved 46 MBA students who had just completed a 3 week study of probability theory and Bayes’ Rule.3 The subjects were randomly divided into two groups. Group 1 played LMAD(3). Winners were paid 5 dollars; losers were paid nothing. Group 2 played LMAD(3) and LMAD(100) simultaneously. They were told that they would be given two sheets, each with a decision problem. Some of them were given LMAD(3) on top, and others were given LMAD(100) on top. The subjects in Group 2 were told that if they got the correct answer to LMAD(3), they received 3 dollars. A correct answer in LMAD(100) paid 2 dollars. Incorrect answers paid nothing. As before, subjects were asked to calculate the probabilities that their choices were correct after both rounds. The findings are summarized in the table below. Group
[ Doors
[ switch
P(switch)
95% C.I.
1 2 3
3 10 100
3 of 24 4 of 22 18 of 22
0.125 0.182 0.818
(0.03,0.29) (0.06,0.37) (0.63,0.94)
Group 1 served as a control group. There is no statistical difference between Group 1 and Group 2 on LMAD(3). In addition, the probability of switching for Group 2 in LMAD(3) (18%) does not differ significantly from that of subjects in the first experiment (12%). Moreover, the probability of switching in LMAD(100) for subjects in Group 2 is also statistically indistinguishable from subjects in the first experiment. The fact that the percentage that correctly solved LMAD(3) does not differ statistically for subjects who did not simultaneously play LMAD(100) substantiates the hypothesis that ‘‘gut feelings’’ are not transferable from LMAD(100) to LMAD(3). A closer look at the data shows that the four subjects in 3
Unlike the first group of subjects, these students were taking a class I was teaching.
S.E. Page / Economics Letters 61 (1998) 175 – 180
179
Group 2 who switched in LMAD(3) also did so in LMAD(100). Unfortunately, all four incorrectly estimated the probability of being correct after switching in both LMAD(3) and LMAD(100). These subjects either recognized the similarity of the problems and took the same action in both, or they understood the logic behind LMAD. Twelve of the 18 subjects (66%) who switched in LMAD(100) calculated the probability of being correct after switching to be ]12 , 5 of the 18 subjects (28%) correctly estimated the probability to be 99 ] , and one subject, one who switched in both problems, could not even write down a probability 100 estimate. A puzzling feature of these data is that none of the five subjects who correctly estimated the probability of being correct in LMAD(100) after switching, were able to make the same calculation in LMAD(3). When questioned as to why they did not apply their mathematical logic derived from LMAD(100) to LMAD(3), one subject said ‘‘even though they were worded identically, they seemed different somehow’’.
4. Variations on a theme Problems similar to LMAD that confuse most people can be constructed easily, though they rarely receive such warm or intelligent responses, save card games, bridge specifically. Fun and games aside (and LMAD is fun) there remains a nagging question as to the relevance of these experiments. Can we learn anything from them? I think we can. First, notice that not only can we make LMAD less difficult by increasing both the number of initial doors and number of opened doors, we can also make it more difficult.4 Second, the tasks confronting people in economic and political environments also often outstrip human abilities. When constructing institutions, we should ask, can we shift incentives so that we (so to speak) ‘‘increase the number of doors and the number of doors revealed’’ and thereby turn LMAD(3) into LMAD(100)? At present, most analyses of mechanisms and institutions focus on the equilibria generated. We include incentive compatibility and individual rationality constraints, but with few exceptions (Mount and Reiter, 1990), we do not gauge the difficulty of calculations confronting economic agents. Equilibria that require super-human logic may have little practical relevance. Effective institutions should structure incentives so that people can use ‘‘gut feelings’’ as well as Bayes’ rule to make optimal decisions.
Acknowledgements The ideas put forth in this paper have been improved through conversations with many colleagues and friends. I would especially like to thank Max Bazerman, Jenna Bednar, Andreas Blume, Dan Friedman, Gordon Green, Michael Kirschenheiter, John Miller, Rebecca Morton, and Roger Myerson. 4
1 1 Consider the problem that for lack of a better name I’ll call LMAD( ] ). In LMAD( ] ) there are 100 cards randomly 100 100 arranged face down on a table. One of the cards has a gold star on its face. If the contestant picks that card, she receives $100. The contestant randomly picks a card. Among the 99 cards she did not choose, at least 98 do not have the gold star. The contestant is randomly shown one of these incorrect, unchosen cards and offered the opportunity to switch to one of the other 98 unchosen cards. If she knows all of the stages of the decision process, should she switch? Yes, just as in LMAD, her odds of winning increase if she switches. Yet, probably even fewer than 10% of subjects would switch when playing 1 LMAD( ] ). 100
180
S.E. Page / Economics Letters 61 (1998) 175 – 180
References Camerer, C., Weber, M., 1992. Recent developments in modeling preferences: uncertainty and ambiguity. Journal of Risk and Uncertainty 5, 325–370. Dawes, R., 1988. Rational Choice in an Uncertain World. Harcourt Brace Jovanovich, Orlando FL. Friedman, D., 1997. Monte Hall’s three doors: construction and deconstruction of a choice anomaly. Manuscript, University of California Santa Cruz, Department of Economics. Gardner, M., 1959. Mathematical recreations. Scientific American, October, pp. 174–182. Ghirardato, P., 1996. Coping with ignorance: unforeseen contingencies and non-additive uncertainty. Forthcoming in Journal of Economic Theory. Kahneman, D., Tversky, A., 1979. Prospect theory an analysis of decision under risk. Econometrica 47, 263–291. Mount, K., Reiter, S., 1990. A model of computing with human agents. Center for Mathematical Studies in Management Science, discussion paper 890. Northwestern University. Nalebuff, B., 1987. Puzzles: choose a curtain, duel-ity, two point conversions, and more. Journal of Economic Perspectives 1 (1), 157–163. Tierney, J., 1991. Behind Monty Halls doors: puzzle debate and answer? New York Times, July 21, p. 1. Vos Savant, M., 1990. Ask Marilyn. Parade Magazine, September 8, 1990, December 2, 1990, February 17, 1991.