Developmental Trends in Decision Making: the Case of the Monty Hall Dilemma

57

Developmental Trends in Decision Making

Chapter 15

Developmental Trends in Decision Making: The Case of the Monty Hall Dilemma

Wim De Neys[1]

University of Leuven, Tiensestraat 102

3000 Leuven, Belgium

Abstract

The Monty Hall Dilemma (MHD) is a notorious brain-teaser where people have to decide whether switching to another option in a game is advantageous. Most adults erroneously believe that chances of winning remain equal or that they should stick to their original choice. The present study tested the impact of cognitive development on MHD reasoning to examine possible differences in the nature of the erroneous intuitions. Twelve to seventeen year old high school students were presented the MHD and selected one of three responses (switch, stick, or chances equal). Results showed that whereas maturation decreased adherence to the erroneous “stick with your first pick” belief, the “chances are equal” belief became more dominant with increasing age. Consistent with predictions, children who selected the latter response also scored better on a syllogistic reasoning task. Results further showed that twelve year old eighth graders selected the correct switching response more frequently than senior high school students. Implications for popular reasoning and decision making theories are discussed.

Developmental Trends in Monty Hall Dilemma Reasoning

Imagine you’re the final guest in a TV quiz. Monty, the show host, is asking you to choose one of three doors. One of the doors conceals a BMW sports car but the two other doors only contain a bunch of toilet paper. If you choose the right door you will be the proud owner of the fancy BMW. However, after you finally select one of the doors, host Monty does not open it immediately. First, he opens one of the doors you did not choose to reveal it contained toilet paper. Monty now actually offers you the possibility to chance your mind and pick the other unopened door. What should you do to have most chance of winning the car of your dreams? Stay with your first choice, switch to the other unopened door, or doesn’t it matter whether you switch or not?

The above switching problem is known as the “Monty Hall Dilemma” (MHD) after the host of the American TV show “Let’s make a deal” where it was introduced. Contrary to most people’s intuition the correct answer is that switching to the other door will actually increase the chance of winning. The counter-intuitive solution hinges on the crucial fact that Monty will never open the door concealing the prize, and obviously, he will not open the door the guest initially picked either. The probability that you initially select the correct door is one out of three. In this case it would be better not to switch. However, in the other two thirds of the cases the non-chosen closed door will hide the prize and switching is advantageous. Hence, switching yields a 2/3 chance of winning.

If you failed to solve the problem correctly you may find some comfort in the fact that you’re in good company. Empirical studies show that typically less than 10% of educated adults give the correct switching response (e.g., Burns & Wieth, 2004; Friedman, 1998; Granberg & Brown, 1995; Krauss & Wang, 2003; Tubau & Alonso, 2003) and even ace mathematicians do not seem to be immune to MHD errors (e.g., Burns & Wieth; vos Savant, 1997). Most people have the strong intuition that whether they switch or not the probability of winning remains 50% either way. Research indicates that this powerful intuition is based on the so called number-of-cases heuristic (“if the number of alternatives is N, then the probability of each one is 1/N”, see Shimojo & Ichikawa, 1989, and Falk, 1992). Since only two doors remain people will automatically assign a 50% chance to each door and fail to take the “knowledgeable host” information into account. The “equal chance” heuristic is so self-evident that it will literally dominate our thinking.

Over the last decades numerous studies have demonstrated that similar intuitive responses are biasing people’s performance in a wide range of reasoning tasks (e.g., Evans & Over, 1996; Kahneman, Slovic, & Tversky, 1982). Influential dual process theories of reasoning and decision making have explained this “rational thinking failure” by positing two different human reasoning systems (e.g., Epstein, 1994; Evans, 2003; Evans & Over, 1996; Sloman, 1996; Stanovich & West, 2000). Dual process theories come in many flavours but generally they assume that a first system (often called the heuristic system) will tend to solve a problem by relying on intuitions and prior beliefs whereas a second system (often called the analytic system) allows reasoning according to normative standards. The heuristic default system is assumed to operate fast and automatically whereas the operations of the analytic system would be slow and heavily demanding of people’s computational resources. Although the fast and undemanding heuristics can provide us with useful responses in many daily situations they can bias reasoning in tasks that require more elaborate, analytic processing (e.g., Sloman, 1996; Stanovich & West, 2000; Tversky & Kahneman, 1983). That is, both systems will sometimes conflict and cue different responses. In these cases the analytic system will need to override the automatically generated intuitive response. Since the inhibition of the heuristic intuitions and the computations of the analytic system draw heavily on people’s limited cognitive resources, most people will be tempted to stick to mere intuitive reasoning. Therefore, correct analytic reasoning would be characteristic of those highest in cognitive capacity (Stanovich & West, 2000).

De Neys and Verschueren (2006) recently examined the relation between people’s cognitive capacities and Monty Hall reasoning. Over 200 participants were presented the MHD and a test to measure their working memory capacity. Consistent with Stanovich and West’s (2000) dual process predictions participants who did manage to solve the MHD correctly were specifically those highest in working memory span. Experimentally burdening the cognitive resources with a secondary task also decreased the rate of correct switching responses. These findings supported the basic claim that the analytic override of heuristic thinking draws on people’s cognitive working memory resources.

Although the dual process framework has been quite influential in the reasoning and decision making community it has also been criticized severely (e.g., Gigerenzer & Regier, 1996; see commentaries on Evans & Over, 1997, or Stanovich & West, 2000, for an overview). One of the critiques focuses on the the framework’s postulation of a single heuristic system that unitarily handles all intuitive processing. It has been argued that different kinds of heuristics, with a different processing nature, need to be differentiated (e.g., De Neys, 2006a; Gigerenzer & Regier, 1996; Moshman, 2000; Newton & Roberts, 2003). De Neys and Verschueren’s (2006) study pointed to an MHD trend that is specifically interesting in this respect. Note that one can distinguish different erroneous MHD responses. Whereas the vast majority of reasoners is biased by the number-of-cases heuristic and beliefs that switching or sticking does not matter, a small group of reasoners is convinced that they should stick to the initially selected door to win the prize. These people are biased by the general belief that when making a decision one should always stick to one’s first choice. Such a bias has long been noted in responses to multiple-choice exams (e.g., Geiger, 1997). Gilovich, Medvec, and Chen (1995) clarified that this “stick with your pick” intuition would be based on an anticipation of regret.

De Neys and Verschueren (2006) found that reasoners in the group of erroneous responders who selected the “equal chances” response tended to have a slightly higher working memory span than those who believed that “sticking” was advantageous. Moreover, the secondary task load specifically boosted the rate of “sticking” responses whereas the selection of “chances equal” responses was hardly affected. Although the trends were not significant, De Neys and Verschueren noted that the two types of heuristic beliefs might indeed have a different processing nature. The number-of-cases heuristic would be based on a cognitive probability estimation (albeit a simple one) whereas the “stick with your pick” heuristic would have a more elementary, affective basis. Hence, the “chances are even” heuristic would be computationally more complex than the “stick with your pick” heuristic. Therefore, the more elementary, least demanding “stick with your pick” response would be the preferred answer under conditions of cognitive load.

De Neys and Verschueren (2006) suggested that in their sample of university students the trends presumably failed to reach significance because the basic number-of-cases computation would be completely automatic for educated adults. Consequently, the distinction with the alleged less demanding “sticking” responses would be blurred. The present study adopts a developmental approach to clarify the issue. The MHD was presented to twelve to seventeen year old adolescents. A first prediction concerned the selection rates of the different MHD responses in the different age groups. Younger reasoners have smaller cognitive resource pools and still lack an important part of the mathematical training that familiarized university students with fractions and probabilities. Therefore, one can expect that computation of the number-of-cases heuristic will be less automated and more demanding in the younger age groups. This should result in a less frequent selection of “equal chances” responses by younger reasoners. Under the assumption that the “stick with your pick” heuristic is more basic one predicts that younger reasoners will show a stronger preference for the “stick” responses. The alleged differential demands of the “stick” and “equal” heuristics should thus result in a specific developmental trend: The rate of “chances equal” responses should increase with age whereas the “sticking” responses should decrease. Given the high computational demands of the correct switching response for adult university students, correct MHD reasoning in the younger age groups was not expected.

A second prediction concerns the overall relation between young adolescents’ responses and their cognitive capacity. After the students had solved the MHD they were also presented a specific syllogistic reasoning task where they had to inhibit automated responses. Previous studies established that this task is a good marker of children’s general cognitive ability (e.g., Kokis, Macpherson, Toplak, West, & Stanovich, 2004). De Neys and Verschueren (2006) argued that the completely automated computation of the “equal chances” response in their sample of university students blurred the distinction with the alleged undemanding “sticking” responses. The less automated and more demanding nature of the number-of-cases heuristic for the younger reasoners (vs. adults) should show a clearer distinction. Therefore, it is predicted that youngsters who manage to give the “chances are equal” response will score better on the syllogisms than those who believe that mere sticking is the best strategy.

The developmental MHD trends will help clarifying possible differences in the nature of different heuristics. In addition, examining children’s MHD performance will also have interesting implications for a fundamental controversy concerning the development of reasoning and decision making itself. The dual process framework and many developmental theories (e.g., Case, 1985; Inhelder & Piaget, 1985) share the assumption that children’s reasoning becomes less heuristic and more logical across the lifespan. Hence, traditionally it has been assumed that analytic thinking simply replaces heuristic thinking with cognitive maturation. Recent developmental reasoning studies have argued against this so-called “illusion of replacement” (e.g., Brainerd & Reyna, 2001; Reyna & Ellis, 1994). Klaczynski (2001), for example, showed that whereas in some tasks (e.g., reasoning about sunk costs) the heuristic appeal indeed decreased with age, other tasks showed the opposite pattern and indicated that heuristic reasoning remained constant or increased with age (e.g., denominator neglect in statistical reasoning). Such findings are already hard to reconcile with the traditional view. The present MHD study allows a direct validation of the replacement claim within one and the same task. Based on the traditional view one would simply expect that both types of erroneous MHD responses will decrease with age and will be replaced by a higer number of correct responses. Evidence for the predicted differential developmental “equal chances” and “sticking” trends will cut the ground under the replacement view.

Method

Participants

A total of 132 high school students in grades 8 to 12 participated in the study. Forty-two students attended eighth grade (mean age = 12.9, SD = .91), 25 attended ninth grade (mean age = 14.64, SD = .64), 20 students were in grade 10 (mean age = 15.75, SD = 1.02), 16 were in grade 11 (mean age = 16.56, SD = .63), and 29 in grade 12 (mean age = 17.66, SD = .69). Students in grades 9/10 and grades 11/12 were collapsed in two age groups for the analyses. All participants were recruited from the same suburban school with socially mixed catchments areas. All spoke Dutch as their first language and had no known behavioral problems or learning difficulties.

Material

Monty Hall Dilemma. Students were presented a version of the MHD based on Krauss and Wang (2003, see De Neys & Verschueren, 2006). The formulation tried to avoid possible ambiguities (e.g., the random placement of the prize and duds behind the doors and the knowledge of the host were explicitly mentioned). As in Tubau and Alonso (2003), participants could choose between three answer alternatives (a. Stick – b. Switch – c. Chances are even). The complete problem format is presented bellow:

Suppose you’re on a game show and you’re given the choice of three doors. Behind one door is the main prize (a car) and behind the other two doors there are dud prizes (a bunch of toilet paper). The car and the dud prizes are placed randomly behind the doors before the show. The rules of the game are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, then opens one of the two remaining doors which always reveals a dud. After he has opened one of the doors with a dud, Monty Hall asks the participants whether they want to stay with their first choice or to switch to the last remaining door. Suppose that you chose door 1 and the host opens door 3, which has a dud.

The host now asks you whether you want to switch to door 2. What should you do to have most chance of winning the main prize?

a.  Stick with your first choice, door 1.