Risky Choice and the Conjunction Error

Risky Choice

If the conjunction fallacy cannot be justified by a reinterpretation of the target events, can it be rationalized by a nonstandard conception of probability? On this hypothesis, representativeness is treated as a legitimate nonextensional interpretation of probability rather than as a fallible heuristic. The conjunction fallacy, then, may be viewed as a misunderstanding regarding the meaning of the word probability. To investigate this hypothesis, we tested the conjunction rule in the following decision problem, which provides an incentive to choose the most probable event, although the word probability is not mentioned.

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequence of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.

1. RGRRR

2. GRGRRR

3. GRRRRR

Note that Sequence 1 can be obtained from Sequence 2 by deleting the first G. By the conjunction rule, therefore, Sequence 1 must be more probable than Sequence 2. Note also that all three sequences are rather unrepresentative of the die because they contain more Rs than Gs. However, Sequence 2 appears to be an improvement over Sequence 1 because it contains a higher proportion of the more likely color. A group of 50 respondents were asked to rank the events by the degree to which they are representative of the die; 88% ranked Sequence 2 highest and Sequence 3 lowest. Thus, Sequence 2 is favored by representativeness, although it is dominated by Sequence 1.

A total of 260 students at UBC and Stanford University were given the choice version of the problem. There were no significant differences between the populations, and their results were pooled. The subjects were run in groups of 30 to 50 in a classroom setting. About one half of the subjects (N = 125) actually played the gamble with real payoffs. The choice was hypothetical for the other subjects. The percentages of subjects who chose the dominated option of Sequence 2 were 65% with real payoffs and 62% in the hypothetical format. Only 2% of the subjects in both groups chose Sequence 3.

To facilitate the discovery of the relation between the two critical sequences, we presented a new group of 59 subjects with a (hypothetical) choice problem in which Sequence 2 was replaced by RGRRRG. This new sequence was preferred over Sequence 1, RGRRR, by 63% of the respondents, although the first five elements of the two sequences were identical. These results suggest that subjects coded each sequence in terms of the proportion of Gs and Rs and ranked the sequences by the discrepancy between the proportions in the two sequences (1/5 and 1/3) and the expected value of 2/3.

It is apparent from these results that conjunction errors are not restricted to misunderstandings of the word probability. Our subjects followed the representativeness heuristic even when the word was not mentioned and even in choices involving substantial payoffs. The results further show that the conjunction fallacy is not restricted to esoteric interpretations of the connective and, because that connective was also absent from the problem. The present test of the conjunction rule was direct, in the sense defined earlier, because the subjects were required to compare two events, one of which included the other. However, informal interviews with some of the respondents suggest that the test was subtle: The relation of inclusion between sequences 1 and 2 was apparently noted by only a few of the subjects. Evidently, people are not attuned to the detection of nesting among events, even when these relations are clearly displayed.

Suppose that the relation of dominance between sequences 1 and 2 is called to the subject’s attention. Do they immediately appreciate its force and treat it as a decisive argument for Sequence 1? The original choice problem (without Sequence 3) was presented to a new group of 88 subjects at Stanford University. These subjects, however, were not asked to select the sequence on which they preferred to bet, but only to indicate which of the following two arguments, if any, they found correct.

Argument 1: The first sequence (RGRRR) is more probable than the second (GRGRRR) because the second sequence is the same as the first with an additional G at the beginning. Hence, every time the second sequence occurs, the first sequence must also occur. Consequently, you can win on the first and lose on the second, but you can never win on the second and lose on the first.

Argument 2: The second sequence (GRGRRR) is more probable than the first (RGRRR) because the proportions of R and G in the second sequence are closer than those of the first sequence to the expected proportions of R and G for a die with four green and two red faces.

Most of the subjects (76%) chose the valid extensional argument over an argument that formulates the intuition of representativeness. Recall that a similar argument in the case of Linda was much less effective in combating the conjunction fallacy. The success of the present manipulation can be attributed to the combination of a chance setup and a gambling task, which promotes extensional reasoning by emphasizing the conditions under which the bets will pay off.