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Contrasting Stochastic and Support Theory Interpretations of Subadditivity

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Contrasting Stochastic and Support Theory Interpretations of Subadditivity Error and Subadditivity 1 A Stochastic Model of Subadditivity J. Neil Bearden University of Arizona Thomas S. Wallsten University of Maryland Craig R. Fox University of California-Los Angeles RUNNING HEAD: SUBADDITIVITY AND ERROR Please address all correspondence to: J. Neil Bearden University of Arizona Department of Management and Policy 405 McClelland Hall Tucson, AZ 85721 Phone: 520-603-2092 Fax: 520-325-4171 Email address: [email protected] Error and Subadditivity 2 Abstract This paper demonstrates both formally and empirically that stochastic variability is sufficient for, and at the very least contributes to, subadditivity of probability judgments. First, making rather weak assumptions, we prove that stochastic variability in mapping covert probability judgments to overt responses is sufficient to produce subadditive judgments. Three experiments follow in which participants provided repeated probability estimates. The results demonstrate empirically the contribution of random error to subadditivity. The theorems and the experiments focus on within-respondent variability, but most studies use between-respondent designs. We use numerical simulations to extend the work to contrast within- and between-respondent measures of subadditivity. Methodological implications of all the results are discussed, emphasizing the importance of taking stochastic variability into account when estimating the role of other factors (such as the availability bias) in producing subadditive judgments. Error and Subadditivity 3 A Stochastic Model of Subadditivity People are often called on the map their subjective degree of belief to a number on the [0,1] probability interval. Researchers established early on that probability judgments depart systematically from normative principles of Bayesian updating (e.g., Phillips & Edwards, 1966). More recently investigators have observed systematic violations of the additivity principle, one of the most basic axioms of probability theory (Kolmogorov, 1933). Consider disjoint events A, B and their union C=A∪B, and let P(.) be a normative probability measure. Additivity requires that P(A) + P(B) = P(C). For instance, if the probability that the home team wins by at least ten points, P(A), is .30 and the probability that the visiting team wins by at least ten points, P(B), is .10, then probability that the margin of victory is at least ten points, P(C), must be .30 + .10 = .40. In contrast to this normative requirement, numerous studies have shown that judged probability, p(.), usually exhibits subadditive so that p(C ) < p(A) + p(B). Evidence of this pattern is reviewed by Tversky and Koehler (1994) and Fox and Tversky (1998). It has been observed among sports fans (e.g., Fox, 1999; Koehler, 1996), professional options traders (Fox, Rogers & Tversky, 1996), doctors (Redelmeier, Liberman, Koehler & Tversky, 1995), and lawyers (Fox & Birke, 2002). Thus far, researchers typically attribute biases in judged probability to heuristic processes based on memory retrieval, similarity judgment, or other cognitive processes (Bearden & Wallsten, in press; Kahneman, Slovic & Tversky, 1982; Gilovich, Griffin & Kahneman, 2002). In contrast, a growing body of research has shown that some commonly observed patterns of bias can occur when otherwise accurate probability judgments are perturbed by random error (e.g., Erev, Wallsten, & Budescu, 1994; Juslin, Olsson, & Björkman, 1997; Soll, 1996). It is important, therefore, to rule out, or at least account for, response patterns due to stochastic Error and Subadditivity 4 variability before positing complex cognitive mechanisms. The aim of this paper is to contrast explanations of subadditive judgments based on random error with those based on the principles of support theory (Tversky & Koehler, 1994; Rottenstreich & Tversky, 1997). It is difficult to distinguish cognitive accounts of subadditivity from stochastic accounts of this phenomenon because past studies have asked participants to make a single probability judgment of events A, B and C = A∪B, in either between- or within-participant experimental designs. Hence, past studies have not allowed researchers to observe the distribution of error surrounding a particular participant’s judgments of a particular event and diagnose the extent to which such error contributes to subadditivity. In the present studies we aim to fill this gap in the literature by providing opportunities for participants to learn the relative frequencies of events in a controlled learning environment and then eliciting multiple judgments of each event. The remainder of this paper is organized as follows. First, we summarize the major cognitive explanations of subadditivity. Next, we show how random error also might explain subadditivity, critically discuss a published test of a particular stochastic model, and develop a very general one. We then present three experiments that examine the extent to which random error can account for subadditivity. (Two of the experiments are described in detail; and one, a replication of Experiment 1, is only summarized.) The paper concludes with a general discussion relating stochastic and support theory explanations of subadditivity. Cognitive interpretations of subadditivity. Traditional accounts of subadditivity associate this phenomenon with the availability heuristic (Tversky & Kahneman, 1973). For instance, Fischhoff, Slovic, and Lichtenstein (1978) asked automobile mechanics to judge the relative frequency of different causes for a car failing to start. Mechanics estimated that a car would fail to start about 22 times in 100 due to causes other than the “battery, fuel system, or engine.” Error and Subadditivity 5 However, when this residual category was broken down into distinct components consisting of “starting system,” “ignition system,” “mischief,” and a new (less inclusive) residual category, a second group of mechanics reported judgments that summed to 44 times in 100. These authors argued that unpacking the catch-all category into specific instances enhanced the accessibility of particular causes and therefore their apparent likelihood (see also Russo & Kolzow, 1994; Ofir, 2000). Likewise, in support theory (Tversky & Koehler, 1994; Rottenstreich & Tversky, 1997) subadditivity is attributed to availability: unpacking a description of an event into more specific constituents may remind people of possibilities that they would have overlooked or enhance their salience. Although availability could contribute to subadditivity in situations where a category of events is partitioned into constituents, it is unlikely to provide a satisfactory account of all instances of subadditivity. First, a number of studies have shown that when descriptions of events (e.g., “precipitation next April 1”) are unpacked into a disjunction of constituents (e.g., “rain or sleet or snow or hail”), judged probabilities sometimes increases, but not as dramatically as the sum of the probabilities of these constituents when they are judged separately (Rottenstreich & Tversky, 1997; Fox & See, 2003). In fact, unpacking descriptions into a disjunction of constituents and a catch-all sometimes leads to a reduction in judged probability. For instance, Sloman, Rottenstreich, Hedjichristidis, Stibel & Fox (2004) found that the median judged probability that a randomly selected death is due to “disease” was .55, the median judged probability that it is due to “heart disease, cancer, stroke, or any other disease” rose to only .60, and the median judged probability that it is due to “pneumonia, diabetes, cirrhosis, or any other disease” actually decreased to .40. Second, subadditivity is typically observed when a dimensional space (e.g., a temperature or a stock market price range) is partitioned into Error and Subadditivity 6 constituents, and it seems implausible that the availability mechanism contributes here. For example, Fox, Rogers, & Tversky (1996) found that most professional options traders in their sample judged the probability of the event “Microsoft Stock (MSFT) will close below $94 per share two weeks from today” to be lower than the sum of his or her estimated probabilities of the events “MSFT will close below $88 per share” and “MSFT will close at least $88 per share but below $94 per share.” Such observations have motivated a second cognitive interpretation of subadditivity: bias toward an “ignorance prior” probability. When respondents assign probabilities simultaneously to each of n events into which a sample space is partitioned, their responses are typically biased toward 1/n for each event (Fox & Clemen, 2004; cf. Van Schie & Van der Pligt, 1994). More generally, when people are asked to judge the probability of single events, their responses are biased toward ½, placing equal credence on the event and its complement, unless a partition of the event space into n > 2 interchangeable events is especially salient (see, Fox & Rottenstreich, 2003; see also Fischhoff & Bruin de Bruine, 1999). Thus, if probabilities of events A, B and A∪B are all biased toward 1/2 then these probabilities will be subadditive. Stochastic interpretations of subadditivity. Despite the appeal of availability and ignorance priors as explanations of subadditivity, it is necessary to consider the extent to which the phenomenon may be an artifact of random perturbations in judgment. Tversky and Koehler (1994) pointed out that subadditivity follows as a consequence of a stochastic model such as Erev, Wallsten, and Budescu’s (1994), which hypothesizes
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