Mathematically Modeling Anchoring Effects

Mathematically Modeling Anchoring Effects Jessica M. Choplin (jchoplin@depaul.edu) DePaul University Department of Psychology 2219 North Kenmore Avenue Chicago, IL 60614 Mark W. Tawney (mtawney@iosolutions.org) Industrial/Organizational Solutions 1127 S Mannheim Road Westchester, IL 60154 integration model formalization was alluded to in at least one anchoring effect paper (see Jacowitz & Kahneman, 's, 1995, discussion of priming models of anchoring effects). In addition, this mathematical formalization has been used to model assimilation effects in phenomena as diverse as impression formation (the domain that originally inspired Anderson's model, see Urada, Stenstrom, & Miller, 2007, for a recent application), physical attractiveness (e.g., Wedell, Parducci, & Geiselman, 1987), product evaluation (e.g., Miyazaki, Grewal, & Goodstein, 2005; Troutman & Shanteau, 1976), risk assessment (e.g., Hampson, Andrews, Barckley, Lee, & Lichtenstein, 2003), and the best timing for lesbian and gay politicians to come out of the closet (Golebiowska, 2003) 1 . Abstract This article proposes a method by which anchoring effects can be mathematically modeled. Anchoring effects are a type of assimilation effect; so this article proposes using Anderson’s (1965; 1981) integration model to model anchoring effects, as it is typically used to model other assimilation effects. The difficulty in using the integration model is that doing so requires that the modeler knows or is able to estimate participants’ unbiased estimates (i.e., what their estimates would have been had they never seen the anchor) and this information is not available from conventional anchoring effect paradigms. A method for estimating unbiased estimates is proposed. This method is used to estimate unbiased estimates for a set of anchoring effect data and the integration model is fit to these data. This article closes by speculating on possible theoretical insights into anchoring effects that might be gleaned by using the proposed methodology and possible practical applications. The Proposed Mathematical Model Anderson’s (1965; 1981) integration model would model the assimilation observed in anchoring effects as a weighted average of the anchor value (A) and the unbiased estimate a participant would have made had she or he never seen the anchor (U: U for Unbiased; see below for how this quantity can be empirically measured): Anchoring Effects The goal of this paper is to propose a method by which anchoring effects can be mathematically modeled. The ability to mathematically model anchoring effects might be useful both for differentiating among theoretical models of anchoring effects and for calculating likely practical applications of anchoring effects in situations such as negotiations (e.g., Chapman & Bornstein, 1996; Galinsky & Mussweiler, 2001), auctions (Ku, Galinsky, & Murnighan, 2006), and pricing (Northcraft & Neale, 1987). These possible applications of the proposed model will be discussed in the General Discussion section. In anchoring effects, estimates of an unknown value are assimilated towards an arbitrary numeric value called the anchor. For example, in a well-known study, Tversky and Kahneman (1974) asked participants to judge whether African nations represented a higher or lower percentage of UN-member nations than an anchor and then to estimate the actual percentage. Estimates were assimilated towards the anchor. When the anchor was 10% of UN-member nations, the median estimate was assimilated downward toward 10% to equal 25%; but when the anchor was 65%, the median estimate was assimilated upward toward 65% to equal 45%. Assimilation effects like these are typically mathematically modeled using Anderson’s (1965; 1981) integration model. A mathematical formalization like the EST = wA + (1-w)U where EST represents a participant’s estimate (i.e., the dependent measure in anchoring estimation tasks) and w is the weight bound between 0 and 1 of the anchor value (A) relative to the unbiased estimate (U). A weight of 0 would represent a case in which estimates were not affected at all by exposure to the anchor. In such a case, unbiased estimates (U) would be equal to participants’ estimates (EST) so that EST = U. A weight of 1 would represent a case in which all participants simply respond with the anchor value. Weights between these two extremes represent estimates that are assimilated toward the anchor value, but are not equal to it. The key problem in using Anderson’s (1965; 1981) integration model to model anchoring effects is that it requires the modeler to know what participants’ unbiased estimates (U) would have been had they never seen the anchor. Measuring these unbiased estimates is made particularly difficult, because it is not possible to ask participants to make a numerical estimate twice (once before and once after being exposed to the anchor value) as