The Dynamics of Anchoring in Bidirectional Associative Memory Networks

The Dynamics of Anchoring in Bidirectional Associative Memory Networks Sudeep Bhatia (sudeepb@andrew.cmu.edu) Department of Social & Decision Sciences, Carnegie Mellon University, 5000 Forbes Ave. Pittsburgh, PA 15232 USA Shereen J. Chaudhry (sjchaudh@andrew.cmu.edu) Department of Social & Decision Sciences, Carnegie Mellon University, 5000 Forbes Ave. Pittsburgh, PA 15232 USA activation (Chapman and Johnson, 1994, 1999; Mussweiler & Strack, 1999). Anchors, according to this view, increase the accessibility of cues supporting the anchor. This evidence subsequently generates final responses that are closer to the anchor than optimal. Is anchoring caused by sequential adjustment or biased activation? Both theories are supported by a large number of empirical findings (discussed in later sections), but neither is able to predict all of these findings by itself. In this paper we provide a simple answer to this question. We show that these processes are not necessarily distinct: sequential adjustment emerges from the dynamics of biased activation. Anchoring, thus, is caused by both these mechanisms simultaneously, and a large range of findings regarding anchoring and its moderators, can be explained within a unitary, parsimonious, theoretical framework. Abstract We formalize the biased activation theory of anchoring using a bidirectional associative memory network. Anchors determine the starting state of this network. As the network settles, we show that the nodes representing numerical responses activate and deactivate consecutively, generating sequential adjustment. By demonstrating that anchoring as adjustment emerges naturally from the dynamics of the biased activation process, we are able to unify the two main theories of the anchoring effect, and subsequently provide a parsimonious explanation for a large range of findings regarding anchoring, and its determinants. Although we focus largely on phenomena related to anchoring, the results of this paper apply equivalently to all judgments under the influence of bidirectional processing, including those involving constraint satisfaction. Keywords: Decision Making, Neural Networks, Dynamic Processes, Anchoring Effect, Constraint Satisfaction Bidirectional Associative Memory Introduction Anchors have a powerful effect on human judgment. Responses to simple questions involving magnitude or time are systematically affected by uninformative numbers, known as anchors, displayed to the decision maker prior to the judgment task. High anchors generate high responses, low anchors generate low responses, and final judgments can be manipulated by selecting the appropriate anchor. The anchoring effect has been shown to emerge in a large number of domains, and is one of the best studied judgment biases in psychology. Yet despite its importance, the cognitive mechanisms responsible for the anchoring effect are still being debated. In their seminal paper on heuristic choice, Tversky and Kahneman (1974) proposed that anchoring is caused by an imperfect sequential adjustment process. At each step in this process, decision makers evaluate the validity of a particular response. The judgment process terminates if the response in consideration is adequate; otherwise it moves on to the next feasible value. Anchors determine the starting point in this process, and adjustment is insufficient. Subsequently responses are closer to the anchor than optimal. This explanation for the anchoring effect has been popular for many decades, and formal models of the anchoring effect have assumed that anchoring operates through sequential adjustment (Johnson & Busemeyer, 2005, but see also Choplin & Tawney, 2010). A more recent approach, however, claims that anchoring is the product of biased Consider a very simple judgment task. The decision maker is asked to select one of N responses based on M cues stored in memory. We assume, for simplicity, that the relationship between the responses and the cues is binary, with each cue either supporting or opposing each response. We can write a response i as r i , and a cue j as c j . If c j supports r i then we can write s ij =+1, and if it opposes r i then we can write s ij =-1. These responses can be numeric, as in typical anchoring tasks, or non-numeric as in more general judgment tasks. For numeric responses, we assume that the N nodes are ordered in a sequence r 1 , r 2 , …, r N , corresponding to the sequence of available responses. For example, when considering the percentage of African countries in the United Nations, with responses in intervals of 1%, r 1 , r 2 , …, r 100 correspond to the responses 1%, 2%, …, 100%. We can implement this structure in a two layer neural network, with the first layer consisting of M nodes representing the M different cues, and the second layer consisting of N nodes representing the N response options. The activation of the node corresponding to c j , at time t, can be written as C j (t), and the activation of the node corresponding to r i , at time t, can be written as R i (t). The connections from the cue layer to the response layer are equal to the strength of support provided by the cues to the responses. As activated response options (such as anchors) also affect the activation of the available cues, these connections can be assumed to be recurrent. Hence the connections from c j to r i and from r i to c j are both simply s ij .