# Parallel Constraint Satisfaction-Plugging in some of the Numbers This is clearly an example of the blind trying to lead when sight is a real advantage. Glockner displays PCS1 and PCS2 in some figures in his January 2014 paper in the Journal of Judgment and Decision Making.  Since I tend to look at the pictures, this got my interest. Was this some different model or some innovation?  I have provided some narrative explanations of Parallel Constraint Satisfaction in earlier posts, but here I am going to look at the difference between PCS1 and PCS2.  I am doing this by cobbling together explanations from a few of Glockner’s papers.  This is a little dangerous since the experiments are different.

According to PCS, the mental representation of a task is modeled as a network including relevant cues and options as interconnected nodes and their subjective validities and cue patterns as connection weights (see Fig. 1).  PCS thus mimics coherence processes in which the interpretation of a decision situation is simplified by a systematic re-evaluation of information. The option with the highest final activation is predicted to be chosen. The number of iterations to find the solution is used as predictor for decision time, and the absolute difference in activation between the two options is used as predictor for confidence.

Figure 1 below is key to understanding.  Cue values for Option 1 or Option 2 were transformed into weights of −.01 (negative prediction) or .01 (positive prediction).  These option cue values  (w c-o) work at the top of Figure 1 between options and cues. (Figure 1 more or less portrays a Brunswik lens model on end.)  “+” signs as shown in the lower left of Figure 1 became 1/100=.01 and “- ” signs became -1/100=-.01.  The transformations are needed to work in the regression simulations. The theoretical cue validities (w) work on the bottom of Figure 1 between the cues and the general validity box. They are computed from individuals’ expressions of cue-use. First  they
ask subjects to indicate how much they relied on the cues when making their decision on a scale from not used (0) to very much used (100).

The two variants (PCS1 and PCS2) of the PCS model used in the simulations differ only in the transformation function for cue usage which is the k parameter of the transformation function (see Fig. 1, lower right). Parameter k can be interpreted as the individual sensitivity to differences between validities. More precisely, according to PCS individuals do not use qualitatively different heuristics as assumed by the adaptive toolbox approach but differ in how sensitive they are to differences in cue validities. Jekel and Glockner write that for k=0 (k→ ∞) the relative weight between differing validities is 1 (→∞), that is, an individual shows no (infinite) sensitivity towards differences in validities. Frankly, I am confused by this since they note that a sensitivity of 0 predicts choices like the Equal Weighting Strategy while a sensitivity approaching infinity converges with the Take the Best strategy. Maybe someone can help me with this.  For k=1 (i.e., PCS1), transformation is linear, whereas for k=2 (i.e., PCS2), transformation is quadratic and thus reflects accentuation of cues due to higher sensitivity.  According to Glockner, PCS2  leads to choices that closely approximate the naïve Bayes solution. Squaring the weights makes them much smaller since they are less than one.

Note that PCS1 and PCS2 should not be considered different strategies or submodels of PCS. They merely capture individual differences in cue perception.  Glockner et al found more subjects used PCS1 than PCS2.

Glockner & his associates note a couple of issues that have been pointed out with respect to the mechanisms of Parallel Constraint Satisfaction Theory and addresses them.

• According to recent findings, people change their subjective valuation of cues in decisions that might be caused by PCS mechanisms Thus, measuring cue validities after the decisions might be considered less than optimal. Glockner argues that the problem is not too severe in the particular experimental setting because, due to the random presentation of decisions, coherence shifts will induce no systematic errors but only increase random variance, and because coherence shifts have been shown to be transient and might at least partially disappear before the rating.
• To rule out the objection that PCS might benefit from the fact that it uses subjects’ cue-usage estimations instead of their cue-validity estimations — as it is the case for the heuristics TTB and RH — Glockner et al recalculated choice predictions of PCS1 and PCS2 based on cue-validity estimations. They used the transformation functions w = ((v−50)/100) and w = ((v−50)/100)² to scale the values between 50 and 100 down to acceptable weights. This is what is shown in the lower right of Figure 1. (The reason for subtracting 50 is that 50 would represent chance. The adherence rates for both strategies remained essentially stable but decreased slightly. Hence, the researchers conclude that the additional analysis demonstrates the robustness of the results.

According to Glockner, the slight drop in predictive performance, however, also indicates that measures of cue-usage should be preferred over measures of cue-validity as input for PCS models, as he says would be expected on theoretical grounds. In the terminology of the classic Brunswik lens model, PCS models the personal weighting of these cues (i.e., the subjective part / the right side of the lens) instead of the objective relation between distal criterion and cues (i.e., the left side of the lens). Both might on the long run converge in Hogarth’s so called “kind environments” with appropriate feedback.

Jekel, M., Glockner, A., Fiedler, S., Broder, A (2012). “The rationality of different kinds of intuitive decision processes.” Synthese  189:147–160.

Glockner, A. and Broder, A.(2014). “Cognitive integration of recognition information and additional cues in memory-based decisions.” Judgment and Decision Making, Vol. 9, No. 1, January 2014, pp. 35–50.

Glockner, A. and Broder, A. (2011). “Processing of recognition information and additional cues: A model-based analysis of choice, confidence, and response time.”  Judgment and Decision Making, Vol. 6, No. 1,  pp. 23–42.