A Range-Normalization Model of Context-Dependent Choice: A New Model and Evidence

Most utility theories of choice assume that the introduction of an irrelevant option (called the decoy) to a choice set does not change the preference between existing options. On the contrary, a wealth of behavioral data demonstrates the dependence of preference on the decoy and on the context in which the options are presented. Nevertheless, neural mechanisms underlying context-dependent preference are poorly understood. In order to shed light on these mechanisms, we design and perform a novel experiment to measure within-subject decoy effects. We find within-subject decoy effects similar to what have been shown previously with between-subject designs. More importantly, we find that not only are the decoy effects correlated, pointing to similar underlying mechanisms, but also these effects increase with the distance of the decoy from the original options. To explain these observations, we construct a plausible neuronal model that can account for decoy effects based on the trial-by-trial adjustment of neural representations to the set of available options. This adjustment mechanism, which we call range normalization, occurs when the nervous system is required to represent different stimuli distinguishably, while being limited to using bounded neural activity. The proposed model captures our experimental observations and makes new predictions about the influence of the choice set size on the decoy effects, which are in contrast to previous models of context-dependent choice preference. Critically, unlike previous psychological models, the computational resource required by our range-normalization model does not increase exponentially as the set size increases. Our results show that context-dependent choice behavior, which is commonly perceived as an irrational response to the presence of irrelevant options, could be a natural consequence of the biophysical limits of neural representation in the brain.

In addition, each option has only two attributes and the overall value of an option is a weighted sum of its values on these attributes. Note that both the CDA and the rangenormalization model can be generalized easily to the case where each option has more than two attributes and the value function is a monotonic function of attribute values.
However, only the range-normalization model can be generalized for the case where the overall value of an option is the product of its values on different attributes.
Based on the above assumptions, the overall subjective value of option T, V(T), can be written as where T i is the option value on attribute i and weight w i quantifies the contribution of this attribute to the overall value function. Moreover, because the original options T and C have equal overall subjective value we have where T 1 > C 1 , T 2 < C 2 . Now consider the introduction of an asymmetrically dominant option to the choice set. That is, the third option (decoy) has larger values than option T in both attributes, but only one of its attribute values is larger than that of option C Using Eq.1 and Eq.S1, for the case of the linear utility function presented here the relative advantage of T with respect to C is equal to However, R(T,D) is equal to zero because D dominates T in both attributes. Similarly, the relative advantage of option C relative to option T and D is equal to values of the original options after the decoy introduction are equal to are the values of option T before and after the decoy introduction, respectively. The last equation shows that in the presence of the decoy, option C has a larger value than option T, and so it is more preferable relative to option T. Therefore, this model accounts for the preference reversal due to introduction of an asymmetrically dominant decoy (also called attraction effect) by an extra relative advantage of C with respect to this decoy.

Range-normalization model with equal representation factors
In this section, we consider a special case of our range-normalization model in which the representation factors are equal and we show that in such case the outcome normalization resembles the "range-adaptation model" of Padoa-Schioppa [1], but only on a trial-bytrial basis and not on a session-by-session basis as in the latter model. More specifically, we show that for an original option set that consists of two options when the representation factors are equal, the ratio of the difference in response to the original options after the decoy is introduced to before the decoy is introduced is inversely proportional to the ratio of the range of option values after the decoy introduction to before it.
First let us consider the case where both f t and f s are smaller than or equal to zero.
In this case the difference between the responses to two original options is equal to one and the slope of the neural response is equal to where ! s " s max # s min is the range of stimulus values in the original set. If the new option (decoy) falls between the original options, then the difference between the responses to original options remains the same, ! r(T ) ! ! r(C) = 1, as the original options will still be below and above the threshold and saturation points. If the decoy introduces a new maximum to the choice set, then the threshold, c t , stays the same (therefore r(C)=0), while the saturation point, c s , increases. Therefore, the difference between the response to the original options, which is proportional to the slope of the neural response after the adjustment, is equal to (using Eq.9) where ! s max and ! ! s are the new maximum and the new range of stimulus values after the decoy introduction, respectively, and ! c t and ! c s are the threshold and saturation points after the decoy introduction. Similarly, if the decoy introduces a new minimum, c s and r(T) stay the same, c t decreases, and the value of r(C) increases from zero. Therefore, the difference between the responses to the original options is equal to Using the Eq.S2 and Eq.S3 we can see that in both cases when the representation factors are equal, f t = f s , the ratio of the difference between the responses to the original options after to the difference in responses before the decoy introduction is equal to One can perform similar calculation for the case in which the representation factors are equal and positive. We find that if the decoy is a new minimum or maximum then But even if the decoy does not introduce new optima, the ratio of the difference in responses after the decoy introduction to the difference before the decoy introduction is equal to (1+f t ). That is, the presence of the decoy can change the preference between the two original options even if its value is between these options.
These results show that if the representation factors are equal, the ratio of the difference in response to the original options after the decoy is introduced to before the decoy is introduced is inversely proportional to the ratio of the range of option values after the decoy introduction to before it. In any case, the change in valuation due to presentation of a new option depends on the configuration of options in the choice set.

Supplementary Figures
Figures S1-S5 and Figure S7 provide additional analyses of the experimental data. Figure  S6 provides the fitting of the CDA and range-normalization models to the main experimental data.  Figure S1: The choice behavior of subjects during the estimation task.
Plotted is the probability of choosing the high-risk gamble as a function of its reward magnitude for individual subjects (the probability of this gamble was fixed at p=0.3).
Note that the magnitude and probability of the low-risk gamble were fixed at M=$20 and p=0.7, respectively. As the magnitude of the high-risk gamble was increased, it was preferred more often over the low-risk gamble. The solid curve on each panel shows the result of the logistic fit for individual subjects from which two quantities are extracted: the indifference value (the magnitude of a high-risk gamble for which the two gambles are selected equally), and the inverse of sensitivity of the each subject to the reward magnitudes (see Figure S2). The relative RT (average RT for a given decoy location minus the average RT over all trials) is plotted for different decoy locations, and separately for trials on which T or C gambles were selected. The relative RT was significantly different from zero (Wilcoxon signed rank test, p<0.05) for all decoys except D2 (the location for which the decoy effect was not significant). For dominant decoys the RT was positive, which shows that subjects were slower when their most preferred gamble was removed before the selection. For dominated decoys the relative RT was negative, which shows that subjects were faster when their least preferred gamble was removed before selection. We found no difference in relative RT for trials on which T or C gambles were selected. Overall, these results suggest that subject used "ranking" to perform the task.