Electronic Supplementary Material 2:

Optimality Analysis for the Linear Response Cost Model

Animal Cognition

Probabilistic Numerical Discrimination in Mice

Dilara Berkay, BilgehanÇavdaroğlu, FuatBalcı

Department of Psychology, Koç University, RumelifeneriYolu, 34450 Sarıyer, Istanbul, Turkey

Corresponding Author:

Fuat Balcı, Ph.D.

Tel: +90 (212) 338 1138

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Optimality Analysis for theLinear Response Cost Model

We conducted the optimality analyses under an alternative model, Linear Response Cost Model, which assumes a linear mapping between the number of responses and energetic costs associated with them (for details of the model see Data Analysis section in the main text). Based on each individual’s CV and the probability of different trial types in a given condition, the expected gain was calculated for hypothetical target switch points ranging between 10 and 20 (i.e. few and many response requirements). The hypothetical target switch point that maximized the expected gain (i.e. optimal switch point) was found for each subject. Under this model, the expected gain function forp(Nfew)=.25conditiondictatedthe optimal switch point to converge to the lower limit for all subject. As it would not be possible to make a reliable evaluation regarding the optimality of subjects in this probability condition, we carried out further analyses only for p(Nfew)= .50 and p(Nfew)= .75conditions.

The median optimal switch points were 10.63and 12.17 for p(Nfew)=.50and p(Nfew)=.75conditions, respectively. The results of the Wilcoxon signed-ranks test revealed that optimal switch points were significantly higher in p(Nfew)=.75 condition comparedto those observed in p(Nfew)=.50 condition, Z = 3.06,P = .002. As this result indicates, optimal strategy dictated significantly shifting the switch points between these two probability conditions.

Next, we calculated the maximum possible expected gain (MPEG) that a given subject could attain in each probability condition and what proportion of this MPEG the subject could actually earn. Figure S1 presents the expected gain curves calculated for average CV values for p(Nfew)=.50and p(Nfew)=.75 conditions (Phase 1).The results of the individual analyses revealed that subjects earned 93.45% and 84.47% (Mdns) of their MPEGsinp(Nfew)=.50and p(Nfew)=.75conditions, respectively.In Phase 2, where the guided trials were eliminated, subjects in the p(Nfew)=.75condition earned 91.57% (Mdn) of their MPEGs.

Fig.S1Expected gain curves for the Linear Response Cost Model calculated using the average CV values obtained from p(Nfew)=.50(a) and p(Nfew)=.75(b) conditions. Filled circles denote the optimal target switch points and dashed lines represent mean empirical switch points for the corresponding condition. Values presented in this figure were gathered from Phase 1

In addition to the Logarithmic Response Cost Model, we also built the OPCs under the Linear Response Cost Model to provide an alternative explanation to the findings of Fetterman, Dreyfus, and Stubbs (1985) which indicated that PSEs were closer to the geometric mean of the references for time-based judgments and were mostly below the harmonic mean of the references for the count-based judgments. Figure S2 presents the OPCs inp(Nfew)=.50built under the Linear Response Cost Model along with the Logarithmic and the No Response Cost Models.As Figure S2 shows, similar to the Logarithmic Response Cost Model, the Linear Response Cost Model (corresponding to the counting tasks) predicts PSEs that are closer to the harmonic mean compared to the model that assumes no response costs (corresponding to the timing tasks). The No Response CostModelpredicts PSEs to be closer to the geometric mean of the references for a CV level of .25 (which corresponds to CV estimates gathered from Weber’s Fractions reported in Fetterman et al. 1985 and from the current study). Thus, these models can explain different localizations of PSEs between timing and counting tasks by assuming the latter to involve response costs. These results are overall consistent with the results attained with the Logarithmic Response Cost Model.

Fig. S2 The optimal performance curves as a function of CV prescribed by three different models of optimality (see Legend) in three different task conditions: 1:2 (a), 1:3 (b), and 1:4 (c) ratio between the larger and the smaller response number requirements. The dashed vertical lines (from left to right) depict the harmonic, geometric, and arithmetic means of the reference values