Electronic Supplementary Material 2:
Optimality Analysis for the Linear Response Cost Model
Animal Cognition
Probabilistic Numerical Discrimination in Mice
Dilara Berkay, Bilgehan Çavdaroğlu, Fuat Balcı
Department of Psychology, Koç University, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey
Corresponding Author:
Fuat Balcı, Ph.D.
Tel: +90 (212) 338 1138
Fax: +90 (212) 338 1415
Email: fbalci@ku.edu.tr
Optimality Analysis for the Linear 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 for p(Nfew) = .25
condition dictated the 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) = .75 conditions.
The median optimal switch points were 10.63 and 12.17 for p(Nfew) = .50 and p(Nfew) = .75 conditions, respectively.
The results of the Wilcoxon signed-ranks test revealed that optimal switch points were significantly higher in p(Nfew)
= .75 condition compared to 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) = .50 and p(Nfew) = .75 conditions (Phase 1). The
results of the individual analyses revealed that subjects earned 93.45% and 84.47% (Mdns) of their MPEGs in
p(Nfew) = .50 and p(Nfew) = .75 conditions, respectively. In Phase 2, where the guided trials were eliminated, subjects
in the p(Nfew) = .75 condition earned 91.57% (Mdn) of their MPEGs.
Fig. S1 Expected 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 in p(Nfew) = .50 built
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 Cost Model predicts 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