Online Resource 1
To bet or not to bet? Decision-making under risk in non-human primates
Journal of Risk and Uncertainty
M. Pelé1,2, M.H. Broihanne3, B. Thierry1,2, J. Call4 & V. Dufour1,2
3
Laboratoire de Recherche en Gestion et Économie, EM Strasbourg Business School, Université de
Strasbourg, Strasbourg, France
4
Max Planck Institute for Evolutionary Anthropology, Leipzig, Germany
Corresponding author:
Valérie Dufour
1
Centre National de la Recherche Scientifique, Département Ecologie, Physiologie et Ethologie,
Strasbourg, France
2
Université de Strasbourg, Institut Pluridisciplinaire Hubert Curien, Strasbourg, France
Email: valerie.dufour@iphc.cnrs.fr
Supplementary information, figures and tables are listed in order of appearance from the manuscript.
1. Supplementary methods
1.1. Subjects
Capuchins and orang-utans were socially housed in enclosures with access to indoor and outdoor
areas. Long-tailed macaques were housed in individual cages allowing visual and physical contact
with others. Water was available ad libitum and subjects were not deprived of food. Species, age and
sex of subjects is described in Table S1.
Table S1. Species, age and sex of subjects.
Species
Subjects
Age (yrs)
Sex
Acc
10
male
Ali
8
female
Arn
8
male
Pao
6
female
Pet
6
female
Pis
5
male
Pop
6
male
Rav
5
male
Cas
12
male
Don
16
male
Jac
16
male
Joe
11
male
Ram
16
male
Dok
19
female
Kil
8
female
Pad
11
female
Pin
20
female
Capuchins
Macaques
Orang-utans
1.2. Supplementary procedure:
Fig. S1 presents the experimental design for each test. The chances of losing or gaining were
manipulated via different combinations of rewards in the cups, giving the subjects a means to assess
the chances of losing or gaining compared to the initial item already in hand. If the subject exchanged
it received the contents of one of the 6 cups (randomly assigned prior to the experiment).
Fig. S1 A tray of six plastic cups corresponding to the combination of rewards # 7. Two cups containing one
piece of cookie 4 x 4 x 0.5 cm in size (left position), two cups containing one piece of cookie 2 x 2 x 0.5 cm
(middle position) and two cups containing one piece of cookie 1 x 1 x 0.5 cm (right position).
2. Supplementary information for theories of decision-making
2.1. Kahneman and Tversky probability weighting function
Fig. S2 gives an example of Tversky and Kahneman’s weighting function where = 0.6
overweights low probabilities (lower than 0.35) and underweights high ones.
Weighting function
KT probability weighting function (distortion parameter 0.6)
45° line
Prob.
0
0.04
0.08
0.12
0.16
0.2
0.24
0.28
0.32
0.36
0.4
0.44
0.48
0.52
0.56
0.6
0.64
0.68
0.72
0.76
0.8
0.84
0.88
0.92
0.96
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Fig. S2 Kahneman and Tversky probability weighting function for Here, = 0.6 overweights low
probabilities (lower than 0.35) and underweights high ones.
2.2. CPT computation example
As an example, under CPT with a power utility function, combination # 4 (containing 2 small, 1
2
1
medium and 3 large rewards) is evaluated 𝑉(# 4) = (𝜋 − (6) × 𝑣(−1.5)) + (𝜋 − (6) × 𝑣(0)) +
3
2
2
1
2
2
1
3
(𝜋 + (6) × 𝑣(6)) = (𝑤(6) × (−2.25 × (1.5)𝛽 )) + ((𝑤 (6 + 6) − 𝑤 (6)) × 0) + ((𝑤 (6 + 6 + 6) −
2
1
𝑤 (6 + 6)) × (6)𝛼 ).
Using
Kahneman
and
Tversky
(1992)’s
weighting
function
and
we obtain 𝑉(# 4) = 0.3334 × (−3.375) + 0.0821 × 0 + 0.4156 × 6 = 1.3683.
2.3 RDEUT computation example
As an example, under RDEUT, combination # 4 (containing 2 small, 1 medium and 3 large rewards) is
2
2
6
1
3
2
𝑉(# 4) = (𝜋(6) × (0.5)𝛿 ) + (𝜋(6) × (2)𝛿 ) + (𝜋(6) × (8)𝛿 ) = (𝑤(6) × (0.5)𝛿 ) +
evaluated
1
6
2
6
2
6
1
6
3
6
2
6
1
6
((𝑤 ( + ) − 𝑤 ( )) × (2)𝛿 ) + ((𝑤 ( + + ) − 𝑤 ( + )) × (8)𝛿 ).
Using Kahneman and Tversky (1992)’s weighting function and = 0.8, we obtain 𝑉(# 4) =
0.3334 × (0.5)𝛿 + 0.0821 × (2)𝛿 + 0.5845 × (8)𝛿 = 3.4194.
3. Supplementary results
3.1. Individual return rates
Fig. S3 presents the average exchange rate of the initial item for each individual (measured on the
left vertical axis over 48 in macaques and capuchins or 24 trials in orang-utans). Fig. S3 also gives a
histogram of the cumulative quantitative outcomes of individuals (measured on the right vertical axis)
for each combination of rewards. Quantitative outcomes are measured in units of small cookies (1 x 1
x 0.5 cm) in relation to the initial medium-sized item (which is equivalent to 4 small ones). For
example, in combination # 4, subjects can receive one of the three different sizes of cookies if they
decide to exchange. If the subject obtains a medium cookie, the outcome is 0 ((-4+4)/4), receiving a
large cookie produces a quantitative outcome of 3 ((-4+16)/4) and a small one results in an outcome of
-0.75 ((-4+1)/4). The histograms shown in Fig. S3 reveal that almost all subjects experienced some
negative cumulative outcomes during trials. The cumulative quantitative outcomes of individuals
decrease as subjects experience losses in unfavorable combinations. Note that five subjects attained a
highly negative cumulative outcome (capuchins, Acc: -11.25, Pis: -12; macaques, Jac: -22.5, Ram: 9.75; orangutans, Pad: -4.5 units of cookie of 1 x 1 x 0.5 cm each; red arrows on the Fig. S3).
Fig. S3 Individual profiles. For each combination of rewards, black lines represent the percentage of initial items
exchanged (% of return), and grey bars the cumulative quantitative outcomes of the subject.
3.2. First and second order stochastic dominance of subjects’ choices
Fig. S4 presents the relative cumulative distribution of combination # 1 (F# 1)), versus the certain
outcome (G(.)). F#
1
is the cumulative distribution function of combination # 1 where the outcome
“medium piece of cookie” (value of 2 on the horizontal axis) is obtained with probability 2/6 and the
outcome “large piece of cookie” (value of 8 on the horizontal axis) with probability 4/6. G(.) is the
cumulative distribution function of the certain outcome of a “medium piece of cookie” (value of 2 on
the horizontal axis). As 𝐹# 1 (𝑥) ≤ 𝐺(𝑥) at all x, F#
1
first-order stochastically dominates G(.) so
subjects should always prefer # 1 to the certain outcome. Fig. S5 indicates the relative distribution for
each lottery. It shows that, in accordance with FOSD, subjects should always exchange in
combinations # 3, # 5 and # 10 and they should not exchange (and prefer the certain outcome) in # 16.
FOSD does not give any decision rule for # 2, # 4, # 6 to # 9 and # 11 to # 15.
Fig. S4: relative cumulative distribution of combination # 1 (F# 1). The dashed area represents the cumulative
distribution function of the combination and the plain grey area is the cumulative distribution function of the
certain outcome of a “medium piece of cookie”.
Fig. S5 Relative cumulative distribution for all lotteries. The dashed area represents the cumulative distribution
function of the combination and the plain grey area is the cumulative distribution function of the certain outcome
of a “medium piece of cookie”.
3.3. Results for individual choices in accordance with FOSD and SOSD
Table S2 presents the number of subjects whose choices are in accordance with FOSD (Panel A)
and SOSD (Panel B and C). For example, Panel A shows that 3 out of 8 capuchins chose “G” in # 1 in
all trials (48 trials). Note that no subject chose the certain outcome in # 1, # 3, # 5 and # 10, nor did
they choose the gamble in # 16. Results in Panel A indicate that choices made by subjects are mainly
consistent with FOSD in 42.35% of all trials. Macaques exhibit a significantly higher average
percentage of FOSD consistent choices (64% of all trials) than capuchins and orang-utans,1 whose
average percentage of FOSD consistent choices do not significantly differ (32.5% and 35% of all trials
choices are consistent with FOSD, Z-stat = 0.66, 10 df). Panel B presents the results for SOSD in # 14:
macaques never choose the certain outcome whereas capuchins and orang-utans sometimes gamble
but exhibit a higher stability in their preference for the certain outcome. In Panel C, choices in # 9 and
# 10 confirm that choices made by macaques are consistent with a preference for gambling, while
capuchins and orang-utans did not show this preference.
1
Z-stats are respectively 8.19 and 7.17, indicating that differences are significant at the higher level with
respectively 11 and 7 degrees of freedom.
Table S2: individual choices according to FOSD (Panel A) and SOSD (Panel B & C) in every trial(a).
Panel A: Number of subjects which choices are in accordance with FOSD in every trials
Combinations
Choice
Capuchins
Macaques
Orangutans
All
G*
3 out of 8
5 out of 5
1 out of 4
9 out of 17
#1
CO
#3
G*
CO
#5
# 10
G*
1 out of 8
4 out of 8
3 out of 5
4 out of 5
-
1 out of 4
5 out of 17
1 out of 4
9 out of 17
CO
-
G*
-
CO
-
-
-
-
G
-
-
-
-
2 out of 5
1 out of 4
3 out of 17
# 16
CO*
Average percentage of FOSD
consistent choices
G in # 1, # 3, # 5,
# 10
& CO in # 16
5 out of 8
32.5%
2 out of 5
64%
3 out of 4
10 out of
17
35%
42.35
%
Panel B: Number of subjects with identical choices in every trial (test of SOSD in # 14)
Combinations
Choice
Capuchins
Macaques
Orangutans
All
# 14
G
1 out of 8
2 out of 5
1 out of 4
4 out of 17
# 14
CO
3 out of 8
1 out of 4
4 out of 17
-
Panel C: Number of subjects with identical choices in every trial (test of SOSD in # 9 and # 10)
(a)
Combinations
Choice
Capuchins
#9
G
-
#9
CO
-
# 10
G
-
# 10
CO
-
Macaques
Orangutans
All
4 out of 5
1 out of 4
5 out of 17
2 out of 5
-
1 out of 4
3 out of 17
-
Column 1 indicates combinations, choices are presented in column 2, a star indicates which choice is
consistent with FOSD. Columns 3 to 6 give the number of subjects choosing G (gamble) or CO (certain
outcome) in every 48 trials.
-
4. Supplementary results for theoretical models
4.1. Estimates of individual parameters for EUT models
Table S3 shows the estimation results for each individual subject. We found some heterogeneity
within species: for example, Ali and Pis (capuchins) and Joe (macaque) exhibit different parameter
estimates from those shown by the other subjects of their species, particularly in the power utility
function. Note that when these subjects were excluded from our estimations, we found that capuchins
are still risk-adverse subjects (0.65 with robust std error of 0.07) whilst macaques exhibit significant
risk-seeking behavior (1.17 with robust std error of 0.08).
Table S3. Estimates of individual parameters for EUT models (b).
Subjects
Cap. 1
Acc
Cap. 2
Ali
Cap. 3
Arn
Cap. 4
Pao
Cap. 5
Pet
Cap. 6
Pis
Cap. 7
Pop
Cap. 8
Rav
Negative
exponential
0.824
Mac. 1
1.224
0.153
***
-393.72
-520.32
Cas
***
-170.98
***
(0.020)
(0.022)
(0.02)
0.321
1.006
0.198
-495.86
Mac. 2
***
-529.49
***
-225.34
***
Don
(0.031)
(0.021)
(0.025)
0.846
1.431
0.136
Mac. 3
***
-315.88
-502.42
***
-162.01
***
Jac
(0.020)
(0.029)
(0.018)
0.753
0.794
0.232
Mac. 4
***
-367.54
-505.01
***
-350.81
***
Joe
(0.020)
(0.019)
(0.033)
0.781
1.275
0.151
Mac. 5
***
-367.17
-509.63
***
-141.61
***
Ram
(0.019)
(0.022)
(0.019)
1.008
0.637
0.412
Or. 1
***
-278.82
-513.43
***
-212.21
***
Dok
(0.020)
(0.031)
(0.092)
0.490
0.443
0.609
Or. 2
***
-509.19
-532.50
***
-257.17
***
Kil
(0.028)
(0.042)
(0.161)
0.699
0.795
0.159
Or. 3
***
-397.99
-505.97
***
-203.44
***
Pad
(0.021)
(0.029)
(0.033)
0.75
0.212
Or. 4
***
-197.28
***
Pin
(0.029)
(0.044)
(b)
This table presents estimates of two utility functions (negative exponential, power) parameters for all 17
Power
Log L
Negative
exponential
0.158
***
(0.023)
1.062
***
(0.125)
0.231
***
(0.031)
0.263
***
(0.038)
0.218
***
(0.031)
0.151
***
(0.02)
0.352
***
(0.07)
0.314
***
(0.048)
Log L
Subjects
Power
Log L
subjects. Standard errors are in parentheses. *, **, *** respectively indicates significance at the 5%, 1% and
0.1% levels. Log L is the pseudo log-likelihood of the estimation.
Log L
-510.52
-500.60
-516.80
-506.10
-510.25
-252.26
-262.75
-260.50
-256.99