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Animal Cognition (2022) 25:385–399
https://doi.org/10.1007/s10071-021-01560-x
ORIGINAL PAPER
Comparing utility functions betweenrisky andriskless choice
inrhesus monkeys
PhilipeM.Bujold1 · LeoChiU.Seak1 · WolframSchultz1 · SimoneFerrari‑Toniolo1
Received: 19 January 2021 / Revised: 10 August 2021 / Accepted: 14 September 2021 / Published online: 27 September 2021
© The Author(s) 2021
Abstract
Decisions can be risky or riskless, depending on the outcomes of the choice. Expected utility theory describes risky choices
as a utility maximization process: we choose the option with the highest subjective value (utility), which we compute con-
sidering both the option’s value and its associated risk. According to the random utility maximization framework, riskless
choices could also be based on a utility measure. Neuronal mechanisms of utility-based choice may thus be common to both
risky and riskless choices. This assumption would require the existence of a utility function that accounts for both risky and
riskless decisions. Here, we investigated whether the choice behavior of two macaque monkeys in risky and riskless decisions
could be described by a common underlying utility function. We found that the utility functions elicited in the two choice
scenarios were different from each other, even after taking into account the contribution of subjective probability weight-
ing. Our results suggest that distinct utility representations exist for risky and riskless choices, which could reflect distinct
neuronal representations of the utility quantities, or distinct brain mechanisms for risky and riskless choices. The different
utility functions should be taken into account in neuronal investigations of utility-based choice.
Keywords Gamble· Preference· Prospect theory· Economic choice· Decision-making
Introduction
Whether we are choosing between fruits or vegetables at
the supermarket, deciding to jaywalk in the face of incom-
ing traffic, or picking the ideal friends to go traveling with,
most of our decisions fall under two categories: some have
certain outcomes, some do not. Economists call these deci-
sions risky and riskless, respectively. Different from eve-
ryday definitions that often view risk as probability of loss
or damage, ‘risk’ in economics refers to higher statistical
moments of probability distributions of choice outcomes,
in particular variance, skewness and kurtosis (Rothschild
and Stiglitz 1970; d’Acremont and Bossaerts 2016; Genest
etal. 2016).
In economics, Expected Utility Theory (EUT) (von Neu-
mann and Morgenstern 1944) served as the dominant model
of risky decision-making until the inception of behavioral
economics in the 1970s. Under EUT, a decision-maker’s
attitude towards risk was fully captured by the curvature of
their utility function: a mapping of reward quantities onto
an internal, subjective metric. A concave utility function
predicted an aversion to risk, while a convex one predicted
risk-seeking behavior. Being a mathematical representation
of economic preferences (Kagel etal. 1995), utility is a fun-
damental decision variable, and it is therefore reasonable to
assume that a single function might be common to all forms
of economic choice. Thus, a risky utility function estimated
according to EUT should also be valid for riskless choice.
However, experimental evidence indicates discrepancies
between risky and riskless utility functions (Barron etal.
1984; Stalmeier and Bezembinder 1999).
Contrasting with EUT, Prospect Theory (PT) highlights
an important difference between risky and riskless choice
Wolfram Schultz and Simone Ferrari-Toniolo: senior coauthors.
* Wolfram Schultz
Wolfram.Schultz@Protonmail.com
* Simone Ferrari-Toniolo
Simone.FerrariToniolo@gmail.com
Philipe M. Bujold
Phbujold@gmail.com
Leo Chi U. Seak
chiuseak@gmail.com
1 Department ofPhysiology, Development andNeuroscience,
University ofCambridge, CambridgeCB23DY, UK
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386 Animal Cognition (2022) 25:385–399
1 3
through the introduction of subjective probability weighting.
As probability weights the utility of each outcome for com-
puting Expected Utility, empirical tests have shown that this
weight may not derive from the true, 'physical' probability
but is distorted at specific probabilities in humans (Kah-
neman and Tversky 1979; Tversky and Kahneman 1992;
Gonzalez and Wu 1999) and monkeys (Stauffer etal. 2015;
Ferrari-Toniolo etal. 2019). Thus, rather than being solely
predicted by an individual’s utility curvature, one’s risk
attitude would also vary with their subjectively weighted
outcome probabilities. In other words, while EUT assumed
that risk attitudes derived exclusively from the way in which
people value rewards as reflected in the curvature of the util-
ity function, PT made the case for an additional subjective
weighting of probability.
Subjective probability weighting has been widely incor-
porated into studies of risky and riskless decision-making
(Kahneman etal. 1990; Lattimore et al. 1992; Camerer
etal. 2002; Hertwig and Erev 2009). With all the studies on
behavior that use subjective probability weighting, there is a
remarkable lack of research validating its predictions in both
risky and riskless choices; the limitation being that risky
utilities (or PT values) are usually measured from choices
between risky options in humans (Stott 2006; Tversky and
Kahneman 1992) and monkeys (Stauffer etal. 2014; Genest
etal. 2016), while this clearly cannot be done in a riskless
context. One interesting avenue has been to compare risky
and riskless preferences via introspective metrics. In a study
by Stalmeier and Bezembinder (1999), medical patients
were asked questions that involved risky outcomes:”would
you rather: live 20years with a migraine on x days per week
(followed by death), or live 20years with a p% chance of
getting migraines y times a week, z times a week otherwise”;
and questions where all options were riskless:” which dif-
ference is larger: the difference between 0days of migraine
and x days of migraine, or the difference between x days of
migraine and 3days of migraine”. Modelling with inclu-
sion of subjective probability weighting resulted in identical
risky and riskless utilities, and that probability weighting
accounted for most of the discrepancy between the risk atti-
tudes measured from risky choices and the risk attitudes pre-
dicted by riskless utilities. A similar approach using money
outcomes (gains) rather than medical outcomes (losses) led
to a similar conclusion: PT successfully reconciled risky and
riskless utilities (Abdellaoui etal. 2007).
As the human participants in these studies were generally
risk-averse (for gains), inclusion of subjective probability
weighting might also reconcile risky and riskless utilities
for risk-seeking decision-makers. Additionally, the results of
these introspective studies have recently been challenged by
a set of studies using a more modern, incentive-compatible
approach: the use of time trade-offs as means to study risk-
less decisions (Cheung 2016). In these studies, humans made
choices between larger rewards delivered in the future (with
certainty) and smaller rewards delivered now; utilities from
intertemporal choices were then compared to those estimated
from risky choices. Unlike introspective experiments, how-
ever, most research on time trade-offs reports discrepancies
between riskless, time-discounted utility functions and risky
ones (Andreoni and Sprenger 2012; Abdellaoui etal. 2013;
Lopez-Guzman etal. 2018, but see Andersen etal. 2008).
These discrepancies cannot even be resolved by probability
weighting, which is not entirely surprising given that tem-
poral aspects add an additional factor to utility estimation.
The lack of clear insight into the inclusion of subjective
probability weighting to reconcile risky and riskless choices
represents a crucial limitation to the interpretation of PT,
particularly as it rapidly became the de facto model of choice
to study behavior and neuro-economics in humans, monkeys
and rats (De Martino etal. 2006; Lakshminarayanan etal.
2011; Marshall and Kirkpatrick 2013; Stauffer etal. 2015;
Chen and Stuphorn 2018; Farashahi etal. 2018; Ferrari-
Toniolo etal. 2019). Simultaneously, since there have been
no attempts at reconciling risky and riskless utilities in non-
human decision-makers, there is no evidence to suggest that
either human interpretations can be used to explain animals’
choice behavior.
The aim of the present study was to compare risky and
riskless utilities in rhesus macaques. Studies on monkeys
are highly relevant to humans, as they belong to an evolu-
tionary close species, have distinguished behavioral capaci-
ties, possess sophisticated brain functions, and allow fine-
grained neuronal analysis of behavioral relationships that
are difficult to obtain in humans. Specifically, monkeys show
similar economic choice behavior as humans, such as risk
preferences (McCoy and Platt 2005; Strait and Hayden 2013;
Stauffer etal. 2014), compliance with first-, second- and
third-order stochastically dominating gambles (Genest etal.
2016) and well-ordered preferences for multi-component
options (Pastor-Bernier etal. 2017). Given the suitability of
monkeys for neurophysiological studies, we respected the
required control of sensory and motor variables and tested
the animals in a well-defined primate laboratory environ-
ment rather than in holding cages or the wild. Rather than
testing the full spectrum of risk choices incurred in everyday
life, we selected specific risk tests as behavioral tools for
eliciting risky and riskless utilities. We therefore tested mon-
keys in two categorical types of binary choice: risky choice
between one certain (riskless) and one uncertain (risky)
juice reward option, and riskless choice between two certain
(riskless) juice magnitudes. We estimated utility functions
statistically from empirically assessed choices using the PT-
based discrete choice model (Tversky and Kahneman 1992)
in combination with the random utility maximization (RUM)
model (McFadden 1974, 2001). Importantly, this risky/risk-
less design addressed two of the most important caveats
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387Animal Cognition (2022) 25:385–399
1 3
in human studies: (i) both risky and riskless choices were
incentive-compatible (relying on revealed preferences rather
than introspection), and (ii) choice options were presented
in the exact same way for both risky and riskless decisions.
This study builds on previous human studies that had
shown closer similarities between risky and riskless utilities
once probability weighting had been accounted for. There-
fore, we parametrically separated the contributions of utility
and probability weighting on the monkeys’ risky choices.
We did not expect identical utilities and even hypothesized
that two different utility quantities or mechanisms could
drive behavior in risky and riskless choices.
Animals, materials andmethods
Ethical note
This research has been ethically reviewed, approved, regu-
lated, and supervised by the following institutions, com-
mittees and individuals in the UK and at the University of
Cambridge (UCam): the Minister of State at the UK Home
Office, the Animals in Science Regulation Unit (ASRU) of
the UK Home Office implementing the Animals (Scientific
Procedures) Act 1986 with Amendment Regulations 2012,
the UK Animals in Science Committee (ASC), the local UK
Home Office Inspector, the UK National Centre for Replace-
ment, Refinement and Reduction of Animal Experiments
(NC3Rs), the UCam Animal Welfare and Ethical Review
Body (AWERB), the UCam Governance and Strategy Com-
mittee, the Home Office Establishment License Holder of
the UCam Biomedical Service (UBS), the UBS Director for
Governance and Welfare, the UBS Named Information and
Compliance Support Officer, the UBS Named Veterinary
Surgeon (NVS), and the UBS Named Animal Care and Wel-
fare Officer (NACWO).
Animals
Two male rhesus macaques (Macaca mulatta; Monkey A:
11.2kg, Monkey B: 15.3kg) participated in this experi-
ment. The animals were born in captivity at the Centre for
Macaques (CFM) in the UK and were pair-housed for most
of the experiment. Monkey A (‘Tigger’) had been surgically
implanted with a headpost under full anesthesia and aseptic
procedures for subsequent neurophysiological recordings; he
was not headposted for the current experiment. Monkey B
(‘Ugo’) had been surgically implanted with a headpost and a
recording chamber for neurophysiological recording; he was
headposted for 2–3h on each test day of the current experi-
ment, which was intermingled with neuronal recordings on
separate days. Both animals had previous experience with
the visual stimuli and experimental setup (Ferrari-Toniolo
etal. 2019). Each animal was seated in a primate chair
(Crist instruments) in which he chose on each trial using a
left–right moveable joystick (Biotronix Workshop, Univer-
sity of Cambridge) between two reward options (reward-pre-
dicting stimuli) presented on an upright computer monitor in
front of them; he received the reward he had selected at the
end of each of these binary choice trials (Fig.1a).
Task design andsetup
The premise of this study was to compare the utility func-
tions estimated from monkeys’ choices in risky or riskless
decisions. To do so, monkeys were presented with sets of
choices that could then be translated into utility metrics.
The utilities derived from risky choices were compared with
utilities measured from riskless choices, first assuming no
subjective weighting of probabilities (EUT utilities), then
accounting for the contribution of probability weighting (PT
utilities). While these tests were specifically tailored to the
requirement of elicitation of utility functions, they did not
encompass the wide spectrum of daily risky choices.
Reward options took the form of various combinations of
reward magnitude and probability and were represented on
the monitor through horizontal lines that scaled, and moved,
relative to two vertical ‘framing’ lines (Fig.1b). Reward
magnitudes were represented by the vertical position of the
horizontal lines: 0ml at the bottom of the vertical frame
(1.5ml at the top, and 0 < m < 1.5 in between), whilst the
probability of receiving said reward was represented by the
width of the horizontal lines within the frame. A single,
horizontal line that touched the frames at both ends signaled
a certain reward (probability p = 1); multiple lines that failed
to touch the frames indicated gambles with probabilistic out-
comes, each with associated probability 0 < p < 1 (Fig.1a).
The monkeys were habituated, trained and tested for asso-
ciations between these two-dimensional visual stimuli and
the blackcurrant juice rewards over the course of two years;
both monkeys had previous experience with the task and
stimuli before this study. Thus, while being different from
behavior measured in holding cages or the wild, the experi-
mental durations assured acceptable standards of reliable
performance with well-controlled stimuli and rewards. Both
animals had experienced reward probabilities that ranged
from 0 to 1 (Ferrari-Toniolo etal. 2019), and reward magni-
tudes that ranged from 0ml to 1.3ml of juice. For this study,
reward magnitudes were held between 0ml and 0.5ml of
blackcurrant juice, and reward probabilities ranged from 0
to 1.
Each binary choice trial began with a white cross at the
center of a black computer monitor, if the monkey were
holding the joystick, a cursor would also appear on the moni-
tor (Fig.1a). Using the joystick, the monkeys initiated each
trial by moving the cursor to the center cross and holding it
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388 Animal Cognition (2022) 25:385–399
1 3
there for 0.5–1s. Following this holding period, two reward
options appeared to the left and to the right of the central
cross (see Fig.1a). The animal had 3s to convey his deci-
sion by moving the joystick to the selected side and holding
his choice for 0.1–0.2s—the unselected option would then
disappear. The selected option lingered on the monitor for
1s after reward delivery—followed by a variable inter-trial
period of 1–2s before the next trial. Errors were defined
as unsuccessful central holds, side selection holds, or tri-
als where no choices were made. Each of these resulted in
a 6s timeout for the animal, after which the trial would
be repeated (ensuring the elicitation of preferences for each
tested option set). Additionally, all reward options were
repeated on both the left and right sides of the computer
monitor, alternating pseudo-randomly to control for any side
preference. Both the joystick position and task event times
were sampled and stored at 1kHz on a Windows 7 com-
puter running custom MATLAB software (The MathWorks
2015a; Psychtoolbox version 3.0.11). We collected on aver-
age 423 ± 91 (SD) trials per session over 22 sessions for
monkey A, and 338 ± 41 trials over 7 sessions for monkey
B. Only one session was run on a given experimental day,
and 3–5 sessions were run on each week. Thus, beyond the
initial general habituation and task performance for similar
tasks for another study mentioned above (Ferrari-Toniolo
etal. 2019), the experiment lasted for several weeks with
each animal after performance in the current task had been
established. Only trials in which a given option set had been
repeated at least four times were analyzed. Data processing
ab
cd
50%
Reward
magnitude
Reward
probability
midpoint
∆Reward magnitude (A-B)
Safe Reward magnitude (A-B)
P(choose safe)
0.00.2 0.4
0
0.2
0.4
0.6
0.8
1
P(chhoose A)
0
0.2
0.4
0.6
0.8
1
Center
Association
Choice + Hold
Choice
0.5–1 s
<3 s
0.1–0.2 s
1 s
EV
Fig. 1 Experimental design and measures of risky and riskless
choices. a Binary choice task. The monkeys chose one of two gam-
bles with a left–right motion joystick. They received the blackcurrant
juice reward associated with the chosen stimuli after each trial. Time,
in seconds, indicate the duration of each of the task’s main events. b
Schema of visual stimuli. Rewards were visually represented by hori-
zontal lines (one or two) set between two vertical ones. The vertical
position of these lines signalled the magnitude of said rewards. The
width of these lines, the probability that these rewards would be real-
ized). c Estimating certainty equivalents from risky choices. Monkeys
chose between a safe reward and a risky gamble on each trial. The
safe rewards alternated pseudo-randomly on every trial—they could
be of any magnitude between 0 and 0.5ml in 0.05ml increments.
Each point is a measure of choice ratio: the probability of choosing
the gamble option over various safe rewards. Psychometric softmax
functions (Eq.1) were fitted to these choice ratios, then used to meas-
ure the certainty equivalents (CEs) of individual gambles (the safe
magnitude for which the probability of either choice was 0.5; black
arrow). The solid vertical line indicates the expected value (EV) of
the gamble represented in the box. d Estimating the strength of pref-
erences from riskless choices. Riskless safe rewards were presented
against one another, the probability of choosing the higher magnitude
option (A) is plotted on the y-axis as a function of the difference in
magnitude between the two options presented (
Δ
magnitude). The
differences in magnitude tested were 0.02ml, 0.04ml, 0.06ml, and a
psychometric curve, anchored with its inflection anchored at a
Δ
mag-
nitude of 0, were fitted on the choice ratios measured (Eq.2). These
functions were fitted to different magnitude levels, and the tempera-
ture of each curve was linked to the strength of preferences at each of
these different levels
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389Animal Cognition (2022) 25:385–399
1 3
and statistical analyses were run in Python (Python 3.7.3,
SciPy 1.2.1).
The progression of testing with on-going reward con-
sumption on a given day may result in satiety. We used sev-
eral measures to control for potential confounding effects.
First, specific satiety tests on other monkeys performing
similar tasks in our laboratory had demonstrated that the
currently used blackcurrant juice reward induced minor sati-
ety compared to other juices (Pastor-Bernier etal. 2021).
Second, we terminated testing for the day when task perfor-
mance degraded rapidly, as indicated by increased errors in
central hold, side selection and no choice; from that point
on the animal would stop performing any task within 30–50
trials. Third, to directly compensate for potential effects on
the utility function, we ran the risky and riskless choice
sequences either on alternating days or started in alternation
on a given day; thus, any potential satiety effect on the esti-
mated utility function would not selectively affect only one
type of utility function. Thus, while satiety may inadvert-
ently affect any repeated tests of stochastic choice to some
extent, the effects in our experiment would apply to both
types of utility function and thus have minor effects on their
comparison, which was the goal of the present experiment.
Revealing preferences forrisky andriskless choice
The monkeys’ daily reward preferences were measured in
risky and riskless choice sequences under the framework
of utility maximization. In risky choice sequences, trials
always pit a risky gamble against a safe option—the utility
of different reward magnitudes was estimated via the ratio of
choices between different gamble and safe rewards. All gam-
bles comprised two equally likely reward outcomes (though
one could be 0ml). In riskless choice sequences, monkeys
were presented with two ‘safe’ options, each with a single
fixed outcome—we used the ratio of choice between the two
rewards to estimate utility. From these empirical data, we
statistically estimated distinct utility functions from risky
and riskless choices using, respectively, a PT-based discrete
choice model (Tversky and Kahneman 1992) and the random
utility maximization (RUM) model (McFadden 1974), as
detailed below.
Estimating utility functions inrisky choice
For risky sequences, utilities were estimated using the
fractile-bisection procedure—a method that involves divid-
ing the range of possible utilities into progressively smaller
ranges and estimating the reward magnitude associated
with each of these utility ranges (Fig.2). The animal chose
between a gamble that was set to two specific and equiprob-
able magnitudes (between 0 and 0.5ml; p = 0.5 each) and a
safe reward whose magnitude was varied across the whole
range of 0ml to 0.5ml. We psychometrically estimated the
choice indifference point at which each option was chosen
with equal, 0.5 probability and determined the ‘certainty
equivalent’ (CE) as the magnitude of safe reward (in ml)
that was subjectively equivalent to the utility of the gamble.
Thus, depending on the animal’s risk-seeking or -avoiding
attitude in a given part of the reward distribution, the CE,
and thus the safe reward, could be, respectively, lower or
higher compared to the mean gamble reward (i.e. the gam-
ble’s Expected Value, EV, defined as summed products of
reward magnitude and probability). The procedure defined
set utility metrics (in this case ½, ¼ and ¾, and 1/8 and 7/8
of the maximum utility, see Fig.2a, b) for which the equiva-
lent safe rewards were derived (Fig.2a).
To estimate the CE, we fitted the following logistic func-
tion to the proportion of safe choices for each gamble/safe
option set:
where the probability that the monkeys would choose a safe
reward over the gamble [P(ChooseSafe)] was contingent on
the safe option’s magnitude (
SafeRewardml)
and two free
parameters: x0, the x-axis position of the curve’s inflection
point, and σ, the function’s temperature. Importantly, this
function’s inflection point represented the exact safe mag-
nitude for which the monkeys should be indifferent between
the set gamble and a given safe reward. The x0 parameter
could thus be used as a direct estimate of the gamble’s CE
or, put simply, the safe reward equivalent to the utility level
being tested. Only sequences that contained a minimum of
three different option sets (repeated at least 4 times) were
used in the elicitation of CEs.
From the first CE, identified as having 0.5 utility, two
new equiprobable gambles were created representing utility
values of 0.25 (¼ of the utility range) and 0.75 (1/4 and ¾
of the utility range, respectively). Of the two new gambles,
one was set between 0ml and the first CE’s ml value, and
the other was set between the first CE and 0.5ml (Fig.2b).
The CE elicitation procedure (logistic fitting, Fig.1c) was
repeated for each of these gambles. All option sets were
interwoven in the same sequence to ensure a similar spread
in the presented rewards. Further details of this procedure
are described in Supplementary Information.
Estimating utility functions inriskless choice
For riskless choice sequences, choice ratios between two safe
options were measured as likelihood of choosing the high-
magnitude option over the lower-magnitude one (Fig.1d).
The range of juice rewards (0.05–0.5ml) was divided into
0.05ml increments and the options were centered on these
magnitude increments. For each increment, we defined
(1)
P(ChooseSafe)=1∕(1+e
−
(SafeReward
ml
−x
0
𝜎
),
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390 Animal Cognition (2022) 25:385–399
1 3
0.0 0.1 0.2 0.3 0.4 0.5
Reward magnitude (ml)
0.0
0.2
0.4
0.6
0.8
1.0
Utility
0 0.5
Certainty equivalent
0.0
1.0
Utility
0 0.5
Certainty equivalent
0.0
1.0
Utility
0 0.5
Certainty equivalent
0.0
1.0
Utility
Step 1 Step 2 Step 3
Step 1 Step 2
0.5 utils 0.75 utils
0.25 utils
0.875 utils
0.125 utils
Time
...
a
b
0.0 0.25 0.5
Reward magnitude(ml)
min
max
Utility
0.0 0.25 0.5
min
max
Utility
Monkey AMonkey B
Reward magnitude (ml)
c
Fig. 2 Estimating risky utilities using the fractile procedure. a Fixed
utilities are mapped onto different reward magnitudes. The gambles
that monkeys experienced are defined from bisections of the range
of possible reward magnitudes. For each step the gambles were held
fixed; safe magnitudes varied by 0.05ml increments. b Estimation
of utility using the stepwise, fractile method. In step 1, the monkeys
were presented with an equivariant gamble comprised of the maxi-
mum and minimum magnitudes in the tested reward range. The CE of
the gamble was estimated and assigned a utility of 50%. In step 2, two
new equivariant gambles were defined from the CE elicited in step
1. The CEs of these gambles were elicited and assigned a utility of
25% and 75%. Two more gambles are defined in step 3, from the CEs
elicited in step 2. Their CEs were then assigned a utility of 12.5% and
87.5%. Parametric utility functions, anchored at 0 and 1, were fitted
on these utility estimates (see methods). c Utility functions estimated
from choices. Data points represent daily CEs (semi-transparent) and
their median values (red filled circles) tied to specific utility levels, as
estimated through the fractile procedure. Both monkeys exhibit risk-
seeking behaviour for low-magnitude rewards, and risk-aversion for
high-magnitude ones. The data represent individual utility estimates
gathered over 22 sessions for monkey A, and 7 sessions for monkey
B. The red curves were obtained by fitting piecewise polynomial
functions to the measured CEs (cubic splines with three knots)
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391Animal Cognition (2022) 25:385–399
1 3
three sets of options, using reward magnitudes differences
of 0.02ml, 0.04ml and 0.06ml, which are hereafter called
‘gaps’. The small size of these differences ensured that
choices would be stochastic. Each gap was anchored at its
respective ‘midpoint’.
The likelihood of choosing the higher magnitude option
in different gap-midpoint option sets was used to infer the
shape of the utility function (Fig.3a, b, c). Specifically, the
difference between the likelihoods of choosing the better
options, at different midpoints, reflected the separability of
the utility of different reward magnitudes. Under RUM, the
degree of certainty with which choices are made (i.e. the
closer choice ratios are to 100%) directly correlates with
the separability of the noisy utilities that correspond to each
option in a choice. This implies that, looking at repeated
choices between two set magnitudes, a decision-maker with
a flatter utility function should exhibit more stochasticity
in their choices (i.e. less precision) than a decision-maker
with a steeper utility (i.e. more precision). Changes in choice
ratios between sequential midpoints, as averaged across
gaps, could therefore be used as a proxy for the slope of the
utility function.
To estimate these RUM-compliant utilities, logistic
curves were fitted to the likelihood of choosing the better
option (for the three gaps) at every midpoint level (Fig.3a):
Unlike for CE estimation, this logistic function captured
the likelihood of choosing the high-magnitude option (in
a safe-safe option set) contingent on the gap between the
two options (
Gapml
) and σ, the logistic function’s ‘tempera-
ture’ parameter. Further analyses of the probability that the
monkeys would pick the better reward (
xi
) employed the
following equations:
In this form, the probability of choosing
xi
rather than
xj
was given by the probability that the difference in the true
utilities of
xi
and
xj
was greater or equal to the noise on
xj
(
𝜀j
) minus the noise on
xi
(
𝜀i
). From this, it followed that the
distribution of noise differences could be used as a predictor
of the distance between the two true utilities [
u(xi)
and
u(xj)
].
Because of the assumption of constant noise, the probability
of choosing
xi
over
xj
would be directly proportional to the
distance between the true utility of two options. In accord-
ance with McFadden’s formulation (McFadden 1974, 2005;
(2)
P
(ChooseHigher)=1∕
1+e−
Gapml
𝜎
.
(3)
P(
x
i)
=P
[
U
(
x
i)
≥U
(
x
j)],
(4)
P(
x
i)
=P
[
u
(
x
i)
+𝜀
i
≥u
(
x
j)
+𝜀
j],
(5)
P(
x
i)
=P
[
u
(
x
i)
−u
(
x
j)
≥𝜀
j
−𝜀
i],
Stott 2006), we assumed that the distribution of error differ-
ences (
𝜀j
−𝜀i
) took a logistic form:
And then used the inverse of this logistic distribution’s
CDF to estimate the difference in utilities (
Δutility
) between
hypothetical 0.03ml reward gaps (Fig.3b)—essentially the
slope of the utility function at every midpoint (Fig.3c). The
cumulative sum of these slopes provided an estimate of the
utility at each midpoint. Further details of this procedure are
described in Supplementary Information.
Estimating utility functions fromrisky andriskless
choices inacommon metric
To directly compare the utility functions between risky and
riskless choices, we re-estimated utilities on a common
scale, compatible with PT. We used the same discrete choice
model to describe both risky and riskless choices, without
the need of two different estimation procedures.
The main assumption of our model is that a random quan-
tity is added to each option’s utility at every trial, using the
PT model as the underlying deterministic choice mechanism.
This model introduced stochasticity in choices and could
readily be applied to both risky and riskless choices without
modification.
In the model, utility functions took the form of the cumu-
lative distribution function of a two-sided power distribution
(Eq.10; Kotz and Dorp 2010), a 2-parameter function that
could easily account for complex risk attitudes (Kontek and
Lewandowski 2018): if
𝛼
< 1, the utility function would be
convex and predict risk-seeking choices up to the inflection
at parameter
𝜅
(predicting risk-averse choices thereafter);
if instead
𝛼
> 1, the utility function would be concave and
predict risk-averse behavior up to the inflection at
𝜅
(predict-
ing risk-seeking behavior afterwards). For risky choices, a
1-parameter power function captured the weighting of prob-
abilities (Eq.9). Since the only probability experienced
was p = 0.5,
𝜌
< 1 implied an overweighting of the prob-
ability of receiving the highest reward whilst
𝜌
> 1 implied
underweighting.
We defined three forms of this discrete choice model,
with different free parameters: the EV model (linear utility
and probability weighting), where only the “noise” param-
eter was free to vary; the EUT model (linear probability
weighting), where the utility parameters could vary; and the
PT model, with both utility and probability weighting free
parameters. In risky choices, we compared the goodness-of-
fit of the three models to identify the one that would produce
the best estimate of a utility function. In riskless choices, we
estimated the utility function using the EUT model.
(6)
P(
xi
)
=
1
(
1+e−Δutility
),
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392 Animal Cognition (2022) 25:385–399
1 3
This analysis placed utility metrics for risky and riskless
choices on a common and comparable scale, and, impor-
tantly, it allowed for the inclusion of probability weighting
as an additional contributor to the monkey’s preferences. As
in most discrete choice models (and in line with the aggre-
gate RUM metric), a logit function (softmax) was used to
0.00.1 0.20.3 0.40.5
min
max
0.00.1 0.20.3 0.40.5
0.6
0.7
0.8
0.9
Prob(choose high)
0.00.1 0.20.3 0.40.5
0.00
0.10
0.20
Softmaxtemperature
0.00.1 0.20.3 0.40.5
Midpoint magnitude (ml)
0.00.1 0.20.3 0.40.5
Midpoint magnitude (ml)
0.7
0.8
0.9
1.0
Prob(choose high)
0.00.1 0.20.3 0.40.5
Midpoint magnitude (ml)
0.00
0.02
0.04
0.06
Softmax temperature
Utility
Utility
Reward magnitude (ml)
Utility
+0.61
Φ-1
0.00.2 0.40.6 0.81.0
P(choose high)
-3
-2
-1
0
1
2
3
∆Utility
0.61
00.04
0.2
0.4
0.6
0.8
1.0
P(choose high)
Monkey A
abc
Monkey B
0.08
min
max
min
max
def
Reward magnitude gap (ml)
0.73
AB1AB2AB3
0.73
0.00.1 0.20.3 0.40.5
Fig. 3 Estimating riskless utilities from the stochasticity in safe–
safe choices. a Measuring stochasticity in choices between safe two
reward options. Example visual stimuli (top) representing choices
between safe rewards (A: low, B: high) resulting in different percent-
age of choices for the high option (bottom; black dots). This was
repeated for different reward option sets, centered at different incre-
ments (midpoints). For each midpoint, the likelihoods were fitted
with a softmax curve (dashed), used to estimate the probability of
choosing the larger option for a gap of 0.03ml (gray dot). b Choice
ratios as differences in utility. The likelihoods that monkeys would
pick the better reward were transformed using the inverse cumulative
distribution function (iCDF) of a logistic distribution. The utility of
different rewards took the form of equally noisy distributions centered
at the true utilities. The output of iCDFs is the distance between these
random utilities (i.e. the marginal utility). c From marginal utilities
to utility. The cumulative sum of marginal utilities approximated a
direct utility measure for each midpoint. These measurements were
normalized whereby the utility of the highest midpoint was 1, and
the starting midpoint had a utility of 0. d Daily strength of prefer-
ence estimates. Each point represented the temperature of the softmax
curve fitted on the choice ratios (blue points: average across days).
The lower the temperature parameter, the steeper was the softmax
curve and the more separable were the random utilities. Lower values
meant higher marginal utility measurement (steeper utility function),
higher ones meant lower marginal utility (flatter function). e Daily
choice ratio estimates from softmax fits. Estimates from the same day
are linked by grey lines. Ratios of 0.5 meant that the random utility
of the two options were fully overlapping (i.e. flat utility function);
choice ratios closer to 1 meant random utilities that were fully disso-
ciated and non-overlapping. f Utility functions. Utilities estimated in
single days (grey lines) and averages (blue), normalized relative to the
minimum and maximum midpoint
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393Animal Cognition (2022) 25:385–399
1 3
represent noise in the decision-making process. Details of
this procedure are described in Supplementary Information.
Statistical comparison ofrisky andriskless choices
Estimating utilities through discrete choice modelling
allowed for the comparison of the functional parameters
that best described decisions in risky and riskless choices,
and to explore the unique contributions of both magnitudes
(through utility) and probabilities (through probability
weighting) in a way that aggregate, non-parametric meas-
ures did not permit.
Because the logit function’s
𝜆
-, and the utility’s α
parameters were asymmetrically distributed (with positive
values < 1 accounting for as much change as values > 1),
these were log-transformed before proceeding with any
comparison. Then, the parameters elicited in risky choice
sequences were compared to those estimated from riskless
sequences using a one-way multivariate analysis of variance
(or MANOVA) whereby the main comparison factor in the
analysis was the risk–riskless choice scenario described
by each set of parameters. Since the probability weighting
parameter for riskless choices was constant and fixed at 1,
we restricted the MANOVA analysis to the softmax and util-
ity parameters. We then ran additional correlation analyses
(Pearson’s R) between risky and riskless utility parameters
to determine if the parameters in one set of choices could
predict those of another.
As utility represents the subjective preferences of individ-
ual monkeys, all parameters were compared independently
for each monkey, results were never pooled across animals,
and the statistics for each monkey are reported separately.
Thus, the results are considered generalizable with limits
given by the variations between individual animals. All sta-
tistical analyses were considered significant at p < 0.05.
Results
Experimental design
Each animal chose between two options presented on the
left and right halves of a computer monitor by moving a joy-
stick towards the chosen side (Fig.1a). The reward options
varied in terms of blackcurrant juice quantity as well as in
the probability that they would be delivered. The monkeys
received the selected rewards after every trial—contingent
on their delivery probability. Choice preferences were elic-
ited in trial sequences in which either both options were
certain and therefore riskless, or in sequences in which one
option was certain (safe option) and the other was a risky
gamble with two possible outcomes (juice magnitudes), each
delivered with probability p = 0.5 (equiprobable gamble).
We separately used these riskless or risky choices to infer
an animal’s utility function.
Utility functions inrisky andriskless choice
Choices were measured during 22 and 7days with monkeys
A and B, respectively. On each of these days, the animals
were tested in both risky and riskless choices, using the cer-
tainty equivalent (CE) with the fractile procedure (Fig.2)
and the random utility maximization (RUM) procedure
(Fig.3), respectively. For both risky and riskless sequences,
a link between utility measurements and reward magnitudes
was confirmed via one-way ANOVA. Both monkeys exhib-
ited a significant main effect of utility on the CEs (Fig.2c) in
risky choices (Monkey A: F(4,124) = 35.482, p = 9.763∙10–20,
Monkey B: F(4,39) = 172.537, p = 3.090∙10–24). In riskless
choices, we contrasted the utilities with the midpoint reward
magnitude (Fig.3f), highlighting a significant main effect
(Monkey A: F(8,232) = 375.763, p = 3.503∙10–128; Monkey B:
F(8,52) = 85.561, p = 3.474∙10–27). These basic results illus-
trated how the utilities associated with different reward mag-
nitudes were significantly different from each other, which
would not have been the case if monkeys selected options
at random.
Importantly, the utility levels were significantly rank-
ordered in relation to the reward magnitudes (Spear-
man rank correlation in Monkey A: risky Rho = 0.7209,
p = 5.853∙10–22; riskless Rho = 0.9628, p = 8.035∙10–138.
In Monkey B: risky Rho = 0.9446, p = 6.092∙10–22; riskless
Rho = 0.9665, p = 1.529∙10–36), in line with the fundamental
principle of utility functions being monotonically related to
the reward magnitudes. In general, utilities appeared to be
non-linear functions of physical reward magnitudes.
In risky choices, the full-elicited risky utility functions
followed an S-shaped pattern in both monkeys, reflecting
the typical risk attitudes observed in macaques: risk-seeking
(convex utility) for relatively low-magnitude rewards and
risk-aversion (concave utility) for relatively high-magnitude
ones (Fig.2c).
In riskless choices, we compared the estimated util-
ity increments to highlight any non-linearity in the util-
ity shape. As increments in utility were proportional to
the temperature parameter (i.e. the slope) of the softmax
curves that described choices around a certain magnitude
level, the softmax temperature could be used as a proxy
for linearity: a constant temperature across magnitude
levels would correspond to a linear utility function, while
a varying temperature would indicate non-linear utility.
We compared the temperature parameter across midpoints
and found that it varied significantly with magnitudes
(Fig.3d; Monkey A: F(8, 232) = 2.663, p = 8.165∙10–3);
Monkey B: F(8, 52) = 4.187, p = 6.370∙10–4) highlighting
the non-linearity in the riskless utility function, in both
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394 Animal Cognition (2022) 25:385–399
1 3
monkeys. The softmax temperature, as a function of the
midpoint, reached a minimum (around 0.30 and 0.15ml
for monkeys A and B, respectively) before increasing
again, suggesting a slight S shape for the riskless utility
function (Fig.3f).
Although these aggregate utility measures were based
on commonly defined economic models, they were not
(i) PT-compatible, and (ii) comparable between the risky/
riskless choice scenarios. In fact, we estimated the risky
utility functions following EUT, which, in contrast with
PT, assumes no subjective weighting of probabilities;
the utility functions had different magnitude ranges in
risky and riskless choices (0–0.5ml and 0.05–0.45ml,
respectively) and different discrete steps. We sought to
overcome these limitations by defining a utility estima-
tion method that allowed for a direct comparison of utility
in risky and riskless choices, compatibly with economic
choice models.
Risky andriskless utility functions onacommon
scale
In risky choices, both the EUT and PT models predicted
S-shaped utility functions (Monkey A EUT: t(22) = −29.0190,
p < 0.00001; Monkey A PT: t(22) = −28.2543, p < 0.00001;
Monkey B EUT: t(22) = −4.2859, p = 0.005172; Monkey B
PT: t(7) = −7.4532, p = 0.000301) (Fig.4a, b). The PT model,
however, relied on concave probability weighting (one-sam-
ple t test, Monkey A: t(22) = −4.2533, p = 3.55× 10×−4;
Monkey B: t(7) = −2.7316, p = 0.0341), rather than a convex
utility function, to explain risk-seeking behavior. For that
reason, PT’s S-shaped utility functions were mostly left-
skewed (more concave than convex), whereas EUT utility
functions captured risk-seeking behavior solely through a
right-skewed S shape (more convex than concave) (Fig.4a).
Overall, the daily best-fitting parameters from the PT and
EUT models were significantly different from each other
(Table1), with the PT model capturing behavior signifi-
cantly more reliably than both EV and EUT models (Fig.4b;
0.0 0.5
Reward probability
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0
Reward probability
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.25
Reward magnitude
0.5
0.0
0.2
0.4
0.6
0.8
1.0
Utility
0.0 0.25 0.5
Reward magnitude
0.0
0.2
0.4
0.6
0.8
1.0
Utility
Monkey A
abc
BIC score
60
80
100
120
140
BIC score
100
150
200
250
300
350
EV
EUT
EUT
EV
PT
PT
Monkey B
Risky utility function
0.0 0.2
50
.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2
50
.5
0.0
0.2
0.4
0.6
0.8
1.0
Riskless utility functionProbability weighting function
Probability weightingProbability weighting
Utility
Utility
Reward magnitude
Reward magnitude
EV
EUT
PT
EV
PT/EUT
Fig. 4 Discrete choice estimates differ between risky and riskless
choices. a Utility functions in risky choice. Median parametric esti-
mates for utility functions and probability weighting functions fit-
ted to risky choices. Shaded area: 95% C.I. on the median of these
functions. Two versions of the discrete choice model were fitted: the
expected utility theory (EUT) model predicted choices solely based
on reward options’ utilities (without probability weighting); the pros-
pect theory (PT) model, predicted choices based on utilities and prob-
ability weighting. An expected value (EV) based model was included
for comparison. Monkeys were risk-seeking, but where the PT model
accounted for this mainly through probability weighting, the EUT
model accounted for it through a more convex utility. b Comparison
of risky choice models. The PT model described individual choices
better than EUT and EV. Bayesian information criterions (BIC) were
calculated from the log likelihoods of the daily best-fitting PT and
EUT discrete choice models. c Utility functions in riskless choice.
Median parametric estimates for utility functions fitted to riskless
choices (shaded area: 95% C.I. on the median). The discrete choice
model predicted choices from the expected utilities of rewards (no
probability weighting). Utilities were mostly linear, though slightly
concave
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395Animal Cognition (2022) 25:385–399
1 3
Wilcoxon rank-sum test; monkey A: p = 1.0 × 10–4
;
mon-
key B: p = 1.8 × 10–2). Through the PT model, we could
separate the contribution of utility and probability weighting
to the risk attitude, obtaining a better estimate of the utility
function underlying choices, compared to the EUT model.
In riskless choices (Fig.4c), the utility function’s α
parameter was not significantly different from one (t
test, Monkey A: t(22) = −0.3267, p = 0.7471; Monkey B:
t(7) = 1.3457, p = 0.2270). This implied that the riskless util-
ity functions were close to linear, suggesting that magni-
tudes were objectively represented, according to the RUM
framework.
Mismatch betweenrisky andriskless utility
functions
We compared the shapes of risky and riskless utilities com-
puted on a common scale, in terms of the two utility param-
eters, α and
𝜅
. The α parameter represented the non-linearity
of the utility function: S-shaped (α < 1), inverse-S-shaped
(α > 1) or linear (α = 1); the
𝜅
parameter represented the
inflection point where the curvature of the utility function
would invert. We found a significant difference in the utility
functions’ shapes in terms of the α parameter, in both mon-
keys (Fig.5a; Monkey A: F(1,42) = 72.717, p = 1.04× 10–10;
Monkey B: F(1,12) = 24.221, p = 3.52× 10–4).
Monkey B’s difference in the utility’s inflection
point between risky and riskless choices (Monkey A:
F(1,42) = 1.282, p = 0.264; Monkey B: F(1,12) = 17.153,
p = 0.00136) was significant, while we found no signifi-
cant difference in either the noise or the side bias param-
eters (noise: Monkey A: F(1,42) = 2.760, p = 0.104; Mon-
key B: F(1,12) = 0.182, p = 0.677; side bias: Monkey A:
F(1,42) = 0.2407, p = 0.626; Monkey B: F(1,12) = 2.338,
p = 0.152).
Overall, these results show that the dissimilarity between
the modeled risky and riskless choices was mainly due to
a difference in the non-linearity of the utility functions,
as expressed by the α parameter. The utility function was
strongly non-linear in risky choices, while it was close to
linear in riskless choices.
The difference in utility functions was also evident when
comparing risky and riskless data from single days, through
a correlation analysis: we found was no significant correla-
tion between any of the parameters of risky utility functions
and those of riskless utility functions across days (Fig.5b).
As a control, we correlated the measured riskless choice
percentages (for the hypothetical 0.03ml gap, grey dot in
Fig.3a) with the modeled ones, separately using the util-
ity function elicited from risky or riskless choices. We
found significant correlation coefficients when predicting
riskless choices using the riskless utility function (Mon-
key A: R = 0.442, p = 1.278× 10–9; Monkey B: R = 0.484,
p = 7.610× 10–5) but not using the risky one (Monkey A:
R = 0.104, p = 0.175; Monkey B: R = 0.087, p = 0.503). Thus,
the riskless utility function captured the behavior in riskless
choices while the risky utility function did not, emphasizing
the difference in risky/riskless utilities.
In summary, estimating utilities by including the subjec-
tive probability weighting of PT, rather than EUT, brought
risky fits more in line with riskless ones (Table. 1; Fig.5a)
in line with previous human studies (Stalmeier and Bezem-
binder 1999; Abdellaoui etal. 2007). However, a direct
comparison between the risky and riskless utility param-
eters revealed significant differences in the utility functions’
shapes between the two choices scenarios (Fig.5b).
Discussion
Using a robust, incentive-compatible task, we showed that
utility functions that describe decisions involving risk more
closely mimicked riskless utility functions when probabil-
ity weighting was considered. To compare utility functions
Table 1 MANOVA tests for pairwise differences between the risky EUT, risky PT, and riskless discrete choice models
The analyses were run on four of the five free parameters, excluding probability weighting. The risky EUT and riskless models had no probabil-
ity weighting parameter to compare with the risky PT model’s probability weighting
Utility type F (1, 42) pWilks λ
Monkey A Riskless,
risky (PT)
28.697 6.158× 10–12 0.209
Risky (EUT),
risky (PT)
5.475 6.856× 10–4 0.581
Utility type F (1, 12) pWilks λ
Monkey B Riskless,
risky (PT)
8.744 4.239× 10–3 0.155
Risky (EUT),
risky (PT)
1.687 0.243 0.487
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396 Animal Cognition (2022) 25:385–399
1 3
between categorically distinct risky and riskless scenarios,
we derived these functions statistically from empirical data
of macaque monkeys’ choices, using stochastic versions of
PT and EUT. These analyses resulted in reliably estimated
functional parameters that best described their empirically
assessed economic choices. Each day, the monkeys were
presented with risky or riskless binary choice sequences.
In risky ones, they made choices between gambles and safe
rewards; in riskless ones, both choices had a single, certain
outcome. We found that analyzing monkeys’ risky choices
by including the subjective probability weighting of PT,
in addition to providing a better fit than EUT, led to deci-
sion parameters that more closely resembled those of risk-
less choice. This trend is in line with the human literature
(Stalmeier and Bezembinder 1999; Abdellaoui etal. 2007).
However, the comparisons between utility functions differed:
the monkeys’ utility functions elicited in risky and risk-
less choices were more alike after including the subjective
weighting of probabilities, while still differing significantly.
The CEs estimated in fractile sequences of risky choices
suggested that both monkeys were risk-seeking for all but
the highest reward magnitudes. The risk-seeking attitude was
robust irrespective of consideration of probability weight-
ing (PT model) or its neglect (EUT model). However, the
way the two models accounted for risk-seeking differed.
The PT model captured risk-seeking by concave probability
weighting (the subjective probability of winning a gamble
was higher than the objective probability). By contrast, the
EUT model captured risk-seeking exclusively by convex util-
ity. Although the inclusion of concave probability weight-
ing in the PT model indicated somewhat less risk-seeking
behavior than the EUT model would predict, both models
were consistent in suggesting risk-seeking behavior. By con-
trast, the riskless choices might suggest a somewhat different
decision mechanism. Here, the relatively linear (if slightly
concave) riskless utility function suggested closer influence
of objective reward magnitude and thus less subjectivity in
choices. It appears that, at least within the confines of our
experiment, the cognitive processes underlying economic
decisions might differ between risky and riskless choices
in a way that is beyond the simple addition of probability
weighting.
While the same binary choice design was used in risky
and riskless choices, the difference between options was
much greater in risky sequences than in riskless ones. To
estimate aggregate riskless utilities, for example, the rewards
that the monkeys experienced differed only by up to 0.06ml
in every trial. In risky sequences, on the other hand, gambles
were pitted against safe rewards spread over the full range
of the gambles’ outcomes. Monkeys experienced a broad
range of magnitudes in each of the sequences, but the dif-
ferences between riskless choices could have required far
Fig. 5 Risky utilities do not
predict riskless ones, and vice
versa. a Median utility func-
tion estimates for risky and
riskless choices. The shaded
area represents the 95% C.I. on
the median of these functions.
For riskless choices, utility
estimates were mostly linear
(though slightly concave). For
risky utilities, the two different
versions of the discrete choice
model predicted S-shaped utili-
ties, but risky EUT utility func-
tions were more convex than PT
utility functions. b Absence of
correlation for utility parameters
in risky vs. riskless choices.
Pearson’s correlations were run
on the parameters from risky
and riskless scenarios. Red
squares highlight Pearson’s R
for the correlation of the α and
inflection parameters between
risky and riskless choices.
Asterisks (*) indicate significant
correlations (p < 0.05)
0.0 0.25 0.5
Reward magnitude (ml)
0.0
0.2
0.4
0.6
0.8
1.0
Utility
0.0 0.25 0.5
Reward magnitude (ml)
0.0
0.2
0.4
0.6
0.8
1.0
Utility
ab
Riskless
α-parameter
Riskless
inflection
Risky (PT)
α-parameter
Risky (PT)
inflection
1
0.0681
-0.26-0.0141
0.34 0.13 0.25 1
-0.8
-0.4
0.0
0.4
0.8
Riskless
α-parameter
Riskless
inflection
Risky (PT)
α
-parameter
Risky (PT)
inflection
Riskless
α-parameter
Riskless
inflection
Risky (PT)
α-parameter
Risky (PT)
inflection
1
-0.76* 1
0.61 -0.23 1
0.24 -0.5 0.11 1
-0.8
-0.4
0.0
0.4
0.8
Correlation coefficient (R)
Correlation coefficient (R)
Riskless
Risky (PT)
Risky (EUT)
Monkey AMonkey B
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397Animal Cognition (2022) 25:385–399
1 3
more attention to dissociate than those in riskless choices
(something we cannot account for; but see, Farashahi etal.
2018). Thus, larger variations of risky options that respect
the wide spectrum of risk preferences could be used for more
complete risk tests and might provide helpful information
for refining current decision models.
As a limitation of the current study, the discrepancy
between risky and riskless utility functions could be influ-
enced by our model specifications. Our study employed
some of the more widely used and well tested fitting mod-
els. However, such models could result in an oversimplifi-
cation, and alternative models should be compared to sup-
port our conclusions. In particular, we assumed a constant
and symmetric noise around each option’s utility, whereas
a more biologically plausible contribution of noise on the
utility measure could include asymmetric and non-constant
noise (especially for activity rates close to the limits of the
neurons’ dynamic range). Also, different noise distributions
could be applied to the option components (magnitude and
probability). Thus, follow-up studies with more refined mod-
els might include different assumptions on the noise shape.
Where these findings fail to replicate the data from risky
and riskless introspective studies (though see Hertwig etal.
2018), they are nonetheless in line with the incentive-com-
patible time trade-off approach. As these types of time dis-
counting tasks are easily adapted to study preferences in
rhesus macaques (Hayden and Platt 2007; Kobayashi and
Schultz 2008; Hwang etal. 2009; Blanchard etal. 2013), it
would be interesting to see how utility functions estimated
using time trade-offs in macaque monkeys correlate with
the present findings. Another approach could be to compare
risky and riskless choices between bundles of outcomes;
here choice indifference arises from the combinations of
rewards. Previous such studies demonstrated variations
between risky and riskless choices when some outcomes
involved losses (Chung etal. 2019). Monkeys show win-
stay–lose-shift strategies (Gilovich etal. 1985; Barron and
Erev 2003; Heilbronner and Hayden 2013) and reverse risk
preferences for gambles depending on previously wins or
losses (Lau and Glimcher 2005; Blanchard etal. 2014; Fer-
rari-Toniolo etal. 2019). Thus, future studies might estimate
utility functions from trial-by-trial changes rather than glob-
ally from an entire experimental procedure.
As typical for scientific studies, our results have gen-
erality constraints (Simons etal. 2017). First, we used
only two monkeys, primarily for reasons of ethics toward
a species close to humans. We did not pool the data from
the two animals as reward functions in general and utility
estimates in particular are subjective and thus should not
be averaged across subjects. Even without pooling, two
rhesus monkeys may not be representative for the whole
species, and other individuals of the same species may
show some behavioral differences. This limitation can be
partly overcome by comparing data across different labo-
ratories. However, despite expected individual variations,
the data from the two animals were reproducible, which
we deemed as most important aspect. Second, monkeys
are not humans, nor rats, and the obtained data should
be compared to those from other species. Third, we used
specific tasks in a laboratory setting. While tasks and
laboratory are typical for well-controlled empirical work,
generality can only be achieved by comparing these data
with those from other, equally well-controlled tasks. The
laboratory situation differs notably from natural, free rang-
ing behavior, and the intuition behind such a traditional
approach is that the observed phenomena would underlie
any natural behavior, which again requires comparisons.
Fourth, our analysis of measured, empirical data applied
current statistical methods to current economic models,
including PT and RUM. As both economic models are less
than 50years old, whereas monkeys seek rewards for mil-
lions of years, future statistics and economic models may
reveal additional, but hopefully not opposite, explanations
of economic choice.
Assuming that the discrete choice model is correct,
the difference in utility functions for risky and riskless
utilities could be used as a quantitative basis for neuronal
tests of utility coding. Monkeys could be using different
strategies for solving the risky and riskless choice prob-
lems, implying different brain mechanisms. In particular,
riskless choices are somewhat similar to perceptual dis-
crimination in which subjective values derive primarily
from individual needs and optimal solutions require per-
ceptual comparisons of predictive stimuli. By comparison,
risky choices require in addition assessment of probability
and computation of risk, both of which involve memory
or inferential processes depending on the frequentist or
Bayesian approach to probability. Given these differences
between risky and riskless choices, utility coding may
occur in neurons in different brain regions, and their pat-
tern should reflect the different utility shapes elicited in
the two choice scenarios.
Overall, the results presented here add to the need for
decision models to account for flexible, context-specific
preferences (Hayden etal. 2008; Heilbronner and Hayden
2016; Farashahi etal. 2018). For decision-theory as a whole,
reconciling dynamic preferences with more traditional eco-
nomic models would go a long way to making more accu-
rate, descriptive predictions. While there are undoubtedly
common cognitive processes underlying the various forms
of economic decisions, there are differences in subjectively
valuing rewarding outcomes, as evidenced by the observed
differences in formal utility functions. The current study
pointed to differences in cognitive processes between risk-
less and risky decisions. It will be interesting to see whether
similar utility differences, indicative of different underlying
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398 Animal Cognition (2022) 25:385–399
1 3
cognitive processes, can be identified with other variations
of economic choice, such as time constraints, time delays
and natural vs. artificial outcomes. Monkeys are particu-
larly suitable for such investigations due to their capacity
to perform thousands of trials, which allows researchers to
test the same agent in different choice scenarios and acquire
sufficient data for thorough model comparisons.
Supplementary Information The online version contains supplemen-
tary material available at https:// doi. org/ 10. 1007/ s10071- 021- 01560-x.
Acknowledgements We thank Aled David and Christina Thompson for
animal and technical support. The work was funded by the Wellcome
Trust (WT 095495, WT 204811) and the European Research Council
(ERC, Advanced Grant 293549).
Declarations
Conflict of interest The authors declare no competing financial inter-
ests.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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