DECISION MAKING AND MOVEMENT PLANNING
UNDER RISK
Julia Trommershäuser Paulina Trzcinka Laurence T. Maloney
Michael S. Landy Department of Psychology and Center for Neural Science, New York University
6 Washington Place, New York, NY 10003, USA
May 13, 2004, submitted
Running head: Decision making and movement planning Keywords: Statistical decision theory, movement planning, optimality,
cumulative prospect theory Abstract word count: 131 words Article word count: 3926 words Figures: 4 (one in color)
Contact information:
Laurence T. Maloney Phone: 212 998-7851 Department of Psychology FAX: 212 995-4349 New York University 6 Washington Place, 8th Floor Email: [email protected] New York, NY 10012 USA
Decision Making and Movement Planning
ABSTRACT
We present an experimental study which compares performance in movement
planning tasks to the predictions of a standard model of human sub-optimality in decision
making, Cumulative Prospect Theory (CPT) (Kahneman & Tversky, 1992). Six naïve
subjects earned monetary rewards and penalties by rapidly touching color-coded objects
distributed at random locations on a monitor. This movement task is formally equivalent
to decision making under risk. In marked contrast to the grossly sub-optimal performance
of human subjects in decision making experiments, our subjects’ performance was often
indistinguishable from optimal. We varied penalty conditions and the uncertainty of
incurring gains and losses and compared subjects’ performance to CPT. Subjects’ near-
optimal performance was disrupted when confronted with uncertainty about rewards and
penalties and the deviations from optimal performance were inconsistent with the
predictions of CPT.
2
Decision Making and Movement Planning
INTRODUCTION
A decision maker chooses among plans of action. Whether it is checking “yes” in
response to a survey question or choosing when to start the swing of a baseball bat, the
chosen plan of action may have serious consequences. While the decision maker can plan
an action, s/he cannot always anticipate the precise outcome of the decision and its
consequences.
If the possible, mutually-exclusive outcomes of a decision are denoted O O
then the effect of any plan is to assign a probability to each outcome. The result is a
lottery
1, , n
( 1 1 2 2, ; , ; ; ,n n )p O p O p O , where ip denotes the probability of outcome O and
(i.e., the list of outcomes is exhaustive). Decision making is, pared down to its
essentials, a choice among lotteries. When the decision maker does not know the
probabilities induced by each action, s/he is engaged in decision-making under
uncertainty. In contrast, if the decision maker knows the probabilities in the lotteries
associated with each action, s/he is engaged in decision making under risk.
i
1
n
ii
p=
=∑ 1
When the outcomes are framed in terms of money, we refer to them as gains,
denoted , and we can assign to each lottery iG ( )1 1 2 2, ; , ; ; ,n nL p G p G p G= , an
expected gain (Arnauld & Nicole, 1662/1992) ,
. (1) ( )1
n
i ii
EG L p G=
=∑
The decision maker who seeks to maximize gain can evidently do so by selecting the
action whose corresponding lottery has the maximum expected gain (MEG). Such a
3
Decision Making and Movement Planning
MEG rule is an example of a normative rule, intended to guide decision making (Bell,
Raiffa & Tversky, 1988). Much research in human decision making under risk is a
catalogue of the many, patterned failures of normative theories, including MEG, to
explain the decisions humans actually make (Bell et al., 1988; Kahneman, Slovic &
Tversky, 1982; Kahneman & Tversky, 2000).
FIGURE 1 ABOUT HERE
The plots in Fig. 1 illustrate a particular theory of how humans make decisions
under risk known as Cumulative Prospect Theory (Kahneman & Tversky, 1992). We
view the theory as a summary of many, but not all, of the deviations from MEG that
human decision makers are prone to. These include a tendency to exaggerate small
probabilities (Allais, 1953; Attneave, 1953; Kahneman & Tversky, 1992; Lichtenstein et
al., 1978; see Fig. 1A), to convert gain into subjective utility (Bernoulli, 1738/1954; von
Neumann & Morgenstern, 1947), and to frame outcomes in terms of losses and gains
with an exaggerated aversion to losses (Kahneman & Tversky, 1979; see Fig. 1B).
There are other well-documented deviations from MEG predictions and the
degree and pattern of deviations depends on many factors. How these factors interact and
affect decision making is controversial. What is not in dispute is that it takes very little to
lead a human decision maker to abandon an MEG rule in decision making tasks.
The typical tasks used in the literature on decision making under risk are paper-
and-pencil choices that one can “meditate on” before responding. Probabilities and values
are arbitrarily chosen by the experimenter and are represented numerically or by simple
4
Decision Making and Movement Planning
graphical devices. In contrast, Trommershäuser, Maloney and Landy (2003a,b)
introduced an experimental paradigm in which subjects were asked to plan and execute
rapid movements in “risky” environments. The task on each trial was formally equivalent
to decision making under risk (Fig. 2), but information about probabilities was not
communicated to the subject “in words”. Despite the lack of explicit descriptions of
probability, subjects in these tasks consistently selected motor plans that came close to
maximizing expected gain. The focus of this article is on the discrepancy between
performance in these tasks and the decision making literature.
MOVEMENT UNDER RISK
In the experiments of Trommershäuser et al. (2003a,b), subjects performed a rapid
pointing movement toward a green target circle on a computer screen. If they hit the
target in time, they earned 100 points. If they accidentally hit a nearby red circle (Fig. 2),
they lost points. In one condition, the penalty associated with the red circle was zero (no
consequences for hitting the red circle). At the other extreme, the penalty for hitting the
red circle was 500 points, five times greater than the reward for hitting the green circle.
Subjects knew that the points they earned over the course of the experiment would be
converted into a monetary bonus.
If subjects were perfectly in control of their movements, they would simply touch
the green circle, avoiding the red whenever it incurred any penalty. However, subjects
were given a time limit, resulting in movement end points with substantial scatter (Meyer
et al., 1988; Murata & Iwase, 2001; Plamondon & Alimi, 1997). Thus, a choice of a
5
Decision Making and Movement Planning
particular movement strategy s effectively selected a probability distribution of
possible end points on the touch screen (Fig. 2A).
( ,sp x y)
FIGURE 2 ABOUT HERE
The instructions to the subject were unusual for an experiment involving a motor
task. Subjects were not instructed to hit the green target nor were they told to avoid the
red penalty circle. Rather, they were instructed to earn as much money as possible, in any
way they saw fit to do so. Probabilities were never mentioned. Yet, with these
instructions, the task that they were asked to perform was precisely equivalent to a choice
among lotteries (Fig. 2B).
To see this, first consider the possible outcomes of their movement when there is
one red and one green circle present on the screen, with gains of -400 and +100 points,
respectively (Fig. 2B). A movement that hits the touch screen within the time limit could
land in one of four regions: red only (Region 1R , gain G1 400= − ), red/green overlap
(Region 2R , gain G ), green only (Region 2 300= − 3R , gain G3 100= ), or outside of
both circles (Region 4R , gain G4 0= ). The probability of each of these outcomes
depends on the subject’s choice of movement strategy . s
In the following, we identify a movement strategy s with a mean movement end
point (i.e., an “aim point”). The diamond in Fig. 2A marks the mean of a Gaussian
distribution of end points (with width σ ) and, given this choice of movement strategy , s
6
Decision Making and Movement Planning
we can compute the probability of each of the four outcomes, denoted 1, , 4p p (see
Trommershäuser et al., 2003a,b, for details on how to compute ip ).
)4
)4
Therefore this choice of movement strategy corresponds to the lottery s
( ) ( 1 1 2 2 3 3 4, ; , ; , ; ,L s p G p G p G p G= , (2)
while an alternative movement strategy s′ (e.g., the small black circle in Fig. 2B)
corresponds to a second lottery
. (3) ( ) ( 1 1 2 2 3 3 4, ; , ; , ; ,L s p G p G p G p G′ ′ ′ ′′ =
Fig. 2 is based on the measured end point distributions for subject JM in the
experiment reported below. For this configuration, the expected value of the lottery
corresponding to movement strategy is less than that corresponding to movement
strategy . However, there are infinitely many other lotteries available to Subject JM,
each corresponding to a particular motor strategy or aim point, and each with an
associated expected value. Fig. 2B lists a subset of these strategies with associated
probabilities. By choosing among these possible motor strategies, subject JM effectively
selects among the possible sets of probabilities associated with each outcome.
s
s′
Given the complexity of the decision making task implicit in Fig. 2B, the outcome
of the experiments of Trommershäuser et al. (2003a,b) is remarkable. They compared the
performance of each subject to that of the movement strategy that maximized expected
value for that subject. They found that subjects’ performance did not differ significantly
from the predictions of expected value theory across three separate experiments.
We are left with a paradox. The movement planning task involves a speeded
choice among infinitely many lotteries and yet, unlike performance in paper-and-pencil
7
Decision Making and Movement Planning
decision making tasks, subjects’ performance is close to that required to maximize
expected gain.
In the study reported here, we manipulated penalty value and the uncertainty with
which subjects incurred gains and losses during a risky motor task. We asked whether
there are conditions under which human movement planners are prone to the same biases
(distortions of probability and utility) typically found in studies on human decision
making.
EXPERIMENT
Methods
Apparatus
The apparatus was as described by Trommershäuser et al. (2003a). Subjects were
seated in a dimly lit room in front of a touch screen (AccuTouch from Elo TouchSystems,
accuracy < ±2 mm SD, resolution 15,500 touchpoints/cm2) mounted vertically in front of
a 21-in computer monitor (Sony Multiscan CPD-G500, 1280×1024 pixels @ 75 Hz). A
chin rest was used to control the viewing distance which was 29 cm. The computer
keyboard was mounted on a table centered in front of the monitor. Subjects started each
trial by holding down the space bar. The experiment was run using the Psychophysics
Toolbox (Brainard, 1997; Pelli, 1997) on a Pentium III Dell Precision workstation. A
calibration procedure was performed before each session to ensure that the touch screen
measurements were aligned with the visual stimuli.
8
Decision Making and Movement Planning
Stimuli
The stimulus configuration consisted of a target region and a penalty region
(Fig. 2A). The penalty region was a filled circle. Its color varied between trials and
indicated the penalty value for that trial (white: 0, pink: -200, red: -400 points). The target
region was also circular, marked by a bright green edge and unshaded so that the overlap
with the penalty circle would be readily visible. The target and penalty regions had radii
of 8.4 mm. The target region appeared in one of four possible positions, horizontally
displaced from the penalty region by ±1 or ±2 multiples of the target radius (“near” and
“far”; Fig. 2 illustrates the near condition). The far configurations were included to keep
subjects motivated through easily scored points, but were not included in the analysis.
The stimulus was displayed at a random location on each trial to prevent subjects
from using pre-planned strategies. Independent x and y displacements from the screen
center were chosen from a uniform distribution with range of ±44 mm. A frame
(114.2 mm × 80.6 mm), centered on the screen, indicated the area within which the target
and penalty regions could appear.
Procedure
Each session began with a test to ensure the subject knew which color was
assigned to which penalty. Following the brief presentation of a single penalty circle for
1 s in the center of the screen, the subject pressed a key according to the corresponding
penalty value. The subject received feedback and had to correctly identify each penalty
type twice. After a subsequent calibration procedure, there was a short block of 12 warm-
up trials with zero penalty. The score was then reset to zero and data collection began.
9
Decision Making and Movement Planning
A trial started with a fixation cross. The subject was required to move the index
finger of the right hand to the starting position. The trial began when the space bar was
pressed. The subject was required to stay at this starting position until after the stimulus
configuration appeared or the trial was aborted. Next, the frame was displayed, delimiting
the area within which the target could appear, and preparing the subject to move shortly.
500 ms later the target and penalty circles were displayed. Subjects were required to
touch the screen within 700 ms of the display of the circles or they would incur a
“timeout” penalty of 700 points. We refer to the point where the observer touched the
touch screen as the end point of the movement ( ),x y . If the end point was within the
penalty or target region, the region(s) that were hit “exploded” graphically. Then, the
points awarded for that trial were shown, followed by the subject’s total accumulated
points for that session (see Trommershäuser et al., 2003a, for further details).
Design
The experiment consisted of one practice session to train subjects on the time
limit of the speeded-response task and five sessions of data collection, run on different
days. In the practice session, there was no response time limit for the first 32 trials, the
next 4 blocks of 20 trials had a time limit of 850 ms and the remaining 8 blocks of 20
trials had the 700 ms time limit used in subsequent blocks. The penalty value alternated
between 0 and –200 in successive blocks. Each practice session 20-trial block consisted
of five repetitions of each of the four spatial configurations, presented in random order.
The actual experiment consisted of five experimental sessions of 372 trials each
(12 warm-up trials and 10 blocks of 36 trials). Blocks alternated between a penalty 200
and a penalty 400 block. The penalty 200 block consisted of 6 repetitions of penalty 200
10
Decision Making and Movement Planning
and 3 repetitions of penalty zero for each of the 4 configurations, and the penalty 400
block was organized analogously. The difference between the sessions was a
manipulation of the probability of incurring a gain or a loss. In some sessions, the penalty
region (session 2), target region (session 3) or both (session 4) were stochastic: when the
subject hit a stochastic region s/he would receive the penalty or reward with probability
0.5.1 Sessions 1 and 5 were certainty sessions during which penalties and rewards were
incurred with certainty each time the respective regions were hit.
In session 2 (penalty 50%), each time the subject hit the penalty region, it
exploded graphically, but the penalty was incurred with a probability of 0.5. In trials in
which the subject hit the penalty region, but did not incur a penalty, the word “lucky”
appeared after the penalty region exploded. Session 3 (reward 50%) was similar. In this
case, in trials in which the subject hit the target region, but did not score a reward, the
word “sorry” appeared after the target region exploded. The fourth session (both 50%)
combined the stochastic reward and penalty.
Subjects and instructions
4 male and 3 female subjects participated in the experiment. All participants had
normal or corrected-to-normal vision and ranged from 18 to 36 years old. All subjects
were unaware of the hypothesis under test. Each gave informed consent prior to testing
and was paid for their participation. All were informed of the payoffs and penalties for
1 A pilot experiment showed that simultaneously manipulating the probabilities of
incurring a gain or loss and the penalty value on a trial-by-trial basis caused subjects to
ignore the manipulation.
11
Decision Making and Movement Planning
each block of trials. Subjects were told that the overall score over the 6 sessions would
result in a bonus payment of 25 cents per 1000 points. They were paid $12 per hour and
received a bonus of $3 - $6 according to their cumulative points at the end of each
session.
Data analysis
For each trial, we recorded reaction time (the interval from stimulus display until
movement initiation), movement time (the interval from leaving the start position until the
screen was touched), the movement end position and the score. Trials in which the subject
left the start position less than 100 ms after stimulus display or hit the screen after the
time limit were excluded from the analysis. Each subject contributed approximately 1800
data points; i.e. 60 repetitions per condition (with data collapsed across spatially
symmetric configurations; 120 repetitions in the certainty conditions). For each subject
individually, a Levene test (Howell, 2002, p. 215) was performed to test for homogeneity
of the variances in the x- and y-directions across spatial (“near” and “far”) and penalty
conditions (0, 200, 400), and to test for isotropy of the variance (i.e., whether 2 2x yσ σ= ).
Then, a single estimate of each subject’s motor variability was computed by averaging
variances across spatial and penalty conditions and across the x- and y-directions. To test
for possible changes in reaction and movement times across conditions, we analyzed both
measures individually for each subject as a 2-factor, repeated-measures ANOVA. The
factors were the target position and the penalty level. Movement end points were tested
for spatial symmetry using a linear contrast comparing targets on the left of the penalty
circle with those on the right (for a given penalty, analyzed individually for each subject).
Movement end points were tested for stability across sessions by comparing the
12
Decision Making and Movement Planning
movement end points recorded in the two certainty sessions using a 3-factor, repeated-
measures ANOVA. The factors were the session, the target position and the penalty level.
Mean movement end points for each condition were compared with optimal
movement end points as predicted by the MEG model of optimal movement planning
(Trommershäuser et al, 2003a,b) based on each subject’s estimated motor uncertainty 2σ .
Subjects’ efficiency was computed as the ratio between the subject’s cumulative score in
a condition and the corresponding expected optimal score predicted by the MEG model.
Subjects’ efficiencies were compared to optimal performance by bootstrap analysis. A
distribution of optimal performance was generated based on Monte Carlo simulation runs
of the equivalent number of experimental trials with each subject’s variance. Efficiencies
outside the 95%, 99% and 99.5% area of this distribution are classified as sub-optimal
and are marked by 1, 2, and 3 asterisks in Fig. 4.
RESULTS
Subjects differed significantly in their motor uncertainty, reaction, and movement
times. Therefore, the data were analyzed individually for each subject. After completion
of practice, reaction and movement times and end point variability had decreased to a
stable level. In both experiments, the results of the statistical analysis for each subject
confirmed that the reaction and movement times did not differ significantly across
conditions (p>0.05 in all cases), and remained constant throughout the experiment. The
distribution of movement end points was symmetric with respect to the y-axis and data
were collapsed across symmetric configurations. Mean movement end points did not vary
systematically across conditions in the y-direction, in the zero penalty or far conditions
(independent of the penalty value; data not shown here).
13
Decision Making and Movement Planning
FIGURE 3 ABOUT HERE
For all subjects, variances in the x- and y-directions were independent of
conditions and isotropic (p>0.05 in all cases). Accordingly, for each subject we computed
a variance estimate 2σ averaged over the x- and y-directions and over all spatial
configuration and penalty conditions. This estimate was used when determining optimal
performance as defined by the MEG model.
For one subject, mean movement end points differed significantly between the
two certainty sessions 1 and 5, indicating that his responses did not remain stable across
sessions. He was excluded from the analysis. Movement end points of the remaining six
subjects were collapsed for session 1 and 5.
Subjects chose optimally among motor strategies when rewards and penalties are certain
We first tested how our subjects’ performance (in points per trial) in the certainty
sessions compared to optimal performance as defined by MEG. Five out of six subjects’
scores were statistically indistinguishable from optimal (Fig. 3A), replicating our
previous findings (Trommershäuser et al., 2003a,b). Only one subject (filled symbols)
differed significantly from optimal performance because he did not shift far enough away
from the penalty region (Fig. 3B). All subjects shifted farther from the target center for
higher penalty values, but in all cases, subjects’ actual movement end points remained
below the line of optimal correspondence with the MEG model (Fig. 3B). That is, all
subjects placed their movement end points closer to the penalty region than predicted by
the MEG model, although this consistent bias did not lead to significant deviations from
optimality in performance (Fig. 3A).
14
Decision Making and Movement Planning
Subjects’ movement endpoints were sub-optimally biased in a direction inconsistent with
Cumulative Prospect Theory
We next tested if the small, patterned biases observed in our subjects’ movement
end points were consistent with the biases found in human decision making and
summarized by Cumulative Prospect Theory (CPT; Kahneman & Tversky, 1992;
Kahneman & Tversky, 2000; see also Fennema & Wakker, 1997, for a comparison of
original and Cumulative Prospect Theory). To do so, we replaced the terms for
probability ip and gain in Eq. 1 by the CPT expressions for the probability weight iG
( )ipπ ,
( ) ( )( )1/1-p p p p
δδδ δπ = + (4)
and utility U G , ( )i
( )
( ) 0
- - 0
U G G G
G G
α
αλ .
= ≤
= < (5)
These are the functions illustrated in Fig. 1. In this version of CPT, biases in human
decision making are captured by the three parameters δ , α and λ . Here δ quantifies
the tendency to exaggerate low and underestimate high probabilities2 (Fig. 1A), λ
parameterizes the typical aversion to losses, and α parameterizes the non-linear
transformation of value into subjective utility (Fig. 1B). Typical values found in studies
on human decision making are 2.25λ = , 0.88α = , and ~ 0 0.7.6 -δ (Kahneman &
Tversky, 1992). Fig. 3C shows that for a variety of parameter values, including these
2 For simplicity we use a single exponent δ for gains and losses.
15
Decision Making and Movement Planning
typical values, CPT predicts a shift of movement end points even farther away from the
target center than predicted by MEG. This effect is largely due to the exaggeration of
small probabilities (Eq. 4). However, all subjects’ movement end points are biased in the
opposite direction, i.e. closer to the penalty region than predicted by MEG (Fig. 3B). The
biases observed in our experiment are not consistent with CPT.
FIGURE 4 ABOUT HERE
Introduction of explicit probabilities disrupted optimal performance
Subjects’ performance dropped significantly below optimal when gains or losses
were incurred with probability of 0.5 (sessions 2-4, Fig. 4A). As shown in Figs. 4B and
4C, in 10 out of 24 cases, subjects shifted their movement end points in response to
manipulations in explicit probabilities in the sub-optimal direction, i.e. closer towards the
penalty when the chance of scoring a reward dropped to 50%, or further away from the
penalty when the chance of scoring a penalty dropped to 50%. In the Both 50%
condition, both the penalty and reward are stochastic, so that the optimal strategy is
identical to that in the certainty condition (Allais, 1953). Yet, four out of six subjects
changed their mean movement end points significantly in this condition (Fig. 4C). (Note
that all subjects returned to their initial optimal strategy in session 5 when rewards and
penalties were scored with certainty). In summary, associating explicit probabilities with
rewards and penalties disrupted optimal performance in our movement planning tasks.
16
Decision Making and Movement Planning
DISCUSSION
We have argued that movement tasks are formally equivalent to decision making
under risk. We conducted an experiment involving a speeded pointing task to explicitly
compare performance in this movement planning task with the predictions of CPT
(Kahneman & Tversky, 1992).
Our subjects’ deviations from optimal performance were not consistent with the
predictions of CPT. While CPT predicted that subjects would avoid hitting close to a
penalty region, the biases observed in our experiment were in the opposite direction. Our
subjects consistently hit too close to the penalty region. A possible explanation for this
bias is that subjects underestimate their own motor uncertainty. This underestimation
would lead them to movement plans whose mean end point is too close to the penalty
region, the outcome we found. Note that this bias in movement end points was too small
to cause significant consequences for our subjects. For five out of six subjects, earnings
were indistinguishable from those expected from an optimal MEG strategy. Performance
dropped significantly below optimal only when subjects were confronted with explicit
uncertainty about whether they would incur a reward or penalty on a single trial.
Our results are consistent with the findings of Gigerenzer and Goldstein (1996)
and of Weber, Shafir and Blais (2004): decision makers have difficulty reasoning with
explicitly-stated probabilities. Weber et al. (2004) find that experience-based choices do
not suffer from the same sub-optimal decisions as pencil and paper tasks involving
explicit probabilities. Human movement planners are capable of choosing a strategy
using the implicit probabilities of hitting a target or penalty region (as a function of aim
17
Decision Making and Movement Planning
point), but also fail when explicit probabilities are required, as in our uncertainty
conditions.
ACKNOWLEDGMENTS
This research was funded in part by NIH grant EY08266, HFSP grant
RG0109/1999-B and by the Emmy-Noether-Programme of the German Science
Foundation (DFG). Part of this research was presented at a conference (Maloney,
Trommershäuser, Trzcinka & Landy, Vision Sciences Society, Sarasota, FL, May 2004)
and in the Intel Science Talent Search.
18
Decision Making and Movement Planning
REFERENCES
Allais, M. (1953). Le comportment de l'homme rationnel devant la risque: critique des
postulats et axiomes de l'école Américaine. Econometrica, 21, 503-546.
Arnauld, A., & Nicole, P. (1662/1992). La Logique ou L’Art de Penser. Paris: Gallimard.
Attneave, F. (1953). Psychological probability as a function of experienced frequency.
Journal of Experimental Psychology, 46, 81-86.
Bell, D. E., Raiffa, H. & Tversky, A. [Eds.] (1988). Decision Making: Descriptive,
Normative and Prescriptive Interactions. Cambridge, UK: Cambridge University
Press.
Bernoulli, D. (1738/1954). Exposition of a new theory on the measurement of risk,
translated by Louise Sommer. Econometrica, 22, 23-36.
Brainard, D. H. (1997). The psychophysical toolbox. Spatial Vision, 10, 433-436.
Fennema, H., & Wakker, P. P. (1997). Original and cumulative prospect theory: A
discussion of empirical differences. Journal of Behavioral Decision Making, 10,
53-64.
Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the fast and frugal way: Models of
bounded rationality. Psychological Review, 103, 650-669.
Howell, D. C. (2002). Statistical methods for psychology, 5th Ed. Australia: Duxbury.
Kahneman, D., Slovic, P., & Tversky, A., Eds. (1982). Judgment Under Uncertainty:
Heuristics and Biases. Cambridge, UK: Cambridge University Press.
19
Decision Making and Movement Planning
Kahneman, D., & Tversky, A. (1979). Prospect Theory: An analysis of decision under
risk. Econometrica, 47, 263-291.
Kahneman, D., & Tversky, A. (1992). Advances in prospect theory: cumulative
representation of uncertainty. Risk and Uncertainty, 5, 297-323.
Kahneman, D. & Tversky, A. [Eds.] (2000). Choices, Values & Frames. New York:
Cambridge University Press.
Lichtenstein, S., Slovic, P., Fischhoff, B., Layman, M. & Combs, B. (1978). Judged
frequency of lethal events. Journal of Experimental Psychology: Human Learning
and Memory, 4, 551-578.
Meyer, D. E., Abrams, R. A., Kornblum, S., Wright, C. E., & Smith, J. E. K. (1988).
Optimality in human motor performance: Ideal control of rapid aimed
movements. Psychological Review, 95, 340-370.
Murata, A., & Iwase, H. (2001). Extending Fitts’ law to a three-dimensional pointing
task. Human Movement Science, 20, 791-805.
Pelli, D. G. (1997). The video toolbox software for visual psychophysics: Transforming
numbers into movies. Spatial Vision, 10, 437-442.
Plamondon, R., & Alimi, A. M. (1997). Speed/accuracy trade-offs in target-directed
movements. Behavioral Brain Sciences, 20, 279-349.
Trommershäuser, J., Maloney, L. T., & Landy, M. S. (2003a). Statistical decision theory
and tradeoffs in motor response. Spatial Vision, 16, 255-275.
Trommershäuser, J., Maloney, L. T., & Landy, M. S. (2003b). Statistical decision theory
and rapid, goal-directed movements. Journal of the Optical Society A, 20, 1419-
1433.
20
Decision Making and Movement Planning
von Neumann, J., & Morgenstern, O. (1947). Theory of Games and Economic Behavior.
Princeton, NJ: Princeton University Press.
Weber, E. U., Shafir S., & Blais, A.-R. (2004). Predicting risk-sensitivity in humans and
lower animals: Risk as variance or coefficient of variation. Psychological Review,
111, 430-445.
21
Decision Making and Movement Planning
FIGURE CAPTIONS
Figure 1: A) Subjective probability as a function of actual probability. The data
are taken from Kahneman & Tversky (1992). The solid curve is the best fit to the data of
a parametric family of functions that they propose to fit distortions of probability, as
specified in Eq. 4 below. The solid curve corresponds to the parameter value 0.61δ = .
B) Subjective utility as a function of gain as modeled by Eq. 5 with 0.88α = , 2.25λ =
(from Kahneman & Tversky, 1979, 1992). Kahneman and Tversky found that human
decision makers tend to overweight losses relative to gains of the same magnitude and
that patterns of risk-seeking and risk-averse behavior found experimentally were readily
explained by a utility function that was concave for gains, convex for losses.
Figure 2: A) This figure shows one of the “near” configurations along with a sub-
optimal strategy (the mean end point indicated by the diamond). 200 simulated responses
are shown based on a bivariate Gaussian distribution and a motor uncertainty
3.89 mmσ =
1
(corresponding to subject JM’s estimated motor uncertainty). B) Six
possible aim points are shown along with the equivalent lottery for each (based on the
probabilities, given the aim point and motor uncertainty, of landing in the penalty (region
R ), the target/penalty overlap ( 2R ), the target ( 3R ), or neither ( 4R ).
Figure 3: A) Results for six subjects, certainty condition. Average scores per trial
are shown as a function of the optimal performance predicted by MEG. Model
predictions were computed based on each subject’s individual variance. Data are
displayed for each subject individually (from top left to bottom right as listed in the key:
22
Decision Making and Movement Planning
23
AM, JM, PL, RF, SK, AL). Error bars indicate the 95% confidence interval. B) Shift of
mean movement end points from the target center as a function of the optimal shift
predicted by the MEG model. Error bars indicate ±1 SEM. Solid diagonal lines in A and
B indicate perfect correspondence of model and experiment. The direction of deviations
from optimal performance as predicted by CPT is indicated by an orange gradient.
C) Comparison of subject RF’s movement end points with the predictions of MEG and
CPT. Subject RF’s movement end points ±1 SEM (red line) are plotted along with the
predictions of the MEG model (solid black line) and the predictions of CPT (blue line) as
a function of the distortions in subject probability δ for four different combinations of
the CPT parameters λ and α (Eqs. 4-5).
Figure 4: Results for six subjects, uncertainty conditions. A) Actual scores,
normalized by the optimal score predicted by the MEG model. The solid horizontal line
indicates perfect correspondence of model and experiment. B) Shift of mean movement
end points with changes in explicit probability. Shifts are displayed as a proportion of the
actual movement end point shift in the certainty condition. C) The difference between the
shifts in the both 50% and certainty conditions.
In all panels, data are displayed for each subject individually and the penalty 200
(left bar) and 400 (right bar) conditions. Model predictions were computed based on each
subject’s variance. In B and C, the solid horizontal lines indicate the optimal shift.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
-400 -200 0 200 400
-400
-200
0
200
400
GainU
tility
p
π(p)
B)A)
Fig. 1
8.4 mm
::::::
(0.8%, -400; 25.3%, -300; 65.5%, 100; 8.4%, 0)(0.2%, -400; 13.1%, -300; 70.9%, 100; 15.8%, 0)(0.1%, -400; 10.8%, -300; 70.7%, 100; 18.4%, 0)(0.0%, -400; 3.9%, -300; 62.4%, 100; 33.7%, 0)(0.0%, -400; 2.1%, -300; 54.2%, 100; 43.7%, 0)(0.0%, -400; 0.9%, -300; 43.1%, 100; 56.0%, 0)
σ = 3.68 mm
...
A) B)
Fig. 2
-400
-400
100
100
R1 R2R3
R4
1 2 3 4 5 6 71
2
3
4
5
6
7
0 20 40 60 80 1000
20
40
60
80
100CPT
CPT
optimal shift (mm)
actu
al s
hift
(mm
)
optimal points per trial
actu
al p
oint
s pe
r tria
l
penalty = 400
penalty = 200
0
4
8
RF, penalty = 400
δ
λ = 1α = 1
λ = 0.5α = 1
λ = 1α = 2.25
λ = 0.88α = 2.25
CPTmaximum gainexperiment
0.6 0.8 1δ
12
0.6 0.8 1
A)
C)
δ0.6 0.8 1
δ0.6 0.8 1
Fig. 3
B)sh
ift (m
m)
AMSK JM PL RFAL
penalty = 200penalty = 400
100
0
-100
-200
-300
-400
******
***
***
***
***
** * * * * * *pe
rform
ance
(% o
f opt
imal
)A)
B)
reward 50% both (50%)
penalty 50%
certainty
rela
tive
chan
ge in
end
poi
nt (%
)
0
100
-100
200
-200 chan
ge in
end
poi
nt (m
m)
1
-1
0
2
-2
3
-3AMSK JM PL RFALAMSK JM PL RFAL
optimal
sub-optimaldirection
AMSK JM PL RFALAMSK JM PLRFAL AMSK JM PL RFAL AMSK JM PL RFAL
direction
both (50%)reward 50%
penalty 50%
Fig. 4
C)
Top Related