Decision Making & Movement Planningmsl/papers/trommeretal04.pdfDecision Making and Movement Planning particular movement strategy s effectively selected a probability distribution

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.

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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


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


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

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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

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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

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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.

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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


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

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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.

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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


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).

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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).

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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.

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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.

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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


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.

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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


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)

Documents

Decision Making & Movement Planningmsl/papers/trommeretal04.pdfDecision Making and Movement Planning particular movement strategy s effectively selected a probability distribution