The skill element in decision making under uncertainty: Control or competence?

Judgment and Decision Making, Vol. 2, No. 3, June 2007, pp. 189–203.
The skill element in decision making under uncertainty: Control or
competence?
Adam S. Goodie? and Diana L. Young
University of Georgia
Abstract
Many natural decisions contain an element of skill. Modern conceptions of the skill component include control
(Goodie, 2003) and competence (Heath & Tversky, 1991). The control hypothesis states that a task’s skill component
(the sensitivity of the task to skill) affects decision making; the competence hypothesis states decision making is affected
only if the participant possesses the skill. Three experiments compared risk taking patterns between two groups. One
group faced bets on random events, and another group faced bets on their answers to general knowledge questions,
which is a task characterized by control. In Experiment 1, control increased risk taking markedly with all statistical
properties held constant. In Experiment 2, decisions made in domains of varying dif?culty, and by individuals of
varying ability, yielded further quali?ed support for the role of competence. In Experiment 3, the role of control was
replicated, and participants’ perceptions of the differences in group treatments aligned more with the implications of
the control hypothesis than with the competence hypothesis. Results offered support for the control hypothesis across a
range of competence.
Keywords: control, competence, decision making, choice, betting, risk, overcon?dence, college students.
1 Introduction
1.1 Ambiguity and skill
Decision researchers know a great deal about the terms of
Ellsberg (1961) and many others have found that people
risk that people will accept and reject on random events
are generally ambiguity averse; in the domain of gains,
such as the drawing of a lottery number, rolling a die, or
people prefer a prospect in which probabilities of possible
pulling a poker chip from a bookbag. Less is known about
outcomes are known to a prospect in which probabilities
how individuals accept or reject risk when they are bet-
of the same outcomes are not stated (ambiguous) but have
ting on their own golf putts, stock picks, organizational
the same average value. The major exception to this is at
decisions or answers to trivia questions.
very low probabilities, where ambiguity is preferred. In
Researchers readily build models of decision making
the domain of losses, these preferences are reversed.
around risky decisions based on random events. Much
decision research is analogous to psychophysical percep-
Examination of the effect of a skill element constitutes
tion research, relating psychological events to objective
a special case of ambiguity. What is Shaquille O’Neal’s
criteria. A bookbag with 70 percent white and 30 per-
probability of making his next free throw? At the con-
cent red poker chips presents a clear objective criterion to
clusion of the 2006–07 season, his career free throw rate
which subjective perceptions may readily be compared.
was 52.5%, but his free throw rate for the season was
Sinking a free throw does not present such a clear crite-
only 42.2%. At his next free throw opportunity, he may
rion with regard to its associated probabilities. For this
be suffering from the ?u, or coming off a terrible game,
reason, researchers have dif?culty in evaluating perfor-
or on a hot streak, or he may merely believe he’s on a
mance relative to a normative criterion when the task is
hot streak (Gilovich, Vallone, & Tversky, 1985). Unlike
assessing the probability of a made free throw, as well
a lottery draw, in which it is easier to construct a reason-
as in establishing valid lawful relationships between rel-
able estimate of the probability of winning (for example,
evant probabilities and decisions.
by reading the ticket), the sample space for a successful
free throw is not clearly de?ned. In other words, the pre-
diction of performance is variable over time in a skilled
?This research was supported by National Institutes of Health
task, hence it is more dif?cult to predict on the basis of
research grant MH067827.
Address correspondence to: Adam S.
past performance. In fact, most de?nitions of skill state or
Goodie, Department of Psychology, University of Georgia, Athens,
GA 30602–3013; Phone 706–542–6624; Fax 706–542–3275; Email
imply that the person exerting skill can change the prob-
goodie@uga.edu
ability of success.
189

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
190
The existing evidence suggests that a skilled task that
fer. Across an assortment of situations, when betting on
determines an uncertain outcome has an effect on prob-
questions drawn from intermixed domains, the propor-
ability assessment and decision making that is distinct
tion of times that participants chose to bet on their knowl-
from that of ambiguity alone. For example, in demon-
edge was a steeply increasing function of the probability
strating the "illusion of control," Langer (1975) showed
of winning (Experiments 1 and 3). Because con?dence
that people responded differently to vague likelihoods
consistently exceeded accuracy in these experiments, bet-
when certain super?cial characteristics of the prospects
ting on a random event whose probability of winning was
were distorted, for example when the familiar symbols of
equal to con?dence was more likely to win than betting
a deck of cards were replaced by unfamiliar symbols, or
on the belief itself, and Heath and Tversky (1991) noted
when participants were permitted to practice on a random
that the acceptance of knowledge-based bets over random
mechanism similar to a roulette wheel. Langer argued
bets resulted in a 15% loss of expected earnings.
that the changes in the appearance of a skill component
Heath and Tversky then (Experiment 4) tested the
caused changes in responses. Con?dence ratings, bet ac-
competence hypothesis by drawing questions from dis-
ceptance and bet amounts were all affected by apparent
crete domains in which participants believed themselves
control, although the illusion of control is not robust to
to be either competent or incompetent. They observed
multi-shot gambles (Koehler, Gibbs, & Hogarth, 1994).
that, with subjective probability held constant, partici-
Participants bet more when given skill-relevant manipu-
pants displayed a consistent behavioral pattern: bets in a
lations such as being able to choose whether to receive
domain of competence were preferred to bets on random
more cards in a simulated blackjack game, but not when
events, which in turn were preferred to bets in a domain
given skill-irrelevant manipulations such as choosing a
of incompetence. They concluded that people seek out
different dealer (Chau & Phillips, 1995). Also, partici-
ambiguity in domains of competence but avoid it in areas
pants high in desire for control bet more than those low in
of incompetence.
desire for control on events over which they had falsely
Fox and Tversky (1995; Fox & Weber, 2002; see also
perceived control. Those high in desire for control bet
Chow & Sarin, 2001) presented a companion to the com-
less than others on events over which they did not have
petence hypothesis, the comparative ignorance hypoth-
illusory control (Burger & Schnerring, 1982).
esis, positing that relative knowledge affects decisions
most strongly when the contrast between conditions of
greater and lesser competence is brought to the decision
1.2 Control and competence
maker’s attention.
These ?ndings are notably contrary to the early ambi-
Recent research has advanced two major conceptions of
guity ?ndings with random events: when evaluating bets
the role of skill in decision making: competence (Heath
on vaguely probable events with a skill component, par-
& Tversky, 1991) and control (Goodie, 2003). These con-
ticipants preferred the ambiguous (skilled) option at high
ceptions have important commonalities, sharing an em-
probabilities but preferred the unambiguous (random) op-
phasis on the role that the skill component of a task plays
tion at low probabilities. However, the evidence speci?-
in shaping decision making under uncertainty (apart from
cally in support of the control hypothesis remains limited
the probability and magnitude of possible outcomes. The
to Heath and Tversky’s Experiment 4 comparing just two
control hypothesis claims that people bet more when skill
domains under unusual selection techniques, which are
makes a difference; the competence hypothesis claims the
discussed at more length below.
same effect but only when an individual possesses the rel-
More recent studies (Goodie, 2003) assessed risk atti-
evant skill. Control is a property of the task: if the task
tude by pitting a bet on knowledge item against no bet
requires actions that can be learned, then it is character-
at all, rather than a bet on a random event of equivalent
ized by control, even if a participant has not yet learned
probability. Goodie constructed bets on knowledge items
the skill. Competence, on the other hand, is an interactive
to be fair, having zero average marginal value if con?-
characteristic of both the task and the person: competence
dence was well calibrated. In the ?rst two experiments,
exists only if the task both can be learned (the task com-
bet acceptance sharply increased as con?dence increased
ponent) and has been learned (the person component).
for knowledge bets, bearing a striking resemblance to the
Heath and Tversky (1991) argued that people prefer to
comparable data obtained by Heath and Tversky (1991)
bet on questions about knowledge topics in which they
when using mixed-domain questions. In Experiment 3,
feel competent rather than incompetent. In their stud-
one group considered bets on their knowledge. The other
ies, participants chose to bet on either the correctness of
groups considered bets on events that appeared random to
their answer to a general knowledge question or a ran-
participants but that Goodie constructed to be identical in
dom event whose probability matched their previously
every statistical way to bets on knowledge. Participants
stated con?dence, with identical payoffs in each bet of-
accepted more bets on random events at low probabilities

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
191
and more bets on their knowledge at high probabilities,
competence and control in decisions under uncertainty.
revealing the anticipated crossover effect.
The present experiments test the competence hypothe-
An important difference arises between studies that uti-
sis against the control hypothesis by eliciting betting de-
lize questions drawn from a single domain (e.g., U.S. his-
cisions within domains of varying dif?culty and among
tory) and those that use questions from mixed domains
participants of varying ability.
(e.g., Greek mythology, U.S. history, and sports). As
The distinction between competence and control is
Heath and Tversky (1991) noted in discussing the dif-
most evident in a skill-based task in which a particular
ferences between single and mixed domains, low con?-
participant has little skill. The control hypothesis sug-
dence items in mixed-domain populations will systemati-
gests people bet more when skill could be attained, the
cally include more questions from low-competence do-
competence hypothesis only when it has been attained.
mains. Similarly, Gigerenzer (1991) noted the impor-
We can best differentiate between these two hypotheses
tance of utilizing single-domain questions in assessing
when skill could be attained but has not. The control
con?dence in answers. In a mixed-domain set of gen-
hypothesis suggests the skill element does alter decision
eral knowledge questions, the methods used by the deci-
making under such conditions, whereas the competence
sion maker to generate con?dence assessments become
hypothesis suggests it does not.
uninterpretable because the decision maker may be using
a different reference set than the experimenter. Asking
1.4 General Method
participants questions in a single domain allows for more
reliable representations of con?dence across all questions
We report three experiments which use the methods de-
asked.
veloped by Goodie (2003; Campbell, Goodie, & Foster,
There is reason to expect that control per se in?uences
2004). The basic task of fair bets on knowledge uses three
decision making. Skinner (1996), in a major review of
kinds of questions, administered in two phases.
the literature, notes that “[w]hen people perceive that they
have a high degree of control, they exert effort, try hard,
initiate action, and persist in the face of failures and set-
1.4.1 Phase 1. General knowledge and con?dence
backs; they evince interest, optimism, sustained attention,
assessment
problem solving, and an action orientation” (p. 556, cf.
The ?rst question type was a two-alternative forced
Seligman, 1975). Where control prevails, a prospect with
choice question. Prior studies (Goodie, 2003) adapted
negative expected value, narrowly conceived, might also
questions from a collection (Nelson & Narens, 1980) that
be an opportunity to learn new skill that will result in fu-
sampled from diverse domains. The present studies ran-
ture prospects with positive value, and might therefore
domly selected questions from ?ve well-de?ned domains.
be worth accepting. This is an interesting complement to
Three question populations selected two of the 50 U.S.
the normative argument made by Frisch and Baron (1988;
states at random and asked for a binary comparison on
Baron, 2000) that other ambiguous prospects, even with
one statistic: population, land area, or population den-
positive expected value, might be worth postponing un-
sity, manipulated between-subjects. The other two ques-
til further information is available to permit better-valued
tion populations randomly selected two of the 50 largest
decisions. We argue that ambiguous prospects character-
U.S. cities and elicited a comparison of the cities on either
ized by control, even with negative expected value, might
population or driving distance to Athens, Georgia.1
be worth pursuing in order to set up better-valued deci-
The second question type asked for an assessment of
sions later. The possibility of accepting bets in order to
con?dence in each question, placed in one of the follow-
increase skill does not apply when competence already
ing categories: 50–52%, 53–60%, 61–70%, 71–80%, 81–
exists, only when the possibility of exerting control to in-
90%, 91–97%, and 98–100%. In a binary task such as
crease competence prevails.
this one, the range of 50%-100% re?ects the full range
of competence, from complete ignorance where accu-
1.3 The present experiments
racy would be 50% and con?dence should not be much
higher, to absolute knowledge where accuracy and con?-
The goals of this paper are: a) to compare across do-
dence are both 100%. Con?dence was taken as the mid-
mains wherein people have different degrees of compe-
1State population was taken as the 1999 Census Bureau estimate,
tence, in order to observe the degree to which variation in
and population density was the ratio of population to land area. Ques-
competence makes a difference in risk attitude; b) to ex-
tions involving city comparisons used the 50 largest metropolitan areas
tend the risk-attitude ?ndings of Goodie (2003) to single-
in the continental U.S., to eliminate the confusion involved in consid-
domain formats, a manipulation that made a considerable
ering driving distance to San Juan, Puerto. City population was taken
as the population of the entire metropolitan area as identi?ed by the
difference in the ambiguity-attitude ?ndings of Heath and
Census Bureau (this was made clear in the instructions), and driving
Tversky (1991); and c) to begin to compare the roles of
distance was the distance to the central city.

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
192
point of the selected con?dence category. We used these
In the Gains Only structure, the certain option was a
categories to assess risk taking across a well-de?ned ar-
gain of 100 points. The bet offered a gain of 100 /con?-
ray of probabilities from chance to certainty, combining
dence points if the answer was correct and no gain if the
equal spacing of categories in the mid-range and greater
answer was wrong. So, if the participant bet on an an-
discrimination near the endpoints. This range confers
swer in which she had 75% con?dence, she won 100/.75
the advantages of re?ecting all binary choices and being
= 133 points if the answer was correct but nothing if the
simple and easily understood, although it also bears the
answer was wrong. She gained 100 points if she rejected
clear limitations of excluding half the probability spec-
the bet. It is easy to show that the average outcome of ac-
trum. These studies adopted con?dence elicitation meth-
cepting a bet in either format is equal to the certain option
ods without alteration from those used by Goodie (2003;
(no change in the Mixed format or a gain of 100 points in
Campbell et al., 2004).
Gains Only) if p(correct) = con?dence, less than the cer-
tain option if p(correct) < con?dence, and greater than
the certain option if p(correct) > con?dence.
1.4.2 Phase 2. Betting on answers
A third question type elicited acceptance or rejection of
1.4.4 “Answers” and “Random” groups
a bet on the correctness of each answer that was given.
Participants played out these bets for point accumula-
In Experiments 1 and 3, we randomly assigned partici-
tions that were not backed by monetary incentives. In
pants to two groups that differed in whether they believed
all conditions, participants faced a two-alternative choice
they were betting on their knowledge or on a random
between a certain outcome and a bet. The bet was al-
event. The Answers group bet on their answers, using
ways fair, having average value equal to the certain op-
either the Mixed or Gains Only format in different exper-
tion if the participant’s con?dence judgment was well-
iments. The Random group’s bets held all statistical prop-
calibrated. Its average value was less than that of the
erties constant, differing from the Answers group’s only
certain option if the participant was overcon?dent and
in appearing to rely on random events rather than partic-
greater than the certain option if the participant was un-
ipants’ answers. Many dimensions of bets on knowledge
dercon?dent. After accepting or rejecting the bet, the par-
are determined by the participants’ responses, such as the
ticipant received feedback, including the correct answer
distribution of subjective probabilities of winning (deter-
to the question, the number of points gained or lost (in-
mined by con?dence), the frequency of winning (deter-
cluding if no points were gained or lost), and the cumula-
mined by accuracy), and any order effects on these di-
tive point total.
mensions (for example, if overcon?dence declines with
experience, cf. Sieck & Arkes, 2005, or accuracy de-
1.4.3 The betting formats
clines with fatigue, or any number of other possibilities).
By basing the apparently random bets on the participant’s
We used two betting formats, with Mixed gains and
responses, we can rule out these and any other alternative
losses, and Gains Only. The Mixed format was used in
explanations based on such statistical properties of the re-
order to re?ect the structure of many risks which contain
sponses of participants in the Answers condition.
the possibility of either gain or loss. The Gains Only for-
Bets that appeared stochastic in fact relied on partici-
mat was used to eliminate the complexity of possibly dif-
pants’ answers and con?dence assessments in the knowl-
fering value and weighting for gains and losses. We de-
edge questions. In the betting phase, each answer was
signed both betting formats to provide average outcomes
converted into a bet on a seemingly random event with
that were equal if the bet was accepted or rejected, as-
the stated probability of winning equal to assessed con?-
suming good calibration. Betting formats were always
dence in a corresponding trivia answer; the correctness of
varied between subjects, or were kept constant within an
the corresponding answer determined the bet’s outcome.
experiment, so that no participant needed to comprehend,
For example, if a participant expressed 75% con?dence
remember, or distinguish between both.
in her answer to the ?rst question, then the ?rst bet she
In the Mixed format, the certain option was no change
encountered in the betting phase instructed: "A number
in points, and the bet provided for a gain of 100 points if
will be chosen at random between 0 and 100, and to win
the answer was correct or a loss of 100 * con?dence/(1-
the bet, the Chosen number must be less than or equal
con?dence) points if the answer was incorrect. For ex-
to the Magic Number. The Magic Number this time is:
ample, if a participant was 75% con?dent in an answer,
75. If the chosen number is LESS THAN or equal to
then she considered a bet wherein she won 100 points if
the Magic Number, you gain 100 points. If the chosen
the answer was correct but a loss of 100 * (.75/.25) = 300
number is greater than the Magic Number, you lose 300
points if the answer was wrong. If she rejected the bet,
points." If the participant accepted the bet, she won the
she did not gain or lose any points.
bet if her answer to the corresponding question was cor-

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
193
Table 1: Structure of the experiments
Experiment
N
Question Types
Betting Format
Survey
1
single domains
mixed and gains-only
No
1a
76
state population
mixed
No
1b
67
city population
mixed
No
1c
48
state population
gains-only
No
1d
35
city population
gains-only
No
2
single domains
mixed and gains-only
No
2a
112
5 groups?
mixed
No
2b
152
5 groups?
gains-only
No
3
185
state population
gains-only
Yes
* 5 groups include: state population, land area, population density, city
population, and driving distance from Athens, GA.
rect and lost the bet if her answer was incorrect. The
2.1 Method
Magic Number, the magnitude of the gain if the bet was
won, and the determination of whether the bet was won
In Experiment 1a (N=76; 37 in Answers and 39 in
or lost changed on each betting trial to re?ect the con-
Random), participants answered binary choices compar-
?dence expressed in the corresponding answer from the
ing states’ populations and faced bets constructed in the
?rst phase and whether it was correct.
Mixed format. In Experiment 1b (N=67; 33 in Answers
and 34 in Random), participants compared cities in pop-
ulation with bets in the Mixed format. In Experiment 1c
1.4.5 Other general facets
(N=48; 23 in Answers and 25 in Random), participants
compared states’ populations with bets in the Gains Only
In all experiments, we recruited participants from the Re-
format. In Experiment 1d (N= 35; 17 in Answers and
search Pool of the Psychology Department at the Univer-
18 in Random), participants compared cities’ populations
sity of Georgia and compensated them with partial credit
with bets in the Gains Only format.
toward lower-division courses. We prevented participants
from participating in more than one of the present exper-
iments or in any additional related experiments. Partic-
2.2 Results
ipants ran in groups of up to three in a room with indi-
vidual computer stations separated by ?ve-foot-tall parti-
2.2.1 Con?dence, accuracy and calibration
tions. We omitted participants’ data from analysis if they
did not use more than three con?dence categories, or if
Average con?dence, accuracy and over/undercon?dence
they showed evidence of not attending to the task (i.e.,
are given in Table 2.
exclusive betting acceptance or rejection, or radical over-
or undercon?dence). Thirty participants were excluded
2.2.2 Bet acceptance
for this reason (13 out of 239 in Experiment 1, 10 out of
274 in Experiment 2, and 7 out of 192 in Experiment 3).
The principal ?nding of these experiments is that, using
See Table 1 for a layout of the structure of our experimen-
questions from single domains with all statistical prop-
tal design.
erties of bets held constant between groups, participants
consistently accepted more bets when betting on their
answers than they did when betting on random events.
2 Experiment 1
These results are presented in Table 3 and show dramati-
cally greater rates of bet acceptance in the Answers group
The ?rst experiment assessed the effect of a skill compo-
in all four sub-experiments, which were statistically sig-
nent using items from single domains, comparing partic-
ni?cant in all cases. Averaged across sub-experiments,
ipants betting on answers with those betting on random
those betting on their answers accepted 75.8% of all bets,
events. Four sub-experiments utilized different question
and those betting on random events accepted 55.5% of all
populations and betting formats.
bets.

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
194
Table 2: Average con?dence, accuracy and overcon?-
Table 3: Overall percentage of bets accepted on answers
dence in Experiment 1, and comparisons between An-
and random events in Experiment 1.
swers and Random groups.
Experiment
Answers
Random
t
df
p
Experiment
Answers
Random
t
df
p
1a
73.9
55.2
4.17
74
.000
Con?dence
1b
73.8
62.1
2.41
65
.019
1a
.751
.753
0.07
74
.95
1c
80.2
55.0
4.90
46
.000
1b
.768
.764
0.21
65
.83
1d
75.2
49.8
2.67
33
.012
1c
.780
.726
2.31
46 .026
Average
75.8
55.5
1d
.778
.756
0.80
33
.43
Accuracy
1a
.745
.757
0.73
74
.47
there is still an increasing rate of bet acceptance as con-
?dence increases, as can be seen in Figure 1. The com-
1b
.713
.713
0.01
65
.99
parison between the present experiments with single do-
1c
.760
.751
0.50
46
.61
mains and past experiments with mixed domains resem-
1d
.665
.694
1.07
33
.29
bles the trend across experiments in Heath and Tversky
(1991). When Heath and Tversky narrowed the focus of
Overcon?dence
questions, they observed a preference to bet on items in
1a
.007
?.004
0.57
74
.57
domains in which participants had competence that did
1b
.055
.051
0.19
65
.85
not depend on probability level. That is, if a participant
1c
.020
?.025
1.52
46
.14
was competent in the domain of politics but felt uncertain
1d
.113
.062
1.33
33 .051
about a particular political question, she still preferred to
bet on that answer rather than an equally uncertain ran-
dom event or item from a domain of incompetence. The
current study incorporates the same narrowing of focus to
Betting rates conditionalized on con?dence are shown
a single domain, relative to the studies of Goodie (2003),
in Figure 1. We found higher betting rates when partic-
and the same trend is observed: the increased risk prefer-
ipants bet on their own knowledge, compared with bets
ence does not depend on probability level.
that were identical in every statistical way but appeared
to rely on random events, at all con?dence levels in all
sub-experiments. Because accuracy did not differ sig-
2.3 Discussion
ni?cantly between groups, neither group experienced a
These results indicate that control affects risk attitude
systematically greater proportion of won bets. In addi-
when extended to the important case of a single domain,
tion, because overcon?dence did not differ signi?cantly
broadly across question populations. This supports the
between groups, neither group bene?ted from a system-
conclusion of Goodie (2003) that control affects risk atti-
atically more favorable outcome for betting. The com-
tude, but these ?ndings suggest that the nature of that ef-
parison of the betting curves is thus an appropriate re-
fect may be different within single domains than in mixed
?ection of the different appearance of betting on knowl-
domains. Whereas in mixed domains participants showed
edge versus betting on random events, rather than a re-
betting proportions that were lower at low judged prob-
?ection of differing probabilities or magnitude of possi-
abilities and higher at high judged probabilities (when
ble outcomes. Had we found a trend of higher accuracy
compared to participants who bet on random events), par-
or overcon?dence in either group, the higher bet accep-
ticipants who bet on their answers in this experiment ac-
tance among those betting on knowledge would have sug-
cepted risk more often at all levels of con?dence than
gested a different explanation for the difference in bet ac-
those who bet on random events.
ceptance other than the difference between groups.
Goodie (2003) discussed the possibility of modeling
The ?ndings in bet acceptance mark a departure from
the effect of control in terms of the probability weight-
what Goodie (2003, Experiment 3) observed with items
ing function proposed by Gonzalez and Wu (1999).
from assorted domains, where participants bet on their
This weighting function includes two parameters that are
knowledge relatively seldom at low levels of con?dence
notable for their psychological plausibility: elevation,
and increasingly often as con?dence increased. In the
which re?ects the overall attractiveness of risk; and cur-
present experiments, rates of bet acceptance are higher
vature, which re?ects the discriminability of different lev-
for bets on answers at all levels of con?dence, though
els of probability. Goodie (2003), interpreting the bet-

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
195
1.0
1.0
(a)
(b)
0.8
0.8
0.6
0.6
0.4
0.4
Bet acceptance
Bet acceptance
Answers
0.2
0.2
Random
0.0
0.0
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
1.0
Confidence
Confidence
1.0
1.0
(c)
(d)
0.8
0.8
0.6
0.6
0.4
0.4
Bet acceptance
Bet acceptance
0.2
0.2
0.0
0.0
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
1.0
Confidence
Confidence
Figure 1: Proportions of bets accepted in (a) Experiment 1a, (b) Experiment 1b, (c) Experiment 1c, and (d) Experiment
1d. Mean bet acceptance was aggregated across all subjects at each level of con?dence, ignoring subject identity. At
all con?dence levels in all sub-experiments, participants accepted bets more frequently on answers than on random
events.
ting proportions that increased with judged probability,
ulation and measure the effect of these competence dif-
speculated that the data could be accounted for by posit-
ferences on performance measures. One way to obtain
ing a more linear weighting function under conditions of
diverse degrees of competence is to rely on naturally oc-
control. The present data suggest that, when answer-
curring variability in competence among participants, ob-
ing questions from well-de?ned domains with random
serving performance differences that depend correlation-
sampling and transparent rules (i.e., under representative
ally on demonstrated knowledge in the content area. An-
sampling), control may increase the attractiveness of risk
other way is to observe differences that arise between
and the elevation of the probability weighting function.
groups when groups are given questions that differ in
However, these data do not permit ?rm conclusions on
dif?culty. We accomplished this by using ?ve question
the mathematical form of the weighting function.
populations: comparisons between pairs of U.S. states
on population, land area and population density; and be-
tween pairs from among the ?fty largest U.S. metropoli-
3 Experiment 2
tan areas on population and driving distance from Athens,
Georgia. We assumed that some of these question popu-
In Experiment 1, the primary aim was to manipulate con-
lations would be more dif?cult than others and could be
trol to determine its effect on risky decisions in single do-
identi?ed by showing a sizable degree of variability in
mains. In Experiment 2, in order to test the competence
average accuracy between groups. We also assumed that
hypothesis, we sought to observe differences in compe-
some participants would be more competent in each con-
tence through both correlation and experimental manip-
tent area than others, which could be identi?ed by differ-

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
196
Table 4: Descriptive statistics of con?dence, accuracy, overcon?dence and betting slope within the groups in Experi-
ment 2.
Accuracy
Con?dence
Overcon?dence
Experiment 2a
City driving distance
.856
.887
.030
State area
.799
.843
.044
State population
.746
.711
?.035
State population density
.707
.742
.035
City population
.706
.758
.053
Experiment 2b
City driving distance
.840
.868
.028
State area
.778
.817
.039
State population density
.764
.730
?.033
State population
.730
.751
.021
City population
.691
.741
.049
Note: Groups are listed in declining order of accuracy in each experiment. The
groups did not show the same ordering of accuracy in both experiments.
ences in accuracy. Would people display different pat-
3.2 Results
terns of betting on a task characterized by control when
they have different degrees of competence?2 If so, the
3.2.1 Con?dence, accuracy and calibration
competence hypothesis would be supported.
Con?dence, accuracy and overcon?dence values for the
two sub-experiments are given in Table 4. Unlike in Ex-
periment 1, differences in accuracy and con?dence were
not only expected but essential as a manipulation check
3.1 Method
for the effect of differential competence on betting.
Average accuracy at the group level ranged from .706
Experiment 2a used the Mixed betting format; Experi-
to .856 in Experiment 2a, and from .691 to .840 in Ex-
ment 2b used the Gains Only betting structure. We ran-
periment 2b (Table 4). In a binary choice task, where
domly divided participants (N=112 for Experiment 2a,
the proportions are constrained to [0.5,1.0], this overall
152 for Experiment 2b) into ?ve groups, with each group
accuracy range of .165 is considerable. The differences
differing in the domain of questions asked. Three groups
among the groups de?ned by question domains were also
answered questions seeking comparisons between pairs
statistically signi?cant. In Experiment 2a, for accuracy,
of randomly selected U.S. states on the dimensions of
F (4,171)=22.2, p<.001; for con?dence, F (4,171)=22.6,
population (n=25 for Experiment 2a, 32 for Experiment
p<.001. In Experiment 2b, for accuracy, F(4,145)=18.9,
2b), land area (n=25 for Experiment 2a, 32 for Experi-
p<.001; for con?dence, F(4,145)=15.0, p<.001. The ro-
ment 2b) and population density (n=25 for Experiment
bustness of the differences in accuracy among the groups
2a, 32 for Experiment 2b). The other two groups made
is re?ected in a robust correlation of .864 between accu-
binary comparisons between U.S. cities on the dimension
racy and con?dence, using group averages in both sub-
of metropolitan area population (n=18 for Experiment 2a,
experiments as the unit of analysis.
28 for Experiment 2b) and driving distance from Athens,
Georgia (n=19 for Experiment 2a, 28 for Experiment 2b).
3.2.2 Bet acceptance
Overall bet acceptance in the two sub-experiments is pre-
2In their empirical studies, Heath and Tversky (1991) established
sented in Figure 2 as a function of groups, shown in de-
two levels of competence, but they did not claim that competence is an
inherently binary construct. We take it to be a variable that can have
scending order of accuracy among groups. Bet accep-
several levels, and that may be continuous.
tance is closely correlated with accuracy at the group

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
197
1.0
1.0
(a)
Bet acceptance
(b)
Bet acceptance
Accuracy
Accuracy
0.9
0.9
Confidence
Confidence
0.8
0.8
0.7
0.7
0.6
0.6
0.5
City
State
State
State pop.
City
0.5
City
State
State
State pop.
City
distance
area
population
density
population
distance
area
population
density
population
Figure 2: Overall bet acceptance, accuracy and con?dence in the ?ve groups in (a) Experiment 2a and (b) Experiment
2b. Bet acceptance correlated strongly with accuracy, but this could be partly attributable to correlations between
accuracy and con?dence.
level — in Experiment 2a, r=.98; in Experiment 2b, r=.96
curves. For this analysis we constructed a linear model
(each correlation based on ?ve pairs). Such a correla-
of each participant’s betting function, using con?dence
tion is the essential claim of the competence hypothesis;
level as the predictor variable and bet acceptance rate as
therefore, these very strong correlations constitute prima
the criterion. A linear model was used because the bet-
facie evidence for the competence hypothesis. However,
ting function has consistently been approximately linear
one must remember that in such settings of betting on
at the group level in both Goodie (2003) and the present
knowledge, betting frequency correlates positively with
studies. This produced a slope and y-intercept for each
con?dence, which is re?ected in increasing betting curves
participant. (Here, the y-axis re?ects bet acceptance.)
such as those in Figure 1. Accuracy also correlates with
Then, in each sub-experiment, we computed a partial cor-
con?dence, as can also be seen in Figure 2 — in Exper-
relation between each individual’s accuracy and the slope
iment 2a, r=.88; in Experiment 2b, r=.84 at the group
and intercept of their betting functions, controlling for the
level. In short, when participants have competence, they
average accuracy observed in the participant’s question
also have high con?dence, which may account for the
domain group. We performed this partial correlation in
increased bet acceptance. Consequently, as Heath and
order to observe only individual differences effects and
Tversky (1991) did in their Experiment 4, domains must
not group treatment effects. In Experiment 2a, neither
be compared at equivalent levels of con?dence.
slope (r(170)=.089 ; p=.247) nor intercept (r(170)=-.038
We achieved this by comparing curves relating bet ac-
; p=.621) correlated signi?cantly with accuracy. Given
ceptance curves to con?dence. Betting proportions across
that slope was not signi?cantly related to competence,
con?dence categories for the ?ve groups in both sub-
the intercept was a reasonable measure of the overall at-
experiments are shown in the two panels of Figure 3;
tractiveness of risk. The absence of a signi?cant cor-
each point on the graph represents the proportion of bets
relation indicates that competence did not increase risk
accepted at a given con?dence level. The graph re?ects
seeking. However, in Experiment 2b, as predicted by the
increasing risk seeking as a function of subjective proba-
competence hypothesis, accuracy did correlate positively
bility in all groups. Larger symbols re?ect domains of
and signi?cantly with intercept (r(147)=.183, p<.025),
greater accuracy. The competence hypothesis predicts
although the magnitude of the correlation is relatively
larger symbols to appear above smaller symbols, and this
small. (Slope and accuracy did not signi?cantly corre-
prediction receives little support. It is clear that any dif-
late; r(147)=-.124, p=.133.)
ferences between groups are small and do not re?ect a
consistent ordering as a function of competence.
3.3 Discussion
3.2.3 Individual variation
The results of these two sub-experiments provide further
quali?ed support for the competence hypothesis. The cor-
We also tested correlationally within groups for the effect
respondence between group-level accuracy and betting
of individual variation in competence on bet acceptance
proportions is strikingly close, which, in addition to sup-

---

Judgment and Decision Making, Vol. 2, No. 3, June 2007
Control and competence in bet acceptance
198
1.0
1.0
(a)
(b)
0.8
0.8
0.6
0.6
City dist.
City dist.
Bet acceptance
State area
Bet acceptance
State area
State pop.
State pop.
0.4
0.4
State dens.
State dens.
City pop.
City pop.
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
1.0
Confidence
Confidence
Figure 3: Bet acceptance among the ?ve groups in (a) Experiment 2a and (b) Experiment 2b. Groups with higher
accuracy are depicted with larger symbols.
porting the competence hypothesis, also bolsters the ro-
now, how much could you do between now and then to
bustness of using binary choice tasks with a half-range
improve your performance?
probability spectrum. However, the correlations between
Participants responded to these questions on a seven-
accuracy and con?dence, and between con?dence and
point Likert scale labeled from 1 (not at all) to 7 (very
betting proportion, needed to be taken into account, and
much). The ?rst two questions re?ected responses to the
this diminished the strength of evidence in favor of the
term “competence” and to a de?nition of competence, re-
competence hypothesis. At the individual level, the par-
spectively. The last two re?ected responses to the term
tial correlation between accuracy and betting curve eleva-
“control” and a de?nition of control utilized by Goodie
tion was small but signi?cant in Experiment 2b and non-
(2003). The de?nition of competence conveyed in Ques-
signi?cant in Experiment 2a.
tion 2 re?ects just one of multiple possible de?nitions.
This de?nition is social in nature, comparing one’s com-
petence with that of others. For both Heath and Tversky
4 Experiment 3
(1991) and Fox and Tversky (1995), this is appropriate.
We framed the survey question in a social-comparative
In Experiment 3, in addition to risk acceptance data,
way because non-comparative questions appeared to of-
we sampled subjective measures of both competence and
fer little more than synonyms of competence. The de?ni-
control. In this experiment, we replicated the methods of
tion of control conveyed in Question 4 represented an at-
the earlier experiments but sought further to examine how
tempt to reduce confusion about possible alternative def-
participants’ perceptions of their competence and control
initions of the term control, such as internal control.
correlated with performance measures.
4.2 Results and Discussion
4.1 Method
4.2.1 Calibration, overall bet acceptance and bet ac-
Participants (N=185) all encountered state population
ceptance curves
comparisons, in the Gains Only betting format. They
Average con?dence across both groups was .759, aver-
were divided into Answers (n=92) and Random (n=93)
age accuracy was .751, and average overcon?dence was
groups, which differed as they did in Experiment 1. In a
.007. The difference between the two groups was less
third phase, all participants answered the following sur-
than .007, and statistically non-signi?cant, for all three
vey questions:
measures.
1. How competent do you feel you are at this task?
Once again, those betting on their own knowledge
2. How do you think your abilities at this task compare
bet considerably more frequently than those betting on
to others?
events that were identical in every statistical way but ap-
3. How much control do you feel you had over this
peared random. The Answers group accepted 71.9% of
task?
all bets, whereas the Random group accepted only 45.6%
4. If you were to do this task again one week from
of all bets. This difference was statistically signi?cant

---