Introduction à la théorie de la
décision
Ferdinand M. Vieider
University of Munich
Home: www.ferdinandvieider.com
Email: fvieider@gmail.com
Université Libre de Tunis, April 6th, 2012
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What is Decision Theory?
• Decision Theory: studies ensemble of human
decision making processes, individual and social
• It mostly becomes relevant in situations with
some complexity (e.g. risk, uncertainty)
• It is closely related to several other fields:
- operations research
- linear programming
- game theory
- experimental economics
- behavioral economics
- cognitive psychology
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- social psychology
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Main similarities and differences
• Cognitive Psychology: the methodology of
investigation and topics is very similar; however:
rationality concepts borrowed from economics
• Experimental Economics: DT methodology is
very often experimental, however not exclusively
so; also historically focus in individual decisions
• Behavioral economics: comes closest, at least
in descriptive aim; however, decision theory also
encompasses rationality models, not only
deviations from such models
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Why experiments?
• All of the disciplines just discussed make
extensive use of experiments
• Experiments allow to reproduce stylized
situations of interest
• Most importantly: one can vary one
independent variable at a time
• This makes it possible to isolate causal
relationships (not just correlation)
• Further distinctions: lab experiments versus
field experiments, artificial experiments versus
natural experiments
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Lecture Overview
• Overview of different approaches: normative,
descriptive and positive
• Origins of decisions theory: expected value
theory to deal with risk
• Introducing subjectivity: expected utility and its
behavioral foundations
• Expected utility's failure as a descriptive theory
of choice
• Descriptive theories of choice: Prospect Theory
(and what it can explain)
• Uncertainty, ambiguity aversion, and other
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puzzles (Wason, Monty Hall)
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Normative, Descriptive, and
Prescriptive approaches:
From Expected
Value to Expected
Utility Theory
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Different Approaches to DT
• Different approaches to decision theory:
normative, descriptive, and prescriptive
• Normative theories describe how a perfectly
rational and well-informed decision maker
should behave
• Descriptive analysis focuses only on actually
observed behavior, and tries to find regularities
• Prescriptive analysis has the aim of helping
real-world decision makers in making better dec.
• Are normative theories also good descriptive
theories?
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Descriptive Issues
• At the outset, normative theories were taken
for descriptive purposes as well
• However: deviations from models soon
emerged (falsification of theory)
• Sprawling of descriptive theories that try to
explain “anomalies”
• Several issues that are often confounded:
evidence from lab produces focus on cognitive
limitations and stability of preferences
• Real world: problems of awareness
(“knowledge about knowledge”), then
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information search and processing
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Prescriptive Analysis
• Prescriptive analysis moves from a limitedinformation and processing perspective
• Goal: helping to reach the best decision given
the information at hand
• In experiments normative and prescriptive
approach often coincide (complete info)
• This means that real-world situations are often
very different (external validity issue)
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The origins of decision theory
• Historically, the concept of probabilities and
how to deal with them is rather recent.
• In the 1600s, Blaise Pascal and Pierre Fermat
developed expected value theory
• According to EVT, a prospect can be
represented as its mathematical expectation:
0.5
DT 100
p*X + (1-p)*Y=
0.5*100= 50
0.5
DT 0
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Example: EV normative?
Choice between 2 known-probability events:
0.9
DT 10
DT 0
0.1
EV: 0.9*10+0.1*0=9 <
0.2
DT 50
DT 0
0.8
0.2*50+0.8*0=10
According to EVT, you should choose the
lottery to the right. Is that your preference?
Does your preference change if we increase
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the amounts *1000, to 10,000 & 50,000 DT?
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From EV to EU
• Expected value may not be a reasonable
theory, even normatively, for large amounts
• Also, these amounts may not be the same for
everybody (wealth situation, preference)
• To deal with this, we need one subjective
parameter: Expected Utility Theory
• In EUT, the value of a prospect is given again
by its mathematical expectation, but instead of
using (objective) monetary amounts we now use
(subjective) utilities of those amounts
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Example: EV versus EU
Choice between 2 known-probability events:
0.3
DT 400
DT 0
0.7
EU: 0.3*u(400)+0.7*0=0.3 ?
0.5
DT 200
DT 0
0.5
0.5*u(200)+0.5*0=?
The extreme outcomes can always be
normalized to 0 and 1. But how about
intermediate outcomes?
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Eliciting Utilities
How can we elicit the missing utility?
p
CE
DT 400
~
1-p
DT 0
We elicit either CE or p such that
U(CE)=p*U(400)+(1-p)*U(0)=p
Let CE=200 and elicit p (in reality easier for
DM to elicit CE!)
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Example reconsidered:
Choice between 2 known-probability events:
0.5
€200
0.3
€400
?
0.5
€0
0.7
€0
U(0)=0, U(400)=1; assume p=0.65, then
U(200)=0.65
This means that now:
0.5*U(200)+0.5*U(0)=0.325 > 0.3*U(400)+0.7*U(0)=0.3
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Subjective Utility and Risk
• Given the non-linearity in the utility function,
preferences can change relative to EV
U(€)
EV
EU
• EUT: concavity=risk aversion. This is not
universally valid!
€
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EV or EU?
• EV is reasonable for small stakes, however
most important decisions deal with large stakes
• Also, many important decisions deal with nonquantitative decisions such as health states
• For the latter EV cannot be defined; also: what
if you have utility over money plus other things?
• Expected Utility is thus generally more useful;
it is however more complex, especially when
combined with unknown probabilities
• For the moment, we consider only utilities over
monetary outcome with known probabilities
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The St. Petersburg Paradox
• Why concave utility? Consider the following
example:
• A bet is proposed to you: a fair coin is flipped
until the first head come up; the amount you win
at first flip is DT2, then DT4, then DT8, so that if
head comes up at the kth flip you get DT2k
• How much would you be willing to pay to play
this game?
• The Expected value of the gamble is infinite:
1/2*2+1/4*4+1/8*8... = 1+1+1... = ∞
• This goes to show that EV does not hold
empirically when large amounts are at stake18
Risk Aversion and Risk seeking
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• Risk Aversion: a prospect is considered inferior
to its expected value
• Risk Seeking: a prospect is preferred to its
expected value
• Risk Neutrality: a prospect and its expected
value are equally valuable
p
p*X+(1-p)*Y
X
?
1-p
Y
•¡Do not confuse risk aversion with concave 19U!
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Behavioral Foundations of EU
• Behavioral foundations are properties of
behavior (axioms) underlying a theory
• They are very helpful in that a theory can be
decomposed into some intuitive rule
• E.g., saying that EU holds is equivalent to
saying that preferences satisfy:
- weak ordering
- standard gamble solvability
- standard gamble dominance
- standard gamble consistency (or the
stronger independence condition)
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Independence
• One of the most discussed issues is the
following independence of common alternatives:
p
p
x
y
≥
x≥y
1-p
C
1-p
C
How intuitive do you find this condition?
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Example: Allais (common consequence)
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• Consider the following two choices:
.10
A
.89
€1,000,000
.01
€0
1
B
€5,000,000
.10
€5,000,000
.90
€0
€1,000,000
C
.11
€1,000,000
D
.89
€0
The most common pattern is BC. This violates
the independence axiom (rational: AC or BD).
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Example: Compound Prospects
• Consider the following two choices:
1/6
1/6
2/3
€200
2/3
€0
€100
1/2
€200
€100
1/3
?
1/3
€0
1/2
2/3
€0
Which one do you prefer?
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EUT and Insurance
• Under EUT, risk aversion coincides with a
concave utility function, and risk seeking with a
convex utility function
• This does not hold generally: shortly we will
see risk seeking with a concave utility function!
• With a concave utility function, the expected
utility of a prospect is lower than the utility of the
expected value:
p*U(x)+(1-p)*U(y)<U(p*x+(1-p)*y)=U(EV)
• The difference between the EV of a prospect
and its Certainty Equivalent is the Risk Premium
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EUT and Insurance
• Under EUT, risk aversion coincides with a
concave utility function.
U(DT)
U
U(y)
U(p*x+(1-p)*y)
p*U(x)+(1-p)*U(y)
U(x)
x
CE
p*x+(1-p)*y
y
What is the risk premium here?
DT
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Insurance example
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• There is a 5% risk that your house may be
flooded, potential damages are – DT100,000
• EV = – DT5,000, However, if you are risk
averse, the CE is lower, e.g. CE = – DT6000
• There is a positive risk premium of DT1000; by
the law of large numbers, the insurance will pay
DT5000 on average, and can thus make up to
DT1000 by ensuring your risk
• Could you represent this problem in a graph?
What changes because of the negative
outcome?
• When is it rational to take out insurance and
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when not?
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Graph Insurance Example
• Nothing changes: implicit reference point
problem (previous wealth)
U(DT)
U
U(y)
U(p*x+(1-p)*y)
p*U(x)+(1-p)*U(y)
U(x)
X
CE
= –€100000
p*x+(1-p)*y
y=0
DT
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Insurance and Lotteries
• We can explain insurance with concave utility
under EUT
• In theory, we can also explain lottery play, but
we need convex utility for that
• However: many people take up insurance and
play lottery at the same time. How can this be
explained?
• Under EUT, we would need convex and
concave sections of the utility function
• We would also need these to hold at different
levels of wealth
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Typical Risk Preferences
• People are typically risk seeking for small
probabilities (± p<0.15): lottery play
• For larger probabilities, people tend to be risk
averse: CE<EV
• For losses, however, these findings are
inverted, with risk aversion for small probabilities
and risk seeking for large probabilities
• EUT cannot explain such preferences, since
probabilities enter the equation linearly
• EUT is thus violated descriptively, so that we
need a more flexible theory to explain these
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phenomena
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Descriptive Theories of Choice:
Prospect Theory
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Experimental Data: Typical CEs
• Using a Prospect offering either €100 or 0 with
different probabilities, I asked choices between
the prospect and different sure amounts
• The switching point between the sure amount
and the prospect indicates a person's CE
• The probabilities were 0.05, 0.5, and 0.9
• Mean CEs obtained from this classroom
experiment in France were:
EV
probability
CE (mean)
CE/EV
€5
0.05
€10.96
2.14
€50
0.5
€46.48
0.93
€90
0.9
€68.37
0.76
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Your (average) utility function
• Remember that U(CE)=p
• Thus: U(11)=0.05; U(46)=0.5; U(68)=0.9, and
we can always set U(0)=0, U(100)=1
U(X)
0.9
0.5
0.05
11
46
68
X
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Prospect Theory
• Kahneman & Tversky (1979; Econometrica)
brought psychological intuition to economics:
• Risk attitudes for small amounts are driven by
feelings about probability, not money
• We can thus let probability be the subjective
parameter, and assume utility to be linear:
PV=w(p)*x+(1-w(p))*y
• Linear utility seems reasonable for small
monetary amounts (but not large!)
• For large amount, we can combine probability
weighting with utility:
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PU=w(p)*u(x)+(1-w(p))*u(y)
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Probability Weighting: Attitudes to Risk
• We have seen that CE=p*U(100); if utility is
linear, then p must be transformed
• Let us thus assume that CE=w(p)*100, where
w represents a weighting function
• From our previous results we get:
- w(0.05)=11/100=0.11
- w(0.5)=46/100=0.46
- w(0.9)=68/100=0.68
• From, this, we can plot a probability weighting
function assuming w(0)=0, w(1)=1
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Probability Weighting Function
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Fig .1 : Probability
Weighting
Function
0.8
w p
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
p
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Insurance and Lottery Play
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• Notice how this function can explain
contemporary insurance and lottery play through
overweighting of small probabilities
• Also, there are jumps at the endpoints: the
possibility and certainty effects
• The latter can explain the Allais paradox
(common consequence effect)
• It also captures common risk attitudes quite
well: fourfold pattern of risk attitudes
• However, with linear utility it may have
problems accommodating decisions over large
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stakes
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Utility: Attitudes towards Outcomes
• We have assumed linear utility above:
however, we have seen that this is not always
reasonable (St. Petersburg paradox)
• Even assuming concave utility, it has problems
dealing with mixed gambles
• Example from Rabin, Matthew (2000). Risk
Aversion and Expected-Utility Theory: A
Calibration Theorem. Econometrica 68 (5):
If a DM turns down (.5:110; -100), then she will
turn down a 50:50 of -1000 and X for all X
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Prospect Theory Utility Function
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• In PT, the utility function describes attitudes
about money only, not probabilities
U(X)
convex
concave
X
kink
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Properties of Utility
• Concave utility for gains means that even for
small probabilities one can be risk averse for
very large outcomes (insensitivity)
• For losses one can be risk seeking for small
probabilities for very large outcomes
• Loss aversion: a loss is felt more than a
monetarily equivalent gain
• Loss aversion has been used to explain the
status quo bias, endowment effect, myopic loss
aversion (equity premium puzzle), etc.
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Loss Aversion
• Under loss aversion, “losses loom larger than
gains”
0.5
DT ?
0 ~
– DT50
0.5
How high would the gain need to be to make
you indifferent between playing and not playing
the prospect?
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Deduction of Loss Aversion
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• Let us assume that DT 100 was elicited as
gain that makes you indifference
• Let us also assume that utility is linear over
gains and losses, but that you are loss averse
Then U(X)=X if X≥0; and U(X)= –λ*X if X<0
u(0)=0.5*u(100)+0.5*U(–50)
0 =0.5*100+0.5*(–λ)*50
λ*25=50
λ=2
What other assumption underlies this elicitation
of the loss aversion parameter λ?
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Some functional forms
• A simple form for the utility that has been
proposed is:
• U(X)=Xα
if X≥0
• U(X)= –λ*Xβ
if X<0
• Can you see why the derivation of loss
aversion as done before is an approximation?
• Some popular functional forms for probability
weighting functions are:
w(p)=pφ/(pφ+(1-p)φ)1/φ
α
w(p)= exp(-ξ (-log p)
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Reference Point: Status Quo Bias
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• Loss aversion is found to be the strongest
phenomenon empirically
• It stands and falls however on the
determination of the reference point
• Most of the time, the reference point is
assumed to be current wealth, or the status quo
• This means that people are often reluctant to
switch from the status quo, no matter what that
status quo is
• This means that changes are perceived as
gains and losses relative to status quo, with
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losses looming larger
Reference Point: Endowment Effect
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• The endowment effect was found by artificially
establishing a reference point
• Some people are randomly given one objects
and others with a different one (e.g. mugs v.
pens)
• People are then given the opportunity to
exchange the object in their possession
• A large majority of people is found not to
exchange their object
• This holds true for both objects; since they
have been randomly assigned, this can however
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not express true (average) preferences
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From known to unknown probabilities:
Subjective expected
utility and the
Ellsberg Paradox
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Unknown Probabilities
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• We have so far only considered the case of
risk, where objective probabilities are known
• Good representation of situations such as
lottery or well-established medical processes
• However: most probabilities are unknown:
stock market, entrepreneurship, education
• In this case one can deduce subjective
probabilities from observed decisions
• Savage (1954) put forth some desirable
attributes for decision making under uncertainty:
Subjective Expected Utility Theory
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Ambiguity Aversion
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You are asked to choose between two urns,
one 50:50, one unknown proportion
of
colors
? 20–?
10 R
10 B
20 R & B
in unknown
proportion
• First you are asked to choose which color you
would like to bet on, then which urn
• Which color would you rather bet on? And
which urn would you prefer to bet on?
• This phenomenon was discovered by
Ellsberg (1961): it violates subjective expected
utility theory since probabilities are the same47 (!)
The Ellsberg Paradox
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When asked for a color preference, most
people are indifferent: prr = prb; par= pab
Most people however have a strict preference
for betting on the known-probability urn, no
matter what which color: prr>par & prb > pab
This implies: prr + prb = 1 > par + pab; however,
probabilities cannot sum to less than 1, hence
the paradox
Prospect Theory has recently been adapted to
deal with this: Source functions, AER 2011
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More realistic decisions under uncertainty49
• Uncertainty has generally been studied in
opposition to risk, not in its own right
• Also: Ellsberg has created strong focus on 5050 prospects
• However: people react differently to different
probability levels (just as for risk)
• Also, people react differently to different
sources of uncertainty (dislike vague
probabilities, but may like uncertainties they
have expertise in-->betting on football)
• Applications: home bias in finance; stock
market participation puzzle;
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Typical Source functions
linear
risk
50
uncertainty
1
0.9
0.8
0.7
w(p)
0.6
0.5
0.4
0.3
0.2
0.1
0
p
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Probability Calculus and Logical
Induction:
Monty Hall's Doors
and the Wason
Selection Task
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Monty Halls Doors
• There are three doors, one of which hides a
car, and two with a goat behind
• You can choose a door. After you have
chosen, the host opens one of the other two
and reveals a goat
• If given the opportunity, should you switch or
stay with your original choice?
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To switch or not to switch: 1
• Imagine that the car is behind door 1, and
the other two doors hide goats
• If you have chosen door 1, the host opens
either door 2 or 3:
In this case, switching loses the prize
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To switch or not to switch: 2
• Imagine again that the car is behind door 1,
and the other two doors hide goats
• If you have chosen door 2, the host opens
door 3 for sure:
Now, switching gives you the prize
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To switch or not to switch: 3
• Imagine again that the car is behind door 1,
and the other two doors hide goats
• If you have chosen door 2, the host opens
door 3 for sure:
Now, switching again gives you the prize
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Summing it up
• We have just seen that switching gets you
the prize in 2 out of 3 cases
• Since the structure is symmetric if we
assume the prize is behind another door, the
probability of winning if switch is 2/3
• This is because the door you pick at first
gives a 1/3 chance; the other two doors
together though give you a 2/3 chance
• Since the removed door is always one of the
other two, you are left with a 2/3 chance by
switching
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Wason's Abstract Selection Task
• There a 4 cards, all of which have a letter on
one side and a number on the other
• Two cards show a number (4 and 7), two
show a letter (O and G):
4
A
7
G
Which card(s) do we need to turn over to test
the logical implication: vowel-->odd (if there is
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a vowel on one side then odd number on other)
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Adding context
• There are 4 rooms with closed doors and
one person in each room
• You know one is older than 18, one younger,
one drinks wine, and one a soda
<18
W
(wine)
>18
S
(soda)
Which door(s) do we need to open to make sure
nobody under drinking age drinks alcohol?
How would you write the problem down in
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logical notation?
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Wason Revealed
• Were your answers to the two questions
above equal? Why, or why not?
• One potential problem lies in the
formulation; different formulations of abstract
task were only partially effective
• The most common answer is to turn around
only the vowel-->confirmation bias
• Confirmation biases are very common, also
in scientific research (how many white swans
do you need to observe to conclude that all
swans are white?)
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