Paradoxes and decisions
PLAN
• Two sets of questions
• Two types of questions in each set:
– Denoted with a number and a letter A or B – these questions are different
for set A and set B, e.g. 1A) i 1B).
• There is no right answer to these questions. Only in connection with the answer to the
respective question in another set, one can say whether these answers are consistent
(in a sense specified later)
– Denoted only with a number e.g. 3) – these questions are the same for both
sets.
• There is usually one right answer to these questions.
• It is neither a test nor an experiment. You will remain
anonymous in your responses if you wish so.
Answers will not be
checked, collected or
used in any way
except for the
purposes of this
lecture
PLAN
• First stage: division into two groups
– Each person in a given group tries to
answer the questions from the set she/he
was assigned to.
– Time: about 20 minutes
– Statistics ??
• Second stage: division into 4-person
groups:
–
–
–
–
–
2 persons with set A, 2 persons with set B
Exchange of sets
Try to answer set-specific questions
Find differences in set-specific questions
Time: about 15 minutes
• Third stage: analysing questions and
discussion
– Time about 40 minutes
1A i 1B
1A) Suppose that you decided to watch a
good theatre performance. The ticket
costs $50. Entering the theatre you
realize that you have lost $50 note
from your wallet. Would you buy the
ticket?
1B) Suppose that you decided to watch a
good theatre performance and
bought a ticket for it for $50. Entering
the theatre you realize that you have
lost the ticket and cannot claim it
back. Would you buy another ticket?
Kahneman, Tversky (1984): (mental accounting)
• 88% answered YES in 1A)
• 46% answered YES w 1B)
2A i 2B
2A) „The Economist” is
offering you two annual
subscribing options:
a) ONLINE access for $37
b) ONLINE access + printed
copy for $77
2B) „The Economist” is
offering you the following
three annual subscribing
options:
a) ONLINE access for $37
b) Printed version for $77
c) ONLINE access + printed
copy for $77
Arieli (?):
• in 2A people choose ONLINE access only more often than in 2B
• And in 2B they ONLINE access + printed more often than in2A
3
3) Suppose that each card has a number on one side and a letter on
the other side. Which cards would you turn over to check whether
the following statement is true: If a card has a vowel on one side, it
has an even number on the other side.
4 U 3 M
Wason (1960) [positive confirmation bias]
• Many people answer: U+4
• The correct answer: U+3
People often try to confirm rather than reject/negate
4
4) You have been invited to take part in an experiment conducted by a
famous psychologist. You are presented with two boxes – one that
is transparent and you see there is $1000 in it, and another one
that is not transparent. You are told that the second box may
contain $1mln or nothing. (…) You are told that the psychologist has
already made her decision and filled the second box according to
her prediction. You are also assured that this psychologist has never
been wrong in her predictions. What will you choose?
• Only the non-transparent box
• Both boxes
Newcomb’s paradox of rationality
• Domination – choose two
• The psychologist is always right:
choosing one box gives always a
higher payoff
5A i 5B
5A) Suppose that you are about to go for a
weekly trip in the Caribbean by ferry and you
would like to buy life insurance. The trip
costs $2000. You are offered an insurance
policy according to which you or your family
will be paid $1mln in case the ferry sunk due
to a terrorist or pirate attack should you die
or be heavily injured. Would you buy this
insurance policy at the following premiums:
5B) Suppose that you are about to go for a weekly
trip in the Caribbean by ferry and you would
like to buy life insurance. The trip costs
$2000. You are offered an insurance policy
according to which you or your family will be
paid $1mln in case the ferry sunk and should
you die or be heavily injured. Would you buy
this insurance policy at the following
premiums:
a)
b)
c)
d)
e)
f)
g)
a)
b)
c)
d)
e)
f)
g)
$39 YES/NO
$59 YES/NO
$79 YES/NO
$99 YES/NO
$119YES/NO
$139YES/NO
$159YES/NO
$39 YES/NO
$59 YES/NO
$79 YES/NO
$99 YES/NO
$119YES/NO
$139YES/NO
$159YES/NO
Johnson (1993)
• In 5A people are willing to pay more than in 5B
6
6) Consider a regular dice with six faces, two of them being
red and four – green. The dice will be thrown 20 times and
the sequence of Green (G) and Red (R) will be recorded.
You should choose one out of three sequences. If the
sequence you chose, occurs in subsequent dice throws, you
will win $100. Indicate the sequence you would like to bet
on:
a. RGRRR
b. GRGRRR
c. GRRRRR
Tversky, Kahneman (1983) [conjuction fallacy]
The correct answer is RGRRR but many people go for GRGRRR
7
7) Linda has just graduated in sociology. During her studies she was a
very active member of several student organizations promoting
political equality and touching upon various social issues. Linda is a
vegetarian and tries to ride a bicycle as often as possible. Based on
what you know about Linda, which statements about her seems
more likely to be true:
a.
b.
Linda is a bank teller (cashier)
Linda is a bank teller and an active member of a feminist
movement
Tversky, Kahneman (1983) [conjuction fallacy]
People often choose b. and it should be a.
8
8) Try to make a ranking of what you think is the lifetime probability of death due to
the following events:
a.
b.
c.
d.
e.
f.
g.
Plane crash
Cancer
Car crash
Terrorist attack
Flood
Heart attack or stroke
Murder
1) Plane crash
1:11 Mln
2) Terrorist attack
1:9.3 Mln
3) Flood
1:30000
4) Car crash
1:8000
5) Murder
1:300
6) Cancer
1:5
7) Heart attack or stroke 1:2.5
How would you place the following two additional events in your ranking?
a.
b.
Terrorist take over the plane (as in WTC)
Terrorist attack or plane crash
[Conjuction vs Disjunction fallacy]
9A i 9B
9A) Which lottery would you choose?
9B) Which lottery would you choose?
a.
b.
a.
b.
(0,0.9;45,0.06;30,0.01;-15,0.03)
(0,0.9;45,0.07;-10,0.01;-15,0.02)
(0,0.9;45,0.06;30,0.01;-15,0.01;-15,0.02)
(0,0.9;45,0.06;45,0.01;-10,0.01;-15,0.02)
Tversky, Kahneman (1986) [cancelation, similarity, framing]
Lotteries in 9A are the same as in 9B, they are presented differently
• In 9A people choose a) more often than in b)
• In 9B people more often choose b) than a)
10
10) You are standing in front of three gates. You know, that there is the
prize (most recent M6 model of BMW) behind one of the gates. You
will get it if you choose the right gate. You have chosen one of the
gates. The host of a TV program, who knows which gate conceals
the prize, opens one of the other gates and shows that is empty.
Subsequently, he offers you the possibility to change your original
choice of gates. Do you agree to his proposal?
• YES
• NO
Famous Monty Hall problem
Conditional probability
The right answer YES, many people answer NO
11
11.1) You are given a new coffee mug (photo below). For what minimal price
would you sell it? Give a price between $1-$50.
11.2) There is a coffee mug for sale. For what maximal price would you buy it?
Give a price between $1-$50.
Kahneman, Knetsch, Thaler (1990) [endowment effect, WTA-WTP disparity]
WTA>WTP
12A i 12B
12A) According to your opinion what 12B) According to your opinion what is
is the probability that in four
the probability that in four weeks
weeks from now on Wednesday it
from now on Wednesday it will be:
will be:
• Sunny
• Sunny
• Shower
• Rainy
• Sprinkle
• Drizzle
• Rainstorm
• Deluge
• Cloudburst
• Squall
• Sleet
P(Rainy) is often reprorted to be smaller than P(shower or sprinkle
or drizzle or rainstorm or deluge or cloudburst or squall or sleet)
13 i 14
13) The taxi drivers in New York sometimes use the following
“targeting” strategy: They work each day until they earn a certain
amount of money for one day (let’s say $200). According to you,
this is a good or bad strategy?
Camerer, Babcock, Loewenstein, Thaler (1997)
The right answer: bad strategy
14) What is the probability of winning the state lottery (6 numbers
between 1-49 has to be the same as the one drawn)? Is the expected
payoff of your winning in the lottery higher or lower is there is an
accumulation in the lottery?
The right answer: There are
possibilities.
In case of accumulation the expected payoff is even smaller than without
accumulation
15
15) Imagine that you have been tested for the presence of HIV and the
test result is “positive”. What is the probability that you are infected
indeed?
(Probability of infection is 0.0769%. Test accuracy is 99.5%)
A = HIV
B = NO HIV
X = test: „positive”
Y = test: „negative”
16
16) You are observing a sequence of tosses of a fair coin. In the last ten
rounds, Head was the result: HHHHHHHHHH. What will you bet on:
•
•
•
•
•
•
•
•
2:1
on Tail
5/3:1
on Tail
4/3:1
on Tail
1:1
Tail vs. Head
4/3:1
on Head
5/3:1
on Head
2:1
on Head
None of the above
Shefrin, Stetman (1985) [disposition effect – hold losers, sell winners]
More people bet on Tail, but the probability is equal
17.1 i 17.2
17.1) Choose one lottery:
P=(1 mln, 1)
Q=(5 mln, 0.1; 1 mln, 0.89; 0 mln, 0.01)
17.2) Choose one lottery:
P’=(1 mln, 0.11; 0 mln, 0.89)
Q’=(5 mln, 0.1; 0 mln, 0.9)
Kahneman, Tversky (1979) [common consequence effect
violation of independence]
Many people choose P over Q and Q’ over P’
18.1 i 18.2
18.1) Choose one lottery:
P=(3000 PLN, 1)
Q=(4000 PLN, 0.8; 0 PLN, 0.2)
18.2) Choose one lottery:
P’=(3000 PLN, 0.25; 0 PLN, 0.75)
Q’=(4000 PLN, 0.2; 0 PLN, 0.8)
Kahneman, Tversky (1979) [common ratio effect, violation of
independence]
Many people choose P over Q and Q’ over P’
19.1 i 19.2
19.1) Choose one lottery:
P-bet = (100 PLN, 0.8; 0 PLN, 0.2)
$-bet = (1000 PLN, 0.1; 0 PLN, 0.9)
19.2) And now imagine that you own the right to play the P-bet. For
what amount of money (minimally) would you be willing to sell this
right. Do the same for the $-bet.
Grether, Plott (1979) [preference reversal, transitivity violation?]
Many people:
• Choose the P-bet over the $-bet in a direct choice
• But assign higher Certainty Equiv. to the $-bet CE(P-bet) < CE($-bet)
20.1 i 20.2
20.1) There is 90 balls in the urn – 30 blue balls and 60 that are either yellow or red.
You pick a colour. Then one ball is drawn randomly from the urn. If the colour of the
ball drawn and the colour of the ball you chose match, you will win $100. Which
coloor do you pick? (One answer)
a)
b)
Blue
Yellow
20.2) Continuation: If the colour of the drawn ball is of one the colours you bet on,
you win $100. Which colours do you pick? (One answer)
a)
b)
Blue and Red
Yellow and Red
Ellsberg paradox (1962?) [uncertainty aversion]
Many people choose:
• Blue in 20.1
• Yellow and Red in 20.2
Why is it strange…
21A i 21B
21A) You are given $40. You can
choose to take it and go or you
can buy for it a lottery which
gives you equal chance of
getting nothing or $100?
Would you buy?
21B) You are offered a lottery
which gives equal chance of
getting nothing or $100 for a
price of $40. Would you buy
it?
Thaler, Johnson (1990) [house money effect]
People answer:
• YES in 21A) more often than in 21B)
22
22) You have three restaurants to choose from: French “La Coupole”, Italian
“Tre Panche” or Polish “Sarmatia”. You selected three equally important
selection criteria: Food quality, Service quality and atmosphere, Price. You can
order the three alternatives according to each of the three criteria. As it is
hard for you to choose the alternative right away, you decided to choose the
best alternative by voting in pairs (which of the two alternatives has higher
sum of ranking positions according to all three criteria)
Is this a good method of restaurant selection? (i.e. for any ranking orderings,
you can reveal the winner)
Condorcet paradox [intransitivity]
Example: three options A,B,C and three choice criteria I,II,III
1
2
3
I
A
B
C
II
C
A
B
III
B
C
A
A vs B 2:1 A>B
B vs C 2:1 B>C
C vs A 2:1 C>A
23A i 23B
23A) Your country is plagued with an
outbreak of an exotic Asian disease,
which may kill 600 people. You are
responsible for making decision
about two programs. Which program
will you choose:
23B) Your country is plagued with an
outbreak of an exotic Asian disease,
which may kill 600 people. You are
responsible for making decision
about two programs. Which program
will you choose:
a)
a)
b)
Program A: 200 people will be
saved for sure
Program B: 600 will be saved with
probability 1/3, nobody will be
saved with probability 2/3.
b)
Program A: 400 people will die for
sure
Program B: Nobody will die with
probability 1/3, 600 people will die
with probability 2/3.
Kahneman, Tversky (1979) [framing, Asian disease]
Lotteries in 23A) are exactly the same as lotteries in 23B).
Framing is different though.
People often:
• Choose program A in 23A
• Choose program B in 23B
24
24) What is the probability that in the room we are currently in, at
least two people have their birthday on the same day?
Birthday problem