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 1B1A) 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 2B2A) „The Economist” is

offering you two annual subscribing options:

a) ONLINE access for $37b) ONLINE access +

printed copy for $77

2B) „The Economist” is offering you the following three annual subscribing options:

a) ONLINE access for $37b) Printed version for $77c) ONLINE access + printed

copy for $77Arieli (?):

• in 2A people choose ONLINE access only more often than in 2B• And in 2B they ONLINE access + printed more often than in2A

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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.

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 U 3 M

44) 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 5B5B) 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) $39 YES/NOb) $59 YES/NOc) $79 YES/NOd) $99 YES/NOe) $119 YES/NOf) $139 YES/NOg) $159 YES/NO

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:

a) $39 YES/NOb) $59 YES/NOc) $79 YES/NOd) $99 YES/NOe) $119 YES/NOf) $139 YES/NOg) $159 YES/NO

Johnson (1993)• In 5A people are willing to pay more than in 5B

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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. RGRRRb. GRGRRRc. GRRRRR

Tversky, Kahneman (1983) [conjuction fallacy]The correct answer is RGRRR but many people go for GRGRRR

77) 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. Linda is a bank teller (cashier)b. 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.

88) Try to make a ranking of what you think is the lifetime probability of death due to the following

events: a. Plane crash 1) Plane crash 1:11 Mlnb. Cancer 2) Terrorist attack 1:9.3 Mlnc. Car crash 3) Flood 1:30000d. Terrorist attack 4) Car crash 1:8000e. Flood 5) Murder 1:300f. Heart attack or stroke 6) Cancer 1:5g. Murder 7) Heart attack or stroke 1:2.5 How would you place the following two additional events in your ranking? h. Terrorist take over the plane (as in WTC)i. Terrorist attack or plane crash

[Conjuction vs Disjunction fallacy]

9A i 9B9A) Which lottery would you choose?

a. (0,0.9;45,0.06;30,0.01;-15,0.03)b. (0,0.9;45,0.07;-10,0.01;-15,0.02)

9B) Which lottery would you choose?

a. (0,0.9;45,0.06;30,0.01;-15,0.01;-15,0.02)b. (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 problemConditional probability

The right answer YES, many people answer NO

1111.1) 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 is the probability that in four weeks from now on Wednesday it will be:

• Sunny• Rainy

12B) According to your opinion what is the probability that in four weeks from now on Wednesday it will be:

• Sunny• Shower• 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 1413) 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

1515) 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 X = test: „positive”B = NO HIVY = test: „negative”

1616) 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 17.217.1) 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 18.218.1) 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 19.219.1) 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 20.220.1) 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) Blueb) 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) c) Blue and Redd) 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 21B21A) 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)

2222) 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 CII C A BIII B C A

A vs B 2:1 A>BB vs C 2:1 B>CC vs A 2:1 C>A

23A i 23B23A) 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) Program A: 200 people will be

saved for sureb) Program B: 600 will be saved

with probability 1/3, nobody will be saved with probability 2/3.

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) Program A: 400 people will die for sureb) 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

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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