ellsberg’s paradoxes: problems for rank- dependent utility explanations cherng-horng lan &...

Ellsberg’s paradoxes: Problems for rank-dependent utility explanations

Cherng-Horng Lan & Nigel Harvey

Department of Psychology

University College London

(Email: [email protected])

mailto:[email protected]

2

Introduction

• Savage’s states-of-nature representation dominates the theoretical analysis of decision under uncertainty.

• Theorists tend to modify Savage’s behavioural axioms to accommodate the Ellsberg paradoxes without considering whether or how people use states of nature.

3

Analysis of decision under uncertainty within Savage’s framework

1. Formulate the states of nature related to a particular decision problem.

2. Draw a decision matrix for this particular decision problem.

3. Apply his decision principles (i.e., behavioural postulates or axioms).

4

Ellsberg’s two-colour problem (I)

Betting on R1 (drawing a red ball from Box 1):

You choose to draw a ball from Box 1. If you draw a red ball, you will win £100 and otherwise nothing. Similarly,

Betting on B1, Betting on R2 and Betting on B2

100 Total Balls100 Total Balls

?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2

5

Ellsberg’s two-colour problem (II)


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2

Which of the following would you choose?

Question 1: Betting on R1, Betting on B1 or Indifference?

Question 2: Betting on R2, Betting on B2 or Indifference?

Question 3: Betting on R1, Betting on R2 or Indifference?

Question 4: Betting on B1, Betting on B2 or Indifference?

Typical Choices:

Indifference

Indifference

Betting on R1

Betting on B1

6

Analysis of the two-colour problemStep 1: Identify states of nature

R1 & R2 R1 & B2 B1 & R2 B1 & B2States

• R1: Drawing a red ball from Box 1.• B1: Drawing a black ball from Box 1.• R2: Drawing a red ball from Box 2.• B2: Drawing a black ball from Box 2.

2 x 2 Cartesian product: Four states

7

Analysis of the two-colour problemStep 2: Formulation of the decision matrix

R1 & R2 R1 & B2 B1 & R2 B1 & B2Betting on R1 £100 £100 £0 £0Betting on R2 £100 £0 £100 £0Betting on B1 £0 £0 £100 £100Betting on B2 £0 £100 £0 £100

States

8

Analysis of the two-colour problemStep 3: Application of the sure-thing principle (STP)

R1 & R2 R1 & B2 B1 & R2 B1 & B2Betting on R1 £100 £100 £0 £0Betting on R2 £100 £0 £100 £0Betting on B1 £0 £0 £100 £100Betting on B2 £0 £100 £0 £100

States

9

Messages theorists have taken from the two-colour problem

Observed preferences: Betting on R1 ~ Betting on B1

> Betting on B2 ~ Betting on R2

1. The obedience to transitivity.2. The violation of the sure-thing principle.3. Non-additive subjective probability.

10

Rank-dependent utility theory

1. Maintain transitivity.

2. Restrict the application of the sure-thing principle to comonotonic sets of acts.

3. Allow non-additive subjective probability (or capacity) that may be induced by insufficient information about likelihood of events (e.g., uncertainty aversion).

11

Comonotonicity and rank-dependent hypothesis

s1 s2 s3 s4

Option A 10 7 6 2Option B 11 8 5 2Option A' 10 7 6 4Option B' 11 8 5 4Option A'' 10 7 6 9Option B'' 11 8 5 9

States

s1 > s2 > s3 > s4

s1 > s2 > s3 > s4

s1 > s4 > s2 > s3

1. STP holds in the set of {A, B, A’, B’} (comonotonic set)

2. STP will be violated in the set of {A, B, A’’, B’’} and in the set of {A’, B’, A’’, B’’} if subjective probability is not additive (uncertainty aversion).

Rank of states

12

Rank-dependent utility explanations for the Ellsberg paradox

• The acts in the two-colour problem are not pair-wise comonotonic.

• Uncertainty aversion may induce non-additive subjective probability.

• Consequence: the violation of the sure-thing principle.

13

Messages theorists have failed to take from the two-colour problem

• The decision matrix of the two-colour problem is not given.

• People do not necessarily formulate the decision matrix by themselves.

• Without the decision matrix, people cannot apply the sure-thing principle.

14

Empirical question

1. If the decision matrix of the two-colour problem is given, or

2. If the states of nature related to the two-colour problem is highlighted by wording,

then will people obey the sure-thing principle?

15

Study 1: Fair-urn version of the two-colour problem

Imagine that there are two boxes on the table. In one box, there are 100 balls composed of 50 red and 50 black balls. In the other box, there are 100 balls but nobody knows how many balls are red and how many balls are black.


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls


?? Red Balls

?? Black Balls

50 Red Balls

50 Black Balls

Box 1 Box 2

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• Three problem presentations:

1. Ellsberg condition: without states of nature.

2. Savage condition: Matrix format (states of nature)

3. Savage condition: Written format (states of nature)

• Two complementary bets:

Question 1: Betting on R1, Betting on R2 or indifference?

Question 2: Betting on B1, Betting on B2 or indifference?

Three equivalent problem presentations and two complementary bets

17

Question 1:

Suppose that you are offered a game that is to be played as follows:1. You are required to choose a box and then to draw a ball from the box

you choose.2. If you draw a Red ball, you will win £100; otherwise, you will win

nothing.

Which box would you prefer to draw a ball from? �Box 1; �Box 2; �No Preference (Please tick one of the answers)

Question 2:

Suppose that the payoff scheme is changed as follows:1. You are still required to choose a box and then to draw a ball from the

box you choose.2. If you draw a Black ball, you will win £100; otherwise, you will win

nothing.

Which box would you prefer to draw a ball from?�Box 1; �Box 2; �No Preference (Please tick one of the answers)

Ellsberg condition

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Question 1:Suppose that you are offered a game that is to be played as follows:• You are required to draw a ball from Box 1 and draw a ball from Box 2.• Before drawing the balls, you have to choose one of the following

options, which define the payoffs depending on the four possible combinations of your drawing.

Which option would you prefer to play? �Option 1; �Option 2; �No Preference (Please tick one of the answers)

Question 2:Suppose that the payoff schemes are changed as follows:


Red from Box 1Red from Box 2

Red from Box 1Black from Box 2

Black from Box 1Red from Box 2

Black from Box 1Black from Box 2

Option 1: £100 £100 £0 £0

Option 2: £100 £0 £100 £0

Red from Box 1Red from Box 2

Red from Box 1Black from Box 2

Black from Box 1Red from Box 2

Black from Box 1Black from Box 2

Option 1: £0 £0 £100 £100

Option 2: £0 £100 £0 £100

Savage condition: the matrix format

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Savage condition: the written formatQuestion 1:Suppose that you are offered a game that is to be played as follows: 1. You are required to draw a ball from Box 1 and draw a ball from Box 2.2. You have to choose one of the following options before drawing the balls:(1) If you draw a Red ball from Box 1 and a Black ball from Box 2 or if

you draw both Red balls, you will win £100; otherwise, you will win nothing.

(2) If you draw a Black ball from Box 1 and a Red ball from Box 2 or if you draw both Red balls, you will win £100; otherwise, you will win nothing.


Question 2:Suppose that the payoff schemes are changed as follows:(1) If you draw a Black ball from Box 1 and a Red ball from Box 2 or if

you draw both Black balls, you will win £100; otherwise, you will win nothing.

(2) If you draw a Red ball from Box 1 and a Black ball from Box 2 or if you draw both Black balls, you will win £100; otherwise, you will win nothing.


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Results of study 1

Question 1: K1= Betting on R1 or A1= Betting on R2

Question 2: K2= Betting on B1 or A2= Betting on B2

N: No preference

Table 1. Fair-Urn VersionOther Four

Conditions K 1 & K 2 A 1 & A 2 K 1 & A 2 A 1 & K 2 N 1 & N 2 Choice combinations n

28* 5 1 1 5 0 40(70.0%) (12.5%) (2.5%) (2.5%) (12.5%) (0%) (100%)

6 5 2 3 16* 9 41(14.6%) (12.2%) (4.9%) (7.3%) (39.0%) (22.0%) (100%)

4 0 3 2 29* 2 40(10.0%) (0%) (7.5%) (5.0%) (72.5%) (5.0%) (100%)

Savage: Written

Note. K i = Known Option; A i = Ambiguous Option; N i = No Preference (i : the question number). The nine choice

combinations are divided into three categories: (1) ambiguity attitudes (avoidance or seeking); (2) consistentbeliefs; (3) other choice combinations. The asterisk indicates that the observed frequency is significantly higher

than the expected value, p < .05§.

Ambiguity Attitudes The Sure-Thing Principle

Ellsberg

Savage: Matrix

Betting on R1 & B1 Indifference

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Summary of study 1

• Ellsberg condition: the most common response (70%) was

(1) Betting on R1 > Betting on R2

(2) Betting on B1 > Betting on B2

(ambiguity aversion)• Savage’s matrix and written conditions: the most common

response (39% & 72.5%) was

(1) Betting on R1 ~ Betting on R2

(2) Betting on B1 ~ Betting on B2

(obeying the sure-thing principle)

22

Discussion of study 1

• Two possible reasons for indifference in Savage’s conditions:

1. Participants believed that the proportion of red to black balls in Box 2 is 50:50;

2. Participants were indecisive or confused by problem presentations.

23

Study 2: Unfair-urn version of the two-colour problem


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls

Box 1 Box 2


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls


?? Red Balls

?? Black Balls

49 Red Balls

51 Black Balls

Box 1 Box 2

Imagine that there are two boxes on the table. In one box, there are 100 balls composed of 49 red and 51 black balls. In the other box, there are 100 balls but nobody knows how many balls are red and how many balls are black.

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Rationale

• If participants believe that the proportion of red to black balls in Box 1 is 49:51 and in Box 2 is 50:50, then we may observe that


(2) Betting on B1 > Betting on B2• If participants are indecisive or confused, then

they may show indifference.

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Results of study 2

Question 1: K1= Betting on R1 or A1= Betting on R2

Question 2: K2= Betting on B1 or A2= Betting on B2

N: No preference

Table 2. Unfair-Urn VersionOther Four


26* 3 0 7 1 3 40(65.0%) (7.5%) (0%) (17.5%) (2.5%) (7.5%) (100%)

5 8 5 17* 5 1 41(12.2%) (19.5%) (12.2%) (41.5%) (12.2%) (2.4%) (100%)

5 3 1 16* 7 8 40(12.5%) (7.5%) (2.5%) (40.0%) (17.5%) (20.0%) (100%)

Ellsberg

Savage: Matrix

Savage: Written





Betting on R1 & B1 Betting on R2 & B1

26

• Ellsberg condition: the most common response (65%) was



(ambiguity aversion)• Savage’s matrix and written conditions: the most common

response (41.5% & 40%) was



(obeying the sure-thing principle)• Indecision or confusion was ruled out.

Summary of study 2

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Discussion of study 1 and 2

• The ‘violation of the sure-thing principle’ in the two-colour problem is caused by the way the problem is presented.

• The states of nature is one of possible frames of the decision problem.

• Did participants apply cancellation as a simple heuristic in the states-of-nature frame?

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Study 3: Replication of Ellsberg’s three-colour problem (which was presented in the matrix format by Ellsberg (1961))

90 Total Balls

30 Red Balls

x Blue Balls

60 x Yellow Balls

90 Total Balls

30 Red Balls

x Blue Balls

60 x Yellow Balls

90 Total Balls

30 Red Balls

x Blue Balls

60 x Yellow Balls

Imagine that there is a box containing 30 red balls and 60 balls that are blue or yellow. However, the proportion of blue balls to yellow balls is unknown.

29

Question 1:Suppose that you are offered a game that is to be played as follows:• You are required to draw a ball from the box.• Before drawing a ball, you have to choose one of the following options,

which define the payoffs depending on the three possible outcomes.




Red Blue Yellow

Option 1: £100 £0 £0

Option 2: £0 £100 £0

Red Blue Yellow

Option 1: £100 £0 £100

Option 2: £0 £100 £100

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Results of study 3

• The most common response (41.3%) was(1) Betting on R > Betting on B(2) Betting on {R or Y} < Betting on {B or Y}• Ambiguity aversion• Violation of the sure-thing principle and cancellation.

R: Drawing a red ballB: Drawing a blue ballY: Drawing a yellow ball

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Summary of study 1, 2 and 3

• Given the decision matrix:

1. Two-colour problem: The obedience to the sure-thing principle or cancellation.

2. Three-colour problem: The violation of the sure-thing principle or cancellation.

• Participants seemed to apply combination as a simple heuristic in the three-colour problem.

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Contrasts between the two-colour problem and the three-colour problem

• Two-colour problem:1.Four states are equally ambiguous. 2.There are two sources of uncertainty (two urns).3.The states of nature is in the form of a Cartesian

product.• Three-colour problem:1.One state is unambiguous and the other two are

ambiguous (I.e., not equally ambiguous).2.There is one source of uncertainty (one urn).

33

Study 4: Four-State (One-Urn) Problems:The Structure of the decision matrix

Option 1 and Option 3 will be known options.

Option 2 and Option 4 will be ambiguous options.

s1 s2 s3 s4Option 1 £100 £100 £0 £0Option 2 £100 £0 £100 £0Option 3 £0 £0 £100 £100

Option 4 £0 £100 £0 £100

States

34

Study 4: Four-State (One-Urn) Problems:Three conditions

• Problem 1: Two states are unambiguous and two states are ambiguous (corresponding to the three-colour problem).

• Problem 2: Four states are all ambiguous (corresponding to the two-colour problem).

• Problem 3: The uncertainty is related to two aspects of a state (Cartesian-product structure).

35

Study 4: Problem 1

200 Total Balls

50 Red Balls

50 Blue Balls

x Yellow Balls

100 – x Green Balls

200 Total Balls

50 Red Balls

50 Blue Balls

x Yellow Balls


200 Total Balls

50 Red Balls

50 Blue Balls

x Yellow Balls


Imagine that there is a box containing 50 red balls, 50 blue balls, and 100 balls that are yellow or green. However, the proportion of yellow balls to green balls is unknown.

36

Questions for Problem 1


which define the payoffs depending on the four possible drawings.




Red Blue Yellow Green

Option 1: £100 £100 £0 £0

Option 2: £100 £0 £100 £0

Red Blue Yellow Green

Option 1: £0 £0 £100 £100

Option 2: £0 £100 £0 £100

37

Study 4: Problem 2

200 Total Balls

x Red Balls

y Blue Balls

x Yellow Balls

y Green Balls

200 Total Balls

x Red Balls

y Blue Balls

x Yellow Balls

y Green Balls

200 Total Balls

x Red Balls

y Blue Balls

x Yellow Balls

y Green Balls

Imagine that there is a box containing x red balls, y blue balls, x yellow balls and y green balls. However, the number of balls of each colour is unknown.

(Same questions as earlier)

38

Study 4: Problem 3

100 Total Balls

Ball # 1 (Red or Black)Ball # 2 (Red or Black)…Ball # 99 (Red or Black)Ball # 100 (Red or Black)

100 Total Balls


100 Total Balls


Imagine that there is a box containing 100 balls, which are numbered from 1 to 100 at random. Besides, these balls are either red or black but the proportion of red balls to black balls is unknown.

39

Questions for Problem 3


which define the payoffs depending on the four possible outcomes.




A Red Ball with an Odd Number

A Black Ball with an Odd Number

A Red Ball with an Even Number

A Black Ball with an Even Number

Option 1: £100 £100 £0 £0

Option 2: £100 £0 £100 £0

A Red Ball with an Odd Number

A Black Ball with an Odd Number

A Red Ball with an Even Number

A Black Ball with an Even Number

Option 1: £0 £0 £100 £100

Option 2: £0 £100 £0 £100

40

Results of study 4

Question 1: K1= Betting on Option 1 or A1= Betting on Option 2

Question 2: K2= Betting on Option 1 or A2= Betting on Option 2

N: No preference

Table 3. Four-State ProblemsOther Four


13* 8 10* 5 3 7 46(28.3%) (17.4%) (21.7%) (10.9%) (6.5%) (15.2%) (100%)

11* 6 8 4 13* 4 46(23.9%) (13.0%) (17.4%) (8.7%) (28.3%) (8.7%) (100%)

17* 10* 4 2 11* 2 46(37.0%) (21.7%) (8.7%) (4.3%) (23.9%) (4.3%) (100%)

50, 50, x, 100-x

x, y, x, y

No. × Colour





Betting on Known Options Indifference

Betting on Known Option in Q1 & Ambiguous Option in Q2

41

Summary of study 4

• The choices obeying ambiguity aversion are observed in all three problems.

• Other choices obeying the sure-thing principle are observed in all three problems:

1. Problem 1: Known Option in Question 1 and Ambiguous Option in Question 2.

2. Problem 2 and 3: Indifference in both questions.

42

Summary of study 1, 2, 3 and 4

• Given the decision matrix:

1.Two-colour problem: the most common response conforms to the sure-thing principle or cancellation.

2.Three-colour problem: the most common response conforms to combination.

3.Four-state problem: some responses conform to the sure-thing principle or cancellation and some responses conform to combination.

43

A challenge to rank-dependent utility theory

• RDU: In all reported problems,

1. all options are not pair-wise comonotonic and

2. the information about likelihood of states are uncertain. • Hence, decision makers will violate the sure-thing

principle.

• Data: The above statement is not always true.

44

Original prospect theory account (I): Editing evaluation choice (Combination ambiguity choice)

Combination: Combine the states associated with the same payoff within a gamble before evaluation.

Combination plus ambiguity aversion will lead to the choice of A and C in both problems.

Problem 1 50 50 x 100 - x

Problem 2 x y x y

Red Blue Yellow GreenOption A 100 100 0 0Option B 100 0 100 0

Option C 0 0 100 100Option D 0 100 0 100

Study 4

45

Cancellation: Cancel the state associated with the same payoff between options before further comparison.

Cancellation plus ambiguity aversion will lead to the choice of A & D in Problem 1 and the choice of no preference in Problem 2.

Problem 1 50 50 x 100 - x

Problem 2 x y x y

Red Blue Yellow GreenOption A 100 100 0 0Option B 100 0 100 0

Option C 0 0 100 100Option D 0 100 0 100

Study 4

Original prospect theory account (II): Editing evaluation choice (Cancellation ambiguity choice)

46

Swing between cancellation and combination

Table 4. A Classification in terms of Combination and CancellationOther Four


6 5 2 3 16* 9 41(14.6%) (12.2%) (4.9%) (7.3%) (39.0%) (22.0%) (100%)

5 8 5 17* 5 1 41(12.2%) (19.5%) (12.2%) (41.5%) (12.2%) (2.4%) (100%)

19* 1 8 3 5 10 46(41.3%) (2.2%) (17.4%) (6.5%) (10.9%) (21.7%) (100%)

13* 8 10* 5 3 7 46(28.3%) (17.4%) (21.7%) (10.9%) (6.5%) (15.2%) (100%)

11* 6 8 4 13* 4 46(23.9%) (13.0%) (17.4%) (8.7%) (28.3%) (8.7%) (100%)

17* 10* 4 2 11* 2 46(37.0%) (21.7%) (8.7%) (4.3%) (23.9%) (4.3%) (100%)

x, y, x, y

No. × Colour

Combination Cancellation

Fair-Urn (Matrix)

Unfair-Urn (Matrix)

Three-colour

50, 50, x, 100-x

47

Concluding remarks

• States-of-nature representation is a way of framing (i.e., not empirically neutral).

• Within the states-of-nature frame, people might select between cancellation and combination before choice.

• Inconsistency of selecting editing rules may be one of the factors for the variety of people’s choices.

48

Time for questions: 5min

Thank you!!

ellsberg’s paradoxes: problems for rank- dependent utility explanations cherng-horng lan &...

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