Expected Value, Expected Utility
& the Allais and Ellsberg Paradoxes
Psychology 466: Judgment & Decision Making
Instructor: John Miyamoto
11/10/2015: Lecture 07-1
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Outline
• Expected Value and Expected Utility: What's the Difference?
• Allais Paradox
♦
Common consequence principle
(a.k.a. Savage’s independence axiom or the sure-thing principle)
♦
Anticipated regret
♦
Nonlinear probability weighting
• Ellsberg Paradox
Psych 466, Miyamoto, Aut '15
Lecture probably
ends here
What Is the Expected Value of a Gamble?
2
Expected Value of a Gamble
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Expected Value of a Gamble (Cont.): More General Version
3
Expected Value of a Gamble (cont.)
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Would It Be Rational to be an Expected Value Maximizer?
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Would It Be Rational to be an Expected Value Maximizer?
Expected Value Maximizer:
Someone who always prefers the gamble that has
the higher expected value.
• Discussions pro and con during the 18th and 19th century.
Rich men wanting to know, which is the better bet?
Psych 466, Miyamoto, Aut '15
Are You an Expected Value Maximizer? Concrete Example
5
Are You an Expected Value Maximizer”
• I offer you a choice:
Higher Risk
Lower Risk
• An expected value maximizer would choose Option 1.
• EV( Option 1 ) > EV( Option 2 )
Psych 466, Miyamoto, Aut '15
Continuation of this Example
6
Are You an Expected Value Maximizer? (Cont.)
• I offer you a choice:
Option 1: 50% chance you win $10,010
50% chance you lose $10,000
Option 2: 50% chance you win $2
50% chance you lose $10
Intuitive argument in favor of Option 2:
♦
The pleasure of winning +$10,010 is smaller in absolute magnitude
than the pain of losing -$10,000.
♦
The worst that can happen with Option 2 is the pain of losing -$10.
♦
What really matters is the subjective value of the outcomes,
+$10,010, +$2, -$10, -$10,000.
and not the objective monetary amounts.
Psych 466, Miyamoto, Aut '15
St. Petersburg Paradox - Introduction
7
St. Petersburg Paradox
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Illustration of the St. Petersburg Game
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St. Petersburg Paradox (cont.)
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Expected Value of the St. Petersburg Game
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Expected Value of St. Petersburg Game is Infinite!
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EU of St. Petersburg Game is Infinite
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Expected Value of St. Petersburg Game is Infinite!
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Does It Feel Right that the St. Petersburg Game is Infinitely Valuable?
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Expected Value of St. Petersburg Game is Infinite!
Would you give your total wealth for the opportunity
to play the St. Petersburg game?
If you are an expected value maximizer, you should be
eager to pay everything you own for the opportunity
to play the St. Petersburg Game just once.
Psych 466, Miyamoto, Aut '15
Bernoulli's Utility Hypothesis
12
Expected Value & Expected Utility
• Nobody is an expected value maximizer.
Nobody always prefers the gamble
with the higher expected value.
• Daniel Bernoulli (1738):
People maximize the expected utility of
their choices; not the expected value of their choices.
• Utility of X = subjective value of possessing or experiencing X
• Next 200 years: Economic theory attempts to get rid of
the concept of subjective value.
Psych 466, Miyamoto, Aut '15
Expected Utility Hypothesis: Simplified Mathematical Statement
13
Expected Utility Hypothesis (Simplified Version)
Let U(X) be the utility of X and let EU(G) be the expected utility of a gamble G.
Expected Utility Hypothesis: There exists a function U such that:
(i) for every pair of gambles G1 and G2 ,
G1 preferred to G2 iff EU(G1) > EU(G2)
(ii) If G = (X1, p; X2, 1-p) is any lottery (for money), then
EU(G) = pU(X1) + (1 - p)U(X2)
• The Expected Utility (EU) Hypothesis is the claim that
a rational agent must satisfy (i) and (ii).
Psych 466, Miyamoto, Aut '15
Example: Calculating the EU of Two Gambles
14
Example: Calculating the EU(Option 1 ) & EU(Option 2 )
• I offer you a choice:
Assume these are the utilities:
Option 1: 50% chance you win $10,010
50% chance you lose $10,000
Option 2: 50% chance you win $2
50% chance you lose $10
U($10,010)
U($2)
U(-$10)
U(-$10,000)
=
=
=
=
+8,000
+1.8
-2.5
-10,000
• Calculate the Expected Utility of Each Option:
♦
EU( Option 1 ) = (½  8000) + ( ½  (-10,000) ) = -1,000 Utils
♦
EU( Option 2 ) = (½  1.8) + ( ½  -2.5 )
♦
EU( Option 1 ) < EU( Option 2 ).
= -3.5 Utils
If you are an EU maximizer, you will choose Option 2.
Psych 466, Miyamoto, Aut '15
Rationality Does NOT Demand that We Be Expected Value Maximizers
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Rationality Does Not Demand
that We Be Expected Value Maximizers
• An insurance company is (approximately) an EV maximizer.
An individual person is not an EV maximizer.
Why?
• Suppose an insurance company sells 10,000 auto insurance
policies for $500/year each.
♦
♦
Insurance company knows that the expected value of each policy is -$420.
An individual auto accident might cost $2,000 to $1,000,000,
but they happen rarely.
• Is it rational to buy auto insurance?
• There are many examples where reasonable people
are NOT EXPECTED VALUE MAXIMIZERS.
Psych 466, Miyamoto, Aut '15
Transition: Risk Aversion Is Related to Shape of Utility Function
16
1944 - 1947: The Birth of Expected Utility Theory
• Daniel Bernoulli (1738):
People maximize the expected
utility of their choices; not
the expected value of their choices.
• Utility of X = subjective value of
possessing or experiencing X
• Next 200 years: Economic theory attempts to get rid of
the concept of subjective value.
• 1944 - 1947: Mathematical work of von Neumann &
Morgenstern leads to the discovery of expected utility theory.
Psych 466, Miyamoto, Aut '15
Preference Axioms - What Are They?
17
Preference Axioms – What Are They?
• Preference Axioms for EU Theory: A set of assumptions about
preference behavior which, if satisfied, imply that a decision
maker is an EU maximizer (conforms to EU theory).
• Transitivity is an example of a preference axiom.
• Sure-thing principle (common consequence assumption)
is another example of a preference axiom. (To be explained next.)
• Preference axioms can be construed as a normative claim:
This is how a rational agent ought to behave.
• Preference axioms can be construed as a descriptive claim:
This is how people actually behave.
Psych 466, Miyamoto, Aut '15
Allais Paradox
18
Allais Paradox
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des
postulats et axiomes de l'école Americaine. Econometrica, 21, 503-546.
typical
Choice 1:
Option A: Receive 1 million for sure.
Option B: Receive
Receive
Receive
typical
2.5 million,
1 million,
0,
Choice 2:
Option A': Receive
1 million,
Option B': Receive 2.5 million,
10% chance
89% chance
1% chance
11% chance, otherwise $0.
10% chance, otherwise $0.
• (Write student responses on the board.)
• Typical choices: Choose A from Choice 1 and choose B' from Choice 2.
Psych 466, Miyamoto, Aut '15
Ellsberg Paradox
19
Tabular Representation of the Allais Choices
Choice 1: Option A: Receive 1 million for sure. ();
Option B: Receive 2.5 million, 10% chance,
Receive 1 million, 89% chance, Receive 0 , 1% chance
Choice 2:
Option A': Receive 1 million, 11% chance, otherwise $0.
Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
Chance of Outcome
Choice 1
Choice 2
Psych 466, Miyamoto, Aut '15
10%
89%
1%
Option A
$1
$1
$1
Option B
$2.5
$1
$0
Option A'
$1
$0
$1
Option B'
$2.5
$0
$0
Same Slide without the Opaque Grey Rectangles
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Allais Paradox Is Based on Common Consequences
Choice 1: Option A: Receive 1 million for sure. ();
Option B: Receive 2.5 million, 10% chance,
Receive 1 million, 89% chance, Receive 0 , 1% chance
Choice 2: Option A': Receive 1 million, 11% chance, otherwise $0.
Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
Chance of Outcome
Choice 1
Choice 2
Psych 466, Miyamoto, Aut '15
10%
89%
1%
Option A
$1
$1
$1
Option B
$2.5
$1
$0
Option A'
$1
$0
$1
Option B'
$2.5
$0
$0
Statement of the Common Consequence Principle
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Common Consequences Principle
(Other Names: Sure-Thing Principle, Savage’s Independence Axiom)
Common Consequence Principle: If two options have the same consequence
given some outcome, then you should ignore this consequence.
♦
Base your choice on the aspects of the options that differ.
Chance of Outcome
Choice 1
Choice 2
Typical
Choice
10%
89%
1%
Option A
$1
$1
$1
Option B
$2.5
$1
$0
Option A'
$1
$0
$1
Option B'
$2.5
$0
$0
Typical
Choice
Psych 466, Miyamoto, Aut '15
Allais Paradox Violates the Common Consequence Principle
22
Psychological Explanations for the Allais Paradox
Choice 1:
Option A: Receive 1 million for sure.
Option B: Receive 2.5 million, 10% chance
Receive 1 million, 89% chance
Receive 0 , 1% chance
Choice 2:
Option A': Receive 1 million, 11% chance, otherwise $0.
 Option B': Receive 2.5 million, 10% chance, otherwise $0.
----------------------------------------------------------------------• Class:
Propose psychological explanations for the Allais Paradox.
Psych 466, Miyamoto, Aut '15
Anticipated Regret – Explanation for Allais Paradox
23
Explaining the Allais Paradox in terms of Anticipated Regret
Choice 1:
Option A: Receive 1 million for sure.
Option B: Receive 2.5 million, 10% chance
Receive 1 million, 89% chance
Receive 0 , 1% chance
Choice 2:
Option A': Receive 1 million, 11% chance, otherwise $0.
Option B': Receive 2.5 million, 10% chance, otherwise $0.
-----------------------------------------------------------------------• If you choose option B in choice 1 and get $0, you will feel intense regret.
Choosing option A avoids the possibility of regret.
• If you choose option B' in choice 2 and get $0, you will not feel regret for your
decision because you could have gotten $0 with option A' as well.
Psych 466, Miyamoto, Aut '15
What Circumstances Cause Feelings of Regret?
24
Comment: Decision-Related Emotion
Negative Emotion
Positive Emotion
Disappointment
Relief
(receive bad outcome when you
hoped for a good outcome)
(receive good outcome when you
feared a bad outcome)
Regret
Self-Congratulation (?)
(receive a bad outcome when a
different choice would have
produced a much better outcome)
(receive a good outcome when a
different choice would have
produced a much worse outcome)
Psych 466, Miyamoto, Aut '15
Allais Paradox & Nonlinear Perception of Probability
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Chance of Outcome
Choice 1
Choice 2
10%
89%
1%
Option A
$1
$1
$1
Option B
$2.5
$1
$0
Option A'
$1
$0
$1
Option B'
$2.5
$0
$0
• Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.
o
o
potential regret
no potential regret
Why Do People Have Allais-Type Preferences?
Regret – comparison between what you have experienced and what you would have
experienced if you made a different choice.
Anticipated Regret – anticipating that a choice will create the possibility of regret.
Psych 466, Miyamoto, Aut '15
Clean Version of This Slide
26
Chance of Outcome
Choice 1
Choice 2
10%
89%
1%
Option A
$1
$1
$1
Option B
$2.5
$1
$0
Option A'
$1
$0
$1
Option B'
$2.5
$0
$0
• Hypothesis: Choices 1 and 2 differ in terms of anticipated regret.
o
o
potential regret
no potential regret
Why Do People Have Allais-Type Preferences?
Regret – comparison between what you have experienced and what you would have
experienced if you made a different choice.
Anticipated Regret – anticipating that a choice will create the possibility of regret.
Psych 466, Miyamoto, Aut '15
Clean Version of This Slide
27
Explaining the Allais Paradox in terms of
Nonlinear Perception of Probability
Choice 1:
Option A: Receive 1 million for sure, 0% chance of receiving 0 dollars.
Option B: Receive 2.5 million,
10% chance
Receive 1 million,
89% chance
Receive 0 ,
1% chance
Choice 2:
Option A': Receive 1 million, 11% chance, otherwise $0. 89% chance of $0
Option B': Receive 2.5 million, 10% chance, otherwise $0. 90% chance of $0
------------------------------------------------------------------------• In choice 1-A, the chance of $0 is 0%; in choice 1-B, it is 1%.
In choice 2-A', the chance of $0 is 89%; in choice 2-B', it is 90%.
• Psychologically, the difference between a 0% and 1% chance of $0
is greater than the difference between an 89% and 90% chance of $0.
Psych 466, Miyamoto, Aut '15
Summary: Explanations of the Allais Paradox
28
Typical Preferences in the Allais Paradox Violate EU Theory
Choice 1: Option A: Receive 1 million for sure. ();
Option B: Receive 2.5 million, 10% chance,
Receive 1 million, 89% chance,
Receive 0 ,
1% chance
Choice 2:
Option A': Receive 1 million, 11% chance, otherwise $0.
Option B': Receive 2.5 million, 10% chance, otherwise $0. ()
Choice 1:
EU( Option A ) = (0.10)U($1 mil ) + (0.89)U( $1 mil ) + (0.01)U($1 mil )
EU( Option B ) = (0.10)U($2.5 mil ) + (0.89)U( $1 mil ) + (0.01)U($0 mil )
Choice 2:
EU( Option A ) = (0.10)U($1 mil ) + (0.89)U( $0 mil ) + (0.01)U($1 mil )
EU( Option B ) = (0.10)U($2.5 mil ) + (0.89)U( $0 mil ) + (0.01)U($0 mil )
Psych 466, Miyamoto, Aut '15
Time Permitting: Present the Ellsberg Paradox
29
Time Permitting: Discuss the Ellsberg Paradox
• Who is Daniel Ellsberg?
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Presentation of the Choices for the Ellsberg Paradox
30
Tuesday, November 10, 2015: The Lecture Ended Here
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