# Decision Theory 2 by pengxiang

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```									Decision Theory

An Introduction of
Psychology Students
History
 Arose in 17th century, with invention of
probability (Pascal, Bernoulli, Bayes)
 Refinement of “common sense”
 Utilitarianism (Bentham, James and John
Stuart Mill  Economics)
 Mathematical development
– Ramsey (1920s), von Neuman and Morgenstern
(1947), Savage (1950s)
   Psychological Interest: Normative versus
descriptive debate (Allais, 1953)
2
Uses

   Decision Analysis
– Business, policy, government, engineering
 Expectancy theory of motivation
 The analysis of “rationality”
 Foundation of Micro-economics
 Some varieties of sociology
– “methodological individualism”
3
Action depends on…

 Set of alternative acts
 Representation of states of the world
(belief)
 Desirability of consequences of acts
(desire)
 (A form of “belief-desire”
psychology, refined for use in decision
analysis)
4
The First Decision Analysis
State of the World

Act               God Exists        God Doesn‟t
Exist
Live Christian Very good            Small
Life           (Saved)              inconvenience
Live Otherwise Very Bad             Normal life
(Damned)
*Options should be more finely divided
* States of the world not exhausted
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Beliefs about the world

 Judgments represented as probabilities
 Connected to reality through “Bayesian
learning Theory”
–   Derives from Bernoulli‟s theorem:
–   Flip a coin, H vs T
–   As we collect more cases
–   H/(H + T) = constant
–   Can use a cut-off criterion, degrees of certainty

6
Types of Decision Theory

   If we can estimate probabilities
(experience or information), then
– Judgement or decision under RISK
   Otherwise
– Decision making under UNCERTAINTY
   Daniel Ellsberg demonstrated: Risk
preferred to uncertainty

7
Assumptions about Beliefs
   In the decision situation
– Judgements (probabilities) must sum to 1
(exhaustive)
– The probability of the state of the world must be
independent of the act chosen
   The latter assumption may in fact be violated
(e.g., the act of smoking influences the
probability of you getting cancer)
– “Causal” decision theory attempts to analyse such
situations

8
Assumptions about Desires

 Value can be captured by an abstract
measure called utility
 Utility is the only information needed
about desires and wants
 Numbers can be assigned to utilities

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Utility

   Utility = abstract measure of goal
attainment
– Example: Win either R50 000 or a Holiday
in Mauritius
– Need to translate into a common measure
to choose
– Utility = Judgement of the desirability of an
outcome (cognitive measure)

10
Measurement of Utility

 Direct scaling (assign numbers between
extremes)
 Difference measurement
 Units do not matter
 Utility is personal, so scale is tailoured
to the specific individual

11
Assigning numbers to utility
   To assign numbers we need “weak
preference ordering”
– Connection: Must make a choice (one, other or
indifference), can‟t opt out
– Transivity
• Oranges > apples
• Apples > pears
•  Oranges > pears
– Invariance or Independence
• Preferences are independent of how they are described
• Preferences relate to the value of the outcome (utility) not
the way they are described
12
Principles of Choice
 Value of a gamble: Probability X Value
 If many outcomes are possible for a single
action?       n
EV   p (i )  V (i )
i 1

Eg, toss a coin according to the rule
H: Loose 50c, T: gain R1
EV = .5 X –50 + .5 X 100 = -25 + 50
= 25c
(Substitute U for V in the above equation to apply to
ultility)                                          13
General Principle of Choice

 Maximize Expected Utility (MEU)
 Select the act which maximizes
EU, except for
– Gambler’s Ruin (choosing an act which
will wipe you out as a player if you lose)

14
Assumptions about MEU

   The Independence/Dominance or Sure thing
Principle
– If there is some state of the world that leads to the
same outcome no matter what choices you
make, then your choice should not depend on that
outcome
– If prospect A (I.e., outcome A) is at least as good
as prospect B in every respect, and better than B
in at least one respect, then A should be preferred
to B
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Applying to Pascal’s Wager
State of the World

Act           God Exists      God Doesn‟t
Exist
Live Christian 1000           -10
Life
Live Otherwise -1000          0

0 = Arbitrary reference point

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Applying to Pascal’s Wager 2

   Calculations
– EU Christ Life = .5 X 1000 +.5 X –10 = 450
– EU Live Other = .5 X –1000 +.5 X 0 = - 500
– Therefore, BY MEU: Choose Christian Life
   Assumption needed for the MEU
Principle:
– The independence, dominance, or sure-
thing principle.

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Relating Utility to (Money) Value

 U(R20) > u(R10), but:
 U(R20) > u(R10) = u(R100) > u(R90)?
 Bernoulli, Bentham, Economists: NO
 Economics: Law of diminishing Marginal
Utility
 Psychology: A power law (S S Stevens)
relates (external) value and (inner)
experienced satisfaction
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S. S. Stevens’ Power Law

V  kA      s

V = Value
A = Amount, quantity
S = subjective sensitivity
K = proportionality constant
Applies to temperature, light brightness, sound, etc
Every psychophysical quality will have its own s
value.
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Power Law when k=1, s=1
Linear Relation Between Utility
and Amount

10
Utility

5

0
0          5             10
Am ount

20
Power Law, s=0.5 (square root)
When Utility = Root of Amount (s
= .5)
Note
3.5                                     The
3
2.5                                     Tapering
Curve!
Utility

2
1.5
1
0.5
0
0   2    4     6     8       10
Amount

(This is what Bernoulli believed the relation between
value and utility to be).                           21
Bentham’s principle – the greatest good
for the greatest number
Utility             Utility

Money                           Money

Poor Person
Rich Person
22
After Redistribution
(Ex-Rich)                   (Ex-Poor)
Utility                       Utility

Money                            Money
Losing half of his money lowers the Ex-rich man’s utility
Less than the rise in the poor man’s utility from receiving
Half the rich man’s money. (But losses loom larger than gains).
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The Reflection Effect 1

 Two Gambles presented to Ss:
 Gamble choice 1
– Which of the following would you prefer?
Alternative A
• 50% chance to win R200
• 50% chance to win nothing
– Or Alternative B
• R100 win for sure

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The reflection Effect, 2

   Gamble Choice 2
– Which of the following would you prefer?
Alternative A
• 50% chance to lose R200
• 50% chance to lose nothing
– Or Alternative B
• R100 loss for sure

25
Reflection effect, 3
   Ss chose B in Gamble choice 1, but A in
Gamble Choice 2. Why?
– Applying power law with s=.5 to transform values
to utilities, and
– Expected utility formula, we get:
– Alternative A:
• (.5 X root(200)) + (.5 X root(0)) = 7.07
– Alternative B
• (1 X root(100)) + (0 X root(0)) = 10
– In Gamble 1, alternative B maximizes gain, but,
– In Gamble 2, alternative A minimizes loss.
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Reflection Effect, 4

 So the choices Ss make in the two
gambles are rational.
 Gains and losses are treated
differently when we substitute utilities
for values
 This effect is called the reflection
effect. (Think of a mirror across the
origin of a graph).

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Gains and Losses are reflections
Utility
Gains

Value

Losses

Mirror
Reflection
28
Alternatives differing in Amount
and Quantity
   Choose between:
– R50 000 and trip to Paris
– Different amounts, different qualities
   Luce‟s Two-Stage model
– 1. Convert qualities and amounts to utility using
power function
– 2. Degree of preference = (values of
alternative)/(sum of values of all alternatives)
   Assume Independence (relative preference
unaffected by presence of another
alternative).                                         29
Partially Similar Alternatives

 We can also get alternatives that
overlap in amount and qualities to
varying degrees
 Luce‟s two-stage model should apply if
we are rational – but it does seem
computationally intensive

30
Luce’s Two Stage Model and
Partially Similar Alternatives
   Choose between:
– R50 000
– Trip to Paris
– Trip to Rome
   If each of the above has a utility of
10, then, by Luce‟s model:
v1       10
     0.33
v1  v2  v3 30
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Tversky’s Alternative

 Amos Tversky suggested an alternative
“heuristic” model of how we choose between
partially similar alternatives
 He called this alternative “Selection by
Aspects”
 Essentially he argued that people will tend to
lump together similar options
(e.g., Paris, Rome) and then choose between
the aggregated options
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Selection by Aspects
p=0.5           (p=0.25)
Trip

P=0.5
p=0.5
(p=0.25)

p=0.5

(p=0.5)           =Decision
Money
33
Choosing by Aspects
   Simplifies choices (that is the idea
behind “heuristics”)
– Instead of having to think about three
things in the choice between a trip to
Paris, a trip to Rome and money) the
person just needs to think about two (trip
versus money)
   Can lead to bad decisions
– Consider case where the lumping process
can obscure a particularly good option
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The World                   The Person

Alternative   A         Perception                   Beliefs
Actions

B
P,A                 P,B

Evaluation                     Desires,

Goals

EU = p * u(A) + q * u(B)

Decision

(MEU)

Postulated Mental Processes in Decision Theory                  35
Some Problems for Decision
Theory
   Indeterminacy
– Search process for alternative acts and
states of the world is assumed – but such
searching is often the most difficult aspects
of a decision
– Sometimes there is no probability (belief)
available (uncertainty versus risk)
– People dislike uncertainty (compared to
Risk) (Ellsberg)
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Limited Rationality

 Most decisions are taken without doing
an exhaustive search of the available
options, etc – our rationality is limited
 Herbert Simon – satisficing
 Kahneman & Tversky (heuristics)

37
Violations of Independence

 Elimination by Aspects
 Reflection effect combined with
cognitive framing (Kahneman and
Tversky)

38
The Allais Paradox
S1             S2             S3
(p(s1)=0.01)   (p(s2)=0.10)   (p(s3)=0.89)
1   act1   R500 000       R500 000       R500 000
act2   R0             R2 500 000     R500 000

2   act3   R500 000       R500 000       R0
act4   R0             R2 500 000     R0

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Commentary on Allais Paradox

 For problems 1 and 2 the choices differ only
in state of world s3
 Yet for act1 and act 2, and for act 3 and act
4, the outcome for s3 are identical (i.e., by
independence principle we should ignore s3)
 Yet people prefer act 1 to act 2, and prefer
act 4 to act 3
 This seems to be a violation of the
independence principle
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Preference Reversal 1
 Which gamble would you prefer:
– 1. Gamble 1
– 34/36 chance to win R3.00
– 2/36 chance to win R1.00
– 2. Gamble 2
– 18/36 chance to win R6.50
– 18/36 chance to win R1.00
 Most people prefer Gamble 1 (because of
higher probability?)
41
Preference Reversal 2
   Now consider two problems:
– 3. What is the most money you would pay
for a 34/36 chance to win R3 and 2/36
chance to lose R2?
– 4. What is the most money you would pay
for a 18/36 chance to win R6.50 and an
18/36 chance to lose R1?
   Most people set a higher value on
alternative 4 than on alternative 3
42
Preference Reversal 3

   The above findings are puzzling for it
seems that
– People prefer what they value less highly
to what they value more highly
 How is this violation of transitivity
possible?
 One possibility is a selection by aspects
heuristic?
43
The Certainty Effect

   People prefer certainty even if it lowers
EU
– “Don‟t gamble. Take all your savings and
buy some good stock and hold it till it goes
up. If don‟t go up, don‟t buy it.”
• Will Rogers

44
The Insurance/Gambling Paradox

   Insurance
– You pay someone to take risk for you
   Gambling
– You pay someone for the privilege of
taking risk
   One and the same individual can both
buy insurance and gamble

45
Framing Effects
   Typical problem (N=132)
   A: Would you accept a gamble that offers a 10%
chance to win R95 and a 90% chance to lose R5?
   B: Would you pay R5 to participate in a lottery with a
10% chance to win R100 and a 90% chance to win
nothing?
   55 subjects responded differently, 42 rejecting A but
accepting B
   It seems that the evaluation of problems may NOT be
independent of how they are described!
46
Heuristics

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Four kinds of reasoning errors

 Errors of logic – especially errors
related to “If...then...”
 Randomness errors – e.g., the
Gambler‟s fallacy
 Errors related to samples of different
sizes – e.g., the hospital problem
 Errors arising from ignoring base rates –
e.g., the lawyer/engineer problem
48
Wason Card Task

A            T            4            7

Rule: If a card has a vowel on one side it has an even
number of the other side

Question: Which cards have to be turned over to test the
Rule?
49
Commentary on Wason Task

 Most Ss turn over the „A‟ card [correct] but
also the „4‟ card [incorrect because the rule
would not be violated whatever is on the back
of the 4 card]
 The Ss should turn over the „7‟ card because
the rule would be violated if there is a vowel
on the other side
 This error has been found to occur in many
conditions...
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Gambler’s Fallacy

 If the ball of a roulette wheel lands on
red ten times in a row it seems
somehow much more likely that the 11th
outcome will be black
 But each spin of the wheel is
independent

51
Hospital Problem
   A certain town is served by two hospitals.
About 45 babies are born each day in the
larger hospital, and about 15 in the smaller
hospital. As you know, about 50% of all
babies are boys. However the exact
percentage varies from day to day. For a
period of one year each hospital recorded the
days on which more than 60% of the babies
born were boys. Which hospital do you think
received more such days?
52
The Hospital Problem, ctd

   The hospital problem was put to 95
undergraduates:
– 21 said the larger, 21 said the smaller, and
53 said both hospitals would have the
same number
   Correct answer: Smaller hospital
– The smaller the sample the more variable it
is
53
Lawyers and Engineers
   Two groups of Ss are given descriptions of people
including their interests and hobbies. Group 1 is told:
The descriptions are drawn randomly from a group of
70 engineers and 30 lawyers. Group 2 is told that the
descriptions are randomly drawn from 30 engineers
and 70 lawyers. All the Ss are asked to estimate the
probability that each person is an engineer or a
lawyer. All were told that a group of experts were also
doing this and they would receive a bonus if their
allocation matched that of the experts

54
Lawyers and engineers, ctd
   Both groups of subjects assigned probabilities almost
identically purely on a basis of the descriptions, not
on the base rate information
   Example of a description:
– Dick is a thirty-year-old man. He is married with no
children. A man of high ability and motivation, he
promises to be quite successful in his field. He is
well liked by his colleagues
– Most people gave this description a .5 probability
of being an engineer – should have used base
rate
55
Heuristics

 A quick and relatively accurate method
of solving a problem where the perfect
solution is difficult or time consuming
 In real life it is too difficult to judge
everything absolutely correctly
– Good enough or satisfice (Herbert Simon)

56
Examples of heuristics

 Selection by aspects
 Representativeness
– Assumption that experience under one set
of conditions will equal experience under
another
   Availability
– Decide on the basis of what is easiest to
call to mind
57
Heuristic examples, ctd
   Anchoring and Adjustment
– Make an initial estimate of information as
the information arrives and then adjust
upwards or downwards
– E.g., what is the product of:
• [A] 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
• [B] 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
– Average answer for A = 512, for B = 2250
but real answer is 40 320
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