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
                                                5
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




                                           9
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
                                                         15
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


                                                 16
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.

                                              17
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
                                      18
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.
                                                   19
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).
                                                            23
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

                                                 24
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.
                                                         26
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).

                                             27
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
                                            31
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
                                               32
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
                                                   34
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)
                                                 36
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




                                                        39
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
                                              40
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




             47
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...
                                              50
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
                                                   58

								
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