Decision Theory 2 by pengxiang


									Decision Theory

An Introduction of
Psychology Students
 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)

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

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

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

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

Assumptions about Beliefs
   In the decision situation
    – Judgements (probabilities) must sum to 1
    – 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

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


   Utility = abstract measure of goal
    – 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)

Measurement of Utility

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

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
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)

Assumptions about MEU

   The Independence/Dominance or Sure thing
    – 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
    – 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
Applying to Pascal’s Wager
              State of the World

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

                 0 = Arbitrary reference point

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
    – The independence, dominance, or sure-
      thing principle.

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
 Psychology: A power law (S S Stevens)
  relates (external) value and (inner)
  experienced satisfaction
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
Power Law when k=1, s=1
            Linear Relation Between Utility
                     and Amount



                 0          5             10
                         Am ount

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

                  0   2    4     6     8       10

(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
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).
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

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

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.
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).

Gains and Losses are reflections



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

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
Tversky’s Alternative

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



                       (p=0.5)           =Decision
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
The World                   The Person

 Alternative   A         Perception                   Beliefs

                   P,A                 P,B

                     Evaluation                     Desires,


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



 Postulated Mental Processes in Decision Theory                  35
Some Problems for Decision
   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)
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)

Violations of Independence

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

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

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
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?)
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
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
 One possibility is a selection by aspects
The Certainty Effect

   People prefer certainty even if it lowers
    – “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

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

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
   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!

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
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
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
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

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?
The Hospital Problem, ctd

   The hospital problem was put to 95
    – 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
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

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

 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)

Examples of heuristics

 Selection by aspects
 Representativeness
    – Assumption that experience under one set
      of conditions will equal experience under
   Availability
    – Decide on the basis of what is easiest to
      call to mind
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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