Decision Theory

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Decision Theory

• An Introduction of Psychology Students

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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)

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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

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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)

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The First Decision Analysis
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Beliefs about the world

• Judgments represented as probabilities
• Connected to reality through Bayesian learning
  Theory
• Derives from Bernoullis 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

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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

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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

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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)

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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

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Assigning numbers to utility

• To assign numbers we need weak preference
  ordering
• Connection Must make a choice (one, other or
  indifference), cant opt out
• Transivity
• Oranges gt apples
• Apples gt pears
• ? Oranges gt 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

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Principles of Choice

• Value of a gamble Probability X value
• If many outcomes possible for a single action?
• EV ?ni1 p(i) X V(i)
• 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)

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General Principle of Choice

• Maximize Expected Utility (MEU)
• Select the act which maximizes EU, except for
• Gamblers Ruin (choosing an act which will wipe
  you out as a player if you lose)

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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 Pascals Wager
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Applying to Pascals 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) gt u(R10), but
• U(R20) gt u(R10) u(R100) gt 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 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 k1, s1
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Power Law, s0.5 (square root)
Note The Tapering Curve!
(This is what Bernoulli believed the relation
between value and utility to be).
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Benthams principle the greatest good for the
greatest number
Utility
Utility
Money
Money
Poor Person
Rich Person
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After Redistribution
(Ex-Rich)
(Ex-Poor)
Utility
Utility
Money
Money
Losing half of his money lowers the Ex-rich mans
utility Less than the rise in the poor mans
utility from receiving Half the rich mans 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

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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
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Alternatives differing in Amount and Quantity

• Choose between
• R50 000 and trip to Paris
• Different amounts, different qualities
• Luces 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).

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Partially Similar Alternatives

• We can also get alternatives that overlap in
  amount and qualities to varying degrees
• Luces two-stage model should apply if we are
  rational

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