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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