Representativeness heuristics

Transcription

1 Representativeness heuristics 1-1 People judge probabilities by the degree to which A is representative of B, that is, by the degree to which A resembles B. A can be sample and B a population, or A can be a person and B a group, or A can be an event/effect and B a process/cause: A < B or A B 1

2 Representativeness heuristics 1-2 E.g. What is the probability that person A (Simon, a shy, introverted man) belongs to Group B (stamp collectors) rather than Group C (BMW drivers)? 2

3 Representativeness heuristics 1-3 E.g. when A is highly representative of (not similar to) B, the probability that A originates from B is judged to be high (low) B can be a class and A can be a variable defined on that class, or an instance of that class, or a subset of that class B can be a causal system and A can be a possible consequence or realization coming from that system. 3

4 Representativeness heuristics 1-4 Behaviors associated with representativeness: Conjunction fallacy Base rate neglect/underweighting Hot hand Gambler s fallacy Overestimating probability 4

5 Conjunction fallacy 1-5 Many people have great difficulty in understanding probability. Simple probabilities vs. joint probabilities Which seems more likely? a. Jane is a lottery winner. b. Jane is happy lottery winner. Many pick b, but a must have a higher probability, as a Venn diagram clearly shows. Problem: conjunction fallacy. 5

6 Conjunction fallacy: Venn diagram 1-7 7

7 Conditional Probability 1-8 The probability of an event given that another event has occurred is called a conditional probability. The conditional probability of A given B is denoted by PP AA BB A conditional probability is computed as follows : PP AA BB = PP(AA BB) PP(BB) 8

8 Bayes Rule 1-9 Often we begin probability analysis with initial or prior probabilities Then, from a sample, special report, or a product test we obtain some additional information Given this information, we calculate revised or posterior probabilities Prior Probabilities New Information Application of Bayes Theorem Posterior Probabilities 9

9 Bayes Rule 1-10 Example: L. S. Clothiers A proposed shopping center will provide strong competition for downtown businesses like L. S. Clothiers. If the shopping center is built, the owner of L. S. Clothiers feels it would be best to relocate to the center. The shopping center cannot be built unless a zoning change is approved by the town council. The planning board must first make a recommendation, for or against the zoning change, to the council. 10

10 Bayes Rule 1-11 Prior probabilities Let A 1 = town council approves the zoning change A 2 = town council disapproves the change Using subjective judgment: P(A 1 ) =.7, P(A 2 ) =.3 11

11 Bayes Rule 1-12 New Information The planning board has recommended against the zoning change. Let B denote the event of a negative recommendation by the planning board. Given that B has occurred, should L. S. Clothiers revise the probabilities that the town council will approve or disapprove the zoning change? 12

12 Bayes Rule 1-13 Conditional Probabilities Past history with the planning board and the town council indicates the following: P(B A 1 ) =.2 P(B A 2 ) =.9 Hence: P(B C A 1 ) =.8 P(B C A 2 ) =.1 13

13 Bayes Rule 1-14 Bayes Theorem To find the posterior probability that event Ai will occur given that event B has occurred, we apply Bayes theorem PA ( B) i = PA ( i) PB ( Ai) PA ( ) PB ( A) + PA ( ) PB ( A) PA ( ) PB ( A) n n 14

14 Bayes Rule 1-15 Posterior Probabilities Given the planning board s recommendation not to approve the zoning change, we revise the prior probabilities as follows: PA ( B) 1 = = PA ( 1) PB ( A1) PA ( ) PB ( A) + PA ( ) PB ( A) (. 7)(. 2) (. 7)(. 2) + (. 3)(. 9) =.34 15

15 Bayes Rule 1-16 Conclusion The planning board s recommendation is good news for L. S. Clothiers. The posterior probability of the town council approving the zoning change is.34 compared to a prior probability of

16 Bayes Rule: Tabular Approach 1-17 (1) (2) (3) (4) (5) Events A i Prior Probabilities P(A i ) Conditional Probabilities P(B A i ) Joint Probabilities P(A i B) Posterior Probabilities P(A i B) A 1.7 A P(B) = /.41 17

17 Base rate neglect 1-18 A base rate is the relative frequency with which an event occurs. Example: Dick is a 30-year-old man. He is married w/ no children. A man of high ability and high motivation, he promises to be quite successfully in his field. He is well liked by his colleagues. subjects were shown personality sketches, allegedly from a group of professionals made up of engineers and lawyers. In one treatment, 70% engineers and 30% lawyers In another, 30% engineers and 70% lawyers 18

18 Base rate neglect and Bayes rule 1-20 pr(b A) = pr(b) * [pr(a B) / pr(a)] This is a way of updating your probability estimate based on new information. You have a barometer that predicts weather. Example: pr(rain) = pr(r) = 40% pr(dry) = pr(d) = 60% pr(rain predicted rain) = pr(rp R) = 90% pr(rain predicted dry) = pr(rp D) = 2.5% 20

19 Base rate neglect and Bayes rule 1-21 Best prediction of tomorrow s weather without looking at barometer is prior (base rate) distribution: you would say 40% chance of rain. What should you predict when barometer predicts rain? That is, what is probability of rain conditional on rain being predicted? pr(r RP) = pr(r) * [pr(rp R) / pr(rp)] 21

20 Base rate neglect and Bayes rule 1-22 We first need to work out pr(rp). This equals: pr(rp R) + pr(rp D) Use conditional probability rule: pr(rp R) = pr(rp R) / pr(r) Re-arrange: pr(rp R) = pr(rp R) * pr(r) =.9 *.4 =.36 22

21 Base rate neglect and Bayes rule 1-23 Next work out pr(rp D). Begin with conditional probability: pr(rp D) = pr(rp D) / pr(d) Re-arrange: pr(rp D) = pr(rp D) * pr(d) =.025 *.6 =.015 Therefore pr(rp) = =.375 Note that the barometer (conservatively) predicts rain less than it actually rains. 23

22 Base rate neglect and Bayes rule 1-24 What should you predict when barometer predicts rain? pr(r RP) = pr(r) * [pr(rp R) / pr(rp)] =.4 * (.9 /.375) =.96 If we know that the barometer is predicting rain, there is a 96% chance that it will rain, vs. only a 40% chance if we don t know the barometer reading. 24

23 Exercise 1-25 Rex is a smart fellow. He gets an A in a course 80% of the time. Still he likes his leisure, only studying for the final exam in half of the courses he takes. Nevertheless when he does study, he is almost sure (95% likely) to get an A. Assuming he got an A, how likely is he studied? If someone estimates the above to be 75%, what error are they committing? Explain. 25

24 Hot hand phenomenon 1-30 As in hot hand phenomenon in sports: In basketball, it is erroneously thought that you should give ball to a hot player A player with a hot hand (also known as a "streak shooter" or an athlete "on a roll") is a player who has a better chance of making a basket after one or more successful shots than after having missed a shot. It is especially true if people aren t sure about nature of distribution/population. 30

25 Hot hand phenomenon 1-31 People thought that the chances of making a basket increased after a player had made several successful shots when in truth the chances of making the next basket were not significantly different from the player's overall probability of making a basket.

26 Gambler s fallacy 1-32 Gambler s fallacy: the belief that a successful outcome is due after a run of bad luck Or, more generally, the belief that a series of independent trials with the same outcome will soon be followed by an opposite outcome Example: Suppose that an unbiased coin is flipped three times, and each time the coin lands on Heads. If you had to bet $100 on the next toss, what side would you choose? 32

27 Gambler s fallacy 1-34 Gambler s fallacy may apply if people are fairly sure about nature of population. They think even small samples should always look like population. Performance has to average out Winning lottery numbers are avoided based on mistaken view that they are not likely to come up again for a while. 34

28 Exercise 1-35 Q: How do hot hand phenomenon and gambler s fallacy relate to representativeness? Provide examples from sports. In what way are they different? 35

29 Overestimating predictability 1-38 Tendency to underestimate regression to mean neglect the diagnosticity of the information on which they base their predictions, and as a result, they make nonregressive predictions GPA example: subjects were asked to predict GPA in college from high school GPA of entrants to the college. 38

30 Overestimating predictability 1-39 Example: Suppose that scores on a high school academic achievement test are moderately related to college grade point averages (GPAs). Given the percentiles below [shown in Figure], what GPA would you predict for a student who scored 725 on the test?

31 Overestimating predictability 1-40 Example: (cont d)

32 1-42 Other biases related to representativeness Recency: Recent evidence is more compelling. Salience: Dramatic evidence is more compelling. Availability: Freely available, easily processed information is more compelling. 42

33 Anchoring 1-43 People are initially anchored on their prior belief. Quickly multiply these eight numbers: 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 Most people will come up with a low estimate: anchored on product of first 4 or 5. A bit better (but still too low) with: 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 43

34 Anchoring bias: Example of anchoring to irrelevant info 1-45 Wheel with numbers was spun. Subjects were asked: 1. Is the number of African nations in the UN more or less than wheel number? 2. How many African nations are there in the UN? Answers were highly influenced by wheel: Median answer was 25 for those seeing 10 from wheel. Median answer was 45 for those seeing 65 from wheel. Grasping at straws! 45

35 Anchoring vs. representativeness 1-46 Anchoring says new information is discounted. Representativeness (base rate neglect variety) says people are too influenced by latest information. Potential conflict between anchoring and representativeness in how people deal with new evidence. Which is right? Perhaps both depending on situation 46

36 Anchoring vs. representativeness ii It is argued that people are coarsely calibrated. Suppose morning forecast is for sun. Day starts sunny. You go on a picnic. Some dark clouds start to move in You are anchored to prior view and discount clouds. More dark clouds: the same thing 47

37 Anchoring vs. representativeness iii Even more dark clouds. Now you coarsely transition thinking that it s going to rain for sure! What is reality? Never 0% or 100%. New information should alter probabilities but a flip-flop doesn t make sense. Coarse calibration has been used to explain tendency for prices to trend and eventually reverse. 48

38 Preview of financial errors from heuristics and biases 1-49 Expectations influence perceptions: If most people are saying good/bad things about company, you will find good/bad things It has been argued that cognitive dissonance can: Explain why people don t exit poorlyperforming mutual funds 49

39 Preview of financial errors from heuristics and biases ii Diversification heuristic Stock-bond menu influences risk taking in DC plans Ambiguity aversion Under-diversification Information overload Lower participation rates for DC plans with more investment choices 50

40 Preview of financial errors from heuristics and biases iii Representativeness (and halo effects) Good companies are good stocks thinking may lead to value advantage Recency May explain chasing winners Anchoring and slow adjustment coupled with representativeness May explain momentum and price reversal 51