Bayesian updating and cognitive heuristics

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1 January 7, 2015

2 The famous Linda problem (Tversky and Kahneman 1983) "Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations."

3 The famous Linda problem (2) Please rank the following statements by their probability of being true: (1) Linda is a teacher in elementary school. (2) Linda works in a book-store and takes Yoga-classes. (3) Linda is active in the feminist movement. (4) Linda is a psychiatric social worker. (5) Linda is a member of the League of Women Voters. (6) Linda is a bank teller. (7) Linda is an insurance salesperson. (8) Linda is a bank teller and is active in the feminist movement.

4 The famous Linda problem (3) Do you think (8) is more likely than (6)? About 90% of subjects do. In a sample of well-trained Stanford decision-sciences doctoral students, 85% do. conjunction fallacy

5 Tversky and Kahneman 1972 A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: (a) 85% of the cabs in the city are Green and 15% are Blue. (b) A witness said the cab was Blue. (c) The court tested the reliability of the witness during the night and found that the witness correctly identied each of the 2 colors 80% of the time and failed to do so 20% of the time. What is the probability that the cab involved in the accident is Blue rather than Green?

6 Tversky and Kahneman 1972 (2) The median and modal response in experiments is 80%. True answer: Pr(Blue/identied as blue) = Base rate neglect 15% 80% 15% 80%+85% 20% = 41%

7 Representativeness What the two examples have in common: Representativeness Representativeness can be dened as "the degree to which [an event] (i) is similar in essential characteristics to its parent population and (ii) reects the salient features of the process by which it is generated" (Kahneman & Tversky, 1982).

8 Representativeness and nance Cooper et al (2001): average abnormal returns of 53 percent associated with adding a dot-com sux to rm names during the internet bubble ( ) independent to the extent to which the rm was actually involved with the internet Investors seemed to view all such rms as representative of dot-com stocks

9 Consequences of representativeness for nance If a sector of the market is doing well, then investors may begin to judge past performance as being representative of future performance. It also explains why individual investors gravitate towards the best performing mutual funds, even though past performance is NOT representative of future results. Representativeness causes us to underestimate the probability of a change in the underlying trend A number of studies indicate that extreme past winners (which continue to win because of representativeness) become signicant underperformers Likewise, extreme losers tend to outperform in the years following such poor performance.

10 Conservatism 2 urns, A and B, look identical from outside. A contains 7 red and 3 blue balls. B contains 3 red and 7 blue balls. One urn is randomly chosen, both are equally likely. Suppose that random draws from this urn amount to 8 reds and 4 blues. What is Prob(A / 8 reds and 4 blues)?

11 Conservatism (2) Typical reply is between 0.7 and 0.8 But Prob(A / 8 reds and 4 blues)=0.97 Consequences: Conservatism bias implies investor underreaction to new information Conservatism bias can generate: short-term momentum (tendency for rising asset prices to rise further, and falling prices to keep falling) in stock returns post-earnings announcement drift, i.e., the tendency of stock prices to drift in the direction of earnings news for three-to-twelve months following an earnings announcement (while the information content should be quickly digested by investors and incorporated into the ecient market price)

12 Conrmatory bias (Wason 1968) You are presented with four cards, labelled E, K, 4 and 7. Every card has a letter on one side and a number on the other side. Hypothesis: "Every card with a vowel on one side has an even number on the other side." Which card(s) do you have to turn in order to test whether this hypothesis is always true?

13 Conrmatory bias (2) Right answer: E and 7 People rarely think of turning 7. Turning E can yield both supportive and contradicting evidence. Turning 7 can never yield supportive evidence, but it can yield contradicting evidence. People tend to avoid pure falsication tests. Conrmatory bias (too little learning) Can lead to overcondence: investors will ignore evidence that their strategies will lose money (Pompian 2006, Zweig 2009)

14 Illusion of control People behave as if they think they have greater control when they roll dice themselves than when someone else rolls for them (e.g., Fleming & Darley, 1986). People prefer to pick their own lottery numbers than to have others pick for them (Dunn & Wilson, 1990; Langer, 1975). Pedestrians in New York push the walk button even though it will not get them across the street any faster. Illusion of control: greater condence in one's predictive ability or in a favorable outcome when one has a higher degree of personal involvement

15 Reaction ro randomness Imagine that you see me ipping a coin, and the outcome is: (Head, Head, Head) Do you believe that it is a fair coin?

16 Reaction ro randomness (2) Consider sequences 1 and 2 of roulette-wheel outcomes. Is sequence 1 less, more or equally likely? Sequence 1: Red-red-red-red-red-red Sequence 2: Red-red-black-red-black-black

17 Reaction ro randomness (2) Consider sequences 1 and 2 of roulette-wheel outcomes. Is sequence 1 less, more or equally likely? Sequence 1: Red-red-red-red-red-red Sequence 2: Red-red-black-red-black-black They are equally likely, but most think, that sequence 1 is less likely.

18 Reaction ro randomness (3) You ip a coin and observe Head-Head. You know that the coin is fair. What do you bet on for the next coin-ip, Head or Tail? Why?

19 Reaction ro randomness (3) You ip a coin and observe Head-Head. You know that the coin is fair. What do you bet on for the next coin-ip, Head or Tail? Why? You should be indierent.

20 Gambler's fallacy Examples: Last week's winning numbers were 2, 10, 1 and 5. Gamblers of the current week avoid betting on these numbers. After Head appeared twice in a fair coin-ip, most people who know that the coin is fair would now bet on Tail.

21 Gambler's fallacy (2) But: In reality, this week it is as likely (or unlikely) as last week that the number 2 (or 10 or 1 or 5) wins. And "Head" is as likely to appear after "Head head" than "Tail Tail". Reason: The events in question are independently distributed.

22 Hot hand (Gilovich et al., 1985) Most people agree with the following statements: "A basket-ball player that scored already three times has a higher probability of scoring with his fourth attempt than a player that failed to score already three times." "One should always pass the ball to a teamplayer who just scored several times in a row."

23 Hot hand (2) But: The data show that the scores of basket-ball players are uncorrelated. The probability to score after three scores is not higher than the probability to score after three failed attempts.

24 Gambler's fallacy and hot hand "Gambler's Fallacy" and "Hot Hand" seem to contradict each other. In the rst case (GF), a sequence (...,x,x,x) inspires the belief that the next event is likely not to be x. In the second case (HH), people expect the exact opposite. How can we reconcile these two biases?

25 Consequences: Momentum and mean reversion "Gambler's Fallacy" => mean reversion: trends suddenly stop working "Hot Hand"=> momentum: a stock which has moved in one direction is likely to continue moving in the same direction Both eects are seen frequently in markets and there seems to be genuine evidence for them over dierent time frames

26 Law of small numbers People tend to believe that each segment of a random sequence must exhibit the true relative frequencies of the events in question. If they see a pattern of repetitions of events, i.e. a segment that violates this "law", they believe that the sequence is not randomly generated.

27 Law of small numbers (2) In reality, representativeness of random sequences holds true only for innite sequences. The shorter the sequence, the less must it represent the true frequencies of events inherent in the random process by which it has been generated. An innite sequence of coin-ips must exhibit 50% Heads and 50% Tails. But this is not true for nite sequences.

28 Law of small numbers (3) Is this all? Would you believe that the following sequence is random? Head, Tail, Head, Tail, Head, Tail... People think that repetitive patterns are not random, even if they are. People think that randomness produces absence of repetitive patterns.

29 The law of small numbers on markets Camerer (1989): Does the Basketball Market Believe in the "Hot Hand"? AER 79, 5, Using a data set containing information about bets on professional basketball games between 1983 and 1986, Camerer nds that a small Hot-Hand-eect exists on the betting market.

30 Heuristics in dealing with probabilities Question: Is there a theory about how people who are not rational in the sense of being Bayesian deal with probabilities? People employ heuristics (Kahneman and Tversky) Heuristic: mental short cuts to ease the cognitive load of making a decision Availability Representativeness Anchoring

31 The availability heuristic Used when people estimate the probability of a given event of type T according to the number of type-t events they can recollect. Example: people overestimate the frequency of rare risks (much publicity) but underestimate the frequency of common ones (no publicity).

32 The availability heuristic (2) Normally, one can remember the more type-t events, the more frequently type-t events occur. But: The ease with which we remember certain events is inuenced by other factors, too, e.g. by the emotional content or salience. Since people do not correct for these other factors, the availability heuristic is biased. (Biased sampling)

33 Anchoring Can be used whenever people start with an initial value that they update in order to reach a nal value. If the nal value is biased into the direction of the initial value, this is called "anchoring" according to Kahneman & Tversky. Example: A person stops too early to collect (costless) information. Closely related to conservatism.

34 Anchoring If traders anchor to an entry point after entering a position, could explain why many traders will refuse to take a loss and wait instead for the market to return to that entry point. Salience plays an important role in anchoring: we are most likely to anchor decisions to criteria that capture our attention. For that reason, traders commonly anchor to high points and low points in market movements A useful behavioral rule is to assume that markets, in probing to establish value, will gravitate toward the price points of highest salience: those anchored by the largest numbers of traders (Brett Steenbarger)

35 Wishful thinking: Mayraz Wishful thinking: the formation of beliefs and making decisions according to what might be pleasing to imagine instead of by appealing to evidence, rationality, or reality Mayraz (2012): Subjects presented with a graph such as this one:

36 Wishful thinking: Mayraz (2) Task: predict the day 100 price Accuracy bonus for good predictions. Farmers: gain if day 100 price is high. Bakers: gain if day 100 price is low. 1 pound for a good prediction.

37 Wishful thinking: Test statistic Treatment eect: mean prediction by farmers - mean prediction by bakers Null: Treatment eect 0 Wishful thinking: treatment eect>0

38 Wishful thinking: Results

39 Wishful thinking: conclusion Bias independent of incentives for accuracy, increases with uncertainty, appears to be increasing with what's at stake. We should expect bias whenever decisions are based on subjective judgment, including environments in which biased beliefs have costly consequences.

40 Monty Hall Problem Named after a 1970s US game show called "Let's Make a Deal" that was hosted by Monty Hall. A contestant is shown 3 closed doors. Behind one is a desirable and high-value grand prize such as a car or a motorboat. The remaining two doors conceal a fake prize: a goat. The contestant is invited to choose one of the doors to be opened Let's say the contestant chooses door a. Now Monty Hall reveals a goat behind one of the remaining 2 doors: let's say it's door b. Does the contestant want to switch to door c or stay with his original choice of a? What would you do?

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