Experimental Economics Lecture 3: Bayesian updating and cognitive heuristics Dorothea Kübler Summer term 2014 1
The famous Linda Problem (Tversky and Kahnemann 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.
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.
Do you believe that (8) is more likely than (6)? About 90% of subjects do. In a sample of well-trained Stanford decision-sciences doctoral students, 85% do. They all commit the conjunction fallacy. Why?
Another bias (Kahnemann & Tversky,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 identified 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?
The median and modal response in experiments is 80%. The true response: Pr Blue identified as Blue 15% *80% 15% *80% 85% *20% 41% Base rate neglect
What do the two examples have in common? Representativeness Representativeness can be defined as the degree to which [an event] (i) is similar in essential characteristics to its parent population and (ii) reflects the salient features of the process by which it is generated (Kahneman & Tversky, 1982, p. 33). Replication in more abstract settings by Grether (1980), Harrison (1989), and Grether (1991)
One more bias (Edwards, 1968) 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 I 8 reds and 4 blues)?
Typical reply is between 0.7 and 0.8 But Prob(A I 8 reds and 4 blues)=0.97 Conservatism
And another 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?
Right answer: E and 7 Why 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 falsification tests. Confirmatory Bias (too little learning)
One more bias (Dittmann, Kübler, Maug and Mechtenberg, 2011) You are one among 5 voters. All have to place a vote for or against a measure M. In state 1, M is good for each voter, and in state 2, M is bad for each voter. All voters observe the state. How likely is it that you decide the outcome of the vote?
Significant group said 50%. Right answer: If voters make no errors, 0%; if observed error rates taken into account, then there is a positive but small probability of being pivotal. Never 50%! Illusion of control
What do we know about randomness? Imagine that you see me flipping a coin, and the outcome is: (Head, Head, Head) Do you believe that it is a fair coin?
Consider sequence 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
Consider sequence 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.
You flip a coin and observe Head-Head. You know that the coin is fair. What do you bet on for the next coin-flip, Head or Number? Why?
You flip a coin and observe Head-Head. You know that the coin is fair. What do you bet on for the next coin-flip, Head or Number? Why? You should be indifferent.
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-flip, most people who know that the coin is fair would now bet on Number.
But: In reality, this week it is as likely (or unlikely) as last weekthatthenumber2 (or10 or1 or 5) wins. And Head is as likely to appear after Headhead than Number. Reason: The events in question are independently distributed.
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 scorealreadythreetimes. One should always pass the ball to a teamplayer who just scored several times in a row.
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.
So what s the bias now? Gambler s Fallacy and Hot Hand seem to contradict each other. In the first 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?
Representativeness/ 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.
In reality, representativeness of random sequences holds true only for infinite 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 infinite sequence of coin-flips must exhibit 50% Heads and 50% Numbers. But this is not true for finite sequences.
Is this all? Would you believe that the following sequence is random? Head, Number, Head, Number, Head, Number People think that repetitive patterns are not random, even if they are. People think that randomness produces absence of repetitive patterns.
What are the consequences of failure to detect randomness? Examples for possible consequences: Cancer Clusters Taxpayer money is spent for investigation although almost all of these clusters are randomly generated. Belief in a Just World / Just-World-Bias Guilt ascriptions to victims (cognitive bias referring to the common assumption that the outcomes of situations are caused or guided by some universal force of justice, order, stability, or desert, not randomness)
Possible psychological reason for failure to detect randomness: Illusion of control Psychological findings: People who accept the influence of randomness on their lives feel less in control and less secure and are more often depressive or anxious. Believing that they have control over their own life, people with illusion of control ascribe the same amount of control to others and deny random causes for failures and successes.
Is Law of small numbers/ Representativeness bias interesting for economists? Mixed Strategies Playing mixed strategies implies the ability to randomize. Adults are not good in writing down or recognizing random sequences. Does this mean that they cannot play mixed strategies?
Methods and critique (Rapoport & Budescu,92) Are the traditional experiments that found the Law of small numbers correct? Traditional Designs: A) Subjects are asked to write down a sequence that could be generated by flipping a coin 300 times and avoid repetitive patterns too much. B) Subjects are asked to bring given events (like drawn cards) into a random order. C) Subjects are asked to identify random sequences among various sequences.
Problems: Randomness is an unobservable property of infinite sequences. However, in the experiments of type A), B) or C), subjects were asked to identify randomness of finite sequences of events. The task already suggested that randomness is expressed in patterns (or absence of patterns) of events.
Problem 1: Experiments A, B and C require the impossible and generate an experimenter-demand effect. Problem 2: The experimenters themselves cannot decide whether the (finite) sequences of events generated by the subjects are truely randomly generated or not. Solutions?
Then, hypothesis that a finite sequence of events is randomly generated can be tested. Problem 2 (unobservability of randomness) is not solvable. However, one can deal with problem 2 as follows: Hypothesis: The finite sequence in question is generated by a Bernoulli-Process. Definition: Random sequence of binary events means sequence that is generated by a Bernoulli-Process. Randomness is equated with certain distributional characteristics.
Problem 3: It is difficult to explain to subjects what a random sequence of events is (problem of definition). Every explanation recurring to infinite sequences does not help to understand the task at hand. But every explanation recurring to finite sequences and the task at hand is misleading. Example: In random sequences there is no recognizable pattern. Avoidance of repetitive patterns like (x,x,x)
Subjects shall put in effort and concentrate. To achieve this, one has to incentivize them. Problem 4: In the experiments A, B and C it is difficult to incentivize subjects. The experimenter has to analyze the data first, before he can pay subjects according to their achievements.
New Methods (Rapoport & Budescu, 1992) Problems 1, 3 and 4 are solvable. Problem 2 (what is a random sequence?) is dealt with as follows: Idea: Don t think, but act! Difference between explicit and implicit knowledge Subjects had to play a repeated zero-sum game in which the unique Nash equilibrium is in mixed strategies.
In the lab, equilibrium will be played only if subjects are able to randomize. Advantages of this idea: a) It is easy to explain the task at hand. The explanation need not mention random sequences or randomness at all. All that has to be explained is the rules of thegameand thepayoffs.
b) Nothing impossible is asked from the subjects. They do not have to identify random sequences. They only have to randomize themselves. (Implicit vs. explicit knowledge) c) Conditional on the opponent s playing the equilibrium strategy, a subject is payed according to how well she / he randomizes (i.e. plays the equilibrium strategy).
The game: 1,2 Red Red 1,-1 Black -1,1 Black -1,1 1,-1 This game is repeated N rounds. In each round, the equilibrium is to play Red or Black with probability ½.
Design of Rapoport & Budescu Treatment D ( Dyad ) Baseline treatment: Random matching of pairs Each pair plays the game on the previous slide for 150 rounds. Thegameiscommonknowledge. Subjects are told that their job is to earn as many points as possible. In each round, each subject makes a hidden choice between Red and Black.
At the end of each round, subjects are informed about the outcome of the game (and can therefore infer the choice of their opponent). Treatment S ( single ) - Control treatment 1: Change: Now, all 150 Choices must be made beforehand. Treatment-comparison D-S isolates the effects of the feedback (if any).
Treatment R ( Randomization ) Control treatment 2: Traditional design: Subjects do not play a gamebuthaveto writedown a sequence that could have been generated by flipping a coin 150 times. Treatment comparison S-R isolates effects of the interactive nature of the game that requires implicit instead of explicit knowledge. Advantage: Direct comparison with old standard experiments becomes possible if treatment R replicates the old results.
All subjects participated in treatment D and one of the control treatments. To exclude order effects, some subjects were assigned first to treatment D and then to the control treatment, and some subjects were assigned first to the control treatment and then to treatment D. Treatment R was always assigned second after D. (Assumption that there would not be any order effects since D and R are sufficiently different -weakness of design?)
Procedure in treatment D: In the Instructions, the game was explained. No detailed explanation of the research question (why?). Instructions said that the experiment served to investigate decision behavior in a competitive situation. True (!), but not too revealing
Each subject got a package of 5 red and 5 black cards and an initial endowment of 20 points. The pairs were seated opposite of each other. Subjects made their choice by placing a card on the table, hiding the colored side. Then, the experimenter turned both cards and determined the winner. The loser of the round then had to transfer one of his points to the winner. The subjects took their cards back, mixed them, and the next round started. Communication was prohibited explicitly.
The game ended either after 150 rounds or after one of the players lost all of his 20 points. Procedures in treatments S and R: In treatments S and R, subjects were not placed opposite to each other but isolated. They filled in decision sheets. The experimenter collected the sheets. Then, in treatment S only, the experimenter determined the winner for each round.
In treatment R, the subject whose sequence was closest to a previously computergenerated sequence (Bernoulli-process) was awarded a prize of 25 points. (This was known beforehand, too.) Main results: In treatment R, standard results were replicated: The hypothesis that subjects generated Bernoulli-sequences had to be rejected. This effect vanished to great extent in treatments D and S. Moreover, subjects mixed with approx. 1/2 most of the time, as predicted.
The bottom line: People can randomize if they have to do it for strategic reasons, interacting with others. They cannot do it only in the head. Does this mean that misperception of randomness is not important for economists?.
The Law of small numbers on markets Camerer (1989): Does the Basketball Market Believe in the Hot Hand? AER 79, 5, 1257-1261. Using a data set containing information about bets on professional basketball games between 1983 and 1986, Camerer finds that a small Hot-Hand-effect exists on the betting market.
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): Availability Representativeness Anchoring
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). Public vs private transport; smoking vs illegal drugs.
(Why) does this make sense? 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 influenced 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)
Anchoring Can be used whenever people start with an initial value that they update in order to reach a final value. If the final 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.
The Representativeness heuristic/ Law of small numbers Often leads to significant deviations from Bayesian rationality. As we have seen before, the representativeness heuristic explains -Hot hand -Gambler s fallacy -Conjunction fallacy -Base-rate neglect.
Question: Are Anchoring/ Conservatism and Representativeness/ Law of small numbers compatible with each other or in conflict? Griffin and Tversky (1992): people focus on the strength or extremeness of the available evidence (e.g., the size of an effect) with insufficient regard for its weight or credence (e.g., the size of the sample). Thus, both heuristics can be reduced to a single one. That is: anchor e.g. on early evidence and disregard its (small) weight; in representativeness, small size of sample is neglected.