Experimental and Behavioral
Lecture 2: Bayesian updating and
Prof. Dr. Dorothea Kübler
Summer term 2019
Thank you for sending them!
Many good summaries, some very very short…but I accepted all of them
unless you received an email from me.
A criticism voiced by one of you: early experiments were designed to
support the theory, results are not surprising
(e.g., regarding indifference curves).
Many pointed out that early experiments laid the foundations in terms of
methodology (paying subjects etc.).
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
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 (3)?
Almost no one does.
Do you believe that (8) is more likely than (6)?
About 90% of subjects do. In a sample of well-
trained Stanford decison-science doctoral students,
They all commit the conjuction fallacy. Why?
Conjunction law of probabilities: P(A&B)≤ P(A)
A. Linda is a bank teller (6)
B. Linda is active in the feminist movement (3)
A&B. Linda is a bank teller and is active in the feminist
From the conjunction law follows that P(8)
(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 at night and
found that the witness correctly identified each of the two
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?
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%.
Base rate neglect
Updating belief X after learning M should follow rule Bayes’ Rule:
𝑃 𝑋 𝑀 = 𝑃 𝑀 𝑋 𝑃(𝑋)
X- Car was Blue
M- Witness identifies the car as blue
𝑃 𝐵𝐵𝐵𝐵 𝐼𝐼𝐵𝐼𝐼. 𝑏𝐵𝐵𝐵 =
𝑃 𝐼𝐼𝐵𝐼𝐼. 𝑏𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 ∙ 𝑃(𝐵𝐵𝐵𝐵)
𝑃 𝐼𝐼𝐵𝐼𝐼. 𝑏𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 ∙ 𝑃(𝐵𝐵𝐵𝐵)
𝑃 𝐼𝐼𝐵𝐼𝐼. 𝑏𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 ∙ 𝑃 𝐵𝐵𝐵𝐵 + 𝑃 𝐼𝐼𝐵𝐼𝐼. 𝑏𝐵𝐵𝐵 𝐼𝑛𝐼 𝑏𝐵𝐵𝐵 ∙ 𝑃 𝐼𝑛𝐼 𝑏𝐵𝐵𝐵
Given that the proportion of blue cabs in the city is only 15%, the
posterior probability is not correctly updated: Base rate neglect
What do the two examples have in common?
Representativeness can be defined as „the degree to which
an event :
(i) is similar in essential characteristics to its parent
(ii) reflects the salient features of the process by which it is
generated“ (Kahnemann& Tversky, 1982, p.33).
Another violation of Bayes’ rule
2 urns, A and B, look identical from outside.
A: 7 red and 3 blue balls.
B: 3 red and 7 blue balls.
One urn is randomly chosen, both are equally likely.
Suppose that random draws with replacement from
this urn amount to 8 reds and 4 blues.
What is the probability that the urn is A?
Typical reply is between 0.7 and 0.8
But Prob(A I 8 reds and 4 blues)=0.967
Conservatism - underweighting of likelihood
Conservatism versus base rate neglect:
Griffin and Tversky (1992): people overweight the
strength of evidence and underemphasize its weight
Obstacles to learn: two more biases
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
People tend to avoid pure falsification tests. Too
If people believe a hypothesis is true, their actions often
produce a biased sample of evidence that reinforces
Example: If waiter has stereotype that young patrons do
not tip well (she may display confirmation bias by
remembering times that young patrons have not tipped
well and ignoring times when young patrons have tipped
well), this stereotype may become a self-fulfilling
prophecy in that she may talk differently and wait less on
young patrons, which in turn may make them tip less
(thereby confirming the stereotype).
Thus both confirmation bias and self-fullfilling
prophecies inhibit learning.
Monty Hall problem.
One door has a car behind it, and behind other doors is a goat. Which
door do you choose?
Would you like to change you choice?
Monty Hall problem
Most people stick to the original choice, but this violates Bayes’ Rule.
𝑃 𝑋 𝑀 = 𝑃 𝑀 𝑋 𝑃(𝑋)
𝑃 𝐶𝐶𝐶1 𝑛𝑜𝐵𝐼𝑜, 𝑐𝑐𝑛𝑐𝐵1 =
𝑃 𝑛𝑜𝐵𝐼𝑜 𝑐𝐶𝐶1, 𝑐𝑐𝑛𝑐𝐵1 𝑃(𝑐𝐶𝐶|𝑐𝑐𝑛𝑐𝐵1)
0.5 ∙ 1𝑜
𝑃 𝐶𝐶𝐶𝐶 𝑛𝑜𝐵𝐼𝑜, 𝑐𝑐𝑛𝑐𝐵1 =
𝑃 𝑛𝑜𝐵𝐼𝑜 𝑐𝐶𝐶𝐶, 𝑐𝑐𝑛𝑐𝐵1 𝑃(𝑐𝐶𝐶|𝑐𝑐𝑛𝑐𝐵1)
1 ∙ 1𝑜
Monty Hall problem in markets
Kluger and Wyatt (2004)
Two types of assets in the experiment: With and without
the right to „switch doors“
Prediction: The asset’s price with right should be twice as
high as asset’s price without the right.
Market aggregation („market magic“): if at least two out of
six participants are rational Bayesians, the prices should
be close to fundamental value.
Result: Only 25% of groups were close to rational prices.
Is randomness intuitive?
Consider sequence 1 and 2 of roulette-wheel outcomes.
Sequence 1: Red-red-red-red-red-red
Sequence 2: Red-red-black-red-black-black
Is sequence 1 less, more or equally likely?
They are equally likely, but most think that sequence 1 is
Representativeness/ Law of Small
1. Representiveness does not respect sample size
2. Law of small numbers (Tversky and Kahnemann 1971):
„All samples will closely resemble the process or populations that
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. „Hot hand“ for example.
Only law of large numbers is true
In reality, representativeness of random sequences holds true
only for infinite sequences („law of large numbers“).
The shorter the sequence, the less it must 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% Tails. But this is not true for finite sequences.
Can people easily generate random
Often people generate sequences with negative
autocorrelation (Wagenaar, 1972). Direct consequence of the
law of small numbers.
Who is doing better than average?
1. Mathematically sophisticated
2. Experienced (trained)
3. Children (Ross and Levy 1958)