Transcript Lecture 8: Psychological approaches to risk

From risk to opportunity
Lecture 8
John Hey and Carmen Pasca
Some of the key psychologists: Kahneman, Tversky and Busemeyer
Kahneman won the Nobel Prize in Economics in 2002 along with
Vernon Smith (an experimental economist).
He is still active. See his book Thinking, Fast and Slow (2011)
Macmillan. ISBN 978-1-4299-6935-2.
Lecture 8 Psychological approaches to risk: Warning
• To fully appreciate this lecture you will need an
understanding of the economists’ favourite theory of
decision-making under risk/uncertainty: (Subjective)
Expected Utility theory.
• This we will examine in detail in Lectures 9 and 10.
• But we need to give you an outline now so that you
appreciate this lecture.
• So we start with that now. Do not worry if you do
not understand it all. You will do later!
• It is based on axioms – which we will explain later.
Lecture 8: EUT
• We denote the n possible outcomes of any gamble by x1,…xn.
• We assume that xi has an objective probability pi (for all i).
• Under the axioms of EUT a decision-maker should evaluate
this risky prospect as
• p1u(x1) + … + pnu(xn)
• where u(.) is the decision-maker’s utility function, and is
fixed.
• The utility function is normalised (like temperature) by
putting the utility of the worst outcome equal to 0 and the
utility of the best equal to 1.
Lecture 8: The axioms of Expected Utility theory
Here the symbol ≿ means prefers or is indifferent to.
Completeness: For every A and B, either A ≿ B or B ≿ A.
Transitivity: If A ≿ B and B ≿ C then A ≿ C.
Continuity: If A ≿ B ≿ C then there is some gamble
which gives A with probability p and C with probability
(1-p) for which the individual is indifferent with B.
• Independence: If A ≿ B then the individual prefers the
gamble which gives A with probability p and C with
probability (1-p) to the gamble which gives B with
probability p and C with probability (1-p), for any p, C.
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The Independence Axiom
• If an individual prefers A to B, then he or she also prefers the
gamble which gives A with probability p and C with probability
(1-p) to the gamble which gives B with probability p and C
with probability (1-p), for any p and any C.
• Note that the only difference between the two gambles is
that you get A in the first and B in the second.
A
p
1-p
C
B
p
1-p
C
Lecture 8 Psychological approaches to risk: First for the non-mathematically inclined!
• Problem 1: choose between
• A which gives you €1000 with probability 80% and €0
with probability 20% and
• B gives you €750 for sure.
• Problem 2: choose between
• C which gives you €1000 with probability 20% and €0
with probability 80% and
• D which gives you €750 with probability 25% and €0
with probability 75%.
Lecture 8 Psychological approaches to risk: K&T
• In what follows we use (x,y;p) to denote a gamble which yields x with
probability p and y with probability (1-p).
• (x) denotes the certainty of x.
• Problem 1: choose between
• A = (€1000,0;0.8) and B = (€750).
• Problem 2: choose between
• C = (€1000, €0;0.2) and D = (€750, €0;0.25).
• According to EUT if you chose A in Problem 1 you should choose C in
Problem 2.
• WHY? Use the expected utility result and calculate expected utilities.
(Note we are normalising by putting u(€0) = 0) and u(€1000) = 1.)
• A preferred to B if and only if 0.8 > u(€750)
• C preferred to D if and only if 0.2>0.25u(€750)
Lecture 8 Psychological approaches to risk: K&T
• With many similar examples (which we show later)
K&T satisfied themselves that EUT is violated.
• Their explanations include:
• Reference Point: evaluations are made with respect
to some reference point, which may change.
• Isolation Effects: the way prospects are evaluated
depends upon the way that they are framed.
• Probability Weighting: people do not use the correct
probabilities but distort (weight) them in some way.
Lecture 8 Psychological approaches to risk: Reference Point
first for the notationally inept
• Problem 3: First I give you €1000 and then ask you to
choose between
• A which gives you €1000 with probability 50% and €0
with probability 50% and
• B which gives you €500 with certainty
• Problem 4: First I give you €2000 and then ask you to
choose between
• C which loses you €1000 with probability 50% and loses
you nothing with probability 50% and
• D which loses you €500 with certainty.
Lecture 8 Psychological approaches to risk: Reference Point
• Problem 3: I give you €1000 and then ask you to choose
between
• A = (€1000, €0;0.5) and B = (€500).
• Problem 4: I give you €2000 and then ask you to choose
between
• C = (- €1000,0;0.5) and D = (- €500). (Note the minus signs =
Losses!!)
• According to EUT if you chose A in Problem 1 you should
choose C in Problem 2.
• WHY? If and only if
• 0.5u(€2000+w)+0.5u(€1000+w)>u(w+ € 1500)
Lecture 8 Psychological approaches to risk: Isolation effect
first for the notationally disfunctional
• Problem 5: This is a two-stage problem; at the first stage there is a
75% of ending the problem without any money and a 25% chance
of moving to the second stage; if you get to the second stage you
have to choose between
• A which gives you €1000 with probability 80% and nothing with
probability 20%
• B which gives you €750 for sure.
• Problem 6: This is a one-stage problem; choose between
• C which gives you €1000 with probability 20% and €0 with
probability 80% and
• D which gives you €750 with probability 25% and €0 with
probability 75%.
Lecture 8 Psychological approaches to risk: Isolation effect
• Problem 5: This is a two-stage problem; at the first stage
there is a 75% of ending the problem without any money and
a 25% chance of moving to the second stage; if you get to the
second stage you have to choose between
• A = (€1000,0;0.8) and B = (€750). (B gives €750 for sure).
• Problem 6: This is a one-stage problem; choose between
• C = (€1000,0;0.2) and D = (€750,0;0.25).
• According to EUT if you chose A in Problem 5 you should
choose C in Problem 6.
• WHY?
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They are exactly the same problem. Framing?
Lecture 8 Psychological approaches to risk: Probability weighting
First for those with notational problems
• Problem 7: choose between
• A which gives you €5000 with probability 1% and €0
with probability 99% and
• B which gives you €5 with certainty.
• Problem 8: choose between
• C which loses you €5000 with probability 1% and
loses nothing with probability 99% and
• D which loses you €5 with certainty.
Lecture 8 Psychological approaches to risk: Probability weighting
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Problem 7: choose between
A = (€5000,0;0.01) and B = (€5).
Problem 8: choose between
C = (- €5000,0;0.01) and D = (- €5).
CARE: choosing A and D is NOT a violation of EUT.
But it illustrates what K& T call an overweighting of
small probabilities.
• They suggest a weighting function w(.) on
probabilities.
Lecture 8 Psychological approaches to risk: weighting of probabilities
• Kahneman and Tversky found empirically that people
underweight outcomes that are merely probable in
comparison with outcomes that are obtained with
certainty; also that people generally discard
components that are shared by all prospects under
consideration.
• They also found that people overweight small
probabilities.
Lecture 8 Psychological approaches to risk: Prospect Theory
• Prospect Theory posits that the generic prospect
(x,y;p) is evaluated after editing with the function
• w(p)v(x) + w(1-p)v(y)
• where w(.) is the probability weighting function and
v(.) is the value function (which depends upon the
reference point). They suggest that v(.) is concave for
gains and convex for losses (but this is not crucial).
• Compare this with EUT’s
• pu(x) + (1-p)u(y)
Lecture 8 Psychological approaches to risk: Comments on PT
• Note, rather trivially, that since PT contains EUT as a special
case (EUT is nested inside PT if the reference point is fixed
and probabilities are not weighted)...
• ... then PT is bound to explain more.
• Note also that PT is based on rather dubious (unincentivated) experimental evidence.
• Note that PT may violate dominance: (€499, €498;0.5)
preferred to (€500).
• Since K&T’s original paper there have been developments
of the theory – partly to eradicate the violation of
dominance.
Lecture 8 Psychological approaches to risk: Cumulative Prospect Theory
• Tversky and Kahneman (JRU 1992)
• We do not have the time to go into detail but the key
difference is that the valuation function, after
editing, is given by
• w(p)v(x) + [1-w(p)]v(y) if v(x)>v(y).
• Compare this with the original
• w(p)v(x) + w(1-p)v(y)
• This no longer violates dominance.
• Note: Cumulative Prospect Theory with a fixed reference point is the same
as Rank Dependent EUT.
Lecture 8 Psychological approaches to risk: Other psychological theories
• There are lots. We only mention a couple.
• Coombs (1975) develops an alternative he calls
portfolio theory – which is rather close to meanvariance analysis (where the valuation of a prospect
depends upon its mean and variance). This latter is
used for the CAPM.
• Busemeyer and his colleagues over the years have
developed what they call Decision Field Theory which is
like EUT plus a theory of errors.
• Errors are important in decision-making, and we
should spend some time talking about them.
Lecture 8: Noise and Decision Field Theory
• It has been observed that subjects make different decisions even
with the same question.
• What are they doing?
• Perhaps they are deliberately randomising?
• Perhaps they just make mistakes from time to time?
• Most theories are deterministic.
• Most do not have errors/noise built into them.
• So either all are wrong or we need to add noise.
• Many experimentalists just add a homoscedastic normal error term
(to the difference in the valuations of the gambles).
• Decision Field Theory builds in heteroscedastic noise into the
theory. It is, if you like, EU plus noise.
Lecture 8: Decision Field Theory (“EU plus noise”)
• Busemeyer and Townsend (1993) Psychological
Review. Here we just give a sketch with 3 events.
• Consider a choice problem between two lotteries:
suppose the lottery on the left (right) leads to
outcomes xa, xb and xc (ya, yb and yc) if the event that
happens is a, b or c respectively .
• va, vb, vc, wa, wb,wc denote the associated utilities.
• To distinguish DFT from EU, Busemeyer and
Townsend use the expression valence instead of
expected utility.
Lecture 8: Decision Field Theory
• Valence of the left is V = Pava + Pbvb + Pcvc
• Valence of the right is W = Pawa + Pbwb + Pcwc
• where the attention weights, Pa, Pb and Pc, are random
variables centred on the individual’s subjective
probabilities pa, pb and pc.
• The decision is taken on the value of V-W, which is a
random variable with mean
• d = (pava + pbvb + pcvc) – (pawa + pbwb + pcwc)
• and variance σ2, which is given by
• s2{pa(va – wa)2 + pb(vb – wb)2 + pc(vc – wc)2 – (V-W)2}.
Lecture 8: Decision Field Theory
• This variance can be interpreted as the weighted variance of the
difference between the utilities of the outcomes conditional on the
events. Busemeyer and Townsend call it the variance of the valence
difference.
• An interesting special case is when both lotteries are certainties and one
dominates the other. In this case, we have that va = vb = vc = v, that wa =
wb = wc = w, and that v = w +d. In this case it follows that the variance σ2 is
zero, so that the subject never makes an error and always chooses the
dominating lottery.
• Note that this property is not implied by the EU specification with a
homoscedastic error term – here violations of dominance are possible.
• More generally, the theory implies that the more dispersed are the
outcomes for particular events, then the more likely it is that the subject
makes a mistake. Interesting.
Lecture 8: Psychological foundations of DFT
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They apply a particular version of approach-avoidance theory. The basic idea is
that the attractiveness of a reward or the aversiveness of a punishment is a
decreasing function of the distance from the point of commitment to an action.
When having to choose between two options, individuals will vacillate between
the two options as they consider them.
Their version of the theory predicts that individuals will vacillate back and forth
during deliberation, but the preference state will always drift far enough to
exceed the inhibitory threshold required to make a decision.
Technically, the process described by the approach-avoidance updating rule is a
Markov chain process.
It is a well-known theorem of Markov chains that the probability of leaving a set of
transient states approaches 1.0 as the number of transitions increases to infinity.
So they will come to a decision eventually.
But it is a random process.
Lecture 8 Psychological approaches to risk: process
• EUT has been generally accepted as a normative
model of rational choice (a model of what people
should do) and widely applied as a descriptive model
of economic behaviour.
• Thus, it is assumed that all reasonable people would
wish to obey the axioms of the theory and that most
people actually do, most of the time.
• It is based on axioms.
• The psychological approach to risk is about decisionmaking under risk as a process.
Lecture 8 Psychological approaches to risk:
• Under Prospect Theory, value is assigned to gains
and losses rather than to final assets; also
probabilities are replaced by decision weights.
• The value function is defined on deviations from a
reference point and is normally concave for gains
(implying risk aversion), commonly convex for losses
(risk seeking) and is generally steeper for losses than
for gains (loss aversion).
• Crucial here is the reference point.
Lecture 8 Psychological approaches to risk:
• Prospect theory helps to explain various paradoxes
found in the finance literature:
• One of them is the notorious reluctance of investors to
sell stocks that lose value, which comes out of loss
aversion (see Odean 1998). People do not want to sell stocks when
they go down in price.
• Another is investors' aversion to holding stocks more
generally, known as the equity premium puzzle (Mehra
and Prescott 1985, Benartzi and Thaler 1995).
• Loss aversion also helps to explain an inability to
ignore sunk costs.
Lecture 8 Psychological approaches to risk:
• Prospect Theory also takes into account the
apparent fact that people underweight outcomes
that are merely probable in comparison with
outcomes that are obtained with certainty.
• This tendency, called the certainty effect,
contributes to risk aversion in choices involving sure
gains and to risk seeking in choices involving sure
losses.
• In addition, people generally discard components
that are shared by all prospects under consideration.
Lecture 8 Psychological approaches to risk: Problems with PT
• However, Prospect Theory has some difficulty explaining why
people both insure and gamble.
• Gambling is modelled as involving low-probability gains, over
which people are risk-averse.
• Insurance involves avoiding low-probability losses, but people
are risk seekers when it comes to losses and the gambling
wins or insurable hazards are evaluated separately from the
ante or premium, which gets overweighted because it is
certain.
• But be careful: being risk-averse for gains and risk-loving for
losses is NOT an essential part of PT.
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The essential points are a reference point, weighting and isolation.
Lecture 8 Psychological approaches to risk: Conclusion
• The standard economics model (SEU) appears from some
experimental evidence to have difficulties in explaining
some behaviour.
• Psychologists (particularly K&T and Busemeyer) have
proposed new theories to rectify these apparent
deficiencies.
• PT, for example, includes reference point effects, isolation
and weighting functions, while DFT includes specifically
randomness.
• These theories obviously explain better...
• ... but do they predict better?
Lecture 8
• Goodbye!