From risk to opportunityLecture 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.

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.

B

C

p

1-p

A

C

p

1-p

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 Pointfirst 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 effectfirst 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? • They are exactly the same problem. Framing?

Lecture 8 Psychological approaches to risk: Probability weightingFirst 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

• 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 (un-

incentivated) 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 mean-variance 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

• 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 decision-

making 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.

• 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!