From risk to opportunityLecture 10

John Hey and Carmen Pasca

Lecture 10 Expected Utility Theory: Overview

• First we have some definitions.• Then we look at static problems under risk.• We generalise briefly to dynamic problems and those

under ambiguity (where probabilities do not exist).

• We discuss the axioms underlying (S)EUT. • We discuss some of the implications.• We briefly mention the circumstances under which

(S)EUT reduces to Mean/Variance theory – which is much used in Finance.

Lecture 10 Expected Utility Theory: Expected Values

• Suppose some variable X is risky and takes possible values x1, …, xN with associated probabilities p1, …, pN,

• where, of course, p1+ … +pN = 1,• then the Expected Value of the variable X, denoted by EX, is

given by• EX = p1x1+ … +pNxN

• It can be interpreted as the average value of X in many repetitions, or the value we expect X to take on average.

• Also the Expected Value of some function f(.) of the variable X, denoted by Ef(X), is given by

• Ef(X) = p1f(x1)+ … +pNf(xN)

Lecture 10 Expected Utility Theory: Static & Risk

• We start by considering static (one-off) decision problems under risk.

• By static we mean that there is one decision, and that the outcome for the decision-maker depends upon the “state of the world”, which is risky in the sense that the possible states can be listed and probabilities attached to them.

• Suppose that there are N mutually exclusive states of the world, n = 1,…,N and suppose that state n has associated probability pn. (p1+…+pN = 1)

Lecture 10 Expected Utility Theory: Notation

• Let us denote a generic risky choice R by [X,Y;p,(1-p)] signifying a choice that leads to outcome X or Y with associated probabilities p and (1-p). Note that the DM gets either X or Y, not both.

• X and Y could themselves be risky choices, though the decision-maker only gets to ‘consume’ a certain outcome.

• EUT is an axiomatic theory. The individual is assumed to have preferences over all risky choices R. We use ≽ to indicate ‘preferred or indifferent to’, and to indicate ∼‘indifferent to’.

Lecture 10 Expected Utility Theory: Axioms on preferences over lotteries (note that there are different possible sets of axioms)

• Completeness: for all R and S either R S ≽ or S R ≽ or both.• Transitivity: if R S ≽ and S T ≽ then R T.≽• Continuity: if R S T ≽ ≽ then ∃ p such that S [R,T;p,(1-p)].∼• Independence: if R S ≽ then, for all T, [R,T;p,(1-p)] [S,T;p,(1-p)] ≽• Existence of Best and Worst: • There exist a best (B) and a worst (W) lottery.• Monotonicity: • [B,W;p,(1-p)] [B,W;q,(1-q)] ≽ if and only if p ≥ q• If these hold then preferences are represented by• u(R) = pu(X) +(1-p)u(Y) • That is, by the Expected Utility of the lottery.

Lecture 10 Expected Utility Theory: The Key axiom (independence)

R

T

1-p

Does the above appear reasonable/compelling?

p

is preferred to

T

1-p

S

if and only if R is preferred to S…

p

… whatever T is

represents a chance node

Lecture 10 Expected Utility Theory: ‘Derivation of utility function’

• Suppose that there are N possible outcomes x1,…,xN.

• Suppose that, in the preferences of the decision-maker x1 is the least preferred and xN the most. (using Best and Worst axiom)

• Let us put u(x1)=0 and u(xN)=1. • Now consider any intermediate

outcome xn.• From continuity, since xN x≽ n x≽ 1 then

∃ un such that xn [x∼ N, x1;un,(1-un)]• Put u(xn)= un (note un is a probability but we interpret it as a utility.)

xN

x1

un

1-un

xn ∼

Lecture 10 Expected Utility Theory: Why it is natural to put u(xn) = un

• The un for which the individual is indifferent between xn and the gamble on the right is a probability.

• Why is it ‘natural’ to call it the utility of xn?

• Because the individual says that he or she is indifferent between it and the gamble... and hence u(xn) must be equal to the expected utility of the gamble; this is given by

• unu(xN) + (1-un)u(x1) which is equal to un

• since u(xN) = 1 and u(x1) = 0.

xN

x1

un

1-un

xn ∼

Lecture 10 Expected Utility Theory: ‘Proof of EU result 1’

This can be re-expressed (replacing each intermediate outcome by a gamble between the best and the worst – using independence.)

xN

x1

pN

p1

…

xn

…

pn

Consider the generic lottery:

xN

xN

x1

pN

p1

…

…

pn

x1

un

1-un

xN

x1

pNuN+…+pnun+…+p1u1

1 minus the above expression

or:

Hence the generic lottery has value (see next slide) pNuN+…+pnun+…+p1u1 – its expected utility!

Lecture 10 Expected Utility Theory: ‘Proof of EU result 2’

xN

x1

1-p

Just a footnote: it follows from monotonicity that:

q

P

is preferred to

x1

1-q

Q

xN

if and only if p is greater than q…

… and hence that p is an indicator of the value of a lottery involving just the best and the worst.

p

Lecture 10 Expected Utility Theory: the EU result

xN

x1

pN

p1

…

xn

…

pn

Consider the generic lottery:

We have shown that the generic lottery is equivalent (for our individual obeying the axioms) to a lottery between the best and the worst outcomes, in which the probability of getting the best outcome is

pNuN+…+pnun+…+p1u1

that is, the Expected Utility of the lottery.

It follows from monotonicity that the individual evaluates the generic lottery by its Expected Utility.

Note that since we put u(x1)=0 and u(xN)=1 it follows that the utility representation is unique only up to a linear transformation (two points fixed – like temperature).

Problems? Remember the Common Ratio Effect?

• (1) Do you prefer (1A) €30 for sure, or (1B) a lottery which yields €40 with probability 0.8 and €0 with probability 0.2?

• (2) Do you prefer (2A) a lottery which yields €30 with probability 0.25 and €0 with probability 0.75, or (2B) a lottery which yields €40 with probability 0.2 and €0 with probability 0.8?

• A preference for 1A in problem 1 and for 2B in Problem 2 is a violation of Expected Utility.

Let us look at this using trees: Note: green squares are decision nodes and red squares are chance nodes.

first second

€30

€400.8

€0

€0

0.25

0.75

Now consider this

€30

€40

£0

0.8

0.2

1A

1B

€30

€50

€0

0.2

0.8

€0

0.25

0.752A

2B

Such an individual would plan to play Down but actually choose Up.

Now recall the Original Allais Paradox (using the notation of Machina)

• Problem 1: Choose between (a1) 1 million for sure and (a2) a lottery which gives 5 million with probability 0.1, 1 million with probability 0.89 and 0 with probability 0.01.

• Problem 2: Choose between (a4) a lottery which gives 1 million with probability 0.11 and 0 with probability 0.89 and (a3) a lottery which gives 5 million with probability 0.1 and 0 with probability 0.9.

• Choosing a1 in Problem 1 and a3 in Problem 2 is a violation of EU.

Lecture 10: Expected Utility theory

• This is closely related to Machina’s arguments for dynamic consistency with EU preferences (taken from Machina JEL 1989). Here squares denote choice nodes and circles chance nodes.

To be consistent with EU if you prefer a1 to a2 you should prefer a4 to a3 sinceu(1) > .1u(5) + .89u(1) + .01u(0) if and only if .11u(1)+.89u(0)> .1u(5) + .9u(0)

Now note that as viewed from the start these are the same as the problems above. But if you get to the square box, the choice problems are the same as each other! Which reinforces the idea that if you prefer a1 to a2 you should prefer a4 to a3. Someone who does not can be dynamically inconsistent.

Lecture 10 Expected Utility Theory: Risk Attitude

• If someone has a linear utility function u(x) = x then they are risk-neutral since the maximisation of Expected Utility is the same as maximisation of Expected Value.

• If someone has a concave utility function u’’(x)<0 then they are risk-averse; the more concave the more risk-averse.

• If someone has a convex utility function u’’(x)>0 then they are risk-loving; the more convex the more risk-loving.

Lecture 10 Expected Utility Theory: Certainty Equivalents

• We define the Certainty Equivalent of a gamble for some individual by CE given by

• U(CE) = EU(X)• so the individual is indifferent between the Certainty

Equivalent and the gamble.• For a risk-neutral person CE = EX• For a risk-averse person CE < EX• For a risk-loving person CE > EX.• Are risk-lovers mad?

Lecture 10 Expected Utility Theory: Measuring Risk Attitude

• The concavity of the utility function tells us the risk-aversion; the more concave the more risk-averse.

• One measure is the Absolute Risk Attitude (ARA) -u’’(x)/u’(x)• Another is the Relative Risk Attitude (RRA) -xu’’(x)/u’(x)• CARA utility function: u(x) = - e-rx ARA = r always.• CRRA utility function: u(x) = x1-r RRA = r always.

Lecture 10 Expected Utility Theory: CE in a particular case

• Suppose an individual has a CARA utility function u(x) = - e-rx and that X is normally distributed with mean μ and variance σ2 then the CE is given by

• U(CE) = E[- e-rx] • Using the properties of moment generating functions

we can show that this implies• -e-CE = - exp(-rμ + r2σ2/2) and hence• CE = μ – r σ2/2 • Which is a rather nice result.

Lecture 10 Subjective Expected Utility Theory

• Now suppose that states of the world 1,…,N but not their probabilities exist.

• Suppose the decision-maker receives xn if state n occurs (n=1,..,N).• Savage showed in 1954 that, if a decision-makers preferences

agreed with certain ‘reasonable’ axioms of behaviour, then the decision-maker would choose among lotteries as if he or she were maximising a function

• p1u(x1) + … + pNu(xN)• where the pn and u(xn) exist and are unique (n=1,..,N).• The axioms can be found, for example, in Leonard Savage’sThe

Foundations of Statistics, 1954.• This is an absolutely breathtaking result!!

Lecture 10 Subjective Expected Utility Theory Axioms

• For those who are interested:

1. the preference relation is transitive and all acts are comparable.2. the preference between acts depend solely on the consequences in states in which

the payoffs of the two acts being compared are distinct.3. the ordinal ranking of consequences is independent of the event and the act that

yields them.4. the betting preferences are independent of the specific consequences that define

the bets.5. it is not the case that the decision maker is indifferent among all acts.6. no consequence is either infinitely better or infinitely worse than any other

consequence.7. if the decision maker considers an act strictly better (worse) than each of the

payoffs of another act on a given non-null event, then the former act is conditionally strictly preferred (less preferred) than the latter.

• Taken from Edi Karni’s Savages’ Subjective Expected Utility Model, 2005.

Lecture 10 Expected Utility Theory: Dynamics

• We do not have the time to study this in depth, but one immensely satisfying property of EU preferences is that someone with such preferences is necessarily dynamically consistent in dynamic decision problems.

• What do we mean by this?• That they do not want to change their minds part way

through a dynamic decision problem.• Or, whichever way they analyse a dynamic decision

problem they come up with the same solution. • In contrast a non-EU person could be dynamically

inconsistent.

Two Problems: what would you choose – Up or Down – in each?

Consider a person who does not violate EU, choosing Up in both problems (you could also examine the implications of Down in both).

Note: green squares are decision nodes and red squares are chance nodes.

£30

£50

£0

0.8

0.2

first

second£30

£50

£0

0.2

0.8

£0

0.25

0.75

What does such a person do in this dynamic problem?Suppose that he/she prefers £30 to the 80% chance of £50 and that he/she

strictly prefers the 25% chance of £30 to the 20% chance of £50.Plays Down – irrespective of the way that he/she looks at the problem.

Note: green squares are decision nodes and red squares are chance nodes.

Suppose that he/she prefers £30 to the 80% chance of £50 and that he/she strictly prefers the 25% chance of £30 to the 20% chance of £50.

Plays Down – irrespective of the way that he/she looks at the problem.Let us look at the strategy method (rather than backward induction)

• There are three possible strategies: Up-Up; Up-Down; Down.

• Up-Up leads to £50 with probability 0.2 and £0 with probability 0.8.

• Up-Down leads to £30 with probability 0.25 and £0 with probability 0.75.

• Down leads to £31 with probability 0.25 and £1 with probability 0.75.

• With these preferences Down is best – the same as found by backward induction.

Now consider a non-EU person

Consider a person who violates EU, choosing Up in the first problem and and Down in the second.

Note: green squares are decision nodes and red squares are chance nodes.

£30

£50

£0

0.8

0.2

first

second£30

£50

£0

0.2

0.8

£0

0.25

0.75

What does such a person do in this dynamic problem?Suppose that he/she prefers £30 to the 80% chance of £50 and that he/she

strictly prefers the 20% chance of £50 to the 25% chance of £30.

Note: green squares are decision nodes and red squares are chance nodes.

What does such a person do in this dynamic problem?Suppose that he/she prefers £30 to the 80% chance of £50 and that he/she

strictly prefers the 20% chance of £50 to the 25% chance of £30.Now look at the strategy method (rather than backward induction)

• There are three possible strategies: Up-Up; Up-Down; Down.

• Up-Up leads to £50 with probability 0.2 and £0 with probability 0.8.

• Up-Down leads to £30 with probability 0.25 and £0 with probability 0.75.

• Down leads to £31 with probability 0.25 and £1 with probability 0.75.

• With these preferences Up-Up initially appears to be the best (though Down at the second node, if reached, looks tempting).

What they do depends on their type

• Resolute: Choose Up at the first and Up at the second (even though they want to choose Down when they get there).

• Naive: Choose Up at the first and Down at the second.

• Sophisticated: Choose Down at the first (because they anticipate moving Down at the second if they choose Up at the first). The latter is what is assumed when backward induction is used.

Lecture 10 Expected Utility Theory=Mean/Variance?

• In finance, preferences are often assumed to be mean/variance; the higher (lower) the former (latter) the better.

• This could arise from a quadratic utility function or from normally distributed outcomes.

• But a quadratic utility function is rather odd (utility decreasing with wealth).

• And normal distributions have tails going to both plus and minus infinity.

Lecture 10 Expected Utility Theory: Conclusion

• (Subjective) Expected Utility Theory is based on very plausible and reasonable axioms.

• It leads to a very elegant solution for problems under risk.

• It leads to an even more elegant solution for problems under ambiguity/uncertainty.

• In dynamic problems it avoids the difficulty of dynamically inconsistent preferences.

• So it is normatively very strong. But positively?

Lecture 10

• Goodbye!