Transcript From risk to opportunity Lecture 10 John Hey and Carmen Pasca Lecture 10 Expected Utility Theory: Overview • First we have some definitions. •

From risk to opportunity
Lecture 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
p
S
p
if and only if R is
preferred to S…
is
preferred
to
1-p
1-p
T
Does the above appear reasonable/compelling?
T
… 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 ≽ xn ≽ x1 then
∃ un such that xn ∼ [xN, x1;un,(1-un)]
• Put u(xn)= un (note u is a probability but we interpret it as a utility.)
n
xN
un
xn ∼
1-un
x1
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
un
xn ∼
1-un
x1
Lecture 10 Expected Utility Theory: ‘Proof of EU result 1’
Consider the
generic
lottery:
This can be re-expressed (replacing
or:
each intermediate outcome by a gamble between the best
and the worst – using independence.)
xN
pN
…
pn
xN
pN
un
1-un
…
…
p1
x1
pNuN+…+pnun+…+p1u1
…
pn
xn
p1
xN
x1
xN
x1
1 minus the above expression
x1
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’
Just a footnote: it follows from monotonicity that:
xN
xN
p
q
is
preferred
to
P
if and only if p is
greater than q…
1-p
… and hence that
p is an indicator of
the value of a
lottery involving
just the best and
the worst.
Q
1-q
x1
x1
Lecture 10 Expected Utility Theory: the EU result
Consider the
generic
lottery:
xN
pN
…
pn
xn
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.
…
p1
It follows from monotonicity that the individual
evaluates the generic lottery by its Expected Utility.
x1
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.
€30
first
second
0.25
€30
0.75
€0
2A
1A
0.8
€40
0.2
2B
1B
€30
0.2
£0
0.8
€50
€0
Now consider this
0.25
0.8
0.75
€40
€0
€0
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 since
u(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 riskaversion; 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).
£30
first
0.8
£50
second
0.25
£30
0.75
£0
0.2
£0
Note: green squares are decision
nodes and red squares are chance
nodes.
0.2
0.8
£50
£0
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; UpDown; 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
£30 Up in the first problem
first
and
and Down in the second.
0.8
£50
second
0.25
£30
0.75
£0
0.2
£0
Note: green squares are decision
nodes and red squares are chance
nodes.
0.2
0.8
£50
£0
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!