Risk & Expected Utility
Modelling Framework
States
There are S states of the world: s=1,2, … , S
Before a state is realised, we agree that state s occurs with probability π s
Outcomes
Possible outcomes measured in income, so y s is money DM has in state s
The set of outcomes in exhaustive
The outcomes are mutually exclusive
States of the world are out of the control of any decision-taker
All decision takers have in the minds the same states of the world, and agree which state is
realised, they are also able to assign probabilities to each state
Choices
DM chooses among lotteries / prospects
Simple lottery is a list of pairs, one per state, each comprising a probability and an outcome:
L=( π 1 , y 1 ; π 2 , y 2 ; … ;π s , y s )
The outcomes of a compound lottery are lotteries themselves: L=( p 1 , L1 ; … ; p K , L K ) , this
can be written as p1 ∘ L1+⋯+ p K ∘ L K
r
For any compound lottery there is a simple reduced lottery, Lr with π s =p 1 π 1 ,s +⋯ p K π K , s
for s=1,2, … , S – as in it factors the outcomes of the second lotteries too 1
Consequentialism: for any risky alternative, the reduced lottery over final outcomes is the
only thing that matters to the DM
Expected Utility (von Neumann-Morgenstern)
Rational agents have a complete and transitive preference ordering over lotteries that is:
o Complete: L1 ≽ L2 or L2 ≽ L1
o Transitive: if L1 ≽ L2 and L2 ≽ L3 then L1 ≽ L3 ie agents are able to order prospects
Continuity: if Lb ≽ L≽ Lw then there is some probability, p, s.t.: L ∼ p ∘ L b+ ( 1− p ) Lw
o Guarantees existence of a utility function representing preferences of a rational agent
over lotteries
o Implication is that if Lb is preferred to L, then a lottery close to Lb will still be
preferred to L ⇒ 3 lotteries: £10 for sure, nothing happens, you die; must be some
α ∈ [ 0,1 ] such that you are indifferent between getting nothing for sure, and getting
£10 with probability α and being killed with probability 1−α (Levin 2004)
Independence: for any L1 , L2 , L , and any p
o L1 ≽ L2 ⇔ p ∘ L1 + ( 1− p ) ∘ L ≽ p ∘ L2+ (1− p ) ∘ L
o In that we prefer higher expected outcomes: ie if I prefer L1 to L2, I also prefer the
possibility of L1 to the possibility of L2 given the other possibility in both cases is the
same ( L)
o If I am comparing p ∘ L1 + ( 1− p ) ∘ L to p ∘ L2 + ( 1− p ) ∘ L, I should focus on the
distinction between L1 and L2 and hold the same preference independently of p and
L – also known as substitution axiom: idea that if L substituted for part of L1 and part
of L2, this shouldn’t change my ranking (Levin 2004)
o Similarly, preference increases with probability – decision-taker prefers the standard
prospect which gives the better chance of achieving the good state of the world ie
could assert p1 L ≻ p2 L ⇔ p1 > p2
Consumers rationally evaluate lotteries and compound lotteries, so a compound lottery is
worth the same as a simple lottery with the same expected value
1
There is an example in the PDF notes
, These axioms provide a procedure for predicting the choices among prospects of a decision-
taker to whom they apply
Expected Utility Theorem
If preference ordering satisfies above axioms, there is a function u ( ⋅ ) that assigns a value
u ( y s ) to each outcome, such that
o L' ≽ L' ' ⇔ π '1 u ( y 1 )+ ⋯+π 'S u ( y S ) ≥ π ''1 u ( y1 ) +⋯ π 'S' u ( y s )
Define expected utility function U ( ⋅) by U ( L )=π 1 u ( y1 ) +⋯ π S u ( y S )
Rational decision makers act as if they were choosing L to maximise U ( L )
Cardinal vs. Ordinal (EU is cardinal)
Uniqueness: if v ( y )=a+bu ( y ) for any a and b> 0, then V ( L )=a+ bU ( L ) is also an EU
representation of preferences: since b> 0, if U ( p' ) ≥U ( p) then V ( p' ) ≥ V ( p)
Properties of the Expected Utility Function
Lottery L=( π , y 1 ;1−π , y 2 ) ⇒ y=π y 1+ ( 1−π ) y 2 ⇒u=πu ( y 1) + (1−π ) u ( y 2 )
Certainty equivalent: u ( y c )=u ie utility of getting this money for sure is equal to expected
utility from partaking in the lottery
Expected utility function is linear in probabilities
Risk premium: difference between expected value and certainty equivalent: r = y− y c
Risk Averse: u ( ⋅ ) is concave, y c < y , r> 0
Risk Neutral: u ( ⋅ ) is linear, y c = y , r=0
Risk Loving: u ( ⋅ ) is convex, y c > y , r< 0
Utility function is unique up to a positive linear transformation (cf ordinal utility function’s
property of being unique up to positive monotonic transformation) – restriction to linear
reflects signficance of the sign of u' ' ( y ) ⇒ expected utility function is cardinal
Risk Aversion
A decision maker is strictly risk averse if for any non-degenerate lottery the decision maker
strictly prefers the expected value of the lottery to the lottery itself: Jensen’s inequality
A decision maker is risk averse iff u is concave
Measures of Risk Aversion
Certainty equivalent can measure risk aversion through risk premium
Degree of risk aversion related to curvature of utility function – one possible measure of
curvature at y is u' ' ( y )
This is not invariant to possible linear transformations v=a+ bu since v' =b u' , v' ' =bu ' '
'' '' ''
u ( y) bu ( y) v ( y)
Simplest modification is to use ' since same as ' ie '
u ( y) bu ( y ) v ( y)
o Change sign to make it positive for functions that are increasing and concave
Arrow-Pratt Coefficient of Risk Aversion
''
−u ( y )
Absolute risk aversion: A ( y )=
u' ( y )
o Can be shown that for lottery with small gambles, an approximation to risk premium
1
is given r ( y ) ≃ A ( y ) σ 2z , where σ 2z is the variance of outcomes
2
Modelling Framework
States
There are S states of the world: s=1,2, … , S
Before a state is realised, we agree that state s occurs with probability π s
Outcomes
Possible outcomes measured in income, so y s is money DM has in state s
The set of outcomes in exhaustive
The outcomes are mutually exclusive
States of the world are out of the control of any decision-taker
All decision takers have in the minds the same states of the world, and agree which state is
realised, they are also able to assign probabilities to each state
Choices
DM chooses among lotteries / prospects
Simple lottery is a list of pairs, one per state, each comprising a probability and an outcome:
L=( π 1 , y 1 ; π 2 , y 2 ; … ;π s , y s )
The outcomes of a compound lottery are lotteries themselves: L=( p 1 , L1 ; … ; p K , L K ) , this
can be written as p1 ∘ L1+⋯+ p K ∘ L K
r
For any compound lottery there is a simple reduced lottery, Lr with π s =p 1 π 1 ,s +⋯ p K π K , s
for s=1,2, … , S – as in it factors the outcomes of the second lotteries too 1
Consequentialism: for any risky alternative, the reduced lottery over final outcomes is the
only thing that matters to the DM
Expected Utility (von Neumann-Morgenstern)
Rational agents have a complete and transitive preference ordering over lotteries that is:
o Complete: L1 ≽ L2 or L2 ≽ L1
o Transitive: if L1 ≽ L2 and L2 ≽ L3 then L1 ≽ L3 ie agents are able to order prospects
Continuity: if Lb ≽ L≽ Lw then there is some probability, p, s.t.: L ∼ p ∘ L b+ ( 1− p ) Lw
o Guarantees existence of a utility function representing preferences of a rational agent
over lotteries
o Implication is that if Lb is preferred to L, then a lottery close to Lb will still be
preferred to L ⇒ 3 lotteries: £10 for sure, nothing happens, you die; must be some
α ∈ [ 0,1 ] such that you are indifferent between getting nothing for sure, and getting
£10 with probability α and being killed with probability 1−α (Levin 2004)
Independence: for any L1 , L2 , L , and any p
o L1 ≽ L2 ⇔ p ∘ L1 + ( 1− p ) ∘ L ≽ p ∘ L2+ (1− p ) ∘ L
o In that we prefer higher expected outcomes: ie if I prefer L1 to L2, I also prefer the
possibility of L1 to the possibility of L2 given the other possibility in both cases is the
same ( L)
o If I am comparing p ∘ L1 + ( 1− p ) ∘ L to p ∘ L2 + ( 1− p ) ∘ L, I should focus on the
distinction between L1 and L2 and hold the same preference independently of p and
L – also known as substitution axiom: idea that if L substituted for part of L1 and part
of L2, this shouldn’t change my ranking (Levin 2004)
o Similarly, preference increases with probability – decision-taker prefers the standard
prospect which gives the better chance of achieving the good state of the world ie
could assert p1 L ≻ p2 L ⇔ p1 > p2
Consumers rationally evaluate lotteries and compound lotteries, so a compound lottery is
worth the same as a simple lottery with the same expected value
1
There is an example in the PDF notes
, These axioms provide a procedure for predicting the choices among prospects of a decision-
taker to whom they apply
Expected Utility Theorem
If preference ordering satisfies above axioms, there is a function u ( ⋅ ) that assigns a value
u ( y s ) to each outcome, such that
o L' ≽ L' ' ⇔ π '1 u ( y 1 )+ ⋯+π 'S u ( y S ) ≥ π ''1 u ( y1 ) +⋯ π 'S' u ( y s )
Define expected utility function U ( ⋅) by U ( L )=π 1 u ( y1 ) +⋯ π S u ( y S )
Rational decision makers act as if they were choosing L to maximise U ( L )
Cardinal vs. Ordinal (EU is cardinal)
Uniqueness: if v ( y )=a+bu ( y ) for any a and b> 0, then V ( L )=a+ bU ( L ) is also an EU
representation of preferences: since b> 0, if U ( p' ) ≥U ( p) then V ( p' ) ≥ V ( p)
Properties of the Expected Utility Function
Lottery L=( π , y 1 ;1−π , y 2 ) ⇒ y=π y 1+ ( 1−π ) y 2 ⇒u=πu ( y 1) + (1−π ) u ( y 2 )
Certainty equivalent: u ( y c )=u ie utility of getting this money for sure is equal to expected
utility from partaking in the lottery
Expected utility function is linear in probabilities
Risk premium: difference between expected value and certainty equivalent: r = y− y c
Risk Averse: u ( ⋅ ) is concave, y c < y , r> 0
Risk Neutral: u ( ⋅ ) is linear, y c = y , r=0
Risk Loving: u ( ⋅ ) is convex, y c > y , r< 0
Utility function is unique up to a positive linear transformation (cf ordinal utility function’s
property of being unique up to positive monotonic transformation) – restriction to linear
reflects signficance of the sign of u' ' ( y ) ⇒ expected utility function is cardinal
Risk Aversion
A decision maker is strictly risk averse if for any non-degenerate lottery the decision maker
strictly prefers the expected value of the lottery to the lottery itself: Jensen’s inequality
A decision maker is risk averse iff u is concave
Measures of Risk Aversion
Certainty equivalent can measure risk aversion through risk premium
Degree of risk aversion related to curvature of utility function – one possible measure of
curvature at y is u' ' ( y )
This is not invariant to possible linear transformations v=a+ bu since v' =b u' , v' ' =bu ' '
'' '' ''
u ( y) bu ( y) v ( y)
Simplest modification is to use ' since same as ' ie '
u ( y) bu ( y ) v ( y)
o Change sign to make it positive for functions that are increasing and concave
Arrow-Pratt Coefficient of Risk Aversion
''
−u ( y )
Absolute risk aversion: A ( y )=
u' ( y )
o Can be shown that for lottery with small gambles, an approximation to risk premium
1
is given r ( y ) ≃ A ( y ) σ 2z , where σ 2z is the variance of outcomes
2
