Decision Making Under Risk

14.123 Microeconomic Theory III Muha met Yildiz

Road map

1. Choice Theory – s ummary

2. Basic Concepts:

1. Consequences

2. Lotteries

3. Expected Utility Maximization

1. Represent ation

2. Characteri zation

4. Indifference Sets under Expected Utility Maximization

Choice Theory – S ummary

1. X = set of alternatives

2. Ordinal Representation: U : X → R is an ordinal representation of ≽ iff:

x ≽ y  U ( x ) ≥ U ( y )  x , y  X.

3. If ≽ has an ordinal representation, then ≽ is complete and transitive.

4. Assume X is a compact, convex subset of a separable metric space. A preference rel a tion has a continuous ordinal representati o n if and only if it is continuous.

5. Let ≽ be continuous and x ′ ≻ x ≻ x ′′ . For any continuous φ :[0,1] → X with φ (1)= x ′ and φ (0 )= x ′′ , there exist s t su ch that φ ( t ) ~ x .

Model

 D M = D e ci sion M a ke r

 DM cares only about consequences

 C = Finite set of consequences

 Ri sk = DM has to choose from alternatives

 whose consequences are unknown

 But the prob ability of each consequence is know n

 Lottery: a probability distribution on C

 P = set of all lotterie s p , q , r

 X = P

 Compounding lotteries are reduced to si mple lotter i es!

Expected Utility Maximization

Von N e umann-Morgenstern representation

A lottery (in P )

p f q   u ( c ) p ( c )   u ( c ) q ( c )

c  C

c  C

U ( p )

≥

U ( q )

 U : P → R is an ordinal representation of ≽ .

 U ( p ) is the expected value of u under p .

 U is linear and hence continuous.

CT

Expected value of

u under p

Expected Utility Maximization Characterization (VNM Axioms)

Axiom A1: ≽ is complete and transitive.

Axiom A2 (Continuity): ≽ is continuous.

VNM

Independence Axiom

Axiom A3: For any p , q , r  P , a  (0,1],

ap +(1- a ) r ≽ aq +(1- a ) r

p

$1000



p ≽ q .

.5

≽

q

.00001

$1M

.99999

.5

.5

$100

.5

$0

≽

.5

.5

A trip to Flo r id a

A trip to Flo r id a

Expected Utility Maximization Characterization Theorem

 ≽ has a von Neumann – M orgenstern representation iff ≽ satisfies Axioms A1-A3;

 i.e. ≽ is a continuous preference relation with Independence Axiom.

 u and v represent ≽ iff v = au + b for some

a > 0 and any b .

Exercise

 Consider a relation ≽ among positive real numbers represented by VNM utility function u with u ( x ) = x 2 .

Can this relation be represented by VNM utility function u * ( x ) = x 1/2 ?

What about u ** ( x ) = 1/ x ?

RT

Implications of Independence Axiom (Exercise)

 For any p,q,r,r ′ with r~ r ′ and any a in (0,1],

ap +(1- a ) r ≽ aq +(1- a ) r ′  p ≽ q .

 B e tw eenness: For any p,q,r and any a , p ~ q  ap +(1- a ) r ~ aq +(1- a ) r .

 Monotonicity: If p ≻ q and a > b , then

ap + (1- a ) q ≻ bp + (1- b ) q .

 Extreme Consequences:  c B ,c W  C :  p  P ,

c B ≽ p ≽ c W .

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Indifference Sets under Independence Axiom

1. Indifference sets are straight lines

2. … and parallel to each other. E x ampl e: C = { x , y , z }

p y

1

1

p x

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