Decision Making Under Uncertainty

14.123 Microeconomic Theory III Muha met Yildiz

Risk v. uncertainty

1. Risk = DM has to choo se from alternatives

 whose consequences are unknown

 But the prob ability of each consequence is given

2. Uncertainty = DM has to choose from alternatives

 whose consequences are unknown

 the probability of conseq uences is n o t given

 DM has to f o rm his own beliefs

3. Von Neumann- Mor genstern: Ri sk

4. Goal:

1. Conve r t unc ertainty to risk by formal izing and eliciting belief s

2. Apply Von Neuman n Mo rgenstern a nalysis

Decision Making Under Risk – S ummary

 C = Finit e set of consequences

 X = P = lotteries (prob. distributions on C )

 Expected Utility Representation:

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

c  C

c  C

 Theorem: EU Representation  continuous preference relation with Independence Axiom :

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

Road map

1. Acts, States, Consequences

2. Expected Utility Maximiza tion – R epresentation

3. Sure- T hing Principl e

4. Conditional Preferences

5. Eliciting Qualitative Beliefs

6. Representing Qualitative Beliefs with Probability

7. Expected Utility Maximiza tion – C haracterization

8. Anscombe & Aumann trick: use indifference between uncertain and risky events

Model

 C = Finit e set of c onsequence s

 S = A set of st at e s (uncountable)

 Act: A mapping f : S → C

 X = F := C S

 DM cares about consequences, chooses an act, without knowing the state

 E x ampl e: Should I take my umbrella?

 E x ampl e: A game from a player’s point of view

Expected-Utility Representation

 ≽ = a relation on F

 Expected-Utility Representation :

 A probabil i ty distribution p on S with expectation E

 A VNM utility function u : C → R such that

 f ≽ g  U ( f ) ≡ E [ u ◦ f ] ≥ E [ u ◦ g ] ≡ U ( g )

 Nec e ssa ry Cond itio ns:

P1 : ≽ is a preference relation

Sure-Thing Principle

 If

 f ≽ g wh en DM kno w s B  S occurs,

 f ≽ g wh en DM kno w s S\B occurs,

 Then f ≽ g

 when DM doesn’t know whether B occurs or not.

P2 : Let f , f ′ , g , g ′ and B be such that

 f ( s ) = f ′ ( s ) and g ( s ) = g ′ ( s ) at each s  B

 f ( s ) = g ( s ) and f ′ ( s ) = g ′ ( s ) at each s  S\B . Then, f ≽ g  f ′ ≽ g ′ .

Sure-Thing Principle – P icture

C

B

S\B

S

ST P

Conditional Preference

 For any acts f and h and e v ent B ,

f 

h

 f  s  if s  B

| B

s  

 h ( s ) otherwise



 Definition : f ≽ g given B if f f | B ≽ g | B .

h h

 Sure-Thing Principle = conditio nal pr eference is well-defined

 Informal Sur e -Thing Prin cip l e, formally:

 f ≽ g given B : f | B ≽ g | B .

f f

 f ≽ g given S \ B : f | S \ B ≽ g | S \ B .

g

g

 Trans i tivity: f = f | f ≽ g | f = f | S \ g ≽ g | S \ g = g .

B B B

B

 B is null  f ~ g given B for all f , g  F.

P3: For any x , x ′  C , f , f ′  F with f ≡ x a nd f ′≡ x ′ , and an y non-null B,

f ≽ f ′ given B  x ≽ x ′ .

ST PP

Eliciting Beliefs

 For an y A  S and x , x ′  C , define f A x,x ’ by

f 

x , x '

A

s  

 x

 x ' otherwise

if s  A



Definition: For any A , B  S ,

A ≽ B  f A x,x ’ ≽ f B x,x ’

for some x , x ′  C with x ≻ x ′ .

 A ≽ B means A is at least as likel y as B .

P4: There exist x , x ′  C such t hat x ≻ x ′ .

P5: For all A , B  S , x , x ′ , y , y ′  C with x ≻ x ′ and y ≻ y ′ ,

f A x,x ’ ≽ f B x,x ’  f A y,y ’ ≽ f B y,y ’ .

Qualitative Probability

Definition: A r e la t i o n ≽ between the events is said to b e a qualit ative probability iff

1. ≽ is complete and transitive;

2. for any B , C , D  S with B ∩ D = C ∩ D = ∅ , B ≽ C  B  D ≽ C  D ;

3. B ≽∅ for each B  S , and S ≻∅ .

Fact: “A t least as likely as” r elation above is a qualitative probability relation.

Quantifying qualitative probability

 For any probability measure p and relation ≽ on events, p is a probability r epresentation of ≽ iff

B ≽ C  p ( B ) ≥ p ( C )  B , C  S .

 If ≽ has a probability representation, then ≽ is a qualitative probability.

 S is infinitely divisible under ≽ iff  n , S has a partition

{ D 1 1 , …, D n n } s uch that D 1 1 ~ … ~ D n n .

P6: For a n y x  C , g , h  F with g ≻ h , S has a partition

{ D ¹,…, D ⁿ } s.t.

g ≻ h i x and g i x ≻ h

for all i ≤ n w h e r e h i x ( s ) = x if s  D i and h (s) otherwise.

 P6 implies that S is infinitely divisible under ≽ .

Probability Representation

Theor em: Under P1- P 6, ≽ has a unique probability representation p .

Proof:

 For any event B and n , define

k ( n , B ) = max { r | B ≽ D 1 1  …  D n r }

 Define p ( B ) = lim n k ( n , B )/2 n .

 B ≽ C  k ( n , B ) ≥ k ( n , C )  n  p ( B ) ≥ p ( C ).

 P6’: If B ≻ C , S has a partition { D ¹,…, D ⁿ } s.t. B ≻ C 

D i for each i ≤ n .

 B ≻ C  p ( B ) > p ( C ).

 Uniqueness: k ( n , B )/2 n ≤ p ′ ( B ) < ( k ( n , B )+1 ) /2 ⁿ

Expected Utility Maximization – Characterization

Theorem: Assume that C is finite. Under P1-P6, there exist a utility functio n u : C → R and a probabilit y measure p on S such that  f , g  F ,