Preference Based Belief Dynamics - Eric Pacuit

April 27, 2006

[email protected]

staff.science.uva.nl/∼epacuit

Eric Pacuit ILLC, University of Amsterdam

Preference Based Belief Dynamics

•

•

•

dene beliefs and utilities in terms of the agent's preferences

utility maximizer.

that the agent can be assumed to act as an

expected

functions and a (state independent) utility function such

properties, then there is a set of conditional probability

If the agents preferences satisfy such-and-such

theorem:

Typically the results come in the form of a representation

the preferences.

Thus logical properties of beliefs are derived from properties of

to

Finetti), there is a tradition in game theory and decision theory

Starting with the work of Savage (based on Ramsey and de

Introduction

Deducing Logical Properties of Beliefs from Properties of

•

•

Dening Beliefs from Preferences (S. Morris)

•

Pointers to future work (with Olivier Roy)

Preferences (S. Morris)

Sketch Savage's Theorem

•

Outline of the Talk

•

Ω

X be a

be a

x ∈ X,

dene

x

is the

objective

y=x otherwise

if

given that the current state is

 1 [x](y|t) = 0

probability of getting prize

f (x|t)

t.

is the set of a probability

: Ω → ∆(X).

nite set of states

The intended interpretation of

For each

∆(A)

nite set of prizes or outcomes.

A.

be an arbitrary set, then

lottery is a funciton f

Let

•

A

Let

•

A

distributions on

Let

•

•

Expected Utility Theorem: Notation

S ∈ Σ and lotteries f and g , f S g is intended to mean that  g is at least as good as f , given that the true state of the world is in S .

• f ∼S g and

f ≺S g

are dened as usual.

If the agent thinks the actual state is in S , then the agent would choose lottery f over g .

Given a set

is a set of events.

•

Σ = {S | S ⊆ Ω & S 6= ∅}

Let

•

Expected Utility Theorem: Notation

•

•

• S∈Σ occurs.

i for all

Ep (u(f )|S) =

t∈S

X

is calculated as

p(t|S)

x∈X

X

u(x, t)f (x|t)

given an even

s, t ∈ Ω, u(x, s) = u(x, t).

expected utility value of a lottery f

Ep (u(f )|S),

The

state independent

S,

It is said

given that an

utility function is any function u : X × Ω → R.

to be

A

event

t

conditional-probability function p : Σ → ∆(Ω) is a

function that gives the probability of a state

A

Expected Utility Theorem: Notation

(Relevance)

(Monotonicity)

(Continuity)

3.

4.

5.

then

γ

such that

f S g and g S h then there exists 0 ≤ γ ≤ 1 and g ∼S γf + (1 − γ)h. If

a number

f ∼S g . then

then

f S h.

t ∈ Ω, f (·|t) = g(·|t)

g S h

f S g and 0 ≤ β < α ≤ 1 αf + (1 − α)g S βf + (1 − β)g . If

If, for all

f S g and

If

(Transitivity)

g S f .

and event.

2.

S or

be lotteries and

(Completeness) f S g

f, g

1.

Let

Expected Utility Theorem

9.

8.

If

(Subjective substitution)

7.

g {r}

and

g(·|r) = g(·|t)

there exist

x, y ∈ X

then

then

such that

r, t ∈ Ω, if f (·|r) = f (·|t), f , then g {t} f .

t∈Ω

For all

For all

(State Neutrality)

[x] {t} [y]

(Non-triviality)

g S∪T f .

S∩T =∅

0 ≤ α ≤ 1,

and

and

g S f , g T f

e S f , h S g αe + (1 − α)h S αf + (1 − α)g . If

(Objective substitution)

6.

Expected Utility Theorem

9.

8.

If

(Subjective substitution)

7.

g {r}

and

g(·|r) = g(·|t)

there exist

x, y ∈ X

then

then

such that

r, t ∈ Ω, if f (·|r) = f (·|t), f , then g {t} f .

t∈Ω

For all

For all

(State Neutrality)

[x] {t} [y]

(Non-triviality)

g S∪T f .

S∩T =∅

0 < α ≤ 1,

and

and

g S f , g T f

e S f , h S g αe + (1 − α)h S αf + (1 − α)g . If

(Objective substitution)

6.

Expected Utility Theorem

9.

8.

If

(Subjective substitution)

7.

g {r}

and

g(·|r) = g(·|t)

there exist

x, y ∈ X

then

then

such that

r, t ∈ Ω, if f (·|r) = f (·|t), f , then g {t} f .

t∈Ω

For all

For all

(State Neutrality)

[x] {t} [y]

(Non-triviality)

g S∪T f .

S∩T =∅

0 ≤ α ≤ 1,

and

and

g S f , g T f

e S f , h S g αe + (1 − α)h S αf + (1 − α)g . If

(Objective substitution)

6.

Expected Utility Theorem

t ∈ Ω, maxx∈X u(x, t) = 1 and

f, g ∈ L and

S ∈ Σ, f S g

i

state-independent.

S 6= ∅,

u

is

Ep (u(f )|S) ≥ Ep (u(g)|S).

and

If, furthermore, axiom 8 is also satised, then

3. For all

2. For all

p : Σ → ∆(Ω)

minx∈X u(x, t) = 0.

and a conditional probability function

R, S, T such that R ⊆ S ⊆ T ⊆ Ω p(R|T ) = p(R|S)p(S|T ).

1. For all

such that

u : X ×Ω → R

Axioms 1 - 8 are jointly satised i there exists a utlity function

Expected Utility Theorem

t ∈ Ω, maxx∈X u(x, t) = 1 and

f, g ∈ L and

S ∈ Σ, f S g

i

state-independent.

S 6= ∅,

u

is

Ep (u(f )|S) ≥ Ep (u(g)|S).

and

If, furthermore, axiom 8 is also satised, then

3. For all

2. For all

p : Σ → ∆(Ω)

minx∈X u(x, t) = 0.

and a conditional probability function

R, S, T such that R ⊆ S ⊆ T ⊆ Ω p(R|T ) = p(R|S)p(S|T ).

1. For all

such that

u : X ×Ω → R

Axioms 1 - 8 are jointly satised i there exists a utlity function

Expected Utility Theorem

There is a large literature surrounding this result!

is w

We write

Let

RΩ

x. the agent prefers

x

over

then the agent

y provided the true state

w,

be the set of all acts.

means that if the true state is

x w y

receives prize

w∈Ω

act is a function x : Ω → R.

An

for

be a nite set of prizes.

X

Let

xw

be a set of states.

Ω

Let

For this Talk

is w

We write

Let

RΩ

x. the agent prefers

x

over

then the agent

y provided the true state

w,

be the set of all acts.

means that if the true state is

x w y

receives prize

w∈Ω

act is a function x : Ω → R.

An

for

be a nite set of prizes.

X

Let

xw

be a set of states.

Ω

Let

For this Talk

b:Ω→2

lists the agents belief state at

w

2Ω

possible at

P : Ω → 2Ω :

E

at state

w

w.

set of states the agent considers

means the agent believes

• B(E ∩ F ) = B(E) ∩ B(F )

• B(Ω) = Ω

is normal if

Possibility function:

B

E ⊆ Ω, w ∈ B(E)

belief operator is a function B : 2Ω → 2Ω

For

A

Belief Operators

B

{w }w∈Ω

x, y, z ∈ RΩ }

denote the new act

for all

E⊆Ω

(xE , y−E )

provided for each

let

B(E) = {w | (xE , y−E ) ∼w (xE , z−E )

reects

For

E ⊆ Ω and two acts x and y , that is x on E and y on −E .

Dening Beliefs from Preferences

then the derived belief operator is normal.

Theorem If the preference relations are complete and transitive,

w0 ∈E

X

p(w0 |w) = 1}

P (w) = {w0 | p(w0 |w) > 0}

B(E) = {w |

representation, then

If the preferences have a state independent expected utility

x >> y

i

xw > yw for each

and

w∈Ω

w∈Ω

for each

xw ≥ yw

x>y

i

w ∈ Ω, xw ≥ yw

i for each

x, y ∈ RΩ ,

x≥y

For

xw0 > yw0

Some Notation

for some

w0 ∈ Ω

x >> y, x, y, z, v ∈ RΩ }

for all

w ∈ Ω.

monotone if x >> y implies x w y and x ≥ y

for all

P ∗ (w) = {w0 | (xw0 , z−w0 ) w y

non-trivial and transitive.

for some

x >> y >> z}

is normal if the preference relations are monotone,

x w y

Theorem B ∗

implies

Preferences are

B ∗ (E) = {w | (xE , z−E ) w (yE , v−E )

Alternative Denitions

J

w0 ∈Ω

X pj (w

0

on

|w)uw (x0w ) j=1

)J

pj (· | w)

is probability 1 for

p1

is belief with probability 1 for each

B∗

B

x w y ⇔

(

w0 ∈Ω

X

j=1

)J 0 pj (w0 |w)uw (yw )

j = 1, . . . , J

≥L

(

w ∈ Ω and j = 1, . . . , J , Ω such that

have a LEU representation if there is

such that for each

{w }w∈Ω

probability distributions

a positive integer

Preferences relation

Take into account probability zero events.

Lexiographic Expected Utility Representation

Epistemic Logic and the Foundations of Decision and Game Theory, eds.

Bacharach et al.

See S. Morris, Alternative Denitions of Knowledge in

x≥y then

xy

| x  y} is closed for all

y ∈ RΩ

x >> y then

xy

P (w) = {w0 | for all

x >> y there

Under these conditions,

3. If

2. If

exists

z yw

1. If

Monotonicity:

Continuity The set {x ∈ RΩ

Additional Properties of Preferences

coherent if choices made at dierent states can be

Ω.

 is a

(xw , z−w )  (yw , z−w ) ⇔ xw ≥ yw

x, y, z ∈ RΩ

meta-ordering if it is complete,

transitive, continuous and for all

A preference relation

seen as reecting a true, metapreference ordering over acts.

Preferences are

dierent states of

A minimal rationality property relating together preferences at

Coherency

is optimal provided for each

D

for all

w∈Ω

for some optimal

w ∈ Ω, f (w) ∈ Cw [D]

C ∗ [D] = {x ∈ RΩ | xw = fw (w)

f

f :Ω→D

A decision rule is a function

y ∈ D}

all

Cw [D] = {x ∈ D | x w y for

A decision problem is a nite set of acts

Decision Problem

f}

y ∈ D.

there exists

x ∈ C ∗ [D]

such that

∗ such x ∗ y , for

introspection (B(E)

⊆ B(B(E))).

⊆ E)

and positive

If preferences are coherent, then the beliefs reecting

D,

them satisfy the knowledge axiom (B(E)

Theorem

all

that for each nite

Preferences are coherent if there exists a meta-ordering

Coherency

x ∗ y ⇔ 16 xa + 23 xb + 61 xc ≥ 16 ya + 23 yb + 61 yc

x c y ⇔ 21 xb + 12 xc ≥ 21 yb + 12 yc

x b y ⇔ xb ≥ yb

is coherent

Ω = {a, b, c}, P (a) = {a, b}, P (b) = {b}, P (c) = {b, c}

x a y ⇔ 12 xa + 21 xb ≥ 12 ya + 12 yb

Let

Example

For



where

suciently small

D = {0, x},

x b y ⇔ xa ≥ ya

0 ∗ x,

x = (, −1)

Ω = {a, b}

x a y ⇔ xa ≥ ya

Suppose

but

C ∗ [D] = {x}

Example 2

Ω = {a, b, c}

where

but

0 ≥∗ (0, 0, −2 )

x = (, −1, −2 )

C ∗ [D] = {(0, 0, −2 )},

D = {0, x}

for any meta-ordering.

x c y ⇔ βxa + (1 − β)xc ≥ βya + (1 − β)yc

x b y ⇔ xb ≥ yb

x a y ⇔ αxa + (1 − α)xb ≥ αya + (1 − α)yb

Suppose

Example 3

for some

f

at time

there is a

∗

t} such that

∗ y ∈ D ∩ C0∗ [D] ∩ · · · ∩ Ct−1 [D]

t = 1, . . . , T ,

there is a meta-ordering

and each

for all

such that

D ⊆ RΩ

x(t) ∗ y

x(t) ∈ Ct∗ [D]

for each nite

Valuable Information:

Ct∗ [D] = {x ∈ RΩ | xw = fw (w)

w,t .

t = 0, . . . , T .

Assume preferences for each time period, denoted

Extend preferences to nite time periods

Dynamic Preferences and Beliefs (Sketch)

for all events

E

and

F

Theoretic Approach,

Journal of Economic Theory, 1996.

See S. Morris, The Logic of Belief and Belief Change: A Decision

renement and historical negative introspection.

induced beliefs satisfy the knowledge axiom, positive introspection,

Theorem If preferences satisfy valuable information, then the

Renement For all s ≤ t, Bs (E) ⊆ Bt (E).

−Bs (E) ∩ Bt (F ) ∩ −Bs (F ) ⊆ Bt (−Bs (E)) and times s ≤ t.

Historical Negative Introspection

Dynamic Properties of Beliefs

probabilistic beliefs.

5. There is a nice survey by Ansheim and Sovik that focuses

logical omniscience.

4. Related approach of B. Lipman: Decision theory without

Perea, Schullte, Ansheim, etc.)

3. How does AGM, updates, etc. t into the above picture? (cf.

Benthem and Liu).

can be lifted to properties of belief change (cf. Hansson, van

2. Properties of preferences change in the face of new information

movement in the set of states.

problem but new information is received; time stamps vs.

1. Adding dyanmics: Dierent decision problems; same decision

Future and Related Work

Happy Birthday Prof. Segerberg!