coherent choice functions in terms of desirability Jasper De Bock - - PowerPoint PPT Presentation

coherent choice functions
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coherent choice functions in terms of desirability Jasper De Bock - - PowerPoint PPT Presentation

Interpreting, axiomatising and representing coherent choice functions in terms of desirability Jasper De Bock & Gert de Cooman ISIPTA 2019 Interpreting, axiomatising and representing coherent choice functions ? in terms of desirability


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SLIDE 1

Jasper De Bock & Gert de Cooman

ISIPTA 2019

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

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SLIDE 2

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

?

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SLIDE 3

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

1 2 3 4

C( ) = ?

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SLIDE 4

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

C( ) = ?

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SLIDE 5

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

C( ) =

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SLIDE 6

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

C( ) = {

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slide-7
SLIDE 7

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

C( ) = ?

v1

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v2

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v3

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v4

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slide-8
SLIDE 8

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

C( ) = ?

v1

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v2

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v3

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v4

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V

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real vector space

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slide-9
SLIDE 9

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

desirable option

elementary concept

∈ V

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v

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strictly preferred

  • ver
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slide-10
SLIDE 10

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

set of desirable options D

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⊆ V

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coherence axioms!

derived notion 1

desirable option

elementary concept

∈ V

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v

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strictly preferred

  • ver
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slide-11
SLIDE 11

sets of desirable options / partial preference orderings lower and upper expectations/previsions
 closed convex sets of probability measures lower and upper probabilities belief and plausibility functions possibility and necessity measures probability measures

set of desirable options D

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⊆ V

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coherence axioms!

derived notion 1

slide-12
SLIDE 12

sets of desirable options / partial preference orderings lower and upper probabilities belief and plausibility functions possibility and necessity measures probability measures lower and upper expectations/previsions
 closed convex sets of probability measures

C( ) = ) = {

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{

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P1

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P2

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set of desirable options D

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⊆ V

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coherence axioms!

derived notion 1

slide-13
SLIDE 13

sets of desirable options / partial preference orderings lower and upper probabilities belief and plausibility functions possibility and necessity measures probability measures lower and upper expectations/previsions
 closed convex sets of probability measures

C( ) = ) = {

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{

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P1

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P2

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set of desirable options D

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⊆ V

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coherence axioms!

derived notion 1

slide-14
SLIDE 14

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

desirable option set

⊆ V

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A

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contains at least

  • ne desirable option!

desirable option

elementary concept

∈ V

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v

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strictly preferred

  • ver
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derived notion 2

slide-15
SLIDE 15

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

desirable option set

⊆ V

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A

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contains at least

  • ne desirable option!

derived notion 2 derived notion 3

! !

set of desirable 


  • ption sets
slide-16
SLIDE 16

coherent choice functions

in terms of desirability Interpreting, axiomatising and representing

choice function set of desirable 


  • ption sets

! !

slide-17
SLIDE 17

choice function

sets of desirable options / partial preference orderings lower and upper probabilities belief and plausibility functions possibility and necessity measures probability measures lower and upper expectations/previsions
 closed convex sets of probability measures

s e t

  • f

set of desirable 


  • ption sets

! !

slide-18
SLIDE 18

sets of desirable options / partial preference orderings lower and upper probabilities belief and plausibility functions possibility and necessity measures probability measures lower and upper expectations/previsions
 closed convex sets of probability measures

C( ) = ) = {

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{

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P1

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P2

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s e t

  • f

choice function set of desirable 


  • ption sets

! !

slide-19
SLIDE 19

Interpreting, Axiomatising and Representing Coherent Choice Functions in Terms of Desirability

Jasper.De Bock Gert.de Cooman @UGent.be

C(

) = ?

{ } { }

a u n i f y i n g f r a m e w
  • r
k f
  • r
m
  • d
e l l i n g s e t
  • v
a l u e d c h
  • i
c e ! V i s t h e s e t
  • f
a l l
  • p
t i
  • n
s : a n a b s t r a c t r e a l v e c t
  • r
s p a c e Q i s t h e s e t
  • f
a l l fi n i t e s u b s e t s
  • f
V a c h
  • i
c e f u n c t i
  • n
C i s a m a p f r
  • m
Q t
  • Q
s u c h t h a t C ( A ) ✓ A f
  • r
e v e r y A 2 Q t h e c
  • r
r e s p
  • n
d i n g r e j e c t i
  • n
f u n c t i
  • n
R i s d e fi n e d b y R ( A ) : = A \ C ( A ) , f
  • r
a l l A 2 Q C O H E R E N T D 1 / 2 D D 2 V ✓ D D 3 i f u , v 2 D a n d ( λ , µ) > , t h e n λ u + µ v 2 D K i f A 2 K t h e n a l s
  • A
\ { } 2 K , f
  • r
a l l A 2 Q K 1 { } / 2 K K 2 { u } 2 K , f
  • r
a l l u 2 V K 3 i f A 1 , A 2 2 K a n d i f , f
  • r
a l l u 2 A 1 a n d v 2 A 2 (λu,v,µu,v) > 0, then also { λ u,v u + µ u,v v : u 2 A 1 , v 2 A 2 } 2 K K 4 i f A 1 2 K a n d A 1 ✓ A 2 2 Q , t h e n a l s
  • A
2 2 K R f
  • r
a l l A 2 Q a n d u 2 A : u 2 R( A) , 2 R( A
  • u)
R 1 R( / 0) = / , a n d R( A) 6= A f
  • r
a l l A 2 Q \ { / } R 2 2 R( { , u }) , f
  • r
a l l u 2 V R 3 i f A 1 , A 2 2 Q , 2 R( A 1 [ { }) a n d 2 R( A 2 [ { }) a n d i f ( λ u,v , µ u,v) > f
  • r
a l l u 2 A 1 a n d v 2 A 2 , t h e n 2 R( { λ u,v u + µ u,v v : u 2 A 1 , v 2 A 2 } [ { }) R 4 i f A 1 ✓ A 2 t h e n R( A 1) ✓ R( A 2) , f
  • r
a l l A 1 , A 2 2 Q T O T A L D T f
  • r
a l l u 2 V \ { } , e i t h e r u 2 D
  • r
  • u
2 D K T { u ,
  • u
} 2 K f
  • r
a l l u 2 V \ { } R T 2 R( { , u ,
  • u
}) , f
  • r
a l l u 2 V \ { } M I X I N G D M i f p
  • s
i( A) \ D 6= / , t h e n a l s
  • A
\ D 6= / , f
  • r
a l l A 2 Q K M i f B 2 K a n d A ✓ B ✓ p
  • s
i( A) , t h e n a l s
  • A
2 K , f
  • r
a l l A , B 2 Q R M i f A ✓ B ✓ c
  • n
v( A) t h e n a l s
  • R(
B) \ A ✓ R( A) , f
  • r
a l l A , B 2 Q A R C H I M E D E A N D A f
  • r
a l l u 2 D , t h e r e i s a n ε 2 R >0 s u c h t h a t u
  • ε
2 D K A f
  • r
a l l A 2 K , t h e r e i s a n ε 2 R >0 s u c h t h a t A
  • ε
2 K R A for all A 2 Q a n d u 2 V s u c h t h a t u 2 R( A [ { u }) , there is some ε 2 R >0 such that u 2 R(( A
  • ε)
[ { u })

D

S e t
  • f
d e s i r a b l e
  • p
t i
  • n
s

K

S e t
  • f
d e s i r a b l e
  • p
t i
  • n
s s e t s

R / C

R e j e c t i
  • n
f u n c t i
  • n
/ C h
  • i
c e f u n c t i
  • n
desirability provides an interpretation for each
  • f
  • u
r m
  • d
e l s a n
  • p
t i
  • n
v 2 V i s d e
  • s
i r a b l e i f i t i s s t r i c t l y p r e f e r r e d t
  • z
e r
  • an option set
A 2 Q is desirable if it is thought to contain at least one d e s i r a b l e
  • p
t i
  • n
a n
  • p
t i
  • n
u 2 A i s r e j e c t e d f r
  • m
A i f a t l e a s t
  • n
e
  • p
t i
  • n
v 2 A i s s t r i c t l y p r e f e r r e d
  • v
e r u , i n t h e s e n s e t h a t v
  • u
i s d e s i r a b l e O T H E R P R O P E R T I E S ? K D : = { A 2 Q : A \ D 6 = / } D K : = { u 2 V : { u } 2 K } l
  • s
s
  • f
i n f
  • r
m a t i
  • n
! ( u n l e s s b i n a r y ) u 2 R ( A [ { u } ) , A
  • u
2 K { v
  • u
: v 2 A } : =

K

i s

, K = \{KD: D 2 D}

f

  • r

s

  • m

e n

  • n
  • e

m p t y s e t

D

  • f

s e t s

  • f

d e s i r a b l e

  • p

t i

  • n

s

a n i n t e r s e c t i
  • n
  • f
s e t s
  • f
d e
  • s
i r a b l e
  • p
t i
  • n
s e t s K a m
  • u
n t s t
  • t
a k i n g t h e u n i
  • n
  • f
t h e c
  • r
  • r
e s p
  • n
d i n g c h
  • i
c e f u n c t i
  • n
s C C O H E R E N T C O H E R E N T T O T A L C O H E R E N T M I X I N G C O H E R E N T A R C H I M E D E A N C O H E R E N T M I X I N G A R C H I M E D E A N s e t
  • f
s t r i c t ( p a r t i a l ) p r e f e r e n c e
  • r
d e r s s e t
  • f
s t r i c t t
  • t
a l p r e f e r e n c e
  • r
d e r s s e t
  • f
l e x i c
  • g
r a p h i c p r e f e r e n c e
  • r
d e r s c l
  • s
e d s e t
  • f
l
  • w
e r p r e v i s i
  • n
s / l
  • w
e r e x p e c t a t i
  • n
s / c r e d a l s e t s c l
  • s
e d s e t
  • f
l i n e a r p r e v i s i
  • n
s / p r
  • b
a b i l i t y m e a s u r e s 1 1 V i s a c
  • n
v e x c
  • n
e i n V \ { } w h
  • s
e e l e m e n t s m u s t b e d e s i r a b l e 2 2 λ
  • ,
µ
  • a
n d λ + µ > 3 3 p
  • s
i( A) := ⇢ n k = 1 λ k u k : n 2 N , λ k 2 R >0 , u k 2 A
  • 4
4 c
  • n
v( A) := ⇢ n k = 1 λ k u k : n 2 N , λ k 2 R >0 , n k = 1 λ k = 1 , u k 2 A
  • 5
5 ( s
  • f
a r )
  • n
l y f
  • r
V = L ( X ) : t h e s e t
  • f
a l l b
  • u
n d e d r e a l
  • v
a l u e d f u n c t i
  • n
s ( g a m b l e s )
  • n
s
  • m
e s e t X V 0 = { u 2 L ( X ) : i n f u > } 6 6 D i s c l
  • s
e d i n t h e A R C H I M E D E A N c a s e s ; h
  • w
s h
  • u
l d w e m
  • d
i f y a r c h i m e d e a n i t y f
  • r
D t
  • n
  • t
b e c l
  • s
e d ?

C ,