Modelling indifference with choice functions

M

ODELLING

I

NDIFFERENCE WITH

C

HOICE

F

UNCTIONS

Arthur Van Camp

1

, Gert de Cooman

1

, Enrique Miranda

2

and Erik Quaeghebeur

3

1

_{Ghent University, SYSTeMS Research Group}

2

_{University of Oviedo, Department of Statistics and Operations Research}

3

_{Centrum Wiskunde & Informatica, Amsterdam}

We want to model

indifference

with choice functions.

Indifference

-

reduces the

complexity

,

We want to model

indifference

with choice functions.

Indifference

-

reduces the

complexity

,

Exchangeability

is an example of both aspects.

In

[De Cooman & Quaeghebeur 2010, Exchangeability and sets of

desirable gambles]

: exchangeability for

sets of desirable gambles

.

Exchangeability

is an example of both aspects.

In

[De Cooman & Quaeghebeur 2010, Exchangeability and sets of

desirable gambles]

: exchangeability for

sets of desirable gambles

.

Why choice functions?

p(T) =

1

₂

=

p(H)

fair coin

p

H

p

T

p

H

(x ) =

(

1

if x = H

0

if x = T

p

T

(x ) =

(

0

if x = H

1

if x = T

coin with identical sides of unknown type

Such an assessment

cannot

be modelled

using sets of desirable gambles!

H

T

X = {H,T}

Why choice functions?

p(T) =

1

₂

=

p(H)

fair coin

p

H

p

T

p

H

(x ) =

(

1

if x = H

0

if x = T

p

T

(x ) =

(

0

if x = H

1

if x = T

coin with identical sides of unknown type

Such an assessment

cannot

be modelled

using sets of desirable gambles!

H

T

X = {H,T}

Why choice functions?

p(T) =

1

₂

=

p(H)

fair coin

p

H

p

T

p

H

(x ) =

(

1

if x = H

0

if x = T

p

T

(x ) =

(

0

if x = H

1

if x = T

coin with identical sides of unknown type

Such an assessment

cannot

be modelled

using sets of desirable gambles!

H

T

X = {H,T}

Why choice functions?

p(T) =

1

₂

=

p(H)

fair coin

p

H

p

T

p

H

(x ) =

(

1

if x = H

0

if x = T

p

T

(x ) =

(

0

if x = H

1

if x = T

coin with identical sides of unknown type

Such an assessment

cannot

be modelled

using sets of desirable gambles!

H

T

X = {H,T}

Why choice functions?

p(T) =

1

₂

=

p(H)

fair coin

p

H

p

T

p

H

(x ) =

(

1

if x = H

0

if x = T

p

T

(x ) =

(

0

if x = H

1

if x = T

coin with identical sides of unknown type

Such an assessment

cannot

be modelled

using sets of desirable gambles!

H

T

X = {H,T}

Choice functions

Consider a

vector space

V

and collect all its

non-empty but finite subsets

in

Q(V )

.

A

choice function

C

is a map

Indifference

The options are

equivalence classes

, rather than gambles.

H

T

Indifference

The options are

equivalence classes

, rather than gambles.

H

T

Indifference

The options are

equivalence classes

, rather than gambles.

H

T

Indifference

The options are

equivalence classes

, rather than gambles.

H

T

Indifference

H

T

We call a choice function

_on

_{Q(V )}

_indifferent

_{if there is}

C

some

representing

choice

func-tion

C

0

on

Q(V /I)

(the

equival-ence classes of

V

), meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

C

selects either all or none of

the options in

red,

orange

, and

Indifference

H

T

We call a choice function

_on

_{Q(V )}

_indifferent

_{if there is}

C

some

representing

choice

func-tion

C

0

on

Q(V /I)

(the

equival-ence classes of

V

), meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

C

selects either all or none of

the options in

red,

orange

, and

Remark the similarity!

Sets of desirable options

A set of desirable options

D ⊆

V

is

indifferent

if and only if there

is some

representing

set of

desir-able options

D

0

⊆

V /I

of

equival-ence classes, meaning that

D = {u : [u] ∈ D

0

}.

Choice functions

We call a choice function

C

on

Q(V )

indifferent

if there is

some

representing

choice

func-tion

C

0

on

Q(V /I)

(the

equival-ence classes of

V

), meaning that

Remark the similarity!

Sets of desirable options

A set of desirable options

D ⊆

V

is

indifferent

if and only if there

is some

representing

set of

desir-able options

D

0

⊆

V /I

of

equival-ence classes, meaning that

D = {u : [u] ∈ D

0

}.

Choice functions

We call a choice function

C

on

Q(V )

indifferent

if there is

some

representing

choice

func-tion

C

0

on

Q(V /I)

(the

equival-ence classes of

V

), meaning that

Some properties

A choice function

C

on

Q(V )

indifferent

if there is some

representing

choice function

C

0

on the equivalence classes of

V

, meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

The representing choice function

C

0

is

unique

and given by

C

0

(O/I) = C(O)/I

for all

O

in

Q(V )

C

is

coherent

if and only if

C

0

is

coherent

.

Some properties

A choice function

C

on

Q(V )

indifferent

if there is some

representing

choice function

C

0

on the equivalence classes of

V

, meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

The representing choice function

C

0

is

unique

and given by

C

0

(O/I) = C(O)/I

for all

O

in

Q(V )

C

is

coherent

if and only if

C

0

is

coherent

.

Some properties

A choice function

C

on

Q(V )

indifferent

if there is some

representing

choice function

C

0

on the equivalence classes of

V

, meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

The representing choice function

C

0

is

unique

and given by

C

0

(O/I) = C(O)/I

for all

O

in

Q(V )

C

is

coherent

if and only if

C

0

is

coherent

.

Some properties

A choice function

C

on

Q(V )

indifferent

if there is some

representing

choice function

C

0

on the equivalence classes of

V

, meaning that

C(O) = {u ∈ O : [u] ∈ C

0

(O/I)}.

The representing choice function

C

0

is

unique

and given by

C

0

(O/I) = C(O)/I

for all

O

in

Q(V )

C

is

coherent

if and only if

C

0

is

coherent

.

M

ODELLING

I

NDIFFERENCE WITH

C

HOICE

F

UNCTIONS

Arthur Van Camp, Gert de Cooman, Enrique Miranda and Erik Quaeghebeur

WHAT?We investigate how to model indifference with choice functions. WHY INDIFFERENCE?

- Adding indifference to the picture typically reduces the complexity of the modelling effort. - Also, knowing how to model indifference opens up a path towards modelling symmetry, which has many important practical applications.

Exchangeability is an example of both aspects. Our treatment here lays the foundation for dealing with, say, exchangeability for choice functions.

WHY CHOICE FUNCTIONS?The beliefs about a random variable may, in an already quite general setting, be expressed using a set of desirable options (gambles). There exists a theory of indifference for sets of desirable options. However, such sets of desirable options might not be expressive enough, as is shown in the next example.

We flip a coin with identical sides of unknown type: either twice heads or twice tails.

?

X = {H,T} H T p(H) = p(T) =1/2

?

X = {H,T} H T p(H) = 1, p(T) = 0

?

X = {H,T} H T p(H) = 0, p(T) = 1

?

X = {H,T} H T vacuous

There is no set of desirable options that expresses this elemen-tary belief. What we want is a more expressive model that can represent the stated belief, being an XOR statement. This belief model should resemble the situation depicted on the right.

X

H T

Sets of desirable options allow only for binary comparison between gambles, whereas choice functions determine “more than binary” comparison.

1. I

NTRODUCTION

VECTOR SPACEConsider a vector spaceV, consisting of options. We assume thatVis equipped with a given vector ordering, meaning that

- is a partial order (is reflexive, antisymmetric, and transitive); - satisfiesu1 u2⇔ λ u1+ v  λ u2+ vfor allu1,u2 andvinVandλinR. With, we associate the strict partial ordering≺asu≺ v ⇔ (u  vandu6= v)for alluandvinV. For anyO⊆V, we letCH(O)be its convex hull.

We defineQ ⊆ P(V )as the collection of non-empty but finite subsets ofV. DEFINITIONA choice functionCis a map

C:Q → Q ∪ {/0}: O 7→ C(O)such thatC(O) ⊆ O.

RATIONALITY AXIOMSWe call a choice functionConQ(V )coherent if for allO, O1, O2 inQ(V ), u, vinVandλinR>0:

C1.C(O) 6= /0; [non-emptiness]

C2. ifu≺ vthen{v} = C({u, v}); [dominance]

C3. a. ifC(O2) ⊆ O2\ O1 andO1⊆ O2⊆ OthenC(O) ⊆ O \ O1; [Sen’sα] b. ifC(O2) ⊆ O1 andO⊆ O2\ O1 thenC(O2\ O) ⊆ O1; [Aizerman] C4. a. ifO1⊆ C(O2)thenλ O1⊆ C(λ O2); [scaling invariance] b. ifO1⊆ C(O2)thenO1+ {u} ⊆ C(O2+ {u}); [independence] C5. ifO⊆ CH({u, v})then{u, v} ∩C(O ∪ {u, v}) 6= /0. [sticking to extremes] THE ‘IS NOT MORE INFORMATIVE THAN’ RELATIONGiven two choice functionsC1 andC2,

C1 is not more informative thanC2⇔ (∀O ∈Q)(C1(O) ⊇ C2(O)). For a collectionCof coherent choice functions, its infimum is the coherent choice function given by

(inf C)(O):=SC(O)for allOinQ.

CONNECTION WITH SETS OF DESIRABLE OPTIONSChoice functions are essentially non-pairwise comparisons of options. Therefore, we can associated a single coherent set of desirable options with a coherent choice functionCby

DC=u ∈V : {u} = C({0,u}) .

Conversely, given a coherent set of desirable optionsD, there are multiple associated coherent choice functions, and the least informative one is given by

CD(O) =u ∈ O : (∀v ∈ O)v − u /∈ D for allOinQ.

D

H T − − − − − − − ∈ CD({ , , , })since -−∈ D/; -−/∈ D; -−∈ D/. ∈ CD({ , , , })since -− /∈ D; -−/∈ D; -−/∈ D. / ∈ CD({ , , , })since∈ D. / ∈ CD({ , , , })since−∈ D.

2. C

OHERENT CHOICE FUNCTIONS

SET OF INDIFFERENT OPTIONSLike a subject’s set of desirable optionsD—the options he strictly prefers to zero—we collect the options that he considers to be equivalent to zero in his set of indifferent options. A set of indifferent optionsIis simply a subset ofV.

We call a set of indifferent optionsIcoherent if for allu, v inVandλinR: I1.0 ∈ I;

I2. ifu∈V0∪V≺0 thenu/∈ I; [non-triviality]

I3. ifu∈ Ithenλ u ∈ I; [scaling]

I4. ifu, v ∈ Ithenu+ v ∈ I. [combination] QUOTIENT SPACEWe can collect all options that are indifferent to an optionuinVinto the equivalence class

[u]:= {v ∈V : v − u ∈ I} = {u} + I.

The set of all these equivalence classes is the quotient spaceV /I := {[u] : u ∈ V }, which is a vector space with vector ordering

[u]  [v] ⇔ (∃w ∈V )u  v + w

for all[u]and[v]inV /I.

INDIFFERENCE AND DESIRABILITYGiven a set of desirable optionsDand a coherent set of indifferent optionsI, we callDcompatible withIif D+ I ⊆ D.

D

H T I Elements of D/I can be identified with elements on this axis.

D0= D/I

AN INTERESTING CHARACTERISATIONWe give an alternative characterisation of indifference: Proposition. A set of desirable optionsD⊆Vis compatible with a coherent set of indifferent optionsIif and only if there is some ( representing) set of desirable optionsD0⊆V /Isuch thatD= {u : [u] ∈ D0} = SD0. Moreover, the representing set of desirable options is unique and given byD0= D/I := {[u] : u ∈ D}. Finally,Dis coherent if and only ifD/Iis.

INDIFFERENCE AND CHOICE FUNCTIONSWe use the same idea as for desirability. We call a choice functionConQ(V )compatible with a coherent set of indifferent optionsIif there is a representing choice functionC0 onQ(V /I)such that

C(O) = {u ∈ O : [u] ∈ C0(O/I)}for allOinQ(V ).

Proposition. For any choice functionConQ(V )that is compatible with some coherent set of indifferent optionsI, the unique representing choice functionC/IonQ(V /I)is given byC/I(O/I) := C(O)/Ifor

allOinQ(V ). Hence alsoC(O) = O ∩SC/I(O/I)

for allOinQ(V ). Finally,Cis coherent if and onlyC/Iis.

Properties:

- Indifference is preserved under arbitrary infima.

- Given a coherent choice functionCthat is compatible withI, thenDC is also compatible withI. - Given a coherent set of desirable optionsDthat is compatible withI, thenCD is also compatible withI.

3. I

NDIFFERENCE

Consider the possibility spaceX := {a,b,c}and the vector spaceV = RX=R3. We want to express indifference betweenaandb, or in other words betweenI{a} andI{b}, whereI{a}:= (1, 0, 0)and

I{b}:= (0, 1, 0). What is the most conservative choice functionCcompatible with this assessment?

Set of indifferent options:

I= {λ (I{a}−I{b}) : λ ∈R} = {(λ,−λ,0) : λ ∈ R}

= {u ∈R3: E1(u) = E2(u) = 0} withE1 andE2 the expectations associated with the mass func-tionsp1:= (1/2,1/2, 0)andp2:= (0, 0, 1). Equivalence class:

[u] = {u} + I = {v ∈R3: E1(u) = E1(v)andE2(u) = E2(v)}. dim(R3/I) = 2.

Vector ordering:

[u]  [v] ⇔ (∃λ ∈R)u  v + λ (I{a}−I{b}) ⇔ (∃λ ∈R)(ua≤ va+ λ,ub≤ vb− λanduc≤ vc) ⇔ (E1(u) ≤ E1(v)andE2(u) ≤ E2(v))

ub uc ua -2 -2 -2 -1 -1 -1 1 1 1 2 2 2

Options inR3/I can be identified with its projection along I on this plane.

set of indifferent options I= {(λ , −λ , 0) : λ ∈R}

I{a}−I{b} The vector ordering on R3/I is the usual one in this two-dimensional vector space ub uc ua -2 -2 -2 -1 -1 -1 1 1 1 2 2 2 v(1, 0, 1) v0(32, −1,12) w(1, 1,1 3) w0(−32,12,32) (0, 0) (1, 1) ( 12 ,1 2 ) (2, 13 ) (−1, 32 ) ua+ ub uc [v] [v0] [w] [w0]

The vacuous—least informative—choice func-tionC/IonR3/Iselects theundominated options:C({[v], [v0], [w], [w0]}) = {[v], [w], [w0]}. That means that the most conservative choice functionCwe are looking for (the one onR3), has the following behaviour on those four op-tions:

C({v, v0, w, w0}) = {v, w, w0}.