Department of Economics, Bilkent...

List rationalizable choice

KEMAL YILDIZ

Department of Economics, Bilkent University

May 7, 2014

We develop and analyze a boundedly rational choice model both indeterministic and stochastic choice setups

General setup

X is an alternative set with n elements

choice sets are S ⊂ X

choice space is the collection of all choice sets; Ω

a choice function is a mapping c : Ω→ X such that for each S,c(S) ∈ S

General setup

X is an alternative set with n elements

choice sets are S ⊂ X

choice space is the collection of all choice sets; Ω

a choice function is a mapping c : Ω→ X such that for each S,c(S) ∈ S

Rational choice model

The decision maker(DM) is endowed with a preferencerelation, , over the alternative set,

and chooses the -best alternative from each choice set,

i.e. a rational DM chooses from each choice set as ifmaximizing a preference relation

LRC has a single main primitive:

a *list* is an ordering of the alternatives

for each x, y ∈ X we write x f y iff x follows y in the list

P denotes a complete asymmetric binary relation on X

For each < f , P >, we define CP,f : Ω→ X recursively as follows.

Let S = x, y, z,

Suppose x P y P z P x

z y xP P

For each < f , P >, we define CP,f : Ω→ X recursively as follows.

Let S = x, y, z,

Suppose x P y P z P x

z y xP P

C1P,f (S) = f1(S)

C2P,f (S) = f1(S) ∨P f2(S)

z y xP

C1P,f (S) = f1(S)

C2P,f (S) = f1(S) ∨P f2(S)

C3P,f (S) = C2

P,f (S) ∨P f3(S)

z y xP P

C1P,f (S) = f1(S)

C2P,f (S) = f1(S) ∨P f2(S)

.CkP,f (S) = Ck−1

P,f (S) ∨P fk(S) for each k ∈ 2, ...s

Finally, let CP,f (S) = CsP,f (S)

z y xP P

C1P,f (S) = f1(S)

C2P,f (S) = f1(S) ∨P f2(S)

.CkP,f (S) = Ck−1

P,f (S) ∨P fk(S) for each k ∈ 2, ...s

Finally, let CP,f (S) = CsP,f (S)

z y xP P

List Rationality

Definition

A choice function c is list rational if there exists a list f such thatfor each S ∈ Ω, if x is the last alternative in S according to f , then

c(S) = c(c(S \ x), x)

motivation

LRC procedure offers an intuition for the choice behavior of a DMwho:

process information sequentially

are constrained by single memory

can exhibit binary choice cycles.

motivation

Salant (2003) argues that list rationality is the unique choiceprocedure that uses a single memory cell.

Russo & Rosen (1975) argues that LRC is followed by thesubjects to minimize the short-term memory load.

Shugan (1980) proposes a formulation based on cost ofthinking, and argues that the lowest choice cost would beachieved by first comparing among an alternative pair with lowthinking cost, and then comparing the winner with the next

Liu & Simonson (2005) argue that decision makers guided bylist rationality are more confident in their choices compared tonon-guided decision makers.

List rationality vs Choice from the lists

Rubinstein and Salant’06 analyze a new choice model whereDMs choose the alternatives out of given observable lists

We model the list as a subjective part of the choice procedure.Even if there is a given list, a DM can follow a different virtualordering.

one can start the comparison from a referencepoint,(Tversky’91)

group the similar items together, (Russo’75) or

can order them to minimize the total thinking cost.(Shugan’80)

Rationalization by a binary game tree

Rationalization by a game tree (Xu & Zhou’07)

Q How to identify the non-observed list from the observed choicesof the DM ?

Answer: This can be achieved if we can conclude from theobserved choices of the DM that an alternative x unambiguouslyfollows another alternative y in his considerations

Random Choice

Random choice

∆(S) is the collection of probability distributions over S

a random choice rule is a mapping c, such that

∀S ∈ Ω, c(S) ∈ ∆(S)

a random binary relation is a mapping P : X ×X → [0, 1]

Random choice

∀S ∈ Ω, c(S) ∈ ∆(S)

Random choice

∀S ∈ Ω, c(S) ∈ ∆(S)

Random choice

∀S ∈ Ω, c(S) ∈ ∆(S)

why modeling random choice?

Random choice accommodates:

repeated decisions of a single individual

choice data of a group of individuals

Consider the LRRC procedure described by < f ,P >, where

z y x1/2 1/2

LRRC which follows f , yields for each S, Pf (S) ∈ ∆(S) :

z y x1/2

⇐ z y x

Pf (S) 1/4

z y x1/2

⇐ z y x

Pf (S) 1/4 1/4

⇐ z y x

Pf (S) 1/4 1/4 1/2

Results

Plott (1973)

Stochastic Path Independence (SPI)

Thm: A choice function is rational iff path independence issatisfied.

Plott (1973)

Thm: A choice function is rational iff path independence issatisfied.

Path Independence

Path Independence: For each S, T such that T ⊂ S, if x = c(T )and y = c(S \ T ), x = c(x, y) implies x = c(S).

x yz w

z w∪ x

Path Independence

x yz w

z w∪ x

Path Independence

x yz w

z w∪ x

⇒ x, y → x

Path Independence

x yz w

z w∪ x

⇒ x, y → x

Path Independence

x yz w

z w∪ x

⇒ x, y → x

Path Independence

x yz w

z w∪ x

⇒ x, y → x

The Stochastic Case

Stochastic Path Independence: For each S ∈ Ω, x ∈ S andy 6∈ S,

cx(S ∪ y) = cx(S) · c(x, y)

The Stochastic Case

Stochastic Path Independence: For each S ∈ Ω, x ∈ S andy 6∈ S,

cx(S ∪ y) = cx(S) · c(x, y)

The Stochastic Case

For each x, y ∈ X, x is revealed to follow y (x Fc y) iff SPI isviolated between x and y, that is : for some S ∈ Ω s.t x ∈ S andy 6∈ S,

cx(S ∪ y) 6= c(x, y) · cx(S)

Proposition

Proposition: c is list rationalizable iff fc is acyclic.

Moreover, the followed list is identified unique up to thecompletions of transitive closure of fc.

The Stochastic Case

cx(S ∪ y) 6= c(x, y) · cx(S)

Proposition

The Stochastic Case

cx(S ∪ y) 6= c(x, y) · cx(S)

Proposition

The Stochastic Case

cx(S ∪ y) 6= c(x, y) · cx(S)

Proposition

The Deterministic Case

For each distinct x, y ∈ X, x is revealed to follow y (x Fc y), if forsome S ∈ Ω, we have either

(i) x = c(S ∪ y) and [y = c(x, y) or x 6= c(S)] or

(ii) x 6= c(S ∪ y) and [x = c(x, y) and x = c(S)].

Proposition

Corollary: c is list rational iff Fc is acyclic.

Followed list is identified unique up to the completions oftransitive closure of fc.

Proposition

Analysis of Stochastic Path Independence

SPI: For each S ∈ Ω, and y 6∈ S, cx(S ∪ y) = c(x, y) · cx(S)

a preference relation is a near linear order if the length of ∼ is atmost two

Proposition

Proposition: c satisfies SPI iff there is a near linear order on Xsuch that; for each S ∈ Ω

c(S) = c(max(S,))

Proposition

c(S) = c(max(S,))

Proposition

c(S) = c(max(S,))

X be a real linear vector space

for each S ∈ Ω, c(S) ∈ ∆(S) ⊂ X

c is continuous if for each x, y ∈ X and xnn≥1 ⊂ X,

lim xn = x ⇒ lim c(xn, y) = c(x, y)

Proposition

Proposition: If X contains the convex hull of three non-collinearpoints, then there is no continuous rcr over X which satisfies SPI

List Rationality and Two-stage choiceprocedures

List Rationality vs Shortlisting (M & M’06)

P1 is transitive

P2 is a linear order

c is shortlisting if for each S, c(S) = max(S, P1)

P1 is transitive

List Rationality vs Shortlisting

P1 is transitive

c is shortlisting if for each S, c(S) = max(max(S, P1), P2)

P1 is transitive

equivalently;

c(S) is the solution to the problem:

maxx∈S u(x)

subject to

there is no y ∈ S s.t y P1 x

P1 is transitive

equivalently;

c(S) is the solution to the problem:

maxx∈S u(x)

subject to

there is no y ∈ S s.t y P1 x

For a given choice function c, and for each distinct x, y ∈ X, x isrelated to y (x Rc y) if for some S ∈ Ω, we have either

(i) x = c(S ∪ y) and x 6= c(S) or

(ii) y = c(S ∪ x) and x = c(x, y).

Proposition

Proposition: A choice function c is shortlisting iff Rc is acyclic.

Note that:

x Ric y ⇒ x Fc y

x Riic y ⇒ y Fc x.

(i) x = c(S ∪ y) and x 6= c(S) or

(ii) y = c(S ∪ x) and x = c(x, y).

Proposition

Note that:

x Ric y ⇒ x Fc y

(i) x = c(S ∪ y) and x 6= c(S) or

(ii) y = c(S ∪ x) and x = c(x, y).

Proposition

Note that:

x Ric y ⇒ x Fc y

conclusion

Choice with limited attention (MNO 2012)

This choice procedure has two primitives

an attention filter Γ and a welfare preference

For each choice set

S, c(S) = maxΓ(S), where Γ is s.t. for each choice set S

and z 6∈ Γ(S), Γ(S \ z) = Γ(S)

LRC ⊆ CLA

an attention filter Γ and a welfare preference For each choice set

S, c(S) = maxΓ(S),

where Γ is s.t. for each choice set S

and z 6∈ Γ(S), Γ(S \ z) = Γ(S)

LRC ⊆ CLA

and z 6∈ Γ(S), Γ(S \ z) = Γ(S)

LRC ⊆ CLA

and z 6∈ Γ(S), Γ(S \ z) = Γ(S)

LRC ⊆ CLA

a related experimental study

Liu and Simonson (2005) compared the behavior of subjects whoare asked to make a choice from a set of ten products according totwo procedures.

people were presented all ten product offers together and areasked to indicate the one they are most interested in.

then, they are asked whether they would like to purchase theselected product.

in the second procedure, subjects are asked to make a choicefrom the set of ten products according to the LRC procedureby following an exogenously specified list.

once the last item in the list is considered, participants areasked to decide whether they want to purchase the winner.

They observe that subjects guided by LRC procedure are morelikely to purchase their selected product (%45) than those inthe simultaneous condition (%34)

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