Rationalizing Irrational Choice

1
Rationalizing Irrational Choice
2
Rational Choice
X
(finite)
X,
A?Ø
A
(complete transitive)
3
Rational Choice
x,y,z
X
x,y,z
, x,y
, x,z
, y,z
A
X
x
y
z
c(A)
Independence of Irrelevant Alternatives (IIA)
if x
then
implies that
A
B ,
x c(B)
x c(A)
4
Proposition
there exists a strict preference relation
on X
such that for every A
X, c(A) is the most
if, and
preferred element in A according to
only if c satisfies IIA
c is rationalizable iff it satisfies IIA
5
Proof
Step 1 If c is rationalizable, then it satisfies
IIA
x
B
A
A
B
x c(B)
x is most preferred in B
x
y
x is most preferred in A
A
B
w
z
6
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) y
y
z
c(x,z) x
x
z
7
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) y
y
z
c(x,z) ?
c(x,y,z) ?
8
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) y
y
z
c(x,z) ?
IIA
c(x,y,z) y
c(x,y) y
9
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) z
y
z
c(x,z) ?
IIA
c(x,y,z) z
c(y,z) z
10
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) z
y
z
c(x,z) ?
c(x,y,z) x
11
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
c(x,y) x
x
y
c(y,z) z
y
z
c(x,z) x
x
z
IIA
c(x,y,z) x
12
Proof
Step 2 If c satisfies IIA, then it is
rationalizable
X
A
c(A)
a
A
?
by definition of
c(A)
c
c(A)
a


,
13
Systematic violations of rationality
Experimental methodology
14
x
x
y
y
z
15
Reference point effects
16
Treatment 1
As a result of an epidemic disease, 600 people
will die
There are 2 mutually exclusive treatments
(A) Exactly 200 people will be saved
(B) With probability 1/3, all 600 people will be
saved,
With probability 2/3, no one will be saved
Which treatment would you choose?
17
Treatment 2
As a result of an epidemic disease, 600 people
will die
There are 2 mutually exclusive treatments
(C) Exactly 400 people will die
(D) With probability 2/3, all 600 people will die,
With probability 1/3, no one will die
Which treatment would you choose?
18
As a result of an epidemic disease, 600 people
will die
(A) Exactly 200 people will be saved
(B) With probability 1/3, all 600 people will be
saved,
With probability 2/3, no one will be saved
(C) Exactly 400 people will die
(D) With probability 2/3, all 600 people will die,
With probability 1/3, no one will die
19
Choosing between a pair of bets
Treatment 1
A
B
20
Treatment 2
C
D
21
Treatment 2
C
D
22
A,C
B,D
23
Days in London
(4,7)
7
(7,4)
4
(6,3)
3
Days in Paris
4
7
6
24
Days in London
(4,7)
7
(7,4)
4
(9,2)
2
Days in Paris
4
9
7
25
Rationalization by multiple rationales
Choice sets which convey information
Chicken
(4,7)
(7,4)
Steak
(6,3)
Frog legs
(4,7)
(7,4)
(9,2)
26
Rationalization by multiple rationales
X 1, ,n
A choice function c is rationalized by K
rationales if there is a collection of K linear
orderings over X,
(R1, ,RK)
such that for every subset A, there exists
k 1, ,K
satisfying that c(A) is the top element in A
according to Rk
Kn
K1
27
Proposition Every choice function can be
rationalized by n-1 rationales
Proof
? c(X) ? y
x y z
1st
2nd
? For x ? y let Rx
? A X
- c(A) x
Rx
- c(A) y
A ? X
x X\A
Rx
28
Proposition The proportion of choice functions
that can be rationalized by less than n-1
orderings tends to 0 as n tends to infinity
29
Q1. Imagine you had to choose between the
following 2 options
(A) Receiving 105 a year and a day from now (B)
Receiving 100 a year from now
Which would you choose?
Q2. Imagine you had to choose between the
following 2 options
(C) Receiving 105 tomorrow (D) Receiving 100
today
Which would you choose?
30
4,000
3,000
0.8
1
A
B
0.2
0
0
0
31
4,000
3,000
0.2
0.25
C
D
0.8
0.75
0
0
32
4,000
3,000
0.8
1
B
A
0.25
0.25
0.2
0
0
0
C
D
0.75
0.75
0
0
33
4,000
3,000
0.8
1
B
A
0.25
0.25
0.2
0
0
0
C
D
0.75
0.75
0
0
34
4,000
3,000
0.8
1
A
B
0.2
0
0
0
4,000
3,000
0.2
0.25
C
D
0.8
0.75
0
0
35
(p,x)P1(q,y) if p q 0.1

A
P1
X
Y
p
q
L
R
1-p
1-q
0
0
36

A
P1
4,000
3,000
0.8
1
A
B
0.2
0
0
0
37

A
P1
4,000
3,000
0.2
0.25
C
D
0.8
0.75
0
0
38
)

(
A
P1
max
4,000
3,000
0.8
1
A
B
0.2
0
0
0
39
)

(
A
P1
max
4,000
3,000
0.2
0.25
C
D
0.8
0.75
0
0
40
)

(

A
P1
max
P2
max
4,000
3,000
0.2
0.25
C
D
0.8
0.75
0
0
41
Shortlisting
A choice function c is a Rational Shortlist
Method (RSM) if there exists an ordered pair of
asymmetric relations such that for every
non-empty A X,
c(A) maxmax(AP1)P2
(P1,P2) is said to sequentially rationalize c
Note order of rationales is the same for every A
X
42
Characterizing RSMs
x
x
z1
y
z2
y
z4
z3
x
z1
y
z2
x
43
Characterizing RSMs
x
x
z1
z2
y
z3
y
z4
x
z1
y
z2
x
z4
z3
44
Weak WARP For any A B X that contain
x,y,
if x c(x,y) c(B), then y?c(A)
Expansion For any A,B X, if x c(A) c(B),
then
x c(A B)
Proposition A choice function c on X is a RSM
iff it satisfies Weak WARP and Expansion
45
Proof 1. Necessity
1.1. If c is RSM, then it satisfies Expansion
x c(A) c(B)
x c(A B)
x
A
B
46
Proof 1. Necessity
1.1. If c is RSM, then it satisfies Expansion
x c(A) c(B)
x ? c(A B)
x
Pi
y
A
B
47
Proof 1. Necessity
1.2. If c is RSM, then it satisfies Weak WARP
x c(x,y) c(B)
y ? c(A)

B
x,y
x
y
A
48
Proof 1. Necessity
1.1. If c is RSM, then it satisfies Weak WARP
x c(x,y) c(B)
y c(A)

B
x,y
x
P1
y
xP1y?
A
49
Proof 1. Necessity
1.1. If c is RSM, then it satisfies Weak WARP
x c(x,y) c(B)
y c(A)

B
x,y
P1
x
P2
z
y
xP1y
xP2y
A
50
2. Sufficiency
If c satisfies Expansion weak WARP, then its a
RSM
xP1y if there is no A X such that x A and
yc(A)
xP2y if xc(x,y)
(Note P1,P2 are asymmetric as required)
x c(S)
Need to show
1. there is no y in S that eliminates x in 1st
round
2. if there is y that could eliminate x in 2nd
round, then that y is eliminated in 1st round
51
x c(S)
yP1x

x is never chosen from a set that contains x,y
P1
x
y
But x,y S and c(S)x
S
yP1x
52
x c(S)
y c(x,y)

P2
x
y
S
53
x c(S)
T1
y c(x,y)

z1
y c(T1)
x
y
S
54
x c(S)
T1
y c(x,y)

z1
y c(T1)
x
y
y c(T2)
z2
y c(TK)
S
T2
55
x c(S)
y c(x,y)

z1
y c(T1)
x
y
y c(T2)
z2
y c(TK)
(expansion)
c(T1 T2 TK x,y)
y
y c(S)
(weak WARP)
56
Violations of rationality (WARP)
Always Chosen For all A X, if x c(x,y) for
all y A\x, then x c(A)
No Binary Cycles For all x1, , xn X, if x
c(xi,xi1) for i 1, , n-1, then x1 c(x1,xn)
Proposition A choice function that is not
rationalizable violates either Always Chosen or
No Binary Cycles
57
Proof By induction on the number of elements in X
Step 1 X x,y,z
Suppose (wlog) c(X)x but c(x,y)y
If c(y,z)y, then Always Chosen violated
If c(y,z)z c(x,z)z, then Always Chosen
violated
If c(y,z)z but c(x,z)x, then No Binary Cycles
is violated
58
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? Consider X, X n1
? Suppose c(X) x, but c(S) y for x,y S
X
- if c on S violates IIA, we are done by
Inductive Step
- suppose c on S satisfies IIA
? let V X\S with c(V) z
- if c on V violates IIA, we are done by
Inductive Step
- suppose c on V satisfies IIA z c(v,z) for
all v in V\z
59
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
(V X\S)
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
y,z
c
z
60
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
y,z
c
z
61
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
(AC) violated
(AC) violated
(AC) violated
(AC) violated
(AC) violated
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
y,z
c
z
62
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
(AC) violated
(AC) violated
(AC) violated
(AC) violated
(AC) violated
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
y,z
c
z
c(z,t) t for some t in S
c(z,t) t for some t in S
63
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
(AC) violated
(AC) violated
(AC) violated
(AC) violated
(AC) violated
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
y,z
c
z
(NBC) violated
(NBC) violated
c(z,t) t for some t in S
c(z,t) t for some t in S
c(z,t) t for some t in S
64
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
(AC) violated
(AC) violated
(AC) violated
(AC) violated
(AC) violated
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
c(z,s) z for all s in S
y,z
c
z
(NBC) violated
(NBC) violated
c(z,t) t for some t in S
c(z,t) t for some t in S
c(z,t) t for some t in S
c(y,z) z, c(t,z) t, c(y,t) y
65
Proof By induction on the number of elements in X
Inductive Step Assume claim is true for X, X
n
? c satisfies IIA on S and on V
? c(X) x but c(S) y
? c(V) z for V X\S
? z c(v,z) for all v in V\z
(AC) violated
c(y,s) y for all s in V
y,z
c
y
(NBC) violated
c(y,t) t for some t in V
c(y,z) y, c(t,y) t, c(z,t) z
66
Proposition A sequentially rationalizable choice
function violates WARP iff it violates No Binary
Cycles
Proof
Step 1 RSM satisfies Always Chosen
? Let A X such that x c(x,y) for all y
A\x
? Then no y A\x satisfies yP1x or yP2x
x c(A)
67
Proposition A sequentially rationalizable choice
function violates WARP iff it violates No Binary
Cycles
Proof
Step 2 A choice function that violates WARP
violates either Always Chosen or No Binary Cycles
By Step 1, RSM satisfies Always Chosen
68
Choice from Lists
? X
? L (a1, ,aK)
? S(L) a1, ,aK
(the choice from the list)
? D(L) S(L)
? L1, L2 are disjoint if S(L1) S(L2) Ø
? ltL1, ,Lmgt
L1 (a1,a2,a3)
ltL1, L2gt (a1,a2,a3,b1,b2)
L2 (b1,b2)
69
Choice from Lists
Partition Independence (PI) For every disjoint
L1,L2
D(ltL1,L2gt) DD(L1),D(L2)
a1
a2
a5
a3
a4
,
,
,
,
(
)
L
70
Choice from Lists
Partition Independence (PI) For every disjoint
L1,L2
D(ltL1,L2gt) DD(L1),D(L2)
a1
a2
a5
a3
a4
,
,
,
,
(
)
L
1.
a1
a2
,
(
)
a5
a3
a4
,
,
)
(
2.
(a2
, a3)
71
Choice from Lists
Partition Independence (PI) For every disjoint
L1,L2
D(ltL1,L2gt) DD(L1),D(L2)
List Independence of Irrelevant Alternatives
(LIIA)
D(a1, ,aK) ai
D(a1, ,aj-1, aj1, ,aK) ai
for all 1 j K, j ? i
Proposition PI LIIA
72
Proposition PI LIIA
Proof
Assume D satisfies LIIA
? L ltL1,L2gt
? D(L) a
? a S(L2) (wlog)
L2 (D(L1),a) are sublists of L that include a
By LIIA, D(L2) DD(L1),a a
D(ltL1,L2gt) D(L) a DD(L1),a)
DD(L1),D(L2)
73
Proposition PI LIIA
Proof
Assume D satisfies PI
D(a1, , aj-1,
aj ,
aj1, ,aK) ai
(w.l.o.g. let j gt i)
By PI, for every h gt i,
ai D(a1, ,aK) DD(a1, ,ah),D(ah1, ,aK)
ai D(a1, ,ah) D(ai,D(ah1, ,aK)
D(a1, ,aj-1,aj1, ,aK)
DD(a1, ,aj-1),D(aj1, ,aK)
74
Proposition PI LIIA
Proof
Assume D satisfies PI
(w.l.o.g. let j gt i)
D(a1, , aj-1,
aj ,
aj1, ,aK) ai
By PI, for every h gt i,
ai D(a1, ,aK) DD(a1, ,ah),D(ah1, ,aK)
ai D(a1, ,ah) D(ai,D(ah1, ,aK)
D(a1, ,aj-1,aj1, ,aK)
Dai,D(aj1, ,aK)
75
Proposition PI LIIA
Proof
Assume D satisfies PI
(w.l.o.g. let j gt i)
D(a1, , aj-1,
aj ,
aj1, ,aK) ai
By PI, for every h gt i,
ai D(a1, ,aK) DD(a1, ,ah),D(ah1, ,aK)
ai D(a1, ,ah) D(ai,D(ah1, ,aK)
D(a1, ,aj-1,aj1, ,aK)
ai
76
Do people satisfy partition independence?
Suppose that you could receive any three of the
following white wines (one bottle each) to take
home with you today. All wines below received
scores of 87 (out of 100) from Wine Spectator
magazine and are in the 15-18 price range.
Which three would you choose?
77
Do people satisfy partition independence?
Chardonnays
Pinot Grigios
Sauvignon Blancs
1. Marchesi di Gresy
1. Bollini
1. Fox Creek
2000 Chardonnay Langhe
1999 Pinot Grigio del Friuli Reserve
2001 Sauvignon Blanc s. Australia
Italy
Italy
Australia
2. Stonehaven
2. Luna
2. Groth
2000 Chardonnay Limestone Coast
1999 Pinot Grigio Napa Valley
2001 Sauvignon Blanc Napa Valley
Australia
California
Italy
78
Do people satisfy partition independence?
Italy
California
Australia
1. Marchesi di Gresy
1. Fox Creek
1. Groth
2000 Chardonnay Langhe
2001 Sauvignon Blanc s. Australia
2001 Sauvignon Blanc Napa Valley
Italy
Australia
California
2. Stonehaven
2. Luna
2. Bollini
2000 Chardonnay Limestone Coast
1999 Pinot Grigio Napa Valley
1999 Pinot Grigio del Friuli Reserve
Australia
California
Italy
79
Do people satisfy partition independence?
All grapes selected
All regions selected
Region partition
Grape partition
Region partition
Grape partition
56
36
32
68
(149 subjects)
80
Do people satisfy partition independence?
? Suppose that you were to advise the financial
aid office on how they should distribute next
years budget among entering freshmen who apply
for financial aid.
? Specifically, you are asked to indicate what
percentage of the budget you would allocate to
aid applicants whose family household incomes
falls in various ranges.
? Besides each of the following income ranges,
please indicate what percentage of the budget
you would recommend to be allocated to that
group of students
81
Do people satisfy partition independence?
82
Do people satisfy partition independence?
83
Do people satisfy partition independence?
95.9
(208 participants)
84
Do people satisfy partition independence?
85
Do people satisfy partition independence?
Treatment 1
For each of the following statements, please
indicate what you think the chances are that the
statement is correct.
the noontime temperature at OHare Airport on
Sunday will be higher than every other day next
week
the noontime temperature at OHare Airport on
Sunday will not be higher than every other day
next week
86
Do people satisfy partition independence?
Treatment 2
For each of the following statements, please
indicate what you think the chances are that the
statement is correct.
Next week at noontime, the highest temperature
of the week at OHare Airport will occur on
Sunday
Next week at noontime, the highest temperature
of the week at OHare Airport will not occur on
Sunday
87
Do people satisfy partition independence?
Treatment 1
Treatment 2
n 53
n 41
median probabilities
median probabilities
(0.3, 0.7)
(0.15, 0.85)
(0.14, 0.86)
(0.5, 0.5)
88
Proposition
a choice function from lists that satisfies PI
is rationalizable as the maximization of a weak
preference
relation on X, where indifferences are resolved
according
to position, i.e., either the 1st or last
maximizer is chosen
?
? d X 1,2
satisfies d(x) d(y) whenever x y
? D( ,d)
89
Proposition
a choice function D satisfies PI iff there exists
a unique over X and a unique d such that D
D( ,d)
Proof
D( ,d) satisfies PI
easy to show
90
Proposition
a choice function D satisfies PI iff there exists
a unique over X and a unique d such that D
D( ,d)
Proof
Assume D satisfies PI
also satisfies LIIA
? a b if D(a,b) D(b,a) a
asymmetric
? a 1 b if D(a,b) a but D(b,a) b
1, 2 symmetric
? a 2 b if D(a,b) b but D(b,a) a
91
Claim 1 is transitive
Proof
Assume a b b c
By PI, D(a,b,c) D ,
a
D(b,c)
D(a,b) a
?
? By LIIA, D(a,c) a
? Similarly, D(c,b,a) a D(c,a) a
a c
92
Similarly, for i 1,2
Claim 2 i is transitive
Claim 3 a b, b i c
a c
Claim 4 a i b, b c
a c
Claim 5 Its impossible that BOTH a 1 b AND b
2 c
Assume a 1 b and b 2 c
Proof of 5
D(c,b)b
? a 1 c symmetry c 1 a transitivity c 1 b
D(c,b)c
? similarly, it cannot be true that a 2 c
c 1 b if D(c,b) c but D(b,c) b
b 2 c if D(b,c) c but D(c,b) b
93
Similarly, for i 1,2
Claim 2 i is transitive
Claim 3 a b, b i c
a c
Claim 4 a i b, b c
a c
Claim 5 Its impossible that BOTH a 1 b AND b
2 c
Assume a 1 b and b 2 c
Proof of 5
a c do not relate to each other by 1
and 2
By Claims 3 4, neither a c nor c a is true
94
Similarly, for i 1,2
Claim 2 i is transitive
Claim 3 a b, b i c
a c
Claim 4 a i b, b c
a c
Claim 5 Its impossible that BOTH a 1 b AND b
2 c
Assume a 1 b and b 2 c
Proof of 5
a c do not relate to each other by 1
, 2 or
contradiction!
95
? Define a b if a b or a 1 b or a 2 b
- By Claims 1-5, is a preference relation
? Define a b if both a b and b a
- if a b, then either a 1 b or a 2 b but not
both
either D(a,b)a, D(b,a)b or D(a,b)b, D(b,a)a
? Let I be some indifference set, a,b X a
b
- By Claim 5, a 1 b for all a,b I or a 2 b
for all a,b I
? if all elements in I relate to each other via
1, then d assigns the value 1 to each such
element
Claim 5 Its impossible that BOTH a 1 b AND b
2 c
96
? Define a b if a b or a 1 b or a 2 b
- By Claims 1-5, is a preference relation
? Define a b if both a b and b a
- if a b, then either a 1 b or a 2 b but not
both
either D(a,b)a, D(b,a)b or D(a,b)b, D(b,a)a
? Let I be some indifference set, a,b X a
b
- By Claim 5, a 1 b for all a,b I or a 2 b
for all a,b I
? if all elements in I relate to each other via
2, then d assigns the value 2 to each such
element
97
(a,b,c)
D
b
LIIA
D(a,b) b
b a
D(b,c) b
LIIA
b c
98
(a,b,c)
D
b
D(a,b) b
b a
D(b,c) b
b c
D(c,b) b
99
(a,b,c)
D
b
D(a,b) b
b a
D(b,a) b
D(b,c) b
b c
D(c,b) b
100
(a,b,c)
D
b
D(a,b) b
b a
D(b,a) a
2
D(b,c) b
b c
D(c,b) b
d(a) d(b) 2
101
(a,b,c)
D
b
D(a,b) b
b a
D(b,a) a
2
D(b,c) b
b c
D(c,b) b
d(a) d(b) 2
102
(a,b,c)
D
b
D(a,b) b
b a
D(b,a) b
D(b,c) b
b c
D(c,b) c
1
d(b) d(c) 1
103
(a,b,c)
D
b
D(a,b) b
b a
D(b,a) b
D(b,c) b
b c
D(c,b) c
1
d(b) d(c) 1
D(L) chooses either the unique best element
according to or, in case of a tie, the first or
last best element
D(L) D( ,d)(L)
104
Uniqueness of and d
( ,d) ? ( ,d)
D( ,d) ? D( ,d)