Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications

1
Behavioral Social ChoiceProbabilistic Models,
Statistical Inference, and Applications
Cambridge University Press, 2006 with B.
Grofman, A.A.J. Marley, I. Tsetlin

• Michel Regenwetter
• Quantitative Division
• Department of Psychology
• ( Department of Political Science)
• University of Illinois at Urbana-Champaign

2
Overview

• Motivation
• 2 Conceptual Distinctions in the Decision
  Sciences
• Criteria for a Unified Theory of Decision
  Making
• Social Choice Theory
• Majority Rule (Condorcet Criterion), Arrow
• The Obsession with Majority Cycles
• Need for Behavioral Social Choice Research
• Behavioral Social Choice Research
• Multiple Representations of Preference/Utility
• A General Concept of Majority Rule
• Model Dependence of Majority Outcomes
• The Problem of Incorrect Majority Outcomes

3
2 Conceptual Distinctions in the Decision Sciences
Normative Theory
Descriptive Theory Data
Individual Judgment and Decision Making
Behavioral Decision Research
Social Choice
4
2 Conceptual Distinctions in the Decision Sciences
Normative Theory
Descriptive Theory Data
Individual Judgment and Decision Making
Social Choice
???
???
5
2 Conceptual Distinctions in the Decision Sciences
Normative Theory
Descriptive Theory Data
Individual Judgment and Decision Making
???
Social Choice
6
Criteria for a Unified Theory of Decision Making
(Inspired by Luce and Suppes, Handbook of
Mathematical Psych.,1965)

• Treat individual group decision making in a
  unified way
• Reconcile normative descriptive work
• Integrate compare competing normative
  benchmarks
• Reconcile theory data
• Encompass integrate multiple choice, rating and
  ranking paradigms
• Integrate compare multiple representations of
  preference, utilities choices
• Develop dynamic models as extensions of static
  models
• Systematically incorporate statistics as a
  scientific decision making apparatus

7
Overview

• Motivation
• 2 Conceptual Distinctions in the Decision
  Sciences
• Criteria for a Unified Theory of Decision Making
• Social Choice Theory
• Majority Rule (Condorcet Criterion), Arrow
• The Obsession with Majority Cycles
• Need for Behavioral Social Choice Research
• Behavioral Social Choice Research
• Multiple Representations of Preference/Utility
• A General Concept of Majority Rule
• Model Dependence of Majority Outcomes
• The Problem of Incorrect Majority Outcomes

8
Majority rule (Condorcet Criterion)

• Majority Winner
• Candidate who is ranked ahead of any other
• candidate by more than 50
• Candidate who beats any other candidate
• in pairwise competition

9
Kenneth Arrows (1951)Nobel Prize winning
Impossibility Theorem

• List of Axioms of Rationality
• Impossibility to simultaneously satisfy all
  Axioms
• For instance Majority permits cycles.
• Democratic Decision Making Impossible?

10
The Obsession with Cycles
11
Majority Cycles
ABC 1 person
BCA 1 person
CAB 1 person
12
Majority Cycles
ABC 1 person
A beats B 2 times
BCA 1 person
B beats A 1 time
CAB 1 person
A is majority preferred to B
13
Majority Cycles
ABC 1 person
B beats C 2 times
BCA 1 person
C beats B 1 time
CAB 1 person
A is majority preferred to B
B is majority preferred to C
14
Majority Cycles
ABC 1 person
A beats C 1 time
BCA 1 person
C beats A 2 times
CAB 1 person
A is majority preferred to B
B is majority preferred to C
C is majority preferred to A
15
Majority Cycles
ABC 1 person
Democratic Decision Making at Risk!?!
BCA 1 person
CAB 1 person
A is majority preferred to B
B is majority preferred to C
C is majority preferred to A
16
State of the Art Shepsle et al. 1997
17
State of the Art Shepsle et al. 1997
18
State of the Art Shepsle et al. 1997
19
State of the Art Shepsle et al. 1997
20
State of the Art Shepsle et al. 1997
GIGO?
21
Shepsle Bonchek (1997)
In general, then, we cannot rely on the method
of majority rule to produce a coherent sense of
what the group wants, especially if there are
no institutional mechanisms for keeping
participation restricted (thereby keeping n
small) or weeding out some of the alternatives
(thereby keeping m small).
22
1,000,000 Question

• Where is the empirical evidence
• for the Concorcet paradox in practice?

Oops. Little evidence that majority cycles have
occurred among serious contenders of major
elections.
Actually, evidence circumstantial at best.
23
Where is the evidence for cycles?

• Majority Winner
• Candidate who is ranked ahead of any other
  candidate by more than 50
• Candidate who beats any other candidate in
  pairwise competition

• Plurality Choose one
• SNTV Limited Vote Choose k many
• Approval Voting Choose any subset
• STV (Hare), AV (IRV) Rank top k many
• Cumulative Voting Give m pts to k many
• Survey Data Thermometer, Likert Scales

Data are incomplete!!
24
Need for Behavioral Social Choice Research

• Real world ballot or survey data hardly ever take
  the form of (deterministic) linear orders.
• Social choice concepts are defined in terms of
  theoretical primitives that are not directly
  observed in social choice behavior.
• We need to bridge this gap in order to study
  social choice theoretical concepts empirically.

25
Overview

• Motivation
• 2 Conceptual Distinctions in the Decision
  Sciences
• Criteria for a Unified Theory of Decision Making
• Social Choice Theory
• Majority Rule (Condorcet Criterion), Arrow
• The Obsession with Majority Cycles
• Need for Behavioral Social Choice Research
• Behavioral Social Choice Research
• Multiple Representations of Preference/Utility
• A General Concept of Majority Rule
• Model Dependence of Majority Outcomes
• The Problem of Incorrect Majority Outcomes

26
Binary Preference Relations

• A binary relation R on a set of objects C is
  a collection of ordered pairs of elements of C
  

27
A General Concept of Majority Rule
Linear Orders complete rankings Weak
Orders rankings with possible
ties Semiorders rankings with (fixed)
threshold Interval Orders rankings with
(variable) threshold Partial Orders
asymmetric, transitive Asymmetric Binary Relations
28
B
Real Representation of Linear Orders
a b c d
42 5 2 1
29
B
Real Representation of Weak Orders
a b c d
7 7 3 1
30
B
Real Representation of Semiorders
a b c d
3 1.9 1.8 1.8
31
Variable/Uncertain Preferences Probability
Distribution on Binary Relations
Variable/Uncertain Utilities Jointly Distributed
Family of Utility Random Variables (Random
Utilities) (parametric or nonparametric)
32
Random Utility Representations
Linear Orders
Weak Orders
(Block and Marschak, 1960, chapter)
33
Random Utility Representations
Semiorders
Interval Orders
(Heyer Niederee, 1992, MSS) (Niederee Heyer,
1997, Luce vol.) (Regenwetter, 1997,
JMP) (Regenwetter Marley, 2002, JMP)
34
A General Definitionof Majority Rule
35
A General Definitionof Majority Rule
For Utility Functions or Random Utility Models
choose a Random Utility Representation and
obtain a consistent Definition
36
Examples
37
1,000,000 Question

• Where is the empirical evidence
• for the Concorcet paradox in practice?

Oops. Little evidence that majority cycles have
occurred among serious contenders of major
elections.
Actually, evidence circumstantial at best.
38
Lets analyze National Survey Data! 1968, 1980,
1992, 1996 ANES Feeling Thermometer
Ratings translated into Weak Orders or
Semiorders
39
.03
H W N
1968 NES Weak Order Probabilities
.04
0
H W N
H W N
H N W
.02
W H N
.32
H N W
W H N
.03
.08
.02
N H W
W N H
.06
W N H
N H W
.27
N W H
.01
.05
No impartial culture!
.07
40
.03
H W N
1968 NES Weak Order Probabilities
.04
0
H W N
H W N
H N W
.02
W H N
.32
H N W
W H N
.03
.08
.02
N H W
W N H
.06
W N H
N H W
.27
N W H
.01
.05
No impartial culture!
.07
41
a b c
a versus b
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
42
a b c
a versus c
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
43
a b c
b versus c
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
44
a b c
Sens NB(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
45
a b c
Net NB(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
46
a b c
Generalized Net NB(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
47
a b c
Sens NM(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
48
a b c
Net NM(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
49
a b c
Generalized Net NM(a)
a b c
a b c
a c b
a c
b a c
b c
a b
a c b
b a c
c b
b a
c a b
b c a
c a
b c a
c a b
c b a
50
.03 (-.04)
H W N
.04 (.03)
ANES 1968
0 (-.05)
H W N
H W N
H N W
W H N
.02 (.-25)
.32 (.26)
.03 (-.05)
H N W
W H N
.08 (.05)
.02 (0)
N H W
W N H
.06 (-.26)
.27 (.25)
W N H
N H W
N W H
.01 (-.03)
.05 (.05)
.07 (.04)
51
A C R
.07 (-.02)
.02 (-.03)
ANES 1980
.05 (-.02)
A C R
A C R
A R C
C A R
.14 (-.02)
.05 (-.07)
.09 (.05)
A R C
C A R
.04 (-.05)
.03 (0)
R A C
C R A
.12 (.07)
.16 (.02)
C R A
R A C
R C A
.05 (.03)
.07 (.02)
.09 (.02)
52
B C P
.13 (.06)
.07 (.03)
ANES 1992
.05 (.02)
B C P
B C P
B P C
C B P
.14 (.09)
.12 (.-04)
.07 (.04)
B P C
C B P
.03 (-.04)
.02 (0)
P B C
C P B
.16 (.04)
.05 (-.09)
C P B
P B C
P C B
.04 (-.03)
.03 (-.02)
.07 (-.06)
53
.17 (.14)
C D P
.10 (.08)
ANES 1996
.05 (.04)
C D P
C D P
C P D
D C P
.09 (.05)
.27 (.08)
.04 (.02)
C P D
D C P
.02 (-.02)
.02 (0)
P C D
D P C
.15 (-.12)
.03 (-.05)
D P C
P C D
P D C
.02 (-.08)
.01 (-.04)
.03 (-.14)
54
B C M
1988 FNES Communists
B C M
B C M
B M C
.02 (0)
C B M
.02 (.02)
B M C
C B M
.02 (0)
M B C
C M B
.53 (.53)
.02 (0)
C M B
M B C
M C B
.30 (.30)
.09 (.09)
55
-.04
H W N
1968 NES Weak Order Net Probabilities
.03
-.05
H W N
H W N
H N W
-.25
W H N
.26
H N W
W H N
-.05
.05
0
Majority
N H W
W N H
-.26
W N H
N H W
.25
N W H
-.03
.05
.04
56
-.02
H W N
1968 NES Semiorder Net Probabilities
Threshold of 10
.03
-.09
H W N
H W N
HNW
H N
WHN
.23
-.19
0
-.01
W N
0
HW
H N W
W H N
-.10
.10
0
NW
WH
.01
Majority
0
NHW
W N H
0
NH
-.23
W N H
N H W
.19
NWH
-.03
.09
.02
57
0
H W N
1968 NES Semiorder Net Probabilities
Threshold of 54
.02
-.04
H W N
H W N
HNW
H N
0
WHN
0
-.01
W N
.12
HW
-.10
-.19
H N W
W H N
0
.19
NW
WH
.10
-.12
Majority
.01
0
NHW
W N H
NH
W N H
N H W
0
NWH
-.02
.04
0
58
ANES Strict Majority Social Welfare Orders
Year 1968
Threshold 0, , 96
SWO Nixon Humphrey Wallace
59
ANES Strict Majority Social Welfare Orders
Year 1992
Threshold 0, , 99
SWO Clinton Bush Perot
60
HoweverThere is no Theory-FreeMajority
Preference Relation
61
ANES Strict Majority Social Welfare Orders
Year 1980
Threshold 0, , 29 30, , 99
SWO Carter Reagan Anderson Reagan Carter Anderson
62
ANES Strict Majority Social Welfare Orders
Year 1996
Threshold 0, , 49 85, , 99 50, ,84
SWO Clinton Dole Perot Dole Clinton Perot
63
State of the Art Shepsle et al. 1997
64
Shepsle Bonchek (1997)
In general, then, we cannot rely on the method
of majority rule to produce a coherent sense of
what the group wants, especially if there are
no institutional mechanisms for keeping
participation restricted (thereby keeping n
small) or weeding out some of the alternatives
(thereby keeping m small).
65
Drawing Random Samplesfrom Realistic
Distributions
What happens if we interview 20 randomly drawn
voters from the 1996 ANES? Do they display
cyclical majorities? Do they display the correct
majority preference order?
66
From now on Individual Preferences are WEAK
ORDERS over three choice alternatives There are
13 possible weak orders There are 27
different possible majority preference relations
67
.09
D C P
1996 ANES
.04
.05
D C P
D C P
D P C
C D P
.15
.26
D P C
C D P
.02
.10
.02
P D C
C P D
.03
.16
C P D
P D C
P C D
.02
.01
.03
68
n5
Intransitivities
D C P
D C P
D C P
D C P
.01
D C P
D C P
.01
D C P
D C P
.01
69
Correct Majority Ordering
n5
D C P
.10
.03
Intransitivities
.06
D C P
D C P
D C P
D C P
.06
DPC
D P
CDP
.01
.01
C P
DC
D C P
D C P
.29
.01
D P C
C D P
.01
.11
PC
CB
.23
D C P
D C P
PDC
C P D
PD
.01
C P D
P D C
.01
PCD
.01
.01
D C P
D C P
Correct Majority Winner
.01
.01
70
32
n5
3
D C P
.10
.03
Intransitivities
.06
D C P
D C P
D C P
D C P
.06
DPC
D P
CDP
.01
.29
.01
C P
DC
D C P
D C P
.01
D P C
C D P
.01
.11
PC
CB
.23
D C P
D C P
PDC
C P D
PD
.01
C P D
P D C
.01
PCD
.01
.01
D C P
D C P
.01
.01
71
Correct Majority Ordering
n50
D C P
.03
Intransitivities
.01
D C P
D C P
D C P
D C P
DPC
D P
CDP
C P
DC
D C P
D C P
.80
D P C
C D P
.03
PC
CB
.13
D C P
D C P
PDC
C P D
PD
C P D
P D C
PCD
D C P
D C P
Correct Majority Winner
72
Correct Majority Ordering
n101
D C P
Intransitivities
D C P
D C P
D C P
D C P
DPC
D P
CDP
C P
DC
D C P
D C P
.92
D P C
C D P
.01
PC
CB
.06
D C P
D C P
PDC
C P D
PD
C P D
P D C
PCD
D C P
D C P
Correct Majority Winner
73
Correct Majority Ordering
n500
D C P
Intransitivities
D C P
D C P
D C P
D C P
DPC
D P
CDP
C P
1
DC
D C P
D C P
D P C
C D P
PC
CB
D C P
D C P
PDC
C P D
PD
C P D
P D C
PCD
D C P
D C P
Correct Majority Winner
74
(No Transcript)
75
.08
S C F
1976 GNES
S C F
S C F
.37
S F C
C S F
.19
S F C
C S F
F S C
C F S
.28
C F S
F S C
.06
F C S
.02
76
Correct Majority Winner
n3
Correct Majority Ordering
S C F
.11
Intransitivities
S C F
S C F
S C F
S C F
SFC
S F
.01
.01
CSF
.24
.35
S C F
S C F
C F
SC
S F C
C S F
FC
CS
S C F
S C F
FSC
C F S
FS
.22
C F S
F S C
.05
FCS
S C F
S C F
.02
77
Correct Majority Winner
n2000
Correct Majority Ordering
S C F
.81
Intransitivities
.01
S C F
S C F
S C F
S C F
SFC
S F
CSF
.18
S C F
S C F
C F
SC
S F C
C S F
FC
CS
S C F
S C F
FSC
C F S
FS
C F S
F S C
FCS
S C F
S C F
78
(No Transcript)
79
B C M
1988 FNES Communists
0
0
B C M
B C M
0
B M C
C B M
.02
0
.02
B M C
C B M
0
.02
M B C
C M B
.02
C M B
M B C
.53
M C B
0
.30
.09
80
B C M
n3
Intransitivities
B C M
B C M
B C M
B C M
BMC
B M
CBM
C M
BC
B C M
B C M
B M C
C B M
MC
CB
MBC
C M B
B C M
B C M
.77
MB
C M B
M B C
MCB
.16
Correct Majority Winner
.07
B C M
B C M
Correct Majority Ordering
81
B C M
n21
Intransitivities
B C M
B C M
B C M
B C M
BMC
B M
CBM
C M
BC
B C M
B C M
B M C
C B M
MC
CB
MBC
C M B
B C M
B C M
1.0
MB
C M B
M B C
MCB
Correct Majority Winner
B C M
B C M
Correct Majority Ordering
82
Behavioral Social Choice Research

• Merge/Synergize
• Individual and Social Choice
• Normative and Descriptive Models
• Theory and Empirical Data

83
Cycles and classical paradoxes are only one
piece of the puzzle surrounding social choice

• Dont Worry So Much About Cycles!
• Arrow
• Impartial Culture
• Domain Restriction

• Model Dependence
• Theory
• Data

Empirical Congruence among Procedures
84
Cycles and classical paradoxes are only one
piece of the puzzle surrounding social choice

• Statistical Analysis
• Sampling
• Inference

Worry about Majority Misestimation!

• Policy Implications for Design of Voting Methods
• Ergonomics
• Empirical Accuracy / Statistical Confidence
• Empirical Congruence

85
Individual Decision Making incompatible
with Normative Theory (HUGE literature, incl.
Kahneman Tversky)
Conjecture Social Choice in Practice better
behaved than predicted by Normative Theory