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Behavioral Social Choice: Probabilistic 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) - PowerPoint PPT Presentation
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Behavioral Social Choice:Behavioral Social Choice:Probabilistic Models, Statistical Inference, Probabilistic Models, Statistical Inference,
and Applicationsand Applications
Michel RegenwetterMichel RegenwetterQuantitative DivisionQuantitative Division
Department of PsychologyDepartment of Psychology(& Department of Political Science)(& Department of Political Science)
University of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign
Cambridge University Press, 2006Cambridge University Press, 2006 with B. Grofman, A.A.J. Marley, I. Tsetlinwith B. Grofman, A.A.J. Marley, I. Tsetlin
OverviewOverview Motivation: Motivation: 2 Conceptual Distinctions in the Decision Sciences2 Conceptual Distinctions in the Decision Sciences
Criteria for a Unified Theory of Decision MakingCriteria for a Unified Theory of Decision Making Social Choice Theory:Social Choice Theory:
Majority Rule (Condorcet Criterion), ArrowMajority Rule (Condorcet Criterion), Arrow
The Obsession with Majority CyclesThe Obsession with Majority CyclesNeed for Behavioral Social Choice ResearchNeed for Behavioral Social Choice Research
Behavioral Social Choice ResearchBehavioral Social Choice ResearchMultiple Representations of Preference/UtilityMultiple Representations of Preference/Utility
A General Concept of Majority RuleA General Concept of Majority Rule
Model Dependence of Majority OutcomesModel Dependence of Majority Outcomes
The Problem of Incorrect Majority OutcomesThe Problem of Incorrect Majority Outcomes
2 Conceptual Distinctions in the Decision Sciences
IndividualJudgment
and Decision Making
Social Choice
NormativeTheory
DescriptiveTheory & Data
Behavioral Decision Research
2 Conceptual Distinctions in the Decision Sciences
IndividualJudgment
and Decision Making
Social Choice
NormativeTheory
DescriptiveTheory & Data
??????
2 Conceptual Distinctions in the Decision Sciences
IndividualJudgment
and Decision Making
Social Choice
NormativeTheory
DescriptiveTheory & Data
???
Criteria for a Unified Theory of Decision Making
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
(Inspired by Luce and Suppes, Handbook of Mathematical Psych.,1965)
OverviewOverview Motivation: Motivation: 2 Conceptual Distinctions in the Decision Sciences2 Conceptual Distinctions in the Decision Sciences Criteria for a Unified Theory of Decision MakingCriteria for a Unified Theory of Decision Making
Social Choice Theory:Social Choice Theory:Majority Rule (Condorcet Criterion), ArrowMajority Rule (Condorcet Criterion), Arrow
The Obsession with Majority CyclesThe Obsession with Majority CyclesNeed for Behavioral Social Choice ResearchNeed for Behavioral Social Choice Research
Behavioral Social Choice ResearchBehavioral Social Choice ResearchMultiple Representations of Preference/UtilityMultiple Representations of Preference/Utility
A General Concept of Majority RuleA General Concept of Majority Rule
Model Dependence of Majority OutcomesModel Dependence of Majority Outcomes
The Problem of Incorrect Majority OutcomesThe Problem of Incorrect Majority Outcomes
Majority rule:(CondorcetCriterion)
Majority Winner• Candidate who is ranked ahead of any other
candidate by more than 50%• Candidate who beats any other candidate
in pairwise competition
Kenneth Arrow’s (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?
The Obsession with Cycles
Majority Cycles
ABC 1 person
BCA 1 person
CAB 1 person
Majority Cycles
ABC 1 person
BCA 1 person
CAB 1 person
A beats B 2 times
B beats A 1 time
A is majority preferred to B
Majority Cycles
ABC 1 person
BCA 1 person
CAB 1 person
B beats C 2 times
C beats B 1 time
A is majority preferred to B
B is majority preferred to C
Majority Cycles
ABC 1 person
BCA 1 person
CAB 1 person
A beats C 1 time
C beats A 2 times
A is majority preferred to B
B is majority preferred to C
C is majority preferred to A
Majority Cycles
A is majority preferred to B
B is majority preferred to C
C is majority preferred to A
ABC 1 person
BCA 1 person
CAB 1 person
DemocraticDemocraticDecision Decision MakingMaking
at Risk!?!at Risk!?!
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
GIGO?
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).”
$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.
Where is the evidence for cycles?
• 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
Majority Winner• Candidate who is ranked ahead of any other candidate by more than 50%
• Candidate who beats any other candidate in pairwise competition
Data are incomplete!!
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.
OverviewOverview Motivation: Motivation: 2 Conceptual Distinctions in the Decision Sciences2 Conceptual Distinctions in the Decision Sciences Criteria for a Unified Theory of Decision MakingCriteria for a Unified Theory of Decision Making
Social Choice Theory:Social Choice Theory: Majority Rule (Condorcet Criterion), ArrowMajority Rule (Condorcet Criterion), Arrow The Obsession with Majority CyclesThe Obsession with Majority Cycles Need for Behavioral Social Choice ResearchNeed for Behavioral Social Choice Research
Behavioral Social Choice ResearchBehavioral Social Choice ResearchMultiple Representations of Preference/UtilityMultiple Representations of Preference/Utility
A General Concept of Majority RuleA General Concept of Majority Rule
Model Dependence of Majority OutcomesModel Dependence of Majority Outcomes
The Problem of Incorrect Majority OutcomesThe Problem of Incorrect Majority Outcomes
Binary Preference Relations A binary relation R on a set of objects C is a
collection of ordered pairs of elements of C
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, transitiveAsymmetric Binary Relations
a|b|c|d
42|5|2|1
Real Representationof Linear Orders
one-to-one is where
)()(),(
u
buauBba
B
a b|c|d
7 7|3|1
Real Representationof Weak Orders
)()(),( buauBba
B
a
b c | d
3
1.9 1.8 | 1.8
Real Representationof Semiorders
1 e.g.,
)()(
)()(),(
buau
buauBba
1B
Variable/Uncertain Preferences:Probability Distribution
on Binary Relations
Variable/Uncertain Utilities:Jointly Distributed Family of
Utility Random Variables(Random Utilities)(parametric or nonparametric)
Random Utility Representations
Bji
Bji
PBP
ji
ji
),(|
and
),(|
)(
UU
UU
Linear Orders Weak Orders
)(
0)(With
ji
P ji
UU(Block and Marschak, 1960, chapter)
Random Utility Representations
Bji
Bji
PBP
ji
ji
),(|
and
),(|
)(
UL
UL
Semiorders Interval Orders
)()(With ii LU
(Heyer & Niederee, 1992, MSS)(Niederee & Heyer, 1997, Luce vol.)(Regenwetter, 1997, JMP)(Regenwetter & Marley, 2002, JMP)
A General Definitionof Majority Rule
'),(),()'()(
ifonly and if
o
relations,binary ofset any on
)(
]1,0[:
ondistributiy probabilit aGiven
BabBbaBPBP
bt preferredmajority strictly isa
BPB
P
B
B
For Utility Functions or Random Utility Models choose a Random Utility Representation
and obtain a consistent Definition
'),(),()'()(
ifonly and if
o
relations,binary ofset any on
)(
]1,0[:
ondistributiy probabilit aGiven
BabBbaBPBP
bt preferredmajority strictly isa
BPB
P
B
B
A General Definitionof Majority Rule
)()()()(
topreferredmajority
iujuNjuiuN
ji
)10()10(
topreferredmajority
ijji PP
ji
UUUU
Examples:
)()( Freq. Rel.)()( Freq. Rel. iujujuiu
$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.
Let’s analyze National Survey Data!1968, 1980, 1992, 1996 ANES
Feeling Thermometer Ratingstranslated into
Weak Orders or Semiorders
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
1968 NESWeak OrderProbabilities
.03
0
.02
.03
.06
.01
.07
.05
.27
.08
.32
.04
.02
No impartial culture!
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
1968 NESWeak OrderProbabilities
.03
0
.02
.03
.06
.01
.07
.05
.27
.08
.32
.04
.02
No impartial culture!
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
a versus b
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
a versus c
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
b versus c
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
Sen’s NB(a)
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
Net NB(a)
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
GeneralizedNet NB(a)
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
Sen’s NM(a)
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
ca
ba
ac
Net NM(a)
cb
bc
abc
cab
acb
bac
bca
cba
b ca
ab c
a cb
ca b
a bc
ba c
ab
cb
ca
ba
bc
ac
GeneralizedNet NM(a)
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
.03 (-.04)
0 (-.05)
.02 (.-25)
.03 (-.05)
.06(-.26)
.01 (-.03)
.07 (.04)
.05 (.05)
.27 (.25)
.08 (.05)
.32 (.26)
.04 (.03)
.02 (0)
ANES 1968
NP N W
H -.05 .55
N .65
ACR
RAC
ARC
CAR
CRA
RCA
C RA
AC R
A RC
RA C
A CR
CA R
.07 (-.02)
.05 (-.02)
.14(-.02)
.09(.05)
.12(.07)
.05 (.03)
.09 (.02)
.07 (.02)
.16 (.02)
.04 (-.05)
.05 (-.07)
.02 (-.03)
.03 (0)
ANES 1980
NP C R
A -.15 -.16
C .06
BCP
PBC
BPC
CBP
CPB
PCB
C PB
BC P
B PC
PB C
B CP
CB P
.12 (.-04)
.03 (-.04)
.05 (-.09)
.03 (-.02)
.07 (-.06)
.04 (-.03)
.16(.04)
.07(.04)
.14(.09)
.05 (.02)
.13 (.06).07 (.03)
.02 (0)
ANES 1992
NP C P
B -.08 .16
C .25
CDP
PCD
CPD
DCP
DPC
PDC
D PC
CD P
C PD
PC D
C DP
DC P
.17 (.14)
.05 (.04)
.09(.05)
.04(.02)
.15(-.12)
.02 (-.08)
.03 (-.14)
.01 (-.04)
.03 (-.05)
.02 (-.02)
.27 (.08)
.10 (.08)
.02 (0)
ANES 1996
NP D P
C .23 .39
D .13
BCM
MBC
BMC
CBM
CMB
MCB
C MB
BC M
B MC
MB C
B CM
CB M
.02(0)
.09 (.09)
.30(.30)
.53(.53)
.02(.02)
.02(0)
.02 (0)
1988 FNES: Communists
NP C M
B .46 -.92
C -.94
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
1968 NESWeak Order
Net Probabilities
-.04
-.05
-.25
-.05
-.26
-.03
.04
.05
.25
.05
.26
.03
0
Majority
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
HW
NW
NH
WH
WN
HN
1968 NESSemiorder
Net Probabilities
-.02
-.09
-.19
-.10
-.23
-.03
.02
.09
.19
.10
.23
.03
.01
-.01
Thresholdof 10
0
0
0
00Majority
HWN
NHW
HNW
WHN
WNH
NWH
W NH
HW N
H NW
NH W
H WN
WH N
HW
NW
NH
WH
WN
HN
1968 NESSemiorder
Net Probabilities
Thresholdof 54
.02 -.04
0
0
-.19
0
-.02
0
.04
0
.19
0 -.01
-.10
-.12.01.10
.12
0
Majority
Year
1968
Threshold
0, …, 96
SWONixon
HumphreyWallace
ANES Strict Majority Social Welfare Orders
Year
1992
Threshold
0, …, 99
SWOClintonBushPerot
ANES Strict Majority Social Welfare Orders
However:There is no Theory-Free
Majority Preference Relation
Year
1980
Threshold
0, …, 29
30, …, 99
SWOCarterReagan
Anderson
ReaganCarter
Anderson
ANES Strict Majority Social Welfare Orders
Year
1996
Threshold
0, …, 4985, …, 99
50, …,84
SWO
ClintonDolePerot
DoleClintonPerot
ANES Strict Majority Social Welfare Orders
Probability of a Cycle: Pr(m, n)Based on Sampling from a Uniform Distribution on Linear Orders
("Impartial Culture")*
number of voters (n)number of
alternatives(m)
3 5 7 9 11 limit
3 .056 .069 .075 .078 .080 .0884 .111 .139 .150 .156 .160 .1765 .160 .200 .215 .2516 .202 .315
limit 1.00 1.00 1.00 1.00 1.00 1.00
*Source: Riker (1982: 122) as reproduced in Shepsle and Bonchek (1997: Table 4.1, 54)
State of the Art: Shepsle et al. 1997
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).”
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?
From now onIndividual Preferencesare WEAK ORDERS
over three choice alternatives
There are 13 possible weak ordersThere are 27 different
possible majority preference relations
DCP
PDC
DPC
CDP
CPD
PCD
C PD
DC P
D PC
PD C
D CP
CD P
.09
.05
.26
.10
.16
.02
.03
.01
.03
.02
.15
.04
.02
1996 ANES
n=5
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
.01
.01
.01
Intransitivities
DCP
PDC
DPC
CDP
CPD
PCD
C PD
DC P
D PC
PD C
D CP
CD P
DC
PC
PD
CB
CP
DP
n=5
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
.01
.01
.01.01
.01
.23
.11
.29
.06
.10.03
.06
.01
.01
.01.01
.01
Correct Majority Ordering
Correct Majority Winner
Intransitivities
DCP
PDC
DPC
PCD
C PD
DC P
D PC
PD C
D CP
DC
PC
PD
CB
CP
DP
n=5
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
.01
.01
.01.01
.01
.06
.10.03
.06
.01
.01
.01.01
.01
Intransitivities
3%32%
CDP
CPD
CD P
.23
.11
.29
DCP
PDC
DPC
CDP
CPD
PCD
C PD
DC P
D PC
PD C
D CP
CD P
DC
PC
PD
CB
CP
DP
n=50
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
.13
.03
.80
.01
.03Correct Majority Ordering
Correct Majority Winner
Intransitivities
DCP
PDC
DPC
CDP
CPD
PCD
C PD
DC P
D PC
PD C
D CP
CD P
DC
PC
PD
CB
CP
DP
n=101
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
.06
.01
.92
Correct Majority Ordering
Correct Majority Winner
Intransitivities
DCP
PDC
DPC
CDP
CPD
PCD
C PD
DC P
D PC
PD C
D CP
CD P
DC
PC
PD
CB
CP
DP
n=500
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
D
C P
1
Correct Majority Ordering
Correct Majority Winner
Intransitivities
1996 ANES
0
0.2
0.4
0.6
0.8
1
1 10 100 1000 10000
sample size (log scale)
prop
ortio
n
proportion transitive
proportion of correctmajority winners
proportion of correctmajority orderings
SCF
FSC
SFC
CSF
CFS
FCS
C FS
SC F
S FC
FS C
S CF
CS F
.08
.19
.28
.02
.06
.37
1976 GNES
SCF
FSC
SFC
CSF
CFS
FCS
C FS
SC F
S FC
FS C
S CF
CS F
SC
FC
FS
CS
CF
SF
n=3
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F.02
.22
.24
.11
.35
.05
.01 .01
Intransitivities
Correct Majority OrderingCorrect
Majority
Winner
SCF
FSC
SFC
CSF
CFS
FCS
C FS
SC F
S FC
FS C
S CF
CS F
SC
FC
FS
CS
CF
SF
n=2000
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F
S
C F
.18
.81Intransitivities
Correct Majority OrderingCorrect
Majority
Winner
.01
1976 Germany
0.00
0.20
0.40
0.60
0.80
1.00
1 10 100 1000 10000
sample size (log scale)
prop
ortio
n
proportiontransitive
proportion ofcorrect majoritywinners
proportion ofcorrect majorityorderings
BCM
MBC
BMC
CBM
CMB
MCB
C MB
BC M
B MC
MB C
B CM
CB M
0
0
0
0
.02
0
.09
.30
.53
.02
.02
0
.02
1988 FNES: Communists
BCM
MBC
BMC
CBM
CMB
MCB
C MB
BC M
B MC
MB C
B CM
CB M
BC
MC
MB
CB
CM
BM
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
Correct Majority Ordering
Correct Majority Winner
Intransitivities
n=3
.77
.16 .07
BCM
MBC
BMC
CBM
CMB
MCB
C MB
BC M
B MC
MB C
B CM
CB M
BC
MC
MB
CB
CM
BM
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
B
C M
Correct Majority Ordering
Correct Majority Winner
Intransitivities
n=21
1.0
Merge/Synergize • Individual and Social Choice• Normative and Descriptive Models• Theory and Empirical Data
Behavioral SocialChoiceResearch:
Cycles and classical paradoxesare only one piece of the puzzle surrounding social choice
Model Dependence• Theory
• Data
Don’t Worry So Much About Cycles!• Arrow
• Impartial Culture• Domain Restriction
Empirical Congruence among Procedures
Statistical Analysis:•Sampling • Inference
Policy Implications for Design of Voting Methods• Ergonomics
• Empirical Accuracy / Statistical Confidence•Empirical Congruence
Worry about Majority Misestimation!
Cycles and classical paradoxesare only one piece of the puzzle surrounding social choice
Individual Decision Makingincompatible with
Normative Theory (HUGE literature, incl. Kahneman & Tversky)
Conjecture:Social Choice in Practice
better behaved than predicted byNormative Theory
