Lecture II: Social Choice

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Lecture II Social Choice Spatial Models
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Recap

• Basic assumptions of rational choice
• transitivity, completeness, no inter-personal
  comparison, optimize
• Expected (Von Neuman-Morgenstern) Utility
• Obtain continuous utility functions by thinking
  about choice over lotteries
• How to describe preferences utility functions
• Convexity, diminishing marginal returns, risk,
  discounting, indifference curves, constraints

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Decision Theory

• How individuals make choice under uncertainty.
• Equate expected marginal utility under different
  courses of action
• Take other actors actions as exogenous

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Decision Theory
Leader i
Unilateral Disarmament
Military Build-up
War
War
Peace
Peace
q
1-p
p
1-q
3
4
1
2
5
Decision Theory
Leader i
Unilateral Disarmament
Military Build-up
War
War
Peace
Peace
1-q
q
1-p
p
3
4
1
2
Expected Payoff Build-up 3(1-p) 2p 3 p
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Decision Theory
Leader i
Unilateral Disarmament
Military Build-up
War
War
Peace
Peace
q
1-p
p
1-q
3
4
1
2
Expected Payoff Disarm 4(1-q) q 4 3q
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Decision Theory
Leader i
Unilateral Disarmament
Military Build-up
War
War
Peace
Peace
q
1-p
p
1-q
3
4
1
2
Build up iff 3 p gt 4 3q or (equivalently) 1
gt 3q - p
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Decision Theory

• Build-up iff 1 gt 3q - p

1
2/3
Disarm
q
1/3
Build-up
0
1
0
p
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Modeling Interaction

• More interested in interaction, strategic or
  otherwise
• How will j react to is build-up?
• How will legislative representatives vote over
  guns butter?
• Game
• Players (utility functions, level of information)
• Rules (set of feasible moves distribution of
  information)
• Strategies (plan action over set of all feasible
  actions)
• Outcomes (payoffs)

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Equilibrium Identification

• Equilibrium a self-enforcing situation
• Existence
• Uniqueness
• Stability
• Comparative Statics
• Equilibrium concept a rule that players can use
  to obtain best outcome given rules of the game

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Equilibrium Identification

• Cooperative assume players can make binding
  coalitions
• Core
• set of feasible allocations (of payoffs) that
  cannot be improved upon
• Pareto efficient
• Non-Cooperative binding coalitions not assumed
• Nash equilibrium
• a best response to a best response, i.e., no
  unilateral change in strategy
• Need not be Pareto efficient

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Pareto Efficiency

• What does Pareto efficiency mean?
• A distribution / allocation is Pareto efficient
  is it does not leave any actor worse off relative
  to the status quo.
• A Pareto improvement is an allocation that makes
  one or more actors better off without making any
  other actor worse off.

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Pareto Efficiency

• Consider intra-party election strategy in
  multi-member electoral systems.
• Assume that within any one district, a party can
  recruit a finite number of volunteer campaign
  workers (say 100 workers).
• Also, assume that a candidates chance of wining
  improves in the number of volunteers working on
  his / her campaign
• How might a partys N 2 candidates share the
  fixed supply of campaign workers?

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Pareto Efficiency
The Pareto Frontier
100
Pareto Improvements
N workers for Candidate B
A Pareto Inefficient distribution
0
100
N workers for Candidate A
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Equilibrium Identification

• Absent structure (e.g., restrictions on
  preferences, exogenous agendas, institutional
  rules), the core of most voting games is empty.
• A Nash Equilibrium always exists, at least in
  mixed strategies.

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Cycling and Social Choice

• Social (i.e., collective) choices may be cyclic
• Amend 2 gt SQ gt Amend 1 gt Amend 2
• Unpredictable and / or dependant on agenda
• Is this a general result?
• Yes Arrows (Impossibility) Theorem

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Arrows (Impossibility) Theorem

• Arrows minimal conditions (CUPID)
• Collective Rationality transitive complete
• Unrestricted (Universal) Domain no restrictions
  on preferences
• Pareto Principle respect unanimity, i.e., if
  xiPyi ? i ? S ? xSPyS
• Independence of Irrelevant Alternatives binary
  preference relations unaffected by other options.
  (This means that only ordering matters.)
• Non-Dictatorship no one individual allowed to
  determine social choice independent of others
  preferences.

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Why are CUPID conditions minimal?
Instructor must translate students exam
performances into an overall ranking
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Why are CUPID conditions minimal?
Collective Rationality unrestricted domain
performance must depend on and only on all exam
results, cant pick and choose results or play
favourites
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Why are CUPID conditions minimal?
Pareto Principle if x gets top mark on every
exam, then x should be top-ranked student.
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Why are CUPID conditions minimal?
IIA If z, drops out, xs ranking relative to y
should not be affected x should not be ranked
above y just because x did better on exams than
every student other than y.
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Why are CUPID conditions minimal?
Non-dictatorship performance on a single exam
cannot determine the grade distribution
irrespective of how students do on the other
exams.
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Arrows (Impossibility) Theorem

• An extended example / illustration
• 100 factory workers voting on a wage offer
• accept
• work to rule
• strike
• Foreman is granted veto power by workers over W
  S (i.e., foreman is decisive
• if wPFs ? wPs

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Arrows (Impossibility) Theorem

• Say that wPFsPFa (though wPFaPFs works too)
• The other workers are more militant prefer a
  strike, i.e.
• sPiwPia
• sPiaPiw
• sPiwIia
• But F is decisive over w s, so we get wPs
• By Pareto, we must have sPa
• By Transitivity, we must also have wPa
• Thus Fs decisiveness over w s, implies
  decisiveness over w a

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Arrows (Impossibility) Theorem

• Can we avoid decisiveness?
• Assume majority rule, so 51 of 100 workers
  decide, and that
• 1 worker aPiwPis
• 50 workers sPjaPjw
• 49 workers wPksPka
• Groups 1 2 imply (i.e., are decisive for) aPw
• sPw inadmissible by the majority rule
• Social choice must have wPs or wIs
• Transitivity demands aPwPs or aPwPs
• But only 1 worker prefers a to s!

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Arrows (Impossibility) Theorem

• Let Q1-6 (majority) be decisive for ranking
  students x y x gt y
• How should we rank y z? Equally? y gt z z gt
  y on 5 exams
• Transitivity then implies x gt z, but z gt x on 9
  exams.
• We have made Question 1 decisive over x z
• If we dont, we violate transitivity.

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Avoiding Cycling

• If preferences are not single-peaked, outcome
  hinges on agenda
• If aggregation device is agenda-free, we get
  cycling

MPc
Utility
MPa
MPb
Defence
SQ
-

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Avoiding Cycling

• Black (1948) if preferences are single-peaked
  a core exists
• Core set of unbeatable (? stable) alternatives
• In one dimension, the core is the median voters
  ideal point

MPc
MPb
MPa
Utility
Defence
SQ
-

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Majority Rule in Multiple Dimensions
Indifference Curve
x
Education
Utility increasing in proximity, i.e., Euclidean
preferences
Health Care
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Majority Rule in Multiple Dimensions
Education
SQ
Health Care
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Majority Rule in Multiple Dimensions
Winset of SQ all points that defeat SQ in
pairwise competition
Education
SQ
A core has an empty winset
Health Care
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Majority Rule in Multiple Dimensions
Education
SQ
Contract curve
Health Care
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Majority Rule in Multiple Dimensions
Education
x1
Health Care
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Majority Rule in Multiple Dimensions
x2
Education
Health Care
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Majority Rule in Multiple Dimensions
x2
Education
Health Care
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Majority Rule in Multiple Dimensions
Education
x3
Health Care
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Majority Rule in Multiple Dimensions
Education
x3
Health Care
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Majority Rule in Multiple Dimensions

• Assembly has cycled back to x1
• No core in ?m !

Education
x1
Health Care
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McKelveys Chaos Theorem

• Chronic condition, indeed chaotic
• Voting can start with any point, and lead to
  any point

Education
x1
Health Care
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McKelveys Chaos Theorem

• Credible coalitions?
• Heresthetics
• Agenda manipulation

Education
x1
Health Care
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The Puzzle of Stability

• Parliaments not typically chaotic
• Institutional rules provide structure
• Structure Induced Equilibria (Shepsle 1979)
• Agenda Control
• Discipline
• Committee jurisdictions

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Structure Induced Equilibria I

• McCubbins, Noll, Weingast (1987)
• Rules on establishment of agencies generate
  drift gridlock

P
S
H
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Structure Induced Equilibria I

• Consider SQ outside Pareto set
• Unanimous agreement to establish new agency
  inside Pareto set

SQ
P
S
Winset of SQ under unanimity
H
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Structure Induced Equilibria I

• Once agency is set up, it unanimity winset is
  empty, i.e., P, S or H cannot agree on
  alternative policy for the agency
• Agency can now move policy as it pleases within
  Pareto set
• Variation on common agency problem (Dixit)

SQ
P
S
H
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Structure Induced Equilibria II

• Laver Shepsle (1990, 1994, 1995) put forward a
  portfolio allocation or lattice approach
• Coalitions off lattice points, e.g., AC, are not
  self-enforcing.

C
Policy Dimension 2
A
B
Policy Dimension 1
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Structure Induced Equilibria II

• Ceding monopoly control over one portfolio to
  coalition partners minister delivers credibility
• Implies that only coalitions on lattice
  intersections are credible
• Tus, here, only cabinets with A B are feasible

C
Policy Dimension 2
A
B
Policy Dimension 1