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EC9A4EC9A4 Social Choice and Social Choice and

VotingVoting Lecture 1Lecture 1

Prof. Francesco SquintaniProf. Francesco Squintani

f.squintani@warwick.ac.uk f.squintani@warwick.ac.uk

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SyllabusSyllabus

1. Social Preference Orders1. Social Preference OrdersMay's TheoremMay's TheoremArrow Impossibility Theorem (Pref. Arrow Impossibility Theorem (Pref. Orders)Orders)

2. Social Choice Functions2. Social Choice FunctionsArrow Impossibility Theorem (Choice Arrow Impossibility Theorem (Choice Funct.)Funct.)Rawlsian Theory of JusticeRawlsian Theory of JusticeArrow Theory of JusticeArrow Theory of Justice

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3. Single Peaked Preferences3. Single Peaked Preferences

Black's TheoremBlack's Theorem

Downsian Electoral CompetitionDownsian Electoral Competition

4. 4. Probabilistic Voting and Ideological Probabilistic Voting and Ideological PartiesParties

5. Private Polling and Elections5. Private Polling and Elections

Citizen Candidate ModelsCitizen Candidate Models

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ReferencesReferences

Jehle and Reny Ch. 6Jehle and Reny Ch. 6 Mas Colell et al. Ch 21Mas Colell et al. Ch 21 Lecture NotesLecture Notes

D. Bernhardt, J. Duggan and F. D. Bernhardt, J. Duggan and F.

Squintani (2009): “The Case for Squintani (2009): “The Case for Responsible Parties”, Responsible Parties”, American Political American Political Science ReviewScience Review, 103(4): 570-587., 103(4): 570-587.

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D. Bernhardt, J. Duggan and F. Squintani D. Bernhardt, J. Duggan and F. Squintani (2009): (2009): “Private Polling in Elections and “Private Polling in Elections and Voters’ Welfare” Voters’ Welfare” Journal of Economic Journal of Economic TheoryTheory, 144(5): 2021-2056., 144(5): 2021-2056.

M. Osborne and A. Slivinski (1996): “A M. Osborne and A. Slivinski (1996): “A Model of Political Competition with Model of Political Competition with Citizen-Candidates,” Citizen-Candidates,” Quarterly Journal of Quarterly Journal of Economics,Economics, 111(1): 65-96. 111(1): 65-96.

T. Besley and S. Coate (1997), “An T. Besley and S. Coate (1997), “An Economic Model of Representative Economic Model of Representative Democracy,” Democracy,” Quarterly Journal of Quarterly Journal of Economics,Economics, 112: 85-114. 112: 85-114.

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What is Social Choice?What is Social Choice?

Normative economics. What is right Normative economics. What is right or wrong, fair or unfair for an or wrong, fair or unfair for an economic environment.economic environment.

Axiomatic approach. Appropriate Axiomatic approach. Appropriate axioms describing efficiency, and axioms describing efficiency, and fairness are introduced. Allocations fairness are introduced. Allocations and rules are derived from axioms. and rules are derived from axioms.

Beyond economics, social choice Beyond economics, social choice applies to politics, and sociology.applies to politics, and sociology.

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Social PreferencesSocial Preferences

Consider a set of social alternatives X, in a Consider a set of social alternatives X, in a society of society of

N individuals.N individuals.Each individual Each individual i,i, has preferences over X, has preferences over X,

describeddescribedby the binary relation R(i), a subset of Xby the binary relation R(i), a subset of X22..The notation `x R(i) y’ means that individual i The notation `x R(i) y’ means that individual i

weakly weakly prefers x to y. prefers x to y. Strict preferences P(i) are derived from R(i): Strict preferences P(i) are derived from R(i): `x P(i) y’ corresponds to `x R(i) y but not y R(i) x.’`x P(i) y’ corresponds to `x R(i) y but not y R(i) x.’

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Indifference relations I(i) are derived from Indifference relations I(i) are derived from R(i): R(i): `x I(i) y’ corresponds to `x R(i) y and y R(i) `x I(i) y’ corresponds to `x R(i) y and y R(i) x.’x.’

The relation R(i) is The relation R(i) is completecomplete: for any x, y in X, : for any x, y in X, either `x R(i) y’ or `y R(i) x’, or both.either `x R(i) y’ or `y R(i) x’, or both.

The relation R(i) is The relation R(i) is transitive: transitive: for any x, y, z in for any x, y, z in X, X, if `x R(i) y’ and `y R(i) z’, then `x R(i) z’.if `x R(i) y’ and `y R(i) z’, then `x R(i) z’.

Social preference relation:Social preference relation: a complete and a complete and transitive relation R = f (R (1), …, R(N)) transitive relation R = f (R (1), …, R(N)) over the set of alternative X, that over the set of alternative X, that aggregates the preferences R(i) for all i = 1,aggregates the preferences R(i) for all i = 1,…, N, and satisfies appropriate efficiency …, N, and satisfies appropriate efficiency and fairness axioms.and fairness axioms.

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Example: Exchange Example: Exchange EconomyEconomy

x22

x11

x12

x21

In the exchange economy with 2 consumers, and 2 goods, x are such that + = + ,where w is the initial endowment, and j is the good.

xj1 xj

2 wj1 wj

2

w

I1

I2

10

x22

x11

x12

x21

w

I1

I2

For individual i, `x I(i) y’ if ( , ) and ( , ) areon the same indifference curve Ii

The relations R(i) and P(i) are described by the contour sets of the utilities ui

x

x2i x2

i y2i y2

i

y

11

x22

x11

x12

x21

w

I1

I2x

The line of contracts describes all Pareto optimalallocations. One possible social preference is R suchthat `x P y’ if and only if x is on the line of contracts,and y is not.

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The case of two The case of two alternativesalternatives

Suppose that there are only two Suppose that there are only two alternatives: alternatives:

x is the status quo, and y is the alternative.x is the status quo, and y is the alternative.

Each individual preference R(i) is indexed Each individual preference R(i) is indexed as as

q in {-1, 0, 1}, where 1 is a strict q in {-1, 0, 1}, where 1 is a strict preference for x.preference for x.

The social welfare rule is a functional The social welfare rule is a functional

F(q(1), …, q(N)) in {-1, 0, 1}.F(q(1), …, q(N)) in {-1, 0, 1}.

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May’s AxiomsMay’s Axioms

AN: The social rule F is AN: The social rule F is anonymousanonymous if for every if for every

permutation p, F(q(1), …, q(N))= F(q(p(1)),…permutation p, F(q(1), …, q(N))= F(q(p(1)),…q(p(N)))q(p(N)))

NE: The social rule F is NE: The social rule F is neutralneutral if F( if F(qq) = - F (- ) = - F (- qq ). ).

PR: The rule F is PR: The rule F is positively responsivepositively responsive if if q q >> q’ q’, , q q == q’ q’

and F(and F(q’q’) ) >> 0 imply that F( 0 imply that F(qq) = 1.) = 1.

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May’s TheoremMay’s Theorem

A social welfare rule is majoritarian A social welfare rule is majoritarian

(i.e. F((i.e. F(qq) =1 if and only if ) =1 if and only if

nn++((qq) ) ==#{i: q(i) = 1} > n#{i: q(i) = 1} > n--((qq) ) ==#{i: #{i: q(i) = - 1} ,q(i) = - 1} ,

F(F(qq) =-1 if and only if n) =-1 if and only if n++((qq) < n) < n--((qq) ,) ,

F(F(qq) =0 if and only if n) =0 if and only if n++((qq) = n) = n--((qq) ),) ),

if and only if it is neutral, anonymous, and if and only if it is neutral, anonymous, and

positively responsive.positively responsive.

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Proof.Proof. Clearly, majority rule satisfies the 3 Clearly, majority rule satisfies the 3 axioms.axioms.

By anonimity, F(By anonimity, F(qq) = G(n) = G(n++((qq), n), n--((qq)). )). If nIf n++((qq) = n) = n--((qq), then n), then n++(-(-qq) = n) = n--(-(-qq), and so:), and so:F(F(qq)= G(n)= G(n++((qq), n), n--((qq))= G(n))= G(n++(-(-qq), n), n--(-(-qq))=))=

=F(-=F(-qq)=-F()=-F(qq), by NE.), by NE.This implies that F(This implies that F(qq)=0.)=0.If nIf n++((qq) > n) > n--((qq), pick ), pick q’ q’ with with q’q’<<qq and n and n++((q’q’) = n) = n--

((q’q’),),Because F(Because F(q’q’) = 0, by PR, it follows that F() = 0, by PR, it follows that F(qq) = 1.) = 1.When nWhen n++((qq) < n) < n--((qq), it follows that n), it follows that n++(-(-qq) > n) > n--(-(-qq), ), hence F(-hence F(-qq) = 1 and by NE, F() = 1 and by NE, F(qq) = -1. ) = -1.

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TransitivityTransitivity

Transitivity is apparently a sound axiom.Transitivity is apparently a sound axiom.

But it fails for the majority voting rule.But it fails for the majority voting rule.

Suppose that X={x, y, z}, and `x P(1) y Suppose that X={x, y, z}, and `x P(1) y P(1) z’, P(1) z’,

`y P(2) z P(2) x’ and `z P(3) x P(3) y’.`y P(2) z P(2) x’ and `z P(3) x P(3) y’.

Aggregating preferences by the majority Aggregating preferences by the majority voting rule yields `x P y’, `y P z’, and voting rule yields `x P y’, `y P z’, and `z P x’.`z P x’.

This is called a Condorcet cycle.This is called a Condorcet cycle.

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Arrow’s Axioms

U. U. Unrestricted Domain.Unrestricted Domain. The domain of f The domain of f must include all possible (R(1), …, R(n)) must include all possible (R(1), …, R(n)) over X.over X.

WP. WP. Weak Pareto Principle. Weak Pareto Principle. For any x, y in X, For any x, y in X, if `x P(i) y’ for all i, then `x P y’.if `x P(i) y’ for all i, then `x P y’.

ND. ND. Non-Dictatorship. Non-Dictatorship. There is no individual There is no individual i such that for all x,y, if `x P(i) y’ i, then `x i such that for all x,y, if `x P(i) y’ i, then `x P y’, regardless of the relations R(j), for j P y’, regardless of the relations R(j), for j other than i.other than i.

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IIA. IIA. Independence of Irrelevant Independence of Irrelevant Alternatives. Alternatives. Let R = f(R(1), …, R(N)), and R’ = Let R = f(R(1), …, R(N)), and R’ = f(R’(1),…, R’(N)).f(R’(1),…, R’(N)).For any x,y, if every individual i ranks x For any x,y, if every individual i ranks x and y in theand y in thesame way under R(i) and R’(i), then the same way under R(i) and R’(i), then the ranking ofranking ofx and y must be the same under R and x and y must be the same under R and R’.R’.

This axiom requires some comments. In This axiom requires some comments. In some sense,some sense,it requires that each comparison can be it requires that each comparison can be taken withouttaken withoutconsidering the other alternatives at considering the other alternatives at play.play.The axiom fails in some very reasonable The axiom fails in some very reasonable voting rules.voting rules.

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Borda CountBorda Count

Suppose that X is a finite set. Suppose that X is a finite set.

Let BLet Bii(x) = #{y : x P(i) y}.(x) = #{y : x P(i) y}.

The Borda rule is: x R y if and only if The Borda rule is: x R y if and only if

BB11(x) + …+ B(x) + …+ BNN(x) (x) >> B B11(y) + …+ B(y) + …+ BNN(y).(y).

This rule does not satisfy IIA.This rule does not satisfy IIA.

Suppose 2 agents and {x,y,z} alternatives.Suppose 2 agents and {x,y,z} alternatives.

x P(1) z P(1) y, y P(2) x P(2) z yields x P yx P(1) z P(1) y, y P(2) x P(2) z yields x P y

x P*(1) y P*(1) z, y P*(2) z P*(2) x yields y x P*(1) y P*(1) z, y P*(2) z P*(2) x yields y P xP x

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TheoremTheorem If there are at least 3 If there are at least 3 allocations in X, then the axioms of allocations in X, then the axioms of Unrestricted Domain, Weak Pareto and Unrestricted Domain, Weak Pareto and Independence of Irrelevant Alternatives Independence of Irrelevant Alternatives imply the existence of a dictator.imply the existence of a dictator.

Corollary Corollary There is no social welfare There is no social welfare function f that satisfies all the Arrow function f that satisfies all the Arrow axioms for the aggregation of individual axioms for the aggregation of individual preferences.preferences.

Arrow Impossibility Theorem

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Proof (Geanakoplos 1996).

Step 1. Consider any social state c. Suppose that

`x P(i) c’ for any x other than c, and for any i.By Weak Pareto, it must be that `x P c’ for all

such x.

Step 2. In order, move c to the top of the ranking of

1, than of 2, all the way to n. Index these orders as (P1(1),…, P1(N)), … (PN(1),…, PN(N)). By WP, as `c PN(i) x’ for all i and x other than c,it must be that `c PN x’ for all x other than c.The allocation c is at the top of the ranking.

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Because c is at the top of the ranking P after raising it

to the top in all individual i’ s ranking P(i), there must

be an individual n such that c raises in P, after raising

c to the top in all rankings P(i) for i smaller or equal

to n. We let this ranking be (Pn(1),… , Pn(N))

We now show that c is raised to the top of Pn forthe ranking (Pn(1), …, Pn(N)), i.e. when raising c tothe top of P(i), for all i < n.

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By contradiction, say that `a Pn c’ and `c Pn b.’ Because c is at the top of Pn(i) for i < n, and at thebottom of Pn(i) for i > n, we can change alli‘s preferences to P*(i) so that `b P*(i) a’. By WP, `b P* a’. By IIA, `a P* c’ and `c P* b.’By transitivity `a P* b’, which is a contradiction.

This concludes that c is at the top of Pn, i.e. whenraising c to the top of P(i) for all individuals i < n.

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Step 3. Consider any a,b different from c. Change the

preferences Pn to P* such that `a P*(n) c P*(n) b’, and for any other i, a and b are ranked in any way,

aslong as the ranking of c (either bottom or top), didnot change. Compare Pn+1 to P*: by IIA, `a P* c’.Compare Pn-1 to P*: by IIA, `c P* b’.By transitivity, `a P* b’, for all a, b other than c.Because a,b are arbitrary, we have: if `a P*(n) b’,

then`a P* b’. n is a dictator for all a, b other than c.

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Step 4. Repeat all previous steps with state a, playing the role of c.Because, again, it is the ranking Pn of n whichdetermines whether d is at the top or at the

bottomof the social ranking, we can reapply step 3.Again, we have: if `c P*(n) b’, then `c P* b’. n is a dictator for all c, b other than a.Because n is a dictator for all a, b other than

c, and n is a dictator for all c, b other than a, we obtain that n is a dictator for all states.

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A diagrammatic proofA diagrammatic proof

We assume that preferences are continuous.We assume that preferences are continuous.

Continuity. For any i, x, the sets {y: y R(i) Continuity. For any i, x, the sets {y: y R(i) x} andx} and

{y: x R(i) y} are closed.{y: x R(i) y} are closed.

Complete, transitive and continuous Complete, transitive and continuous preferences R(i)preferences R(i)

can be represented as continuous utility can be represented as continuous utility functions ufunctions uii

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We aggregate the utility functions u into a We aggregate the utility functions u into a socialsocial

welfare function V (x) = f (uwelfare function V (x) = f (u11(x), …, u(x), …, uNN(x)).(x)).

PI. PI. Pareto indifference. Pareto indifference. If uIf uii(x)= u(x)= uii(y), for all i, (y), for all i, then V(x)=V(y).then V(x)=V(y).

If V satisfies U, IIA, and P, then there is aIf V satisfies U, IIA, and P, then there is a

continuous function W such that:continuous function W such that:

V(x) V(x) >> V(y) if and only if W( V(y) if and only if W(uu(x)) (x)) >> W( W(uu(y)).(y)).

The welfare function depends only on the The welfare function depends only on the utility utility

ranking, not on how the ranking comes about.ranking, not on how the ranking comes about.

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The axioms of Arrow’s theorem require thatThe axioms of Arrow’s theorem require that

the utility is an the utility is an ordinal concept (OS) ordinal concept (OS) and that and that utility isutility is

interpersonally interpersonally noncomparable (NC).noncomparable (NC).

If uIf uii represents R(i), then so does any increasing represents R(i), then so does any increasing

transformation vtransformation vi i (u(uii). All increasing ). All increasing transformations transformations

vvi i must be allowed, independently across i.must be allowed, independently across i.

The function W aggregates the preferencesThe function W aggregates the preferences

(u(uii))i=1i=1 if and only if the function g(W) aggregates if and only if the function g(W) aggregates

the preferences (vthe preferences (vii(u(uii))))i=1i=1, where g is an , where g is an increasing increasing

transformation.transformation.

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Suppose that N =2 Suppose that N =2

(when N>2 the analysis is a simple (when N>2 the analysis is a simple extension)extension)

Pick an arbitrary utility vector u.Pick an arbitrary utility vector u.

u

III

III IV

u1

u2

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By weak Pareto, W(By weak Pareto, W(u) > W(w) for all utility indexes

w in III, and W(W(w)>W(u) for all utility indexes w in I.

u

III

III IV

u1

u2

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Suppose that W (u) > W (w) for some w in II.

Applying the transformation vv1 1 (w11) = w*11 and

vv2 2 (w22) = w*22, the OS/NC principle implies that

W (u) > W (w*).u

III

III IV

u1

u2 w*w

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This concludes that for all w in II, either W (u)<W (w), W (u) = W (w), or W (u)

> W (w).It cannot be that W (u) = W (w)

because transitivitywould imply W (w*) = W (w).

u

III

III IV

u1

u2 w*w

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Suppose that W (u) > W (w). In particular, W (u) > W (u1-1, u2+1).

Consider the transform v1(u1) = u1+1, v2(u2) = u1-1.

By the OS/NC principle, W (v(u)) > W (v1 (u1-1), v2(u2+1)) = W

(u). u

III

III IVu1

u2 w

v(u)v(w)

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By repeating the step of quadrant II, the transform

vv1 1 (w11) = w*11 and vv2 2 (w22) = w*22, the OS/NC

principle implies W (u) < W (w*) for all w* in IV.

u

III

III IVu1

u2

w*v(u)

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The indifference curves are either horizontal or

vertical. Hence there must be a dictator.In the figure, agent 1 is the dictator.

u

III

III IVu1

u2

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ConclusionConclusionWe have defined the general set up of the We have defined the general set up of the

social choice problem.social choice problem.

We have shown that majority voting is We have shown that majority voting is particularly valuable to choose between particularly valuable to choose between two alternatives.two alternatives.

We have proved Arrow’s theorem: We have proved Arrow’s theorem:

The only transitive complete social rule The only transitive complete social rule satisfying weak Pareto, IIA and satisfying weak Pareto, IIA and unrestricted domain is a dictatorial unrestricted domain is a dictatorial rule.rule.

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Preview next lecturePreview next lecture

We will extend Arrow’ theorem to social We will extend Arrow’ theorem to social choicechoice

functions.functions.

We will introduce the possibility of We will introduce the possibility of interpersonalinterpersonal

utility comparisons.utility comparisons.

We will axiomatize the Rawlsian Theory We will axiomatize the Rawlsian Theory of Justiceof Justice

and the Arrowian Theory of Justiceand the Arrowian Theory of Justice