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EC9A4EC9A4 Social Choice and Social Choice and

VotingVoting Lecture 3Lecture 3

Prof. Francesco SquintaniProf. Francesco Squintani

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

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Summary from previous Summary from previous lectureslectures

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

choice problem.choice problem.

We have shown that majority voting is We have shown that majority voting is particularlyparticularly

valuable to choose between two alternatives.valuable to choose between two alternatives.

We have proved Arrow’s theorem: the onlyWe have proved Arrow’s theorem: the only

transitive complete social rule satisfying weaktransitive complete social rule satisfying weak

Pareto, IIA and unrestricted domain is a Pareto, IIA and unrestricted domain is a

dictatorial rule.dictatorial rule.

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We have extended Arrow’ theorem to social We have extended Arrow’ theorem to social

choice functions.choice functions.

We have introduced the possibility ofWe have introduced the possibility of

interpersonal comparisons of utility.interpersonal comparisons of utility.

We have described different concept of We have described different concept of social social

welfare: the utilitarian Arrowian welfare: the utilitarian Arrowian representation representation

and the maximin Rawlsian representation.and the maximin Rawlsian representation.

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Preview of the lecturePreview of the lecture

We will introduce single-peaked utilities.We will introduce single-peaked utilities.

We will prove Black’s theorem: Majority We will prove Black’s theorem: Majority voting isvoting is

socially fair when utilities are single-peaked.socially fair when utilities are single-peaked.

We will prove Median Voter Convergence in We will prove Median Voter Convergence in thethe

Downsian model of elections.Downsian model of elections.

We will introduce the probabilistic voting We will introduce the probabilistic voting model.model.

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Single Peaked Single Peaked Preferences Preferences

A binary relation A binary relation >> on the set X is a linear order on the set X is a linear orderon X if it on X if it reflexivereflexive (i.e. x (i.e. x >> x for all x), x for all x), transitive transitive andandtotal total (for any x,y in X, either x (for any x,y in X, either x >> y or y y or y >> x, not x, not

both).both).

The preference R(i) is single peaked wrt the linearThe preference R(i) is single peaked wrt the linearorder order >> if there is x such that if x if there is x such that if x>>z>y, then z P(i) z>y, then z P(i)

y;y;and if y’>z’and if y’>z’>>x, then z’ P(i) y’x, then z’ P(i) y’

The alternative x is the peak of satisfaction relative The alternative x is the peak of satisfaction relative to the linear order to the linear order >>

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x

u

y z x z’ y’The single peak of this preference profile is x.

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x

u

This preference profile does not have a single peak.

y x z z’ y’

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The complete and reflexive R is The complete and reflexive R is acyclicacyclic if for if for

every subset X’ of X, there is a maximal every subset X’ of X, there is a maximal elementelement

{x in X’ : x R y for all y in X’} is nonempty.{x in X’ : x R y for all y in X’} is nonempty.

Every transitive relation R is acyclic. Every transitive relation R is acyclic.

The Condorcet paradox violates acyclicity.The Condorcet paradox violates acyclicity.

There are rules that violate transitivity but There are rules that violate transitivity but not not

acyclicity.acyclicity.

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An oligarchy is a set agents S, subset of N, An oligarchy is a set agents S, subset of N, suchsuch

that x P y if and only if for every i in S, x P(i) y.that x P y if and only if for every i in S, x P(i) y.

Whenever #S>1, this is not transitive.Whenever #S>1, this is not transitive.

Example: `z P(1) x,’ `P(1) y’ and `y P(2) z P(2) Example: `z P(1) x,’ `P(1) y’ and `y P(2) z P(2) x.’x.’

We obtain that `x R y’, `y R z,’ and `z P x’.We obtain that `x R y’, `y R z,’ and `z P x’.

The oligarchic rule is acyclic because there is The oligarchic rule is acyclic because there is

always a set of allocations for which there is always a set of allocations for which there is no no

unanimous better allocation in S.unanimous better allocation in S.

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Consider the majoritarian social rule:Consider the majoritarian social rule:

x R y if n(x,y) x R y if n(x,y) == #{i : x R(i) y} #{i : x R(i) y} >> N/2. N/2.

This rule induces a complete relation, This rule induces a complete relation, but we havebut we have

seen that the relation may be seen that the relation may be intransitive.intransitive.

In the domain of single-peaked In the domain of single-peaked preference, we willpreference, we will

show that it is acyclic.show that it is acyclic.

Denote by x*(i) the peak for agent i.Denote by x*(i) the peak for agent i.

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Agent n is the median agent for the single Agent n is the median agent for the single peakedpeaked

profile of preferences (R(1), …, R(N)) if profile of preferences (R(1), …, R(N)) if

#{i : x*(i)#{i : x*(i)>>x*(n)}x*(n)}>> N/2; #{i : x*(n) N/2; #{i : x*(n)>>x*(i)}x*(i)}>>N/2. N/2.

Black’s TheoremBlack’s Theorem: If : If >> is a linear order and R(i) is is a linear order and R(i) is

single peaked wrt single peaked wrt >> for all i, then the peak x*(n) for all i, then the peak x*(n)

cannot be defeated by majority by any cannot be defeated by majority by any alternative.alternative.

The peak x*(n) is called the Condorcet winner, The peak x*(n) is called the Condorcet winner, andand

it makes majority voting rule acyclic.it makes majority voting rule acyclic.

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Proof.Proof. Take any y < x*(n). Take any y < x*(n). Consider any agent i with peak x*(i) Consider any agent i with peak x*(i) >> x*(n). x*(n). Because X is linearly ordered by Because X is linearly ordered by >>, ,

x*(i)x*(i)>>x*(n)> y.x*(n)> y.Because R(i) is single peaked with respect to Because R(i) is single peaked with respect to >>, , we obtain that x*(n) R(i) y for all such agents i.we obtain that x*(n) R(i) y for all such agents i.By definition of median voter,By definition of median voter,#{i : x*(i) #{i : x*(i) >> x*(n)} x*(n)}>> N/2; N/2;hence, x*(n) cannot be rejected by majority hence, x*(n) cannot be rejected by majority

vote.vote.

The case for y > x*(n) is analogous.The case for y > x*(n) is analogous.

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x

u

x*(1) x*(2) x*(3) x*(4) x*(5)

An example with n odd.

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x

u

x*(1) x*(2) x*(3) x*(4)

An example with n even.

All the allocations between x*(2) and x*(3) cannot be rejected by majority voting.

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The key feature of the above example is the The key feature of the above example is the

unidimensionality of the set of allocations.unidimensionality of the set of allocations.

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The space of alternative is the unit square X = [0,1]2

There are 3 agents, withpreferences:u1(x) = - 2x1 - x2

u2(x) = x1 + 2x2

u3(x) = x1 – x2

x1

x2

u1

u2

u3

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If x1=0, then agents 1 and 3 prefer y = (1/2, x2) to x.

0 x1

x2

u2

u3

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0

If x2=1, then agents 1 and 3 prefer y = (x1, 1/2) to x.

x1

x2

u1

u3

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0 x1

x2

u1

u2

If x1>0 and x2<1, then agents 1 and 2 prefer y = (x1- a, x2 + a) to x.

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Voting ModelsVoting Models

Voting models are non-cooperative games that Voting models are non-cooperative games that

model elections.model elections.

They can study one-shot elections, or repeated They can study one-shot elections, or repeated

elections. There may be 2 or more candidates.elections. There may be 2 or more candidates.

Candidates’ strategic decisions may include Candidates’ strategic decisions may include whether whether

and when to run in the election, with which and when to run in the election, with which promised promised

policy platform, campaign spending, and so on.policy platform, campaign spending, and so on.

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Downsian ElectionsDownsian Elections There are 2 candidates, i=1,2. Candidates care only about winning

the election. Candidates i=1,2 simultaneously

commit to policies xi if elected. Policies are real numbers.

There is a continuum of voters, with diverse ideologies k, with cumulative distribution F.

The utility of a voter with ideology k if policy x is implemented is u(x,k)= L(|x-k|), with L’<0.

After candidates choose platforms, each citizen votes, and candidate with the most votes wins. If x1 = x2 , then the election is tied.

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For any ideology distribution F, let the median

policy m be such that 1/2 of voters' ideologies y > m & 1/2 of ideologies y < m : F (m) = 1/2.

Median Voter Theorem: The unique Nash Median Voter Theorem: The unique Nash

Equilibrium of the Downsian Election model Equilibrium of the Downsian Election model is is

such that candidates i choose such that candidates i choose xi = m,

and tie the election.

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Proof. Fix any x1 = x2. Because L’<0, each voter

with ideology k votes for the candidate i that

minimizes |xi-k|. Hence, if xi < xj, candidate i’s

vote share is F(½ ( x1 + x2 ) ), and candidate j’s

is 1- F(½ ( x1 + x2 ) ).

For either candidate i, if xi = m, then candidate j’s

best response is BRj = {xj : |xj - m|< |xi - m|}.

Candidate j wins the election.

Hence, there cannot be any Nash equilibrium

where either candidate i plays xi = m.

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Suppose both candidates play x1 = x2

= m, then allvoters are indifferent between x1 and

x2, and the

election is tied. If either candidate i deviates and plays

xi = m, then

she loses the election.

So, there is unique Nash equilibrium: x1 = x2 = m.

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Downsian Elections with Downsian Elections with Ideological Candidates Ideological Candidates

Suppose that candidate i’s ideology is kSuppose that candidate i’s ideology is ki , with

kk1 < m < k2. . The utility of candidate i if policy x is

implemented is u(x, kki)= L(|x- kki|), with L’<0.

Theorem: The unique Nash Equilibrium is Theorem: The unique Nash Equilibrium is

such that candidates i choose such that candidates i choose xi = m, and tie.

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Proof. Proof. Again, for any x1 = x2, if xi < xj, candidate

i’s vote share is F(½ ( x1 + x2 ) ), and candidate j’s

is 1- F(½ ( x1 + x2 ) ).

Suppose that x1 < m, then candidate 2 wins and implement x2 by choosing x2 in (x1, 2m - x1).

Hence, if x1 < 2m- k2, then BR2 (x1)={k2}, and if 2m- k2 < x1 < m, then BR2 (x1) is empty.

If m < x1 < k2, then BR2 (x1)=[x1,infinity).

If x1 > k2, then , then BR2 (x1)={k2}.

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The best response of candidate 1 is symmetric.

Hence, we conclude that there cannot be any Nash

Equilibrium with x1 = m or x2 = m.

Suppose that candidate i chooses xi = m. Then, regardless of the choice xj, the

implemented policy is m. Hence, BRj (xi)=(-infinity, infinity).

We conclude that the unique Nash Equilibrium is

such that x1 = x2 = m, and the election is tied.

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Probabilistic VotingProbabilistic VotingSuppose that candidates do not know

the voter’s preferences. The median policy m is

randomly distributed, with cdf G. Let M be the

median of G.

Theorem. If candidates care only about winning

the election, then the unique Nash Equilibrium is

such that both candidates i choose candidates i choose xi = M, and tie.

Theorem. If candidates are ideological, with

kk1 < M < k2, then there is no Nash Equilibrium

such that both candidates i choose candidates i choose xi = M.

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Proofs. For any x1 = x2, candidate i wins the

election if and only if |xi - m|<|xj - m|.

If xi < xj, then candidate i wins with probability

Pr(½ ( x1 + x2 ) > m) = G (½ ( x1 + x2 ) ), and

Pr(j wins) = 1 - G (½ ( x1 + x2 ) ).

The probability that i wins increases in xi

on (-infinity, xj), decreases in xi on (xj, infinity),

and is discontinuous at xi = xj unless xi = xj =m.

Hence, if candidates only care about winning,

there is a unique NE: x1 = x2 = M.

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If instead candidates are ideological, there cannot

be any equilibrium such that x1 = x2 = M.

Suppose in fact that x2 = M.

If candidate 1 plays x1 = M, then the implemented

policy is M with probability 1.

If 1 plays any x1 in (k1, M), she increases her

expected utility, because x1 is implemented with

positive probability, and she prefers x1 to M.

Also, note that a Nash Equilibrium exists in this

game, by “standard” general existence results.

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ConclusionConclusion

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We have introduced single-peaked We have introduced single-peaked utilities.utilities.

We have proved Black’s theorem: We have proved Black’s theorem: Majority voting Majority voting

is socially fair when utilities are is socially fair when utilities are single-peaked.single-peaked.

We have proved Median Voter We have proved Median Voter Convergence in theConvergence in the

Downsian model of elections.Downsian model of elections.

We have shown that Median We have shown that Median Convergence persists Convergence persists

in the presence of either ideological in the presence of either ideological parties or parties or

uncertainty on voters’preferences.uncertainty on voters’preferences.

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Preview of the next Preview of the next lecturelecture

We will introduce ideology in the We will introduce ideology in the probabilistic probabilistic

voting model, modelling `responsible voting model, modelling `responsible parties’.parties’.

We will derive the equilibrium in the We will derive the equilibrium in the model with model with

responsible parties uncertain of the responsible parties uncertain of the voters’ voters’

preferences.preferences.

We will prove that responsible parties We will prove that responsible parties yield higher yield higher

welfare to the electorate than welfare to the electorate than opportunistic parties.opportunistic parties.