PhD Course
Foundations of Social Choice
Lecture 5
Francesco Squintani
University of Warwick
email: [email protected]
1
2.) The Median Voter Theorem
Literature:
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Peter C. Ordeshook,
Game Theory and Political Theory,
Chapter 4
The Model
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2 candidates A and B run for office.
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One-dimensional issue in the election campaign: policy space
X ⊂ R.
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Example: The weighting of military expenditures versus social
spendings.
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We assume that voters have single-peaked preferences
concerning this issue.
Single-peaked Preferences
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Single-peaked preferences:
Consider a (linearly) ordered policy space X . The rational
preference order R is single peaked with respect to the linear
order on X , if there is an alternative x ∈ X with the property that
for any y, z ∈ X ,
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◮
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if x ≥ z > y, then z ≻ y,
if y > z ≥ x, then y ≻ z.
Consider policy space X ⊂ R.
Single-peaked preferences imply that for each voter i there is an
xi such that for all y, z ∈ X ,
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if xi ≥ z > y, then z ≻i y,
if y > z ≥ xi , then y ≻i z.
Conditions - I
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All elective citizens actually vote.
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All voters know the candidates’ positions in the policy space.
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Party affiliation, sympathy etc. do not matter. Voters decide only
on the basis of the candidates’ position in the policy space.
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Both candidates have the same strategic possibilities, i.e. no
candidate is constrained to a “left” or “right” position.
Conditions - II
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The candidates try to maximize their votes. Candidates have
utility +1/0/-1 in the case of victory/tie/defeat.
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More generally, they have payoff functions ΠA (ZA ) and ΠB (ZB ),
where ZA and ZB are the percentages of vote (with ZA + ZB = 1).
ΠA and ΠB are strictly monotonically increasing and normally not
continuous at ZA = 12 , ZB = 12 .
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Minimal requirement for the information of the candidates:
They know the ideal points of the voters and that their
preferences are single-peaked.
The Median Voter Theorem
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Definition: The ideal point of the median voter is a point xm ,
such that at least half of the voters have their ideal point at xm , or
left of xm , and such that at least half of the voters have their ideal
point at xm , or right of xm .
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If the number of voters is odd and if preference curves have no
flat spots, then xm will be unique. Otherwise, xm will be a region.
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Theorem (Downs, 1957): In a 2-candidate-majority-vote
decision with only 1 decisive issue and voters who have
single-peaked preferences concerning this issue, both
candidates will occupy the ideal point of the median voter.
Notation
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N: Set of voters, N = (1, ...n) with n odd.
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x1 , ..., xn are the ideal points of the voters.
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xm is the ideal point of the median voter.
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The candidates A and B choose positions PA and PB .
Proof - I
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We want to show that both candidates choose xm , i.e., {xm , xm } is
the unique Nash-equilibrium of the game.
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Existence of the Nash-equilibrium?
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Both candidates obtain 2n votes in the expected value at the
point {xm , xm }.
Assumption: Candidate A chooses PA = xm .
′
′
If candidate B chooses PB with PB < xm , he will obtain less
n
than 2 votes.
⇒ B is not elected.
′′
′′
If candidate B chooses PB with PB > xm , he will obtain less
n
than 2 votes.
⇒ B is not elected.
In xm , B has an election probability of 12 , in all other points
he is not elected. Thus, B has no incentive to deviate from
xm .
⇒ {xm , xm } are mutual best strategies, i.e., {xm , xm } is a
Nash-equilibrium.
Proof - II
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Uniqueness of the Nash-equilibrium? (Could there exist a
Nash-equilibrium {PA , PB } with PA 6= xm , PB 6= xm ?)
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Case I: PA < PB (or PA > PB )
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Case II: PA = PB 6= xm
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If B has more than n2 votes ⇒ PA is not a best response, as A
could win more votes than n2 by PA = xm .
If A has more than n2 votes ⇒ PB is not a best response, as B
could win more votes than n2 by PB = xm .
If both candidates have n2 expected votes ⇒ PA and PB are
no best responses. E.g., B can obtain more than 2n votes by
PB = xm .
Both candidates expect n2 votes.
PA and PB are no best responses. E.g., B can obtain more
than 2n votes by PB = xm .
⇒ The Nash-equilibrium {xm , xm } is unique.
q.e.d.
An Alternative Formulation of the Theorem
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If all voters have single-peaked preferences concerning a
one-dimensional issue,
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then the preference of the median voter will be chosen in
each pairwise election (i.e. it is the Condorcet-winner)
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and the social preference order is transitive with the ideal
policy of the median voter in the first place.
In the median voter theorem, Arrow’s axiom of unrestricted
domain (A2) is violated by the assumption of single-peaked
preferences.
Relevance of the Theorem
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The theorem states that both candidates choose the ideal point
of the median voter.
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In reality there is a tendency to move towards the center,
however, not entirely.
Candidates often choose different positions in crucial issues.
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Thus, some assumptions of the theorem seem to be critical.
Critical Assumptions of the Theorem
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The assumption that voters have complete information about the
positions of the candidates may not be realistic.
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In reality, there are often multi-dimensional problems.
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The assumption of single-peaked preferences is critical.
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In which cases do we have single-peaked preferences, and
in which cases do we have preference profiles with more
than one peak?
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An empirical answer is difficult, but the tendency is as
follows: There will be single-peaked preferences if voters
use a common criterion to valuate alternatives. If voters use
several criteria, there will be preferences with more than
one peak.
Further Criticism to the Theorem I
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Note, however, that assumption of single peaked preferences
can be “weakened” somewhat by assuming single-crossing
property
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Let αi denote the specific features of voter i (i.e. idiosyncratic
preferences, socio-economic attributes...)
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Suppose that parameter αi is unidimensional
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Def: Consider an ordered policy space X and order voters
according to their α′i s. Then the preferences of voters satisfy the
single-crossing property over the policy space X when the
following statement is true:
if x > x′ and α′i > αi or x < x′ and α′i < αi
then x ≻αi x′ implies x ≻α′i x′ .
Single Crossing vs Single Peakedness I
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Single-crossing property does not imply single-peaked
preferences
1 x≻y≻z
2 x≻z≻y
3 z≻y≻x
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These preferences are not single peaked, but satisfy single
crossing.
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Natural ordering x < y < z:
αi = 2 :z ≻ y
⇒ αi′ = 3 : z ≻ y
αi = 2 :x ≻ z
x≻y
⇒ αi′ = 1 : x ≻ z
⇒ αi′ = 1 : x ≻ y
Single Crossing vs Single Peakedness II
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Following preferences are single-peaked but do not satisfy
single-crossing property.
1 w≻x≻y≻z
2 x≻y≻z≻w
3 y≻x≻w≻z
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Order policies: w < x < y < z
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Order individuals 3 > 2 > 1.
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Single crossing requires that z ≻2 w should imply z ≻3 w, which
is not the case.
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Similar contradiction with different ordering of individuals.
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⇒ Single-peakedness does not imply single-crossing property.
Further Criticism to the Theorem II
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Differences in competence and media skills do not matter in the
model.
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In reality, the incumbent often has an advantage in the election.
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The support during the election campaign may be larger if the
incumbent has a left/right position. This might be an incentive to
deviate from the median voter’s position.
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There is no equilibrium in pure strategies if there are three
politicians!
4.) Multi-dimensional Policy Spaces
Problems with Multi-dimensional Policy Spaces
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Many policy problems multi-dimensional: Public good project
involves financing, exact location (of bridge, or military bases, or
disposal site for nuclear waste), maybe ethical considerations,
ideology...
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First check: Is the political conflict really multidimensional?
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Sometimes multidimensional policy can be projected on a
unidimensional space (often via budget constraint) in which
voters can be ordered by type.
⇒ Intermediate preferences (Grandmont, 1978).
Multi-dimensional Policy Spaces
Intermediate Preferences
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Let q be an n-dimensional policy vector.
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Def.: Voters have intermediate preferences, if their (indirect)
utility function W(q; αi ) can be written as
W(q; αi ) = J(q) + K(αi )H(q),
where K(αi ) is monotonic in αi , for any H(q) and J(q) common
to all voters.
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Let q(αm ) be the policy preferred by the median value of αi in set
of voters.
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Prop: If voters have intermediate preferences, a Condorcet
Winner exists and is given by q(αm ).
Multi-dimensional Policy Spaces
McKelvey’s Theorem
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Can we make predictions of voting outcome for genuine
multi-dimensional policy conflict?
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⇒ McKelvey’s “Chaos Theorem”
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Define Top Cycle: Set of all alternatives a’s such that for each
b 6= a there is a sequence c1 , ..., cK such that
a = c1 ≻ c2 ≻ c3 ... ≻ cK = b.
(1)
where ≻ means that a majority prefers (votes) for c2 over c3 .
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McKelvey: In a multi-dimensional policy space X = Rm with
utility monotonically decreasing in Euclidean distance from each
voter’s ideal point, if there is no Condorcet Winner then the Top
Cycle of the majority rule system is the set of all alternatives.
Multi-dimensional Policy Spaces
McKelvey’s Theorem – Example I
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Divide the dollar game with n = 3:
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Set of alternatives X = {x = (x1 , x2 , x3 ) ≥ 0 with x1 + x2 + x3 = 1}
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Voter i prefers larger xi to smaller
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Top Cycle: TC = X \ {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
Multi-dimensional Policy Spaces
McKelvey’s Theorem – Example II
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If all entries in x greater 0 then majority voting implies
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x ≻ (0, ε, 1 − ε) ≻ (1/2, 0, 1/2) for some ε and similarly for
(1/2, 1/2, 0) and (0, 1/2, 1/2).
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Every alternative other than the following three: (1/2, 0, 1/2),
(1/2, 1/2, 0), (0, 1/2, 1/2), is beaten by one of these three
[(1/2, 0, 1/2), (1/2, 1/2, 0), (0, 1/2, 1/2)]
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If exactly two entries of x are positive, then that beats some
interior point, which then indirectly beats all others (by first point).
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⇒ With finite (closed) agenda, agenda-setter can determine
outcome.
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But what if voters know that they vote on a sequence? ⇒
Incentives for strategic voting. (Recall Gibbard-Satterthwaite)
Multi-dimensional Policy Spaces – Median in all
directions
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Suppose agents preferences can be represented by U(q − αi ),
where αi is vector describing i’s bliss point in this policy space.
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U decreasing and concave in the distance ||q − αi ||.
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Condorcet Winner in multidimensional policy spaces exists if
there is a median in all directions
(Plott, 1967; Davis, de Groot, and Hinich, 1972).
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Median in all direction requires strong symmetry assumptions on
the distribution of individual bliss points.
Multi-dimensional Policy Spaces – Sophisticated
Voting
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If agenda is known solve voting game by backward induction
⇒ Shepsle-Weingast (1984) - Algorithm: Voting under
successive elimination.
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Banks Set: includes all alternatives in X that can survive
successive elimination. I.e. subsumes all alternatives that can
be the outcome under sophisticated/strategic voting.
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Banks Set smaller than Top-cycle: Set of potential outcomes
under sophisticated voting smaller than under sincere voting.
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However, Banks Set can still be very large.
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Further reduction of Banks set possible when a particular status
quo is given.