EC9A4 Social Choice and Voting Lecture 1

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EC9A4 Social Choice and Voting Lecture 1

• Prof. Francesco Squintani
• f.squintani_at_warwick.ac.uk

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Syllabus

• 1. Social Preference Orders
• May's Theorem
• Arrow Impossibility Theorem (Pref. Orders)
• 2. Social Choice Functions
• Arrow Impossibility Theorem (Choice Funct.)
• Rawlsian Theory of Justice
• Arrow Theory of Justice

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• 3. Single Peaked Preferences
• Black's Theorem
• Downsian Electoral Competition
• 4. Probabilistic Voting and Ideological Parties
• 5. Private Polling and Elections
• Citizen Candidate Models

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• References
• Jehle and Reny Ch. 6
• Mas Colell et al. Ch 21
• Lecture Notes
• D. Bernhardt, J. Duggan and F. Squintani (2009)
  The Case for Responsible Parties, American
  Political Science Review, 103(4) 570-587.

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• D. Bernhardt, J. Duggan and F. Squintani (2009)
  Private Polling in Elections and Voters
  Welfare Journal of Economic Theory, 144(5)
  2021-2056.
• M. Osborne and A. Slivinski (1996) A Model of
  Political Competition with Citizen-Candidates,
  Quarterly Journal of Economics, 111(1) 65-96.
• T. Besley and S. Coate (1997), An Economic Model
  of Representative Democracy, Quarterly Journal
  of Economics, 112 85-114.

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What is Social Choice?

• Normative economics. What is right or wrong, fair
  or unfair for an economic environment.
• Axiomatic approach. Appropriate axioms describing
  efficiency, and fairness are introduced.
  Allocations and rules are derived from axioms.
• Beyond economics, social choice applies to
  politics, and sociology.

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Social Preferences

• Consider a set of social alternatives X, in a
  society of
• N individuals.
• Each individual i, has preferences over X,
  described
• by the binary relation R(i), a subset of X2.
• The notation x R(i) y means that individual i
  weakly
• prefers x to y.
• 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.

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• Indifference relations I(i) are derived from
  R(i)
• x I(i) y corresponds to x R(i) y and y R(i)
  x.
• The relation R(i) is complete for any x, y in X,
  either x R(i) y or y R(i) x, or both.
• The relation R(i) is transitive for any x, y, z
  in X,
• if x R(i) y and y R(i) z, then x R(i) z.
• Social preference relation a complete and
  transitive relation R f (R (1), , R(N)) over
  the set of alternative X, that aggregates the
  preferences R(i) for all i 1,, N, and
  satisfies appropriate efficiency and fairness
  axioms.

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Example Exchange Economy
x12
x21
I1
I2
w
x11
x22
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
xj2
wj1
wj2
10
x12
x21
I1
I2
x
y
w
x11
x22
For individual i, x I(i) y if ( , ) and
( , ) are on the same indifference curve
Ii The relations R(i) and P(i) are described by
the contour sets of the utilities ui
x2i
x2i
y2i
y2i
11
x12
x21
I1
I2
x
w
x11
x22
The line of contracts describes all Pareto
optimal allocations. One possible social
preference is R such that 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 alternatives

• Suppose that there are only two alternatives
• x is the status quo, and y is the alternative.
• Each individual preference R(i) is indexed as
• q in -1, 0, 1, where 1 is a strict preference
  for x.
• The social welfare rule is a functional
• F(q(1), , q(N)) in -1, 0, 1.

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Mays Axioms

• AN The social rule F is anonymous if for every
• permutation p, F(q(1), , q(N))
  F(q(p(1)),q(p(N)))
• NE The social rule F is neutral if F(q) - F (-
  q ).
• PR The rule F is positively responsive if q gt
  q, q q
• and F(q) gt 0 imply that F(q) 1.

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Mays Theorem

• A social welfare rule is majoritarian
• (i.e. F(q) 1 if and only if
• n(q) i q(i) 1 gt n-(q) i q(i) - 1
  ,
• F(q) -1 if and only if n(q) lt n-(q) ,
• F(q) 0 if and only if n(q) n-(q) ),
• if and only if it is neutral, anonymous, and
• positively responsive.

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• Proof. Clearly, majority rule satisfies the 3
  axioms.
• By anonimity, F(q) G(n(q), n-(q)).
• If n(q) n-(q), then n(-q) n-(-q), and so
• F(q) G(n(q), n-(q)) G(n(-q), n-(-q))
• F(-q)-F(q), by NE.
• This implies that F(q)0.
• If n(q) gt n-(q), pick q with qltq and n(q)
  n-(q),
• Because F(q) 0, by PR, it follows that F(q)
  1.
• When n(q) lt n-(q), it follows that n(-q) gt
  n-(-q),
• hence F(-q) 1 and by NE, F(q) -1.

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Transitivity

• Transitivity is apparently a sound axiom.
• But it fails for the majority voting rule.
• Suppose that Xx, y, z, and x P(1) y P(1) z,
• y P(2) z P(2) x and z P(3) x P(3) y.
• Aggregating preferences by the majority voting
  rule yields x P y, y P z, and z P x.
• This is called a Condorcet cycle.

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Arrows Axioms

• U. Unrestricted Domain. The domain of f must
  include all possible (R(1), , R(n)) over X.
• WP. Weak Pareto Principle. For any x, y in X,
• if x P(i) y for all i, then x P y.
• ND. Non-Dictatorship. There is no individual 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 other
  than i.

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IIA. Independence of Irrelevant Alternatives.
Let R f(R(1), , R(N)), and R f(R(1),,
R(N)). For any x,y, if every individual i ranks
x and y in the same way under R(i) and R(i),
then the ranking of x and y must be the same
under R and R. This axiom requires some
comments. In some sense, it requires that each
comparison can be taken without considering the
other alternatives at play. The axiom fails in
some very reasonable voting rules.
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Borda Count

• Suppose that X is a finite set.
• Let Bi(x) y x P(i) y.
• The Borda rule is x R y if and only if
• B1(x) BN(x) gt B1(y) BN(y).
• This rule does not satisfy IIA.
• 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 y
• x P(1) y P(1) z, y P(2) z P(2) x yields y P
  x

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Arrow Impossibility Theorem

• Theorem If there are at least 3 allocations in X,
  then the axioms of Unrestricted Domain, Weak
  Pareto and Independence of Irrelevant
  Alternatives imply the existence of a dictator.
• Corollary There is no social welfare function f
  that satisfies all the Arrow axioms for the
  aggregation of individual preferences.

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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 for
• the ranking (Pn(1), , Pn(N)), i.e. when raising
  c to
• the top of P(i), for all i lt 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 lt n, and
  at the
• bottom of Pn(i) for i gt n, we can change all
• is 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.
  when
• raising c to the top of P(i) for all individuals
  i lt 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, as
• long as the ranking of c (either bottom or top),
  did
• not change.
• Compare Pn1 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 which
• determines whether d is at the top or at the
  bottom
• of 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 proof

• We assume that preferences are continuous.
• Continuity. For any i, x, the sets y y R(i) x
  and
• y x R(i) y are closed.
• Complete, transitive and continuous preferences
  R(i)
• can be represented as continuous utility
  functions ui

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• We aggregate the utility functions u into a
  social
• welfare function V (x) f (u1(x), , uN(x)).
• PI. Pareto indifference. If ui(x) ui(y), for all
  i, then V(x)V(y).
• If V satisfies U, IIA, and P, then there is a
• continuous function W such that
• V(x) gt V(y) if and only if W(u(x)) gt W(u(y)).
• The welfare function depends only on the utility
• ranking, not on how the ranking comes about.

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• The axioms of Arrows theorem require that
• the utility is an ordinal concept (OS) and that
  utility is
• interpersonally noncomparable (NC).
• If ui represents R(i), then so does any
  increasing
• transformation vi (ui). All increasing
  transformations
• vi must be allowed, independently across i.
• The function W aggregates the preferences
• (ui)i1 if and only if the function g(W)
  aggregates
• the preferences (vi(ui))i1, where g is an
  increasing
• transformation.

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• Suppose that N 2
• (when Ngt2 the analysis is a simple extension)
• Pick an arbitrary utility vector u.

u2
I
II
u
III
IV
u1
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• By weak Pareto, W(u) gt W(w) for all utility
  indexes
• w in III, and W(w)gtW(u) for all utility indexes w
  in I.

u2
I
II
u
III
IV
u1
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• Suppose that W (u) gt W (w) for some w in II.
• Applying the transformation v1 (w1) w1 and
• v2 (w2) w2, the OS/NC principle implies that
• W (u) gt W (w).

u2
w
I
II
w
u
III
IV
u1
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• This concludes that for all w in II, either
• W (u)ltW (w), W (u) W (w), or W (u) gt W (w).
• It cannot be that W (u) W (w) because
  transitivity
• would imply W (w) W (w).

u2
w
I
II
w
u
III
IV
u1
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• Suppose that W (u) gt W (w).
• In particular, W (u) gt W (u1-1, u21).
• Consider the transform v1(u1) u11, v2(u2)
  u1-1.
• By the OS/NC principle,
• W (v(u)) gt W (v1 (u1-1), v2(u21)) W (u).

u2
w
I
II
u
v(w)
v(u)
III
IV
u1
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• By repeating the step of quadrant II, the
  transform
• v1 (w1) w1 and v2 (w2) w2, the OS/NC
• principle implies W (u) lt W (w) for all w in
  IV.

u2
I
II
u
w
v(u)
III
IV
u1
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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.

u2
I
II
u
III
IV
u1
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Conclusion

• We have defined the general set up of the social
  choice problem.
• We have shown that majority voting is
  particularly valuable to choose between two
  alternatives.
• We have proved Arrows theorem
• The only transitive complete social rule
  satisfying weak Pareto, IIA and unrestricted
  domain is a dictatorial rule.

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Preview next lecture

• We will extend Arrow theorem to social choice
• functions.
• We will introduce the possibility of
  interpersonal
• utility comparisons.
• We will axiomatize the Rawlsian Theory of Justice
• and the Arrowian Theory of Justice