Social Choice
Lecture 16
John Hey
More Possibility/Impossibility Theorems
• Sen’s Impossibility of a Paretian
Liberal
• Gibbard’s theory of alienable
rights
• Manipulability
• Gibbard-Satterthwaite theorem
Sen’s Impossibility of a Paretian Liberal
• The idea is that each individual has the right to
determine things ‘locally’ – that is, those things that
concern only him or her. So individuals are decisive over
local issues.
• For example, I should be free to choose whether or not I
read Lady Chatterley’s Lover.
• He gives a nice example. Three alternatives, a, b and c,
and two people A and B.
• a: Mr A (the prude) reads the book;
• b: Mr B (the lascivious) reads the book;
• c: Neither reads the book.
Lady C.
• Mr A (the Prude): c > a > b
• Mr B (the Lascivious): a > b > c
• Now assume that Mr A is decisive over (a,c) and
that Mr B is decisive over (b,c).
• So from A’s preferences c > a and from B’s
preferences b > c. From unanimity a > b.
• Hence we have
b > c (B) and c > a (A) and a > b (unanimity)!
• WEIRD! (Intransitive).
Sen’s Theorem
• Condition U (Unrestricted domain): The domain of the
collective choice rule includes all possible individual
orderings.
• Condition P( Weak Pareto): For any x, y in X, if every
member of society strictly prefers x to y, then xPy.
• Condition L* (Liberalism): For each individual i, there is
at least one pair of personal alternatives (x,y) in X such
that individual i is decisive both ways in the social choice
process.
• Theorem: There is no social decision function
that satisfies conditions U, P and L*.
Proof
• P indicates Society’s preference and Pi that of individual i.
• Suppose i is decisive over (x,y) and that j is decisive
over (z,w). Assume that these two pairs have no element
in common.
• Let us suppose that xPiy, zPjw, and, for both k=i,j that
wPkx and yPkz.
• From Condition L* we obtain xPy and zPw.
• From Condition P we obtain wPx and yPz.
• Hence it follows that
• xPy
yPz
zPw and wPx.
• Cyclical.
Gibbard’s Theory of Alienable Rights
• Background...
• Going back to the Lady C example, Mr A may realise
that maintaining his right to decisiveness over (a,c) leads
to an impasse/intransitivity.
• He cannot get c (his preferred option) because Mr B has
rights over that and renouncing his right to decisiveness
over (a,c), society will end up with a (which is preferred
by Mr A to b – his least preferred).
• (Might Mr B think similarly (mutatis mutandis) and give
up his right to decisiveness?)
Gibbard’s own example
• Three persons: Angelina, Edwin and the ‘judge’.
• Angelina prefers marrying Edwin but would marry the judge.
• Edwin prefers to remain single, but would prefer to marry
Angelina rather than see her marry the judge.
• Judge is happy with whatever Angelina wants.
• Three alternatives:
• x: Edwin and Angelina get married
• y: Angelina and the judge marry (Edwin stays single)
• z: All three remain single
• Angelina has preference: x PA y PA z
• Edwin has preference: z PE x PE y
The problem and its solution
•
•
•
•
•
Angelina has a libertarian claim over the pair (y,z).
Edwin has a claim over (z,x).
Edwin and Angelina are unanimous in preferring x to y.
So we have a preference cycle: yPz, zPx, xPy.
If Edwin exercises his right to remain single, then
Angelina might end up married to the judge, which is
Edwin’s least preferred option.
• ‘Therefore’ it will be in Edwin’s own advantage to waive
his right over (z,x) in favour of the Pareto preference
xPy.
Gibbard’s Theory of Alienable Rights
• Condition GL: Individuals have the right to waive their
rights.
• Gibbard’s rights-waiving solution: There exists a
collective choice rule that satisfies conditions U, P and
GL.
• The central role of the waiver is to break a cycle
whenever there is one ...
• ... but the informational demands are high.
Manipulation
• Suppose there are three people in society A, B and C
and three propositions a, b and c.
• A‘s preferences: a > b > c
• B‘s preferences: b > c > a
• C‘s preferences: c > a >b
• There is clearly a problem with choosing by majority rule:
a majority (A and C) prefer a to b, a majority (A and B)
prefer b to c and a majority (B and C) prefer c to a.
• Suppose however that we organise the voting in stages:
first between two alternatives and then between the
winner of the first stage and the third alternative.
Depends who chooses the order
• A proposes a first vote between b and c; and then between
the winner of that and a. Which will win? If no strategic voting,
clearly a - A’s preferred option.
• B proposes a first vote between a and c; and then between
the winner of that and b. Which will win? If no strategic voting,
clearly b - B’s preferred option.
• C proposes a first vote between a and b; and then between
the winner of that and c. Which will win? If no strategic voting,
clearly c - C’s preferred option.
• So the person who chooses the order can manipulate the
voting to get what he/she wants.
• But what happens if people vote strategically...
Strategic Voting
• A proposes a first vote between b and c; and then between
the winner of that and a. Which will win? If no strategic
voting, clearly a - A’s preferred option.
• But suppose B realises this and hence knows that his least
preferred option is going to win, then at the first stage B will
vote for c thus ensuring that c will win at the second stage.
• C is very happy to go along with this, but A is clearly not (c
is A’s least preferred). Can A do anything about it?
• A can propose that voting is first over (a,c) and then over
the winner of that and b. With strategic voting by C then a
will win. But A relies on C to vote strategically!
The Gibbard-Satterthwaite Theorem
• Result (a): If there are at least three alternatives and if
the social choice function h is Pareto Efficient and
monotonic, then it is dictatorial. (Very similar proof to that
of Arrow presented in the lectures.)
• Result (b): If h is strategy-proof and onto, then h is
Pareto efficient and monotonic.
• Theorem: Let h is a social choice function on an
unrestricted domain of strict linear preferences. It the
range of h contains at least three alternatives and h is
onto and strategy proof, then h is dictatorial.
• (onto: every element of choice set is chosen for some profile.)
Conclusions
• It seems difficult to avoid dictatorship...
• ... even with individuals protected by their own rights.
• Manipulation is a serious problem, but can be selfdefeating. (Also requires information about
motives/intentions/behaviour.)
• Giving people the right to waive their rights simplifies in
some senses and complicates in others, but does not
remove the fundamental problem of information.
• But if preferences differ, it seems inevitable that conflicts
exist, and that politicians do also.