Transcript How Can Societies Choose? (part 2)

Social Choice
Session 3
Carmen Pasca and John Hey
Plan for this session
• We start with a review of Arrow’s theorem – using
now Jen’s preferences to give completeness and
showing that this leads to her being the dictator.
• Then, within the same framework (ordinal preferences)
we explore how we may weaken Arrow’s assumptions
and possibly get some possibility.
• After a short break we will invite comments from you
as to other ways to get possibility.
• (Note: in session 6 we explore the implications of
cardinality/measurability of preferences – so we do not
do this today.)
Arrow’s Impossibility Theorem
• In social choice theory, Arrow’s impossibility theorem ...
states that, when voters have three or more discrete
alternatives (options), no voting system can convert the
ranked preferences of individuals into a community-wide
ranking while also meeting a certain set of criteria. These
criteria are called unrestricted domain, non-dictatorship,
Pareto efficiency, and independence of irrelevant
alternatives. (Wikipedia)
• We remind you of Arrow’s famous result and explore more
its implications.
• We again draw inspiration from John Bone at York and
borrow some of his slides.
Arrow
• Arrow’s conditions are these:
• Universal Domain: is applicable to any profile.
• Consistency: produces a complete, transitive ordering
of available alternatives.
• Pareto: if everyone prefers x to y then so should
society.
• Independence of Irrelevant Alternatives: orders x and
y on the basis only of individual preferences on x and y.
• Can we replace/eliminate/weaken any of these?
• This is the question we address today.
Back to session 2 briefly
• We go back to where we had to assume some
society’s preferences in order to get
completeness...
• ...and we take an alternative way.
• We go to slide number 12 of the PowerPoint
presentation of session 2...
• ... after which we once again shamelessly use
John Bone’s thoughts and slides.
Can we plausibly relax Arrow’s conditions?
Universal Domain: is applicable to any profile
Perhaps there are some profiles at which we might not require a social
ordering at all?
Especially for profiles of (e.g. malicious) preferences.
But perhaps also for profiles of welfare rankings.
For example if x and y are different allocations of economic goods, and J
is allocated more of every good under x than under y,
then it might be reasonable to ignore any hypothetical profile in which J
ranks y above x.
Might this be enough to escape the problem?
Can we plausibly relax Arrow’s conditions?
Pareto: if everyone prefers x to y then so should society
Perhaps for similar reasons as with Universal Domain
A Sen, “The impossibility of a Paretian Liberal”,
Journal of Political Economy, vol 78 (1970), 152-7
A single copy of Lady Chatterley’s Lover
a: Ken (Prude) reads it b: Jen (Liberal) reads it c: neither reads it
Jen
Ken
Pareto →
abc
cab
Liberalism →
aPb
bPc
cPa
Ignore “meddlesome” preferences?
(We will return to Sen shortly.)
Can we plausibly relax Arrow’s conditions?
Independence of Irrelevant Alternatives
Whatever determines the relative merits of x and y should not include
individual welfare data regarding z …
... but should instead be intrinsic to x and y.
So at any two profiles in which the individual welfare data regarding (only) x
and y is the same, the social ordering of x and y should also be the same.
This becomes problematic when the individual welfare data comprises
only rankings
that is, comprises only ordinal welfare data
Can we plausibly relax Arrow’s conditions?
Consistency: complete transitive ordering
We may only have to decide which of the available alternatives should be
(socially) chosen.
So why require a complete, transitive social ordering of all available
alternatives ?
This is equivalent to requiring the social choice to be consistent across
different (hypothetical) sets of alternatives
Whatever determines the relative merits of x and y should not include the
availability or otherwise of z …
... but should instead be intrinsic to x and y.
A single copy of Lady Chatterley’s Lover
a: Ken reads it
Jen
abc
Ken
cab
b: Jen reads it
choose:
c: neither reads it
a from {a,b}
b from {b,c}
c from {a,c}
? from {a,b,c}
Whatever is the choice from {a,b,c}, there is inconsistency.
But so what?
Majoritarianism
Choose x from {x,y} iff more people prefer x to y than prefer y to x.
Jen
abc
Ken
cab
Len
bca
choose:
a from {a,b}
b from {b,c}
c from {a,c}
? from {a,b,c}
Whatever is the choice from {a,b,c}, there is inconsistency.
But so what?
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, which we have already seen. 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/liberal) 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.
Now a short break
• After the break we invite you to suggest other ways of
breaking Arrow’s impossibility...
• ...within the ordinal preferences framework.
• Note that we could dispense with the notion of trying
to find society’s preferences...
• ... we could immediately think about a decision rule of
society which depends upon individual preferences....
• ... Like doing something if a majority of the population
want to do it.
• Problems with this? Intransitivity?
Conclusions
• So what do we conclude from all of this?
• Essentially that, whatever way we do the aggregation,
there is no method, that does not have objections, to
aggregate individual preferences into a set of social
preferences.
• Is this surprising?
• What are its implications?
• That we cannot leave a computer in charge of society.
• We need decision-making bodies with flexibility and some
autonomy but also somehow under the control of the
members of society.