Social Choice
Session
6
Carmen Pasca and John Hey
Plan for today
•
Remember that we are trying to find/express Society’s preferences.
•
They come from individuals in society. So far we have shown that
aggregating individual
ordinal
preferences is impossible.
•
We have shown that getting agreement on
principles
is impossible.
•
Today we are going to talk about aggregating individual
cardinal/measurable
preferences. Is this possible?
•
First we need to talk about
existence
of cardinal preferences.
•
We do this through Neumann

Morgenstern utility theory which
seems to provide cardinality.
•
Then we talk about comparability.
•
We finish looking at
Nash Bargaining Theory.
•
We conclude…?
Cardinal preferences
•
So far we have maintained the assumption that preferences
are
ordinal
–
just the
ordering
can be expressed.
•
This appears to have seriously constrained what we can do.
•
Surely we can measure
intensity
of preferences? And how
increases in goods/consumption increases happiness?
•
Neumann

Morgenstern utility theory seems to let us do this.
•
So we can cardinally measure the utility of individuals.
•
The next tricky question is whether we can
compare
the
happiness of different individuals. If we can, fine..
•
…if not, perhaps we can borrow results from Nash Bargaining
theory…
Our framework and assumptions
•
Denote Society’s utility/happiness by
U
.
•
Individuals in society receive money income
x
.
•
Denote
i
’s utility by
u
i
= u
i
(x
i
).
•
This could just be
u
i
(x
i
) = x
i
but we keep it general.
•
Then society’s happiness is some function of the
u
i
.
•
We could have
U = (u
1
+ … + u
I
)
Egalitarian.
•
Or
U = min(u
1
, … ,u
I
)
Rawlsian.
•
Or
U = max[(u
1

u
d
)(u
2

u
d
)]
Nash special case
•
Or…
•
To make this operative the
u
i
must be cardinal/measurable.
•
(We concentrate on this and ignore in this session the issue of the
form
of the function.)
Notice the advantages of this
•
If society has a sum of money to distribute, then it can be
done optimally through maximisation of Society’s welfare
function.
•
We just have to agree on the form of the function.
•
Here we have assumed that the happiness of the members
of society depends only on their money income…
•
… but if this method works it can be generalised…
•
…so that, for example, the happiness depends on goods
and services consumed (this is not a big generalisation) and
also depends on the consumption of goods and services by
others
in society.
Neumann

Morgenstern utility theory
•
Is essentially a theory of decision

making under risk…
•
…but does lead to a
cardinal
utility function.
•
We constrain ourselves to
money
outcomes (as above).
•
Fix end

points
B
and
W (B>W)
–
the
B
est and
W
orst.
•
Put
u(W) = 0
and
u(B) = 1
.
•
Like temperature
–
centigrade freezing is 0
0
boiling 100
0
.
•
Let
X
be some amount of money between the best and the
worst
–
W < X < B.
•
We are going to find
u(X).
•
We can do this for
any
X
–
so we can find the utility function.
•
Note that this is individual specific.
How do we find utility values?
•
Easy peasy!
•
Ask “what must the probability
p
be
to make you indifferent
between getting
X
with certainty and playing out a lottery
which gives you
B
with probability
p
and
W
with probability
(
1

p)
?”
X
for sure
indifferent with
B
W
p
1

p
So?
•
Put
u(B)=
1
and
u(W)=
0
.
•
Then the utility of the left equals expected utility of the right:
•
u(X) = pu(B) + (
1

p)u(W) = p*
1
+ (
1

p)*
0
= p
•
We have found the utility of
X
!!! Precisely
p
.
•
Note that we can do this for any
X
(between
B
and
W
).
•
Clever?!
X
for sure
indifferent with
B
W
p
1

p
Example with
W=
€
0
,
B=
€
1000
and
X=
€
500
•
Put
u(
€
1000
)=
1
and
u(
€
0
)=
0
.
•
Then the utility of the left equals expected utility of the right:
•
u(
€
500
) =
0.75
u(
€
1000
) +
0.25
u(
€
0
) =
0.75
*
1
+
0.25
*
0
=
0.75
•
We have found the utility of
€
500
for an individual with the
above preferences.
•
Clever?!
€
500
for sure
indifferent with
€
1000
€
0
0.75
0.25
Implications
•
Using this method we can find the utility of any amount of
money between
W
and
B
for any individual.
•
The shape of the function is individual specific.
•
It reflects the attitude to risk of the individual.
•
[Ask yourself: what is the form if the individual is
risk

neutral
(that is does not care/worry about risk)?]
•
It is
cardinal
.
•
It depends upon the end points
W
and
B
.
•
If they change the function changes linearly.
•
This is
exactly
like temperature. Freezing and boiling temperatures:
0
and
100
Celsius,
32
and
212
Fahrenheit.
•
Temp Fahrenheit =
32
+ (
180
/
100
)*Temp Celsius. Is this a problem?
Comparability?
•
Now let us ask about comparability.
•
We note that the utility function that we have derived is
unique
only up to
a linear transformation
•
We need to fix end

points if we want to use these functions
to find social welfare/happiness as in the previous slides.
•
Can we have the same end

points for everyone?
•
“Is Socrates dissatisfied happier than a pig satisfied?”
•
During the
19
th
and
20
th
centuries most economists argued
that one cannot measure happiness on an absolute scale.
•
But the idea is coming back into fashion. Andrew Oswald is a
particularly energetic proponent. See his
article
on measuring
and comparing International Happiness.
John Hey’s view
•
The idea of measuring happiness is cr*p.
•
To say that I am happier than Professoressa Pasca is
meaningless.
•
But to say that I am happier today that yesterday has some
merits.
•
And to say that B*rl*sc*n* is happier than Amanda Knox (and
indeed most people in Italy) seems reasonable.
•
Perhaps it is OK to say that the Pope is happier than Ruby?
•
A hermit happier than a Ghedaffi?
•
Perhaps society
needs
to take a view?
•
But what is society? The Great and the Good? The
Disinterested Few? Grand Old Men? We will see later…
One way out?
•
Nash Bargaining Theory.
•
Let us consider this with just two members in society
u
and
v
.
•
Suppose they are bargaining over money. If they do not reach
agreement then they get some
default d
.
•
Suppose
u
gets
x
and
v
gets
y
if agreement is reached.
•
Nash showed that under some axioms (see the next slide) the
solution is the allocation
x
to
u
and
y
to
v
such that the
expression
•
[u(x)

u(d)][v(y)

v(d)]
•
is maximised.
Nash’s axioms
•
[See the Wikipedia
article
.]
•
Invariant to affine transformations or Invariant to
equivalent utility representations
: so we do not
worry about comparibility.
•
Pareto optimality:
so there is no other solution
which is better for both than this.
•
Independence of irrelevant alternatives:
we have
come across this before with Arrow.
•
Symmetry:
the two individuals are treated
symmetrically (by the state).
Extensions and problems
•
Can be extended to more than two people.
•
The same conclusions apply.
•
Its implementation requires all people in society to agree to
the axioms, including symmetry. (Note this latter does
not
imply that
x = y
).
•
Problems?
•
We need to know the utility functions?
•
We already know how to find them.
•
But our method requires that people answer honestly.
•
Is it in their own interests so to do?
•
Question for the nonna.
Conclusions
•
We started this lecture with a long liturgy of impossibilities…
•
…but with hope in our hearts.
•
We ended it with less hope and more impossibilities…
•
…but greater clarity about what ‘The State’ needs to know.
•
We need to judge what the State does, not from our own
selfish perspectives, but as Grand Old Men taking a
benevolent and disinterested view of Society.
•
But another question strikes us at this stage:
•
“Why do we need a State at all? Why can we not just let the
individuals in society get on and run it by themselves?”.
•
What is wrong with
anarchy
? Or perhaps a little anarchy?
•
We move on to this in the next session.
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