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Feb 5, 2013 (4 years and 9 months ago)

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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.