Social Choice Session 6

1
Social Choice Session 6

• Carmen Pasca and John Hey

2
Plan for today

• Remember that we are trying to find/express
  Societys 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?

3
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

4
Our framework and assumptions

• Denote Societys utility/happiness by U.
• Individuals in society receive money income x.
• Denote is utility by ui ui(xi).
• This could just be ui(xi) xi but we keep it
  general.
• Then societys happiness is some function of the
  ui.
• We could have U (u1 uI )
  Egalitarian.
• Or U min(u1, ,uI ) Rawlsian.
• Or U max(u1-ud)(u2-ud) Nash special
  case
• Or
• To make this operative the ui must be
  cardinal/measurable.
• (We concentrate on this and ignore in this
  session the issue of the form of the function.)

5
Notice the advantages of this

• If society has a sum of money to distribute, then
  it can be done optimally through maximisation of
  Societys 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.

6
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 (BgtW) the Best and
  Worst.
• Put u(W) 0 and u(B) 1.
• Like temperature centigrade freezing is 00
  boiling 1000.
• Let X be some amount of money between the best
  and the worst W lt X lt 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.

7
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)?

8
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) p1 (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?!

9
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) 0.75u(1000) 0.25u(0) 0.751
  0.250 0.75
• We have found the utility of 500 for an
  individual with the above preferences.
• Clever?!

10
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?

11
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 19th and 20th 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.

12
John Heys view

• The idea of measuring happiness is crp.
• 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 Brlscn 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

13
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.

14
Nashs 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).

15
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.

16
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.