Social ChoiceLectures 14 and 15

John Hey

Lectures 14 and 15: Arrow’s Impossibility Theorem and other matters

Plan of lecture:• Aggregation of individual preferences into social

preferences.• Just two alternatives.• More than 2 alternatives.• Arrow’s axioms and his Impossibility Theorem.• Possibilities (relaxing some axiom):• (1) Borda Count,• (2) Single-peaked preferences/Median voter.• Conclusions

What are we trying to do in this lecture?

• Starting with individual preferences over social alternatives, we will try and aggregate them into social preferences.

• Let x ≥i y mean that individual i ( = 1, .., I) prefers alternative x to alternative y.

• A social welfare function must assign a rational preference relation F(≥1, ... ≥I) to any set ≥1, ... ≥I.

Just two alternatives

• Alternatives x and y. (One could be the status quo.)

• I individuals. Preferences given by

(α1, ... αI) where each α takes the value 1, 0 or -1 depending whether the individual prefers x, is indifferent, or prefers y.

• A Social Welfare Functional is a rule that assigns a social preference, that is a number -1, 0 or 1, to each possible profile of individual preferences.

Just two alternatives: a simple example

• F(α1, ... αI) = 1 if Σiβiαi > 0,

= 0 if Σiβiαi = 0 and

= -1 if Σiβiαi < 0.

• A particularly important case is when βi=1 for all i. This is just majority voting.

• Dictatorship if αh = 1 (0, -1) implies F(α1, ... αI) = 1 (0, -1).

• Anonymity is implied by βi=k all i.

• Neutral between alternatives if F(α1, ... αI) = - F(-α1, ... -αI)

• Positively responsive if ....

• May’s Theorem: A SWF is a majority voting SWF if and only if it is symmetric, neutral between alternatives and positively responsive.

Arrow’s Impossibility Theorem

• There are at least three alternatives.• There are N individuals with transitive (perhaps different)

preferences.• unanimity (or weak pareto): society ranks a strictly above

b if all individuals rank a strictly above b.• independence of irrelevant alternatives: the social

ranking of two alternatives a and b depends only on their relative ranking by every individual.

• The Theorem: Any constitution that respects completeness, transitivity, independence of irrelevant alternatives and unanimity is a dictatorship.

Arrow’s Impossibility Theorem: Proofs

• Of course, there is the original proof.• There is a nice example (with just 2 voters and 3 alternatives) on the

site at

www.luiss.it/hey/social choice/documents/arrow impossibility theorem.ppt

• There is another nice example at www.luiss.it/hey/social choice/documents/john bone and arrow.ppt

• There are three simple proofs in the paper by Geanakoplis which I have also put on the site:

• www.luiss.it/hey/social choice/documents/geanakoplis 3 proofs of arrow.pdf.

• There is also a proof in the book by Wulf Gaertner A Primer in Social Choice Theory, LSE Perspectives in Economic Analysis. I will follow this and perhaps look briefly at the nice example above.

An important preliminary

• Let b be some arbitrary alternative.• We show: if every voter puts b either at the top or the bottom of

his or her ranking, then so must society.• Proof: suppose to the contrary that for such a profile, then for

distinct a, b and c, the social

preference has a≥b and b≥c.• By independence this must

continue even if all individuals

move c above a. (No ab or bc

votes would be disturbed.) • By transitivity a≥c but by

unanimity c>a. Contradiction.

Proof of Arrow. We start with Unanimity and then move b up place by place and person by person

R1 .. Rm-1 Rm Rm+1 ... Rn R social order

a ... a a a ... a a

. ... . . . ... . .

. ... . . . ... . .

. ... . . . ... . .

b ... b b b ... b b

Looking for the Pivotal Voter (Keeping all the other alternatives fixed)

R1 ... Rm-1 Rm Rm+1 ... Rn R social order

b ... a a a ... a a

a ... . . . ... . .

. ... . . . ... . b

. ... . . . ... . .

. ... b b b ... b .

Looking for the Pivotal Voter

R1 .. Rm-1 Rm Rm+1 ... Rn R social order

b ... b a a ... a a

a ... a b . ... . .

. ... . . . ... . b

. ... . . . ... . .

. ... . . b ... b .

The Pivotal Voter (m)

R1 .. Rm-1 Rm Rm+1 ... Rn R social order

b ... b b a ... a b

a ... a a . ... . a

. ... . . . ... . .

. ... . . . ... . .

. ... . . b ... b .

After the Pivotal Voter

R1 .. Rm-1 Rm Rm+1 ... Rn R social order

b ... b b b ... a b

a ... a a a ... . a

. ... . . . ... . .

. ... . . . ... . .

. ... . . . ... b .

Unanimity again

R1 .. Rm-1 Rm Rm+1 ... Rn R social order

b ... b b b ... b b

a ... a a a ... a a

. ... . . . ... . .

. ... . . . ... . .

. ... . . . ... . .

Table 1 (top) before and Table 2 (bottom) after the pivotal voter

R1 .. Rm-1 Rm Rm+1 ... Rn R

b .. b a a ... a a

a .. a b . ... . .

. .. . . . ... . b

. .. . . . ... . .

. .. . . b ... b .

R1 .. Rm-1 Rm Rm+1 ... Rn R

b ... b b a ... a b

a ... a a . ... . a

. ... . . . ... . .

. ... . . . ... . .

. ... . . b ... b .

Now we move alternative a

• We move alternative a to the lowest position of individual i‘s ordering for i<m...

• We move alternative a to the second lowest position of individual i‘s ordering for i>m...

• We keep individual m as is...

• ... Look at the bottom graph. Because of Independence social ranking does not change...

Table 1 (top) before and Table 2 (bottom) after the pivotal voter

R1 .. Rm-1 Rm Rm+1 ... Rn R

b .. b a a ... a a

a .. a b . ... . .

. .. . . . ... . b

. .. . . . ... . .

. .. . . b ... b .

R1 .. Rm-1 Rm Rm+1 ... Rn R

b ... b b a ... a b

a ... a a . ... . a

. ... . . . ... . .

. ... . . . ... . .

. ... . . b ... b .

Table 1’ (top) before and Table 2’ (bottom) after the pivotal voter- see Gaertner pages 26/7.

R1 .. Rm-1 Rm Rm+1 ... Rn R

b .. b a . ... . a

. .. . b . ... . b

. .. . . . ... . .

. .. . . a ... a .

a .. a . b ... b .

R1 .. Rm-1 Rm Rm+1 ... Rn R

b ... b b . ... . b

. ... . a . ... . a

. ... . . . ... . .

. ... . . a ... a .

a ... a . b ... b .

What is crucial is the Independence of Irrelevant Alternatives Axiom

• The relative positions of a and b do not change for anyone going from table 1 to table 1’.

• Note that the relative rankings differ from individual to individual (“People are Different”) but we have the same relative rankings for each individual in the two tables.

• So a, being socially best in Table 1 remains so in Table 1’.

We can begin to see why the Pivotal Voter is a dictator – because a is socially chosen here.

R1 .. Rm-1 Rm Rm+1 ... Rn R

b .. b a . ... . a

. .. . b . ... . b

. .. . . . ... . .

. .. . . a ... a .

a .. a . b ... b .

Now move b downwards – a remains top.(Note that in Tables 1 and 1’ b is above a for 1 to m-1 and a is above b for m+1 to N)

R1 .. Rm-1 Rm Rm+1 ... Rn R

. .. . a . ... . a

. .. . . ... . .

.. b ... .

b .. b . a ... a .

a .. a . b ... b .

Now identify a third alternative c – above b – a remains top, because all we have done is to identify c. (Step 3)

(Note that in Tables 1 and 1’ b is above a for 1 to m-1 and a is above b for m+1 to N)

R1 .. Rm-1 Rm Rm+1 ... Rn R

. .. . a . ... . a

. .. . c . ... . .

c .. c b c ... c .

b .. b . a ... a .

a .. a . b ... b .

Now Reverse a and b for i >m Can b become best? NO because c is preferred to b by all. And c cannot be preferred to a since we have not

changed the rankings of a and c.

R1 .. Rm-1 Rm Rm+1 ... Rn R

. .. . a . ... . a

. .. . c . ... . .

c .. c b c ... c c

b .. b . b ... b .

a .. a . a ... a b

Penultimately consider this (Step 5 first part)

Pivotal Voter m is dictatorial. (Note that c cannot effect the social ranking between a and b)

R1 .. Rm-1 Rm Rm+1 ... Rn R

c .. c a c ... c a

. .. . c . ... . .

. .. . b . ... . c

b .. b . b ... b .

a .. a . a ... a b

... and finally (Step 5 second part) ...Pivotal Voter m is dictatorial wrt a versus any other option.

R1 .. Rm-1 Rm Rm+1 ... Rn R

b .. b a b ... b a

. .. . b . ... . .

. .. . c . ... . b

c .. c . c ... c .

a .. a . a ... a c

More than one dictator?!

• Note that a was chosen arbitrarily at the start of this argument.

• Hence there is a dictator for every a.• Can there be different dictators for different alternatives?• Obviously not – otherwise we would get contradictions

(in the construction of the social ordering whenever these ‘potential dictators’ have individual orderings that are not the same).

• Therefore there can only be one dictator.

• FASCINATING!

Possibilities

• Must relax some axiom to get a SWF:

• (1) Borda Count,

• (2) Single-peaked preferences/Median voter.

• We note that the Borda count does not satisfy the pairwise independence condition. The reason is simple: the rank of any alternative depends upon the placement of every other alternative.

• Single-peaked preferences put strong restrictions on the domain of preferences.

Borda Count

• Suppose number of alternatives is finite. Denote generic alternatives by x and y.

• For individual i, define the count ci(x) = n if x is the n’th ranked alternative in the order of i. (Indifference....)

• Now define a SWF by adding up these counts – so

• This preference relation is complete and transitive and Paretian.

• However it does not satisfy the pairwise independence condition.

( ) ( )ii

c x c x

Single-Peaked Preferences

• Let decision variable be x.• Suppose the utility of decision-maker i is u(x).• Suppose u(.) is single-peaked for all i, for example:• Not like this:

• Then pairwise majority voting generates a well-defined social welfare functional.

• See next slide.

Single-Peaked Preferences

• Suppose all utility functions are single-peaked.• Here Agent 5 is the Median Agent

• The value x5 will beat any other value in majority voting.

Lectures 14 and 15

• SWFs are generally impossible (in the sense that unamity, independence of irrelevant alternatives and non-dictatorship are mutually inconsistent)

• However in special cases: Borda rule; Single-peaked preferences; they are possible. These relax the restrictions implied above: the Borda count relaxes IIR and single-peaked preferences restrict the domain.

• Is all of this surprising?

• Why do we have politicians?