Social Choice Lectures 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 (α 1,... α I ) 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(α 1,... α I ) = 1 if Σ i β i α i > 0, = 0 if Σ i β i α i = 0 and = -1 if Σ i β i α i < 0. A particularly important case is when β i =1 for all i. This is just majority voting. Dictatorship if α h = 1 (0, -1) implies F(α 1,... α I ) = 1 (0, -1). Anonymity is implied by β i =k all i. Neutral between alternatives if F(α 1,... α I ) = - F(-α 1,... -α I ) 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 choice/documents/arrow impossibility theorem.pptwww.luiss.it/hey/social choice/documents/arrow impossibility theorem.ppt There is another nice example at choice/documents/john bone and arrow.pptwww.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: choice/documents/geanakoplis 3 proofs of arrow.pdf. 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 R1R1..R m-1 RmRm R m+1...RnRn R social order a...aaa aa b bbb bb

Looking for the Pivotal Voter (Keeping all the other alternatives fixed) R1R1...R m-1 RmRm R m+1...RnRn R social order b...aaa aa a b bbb b.

Looking for the Pivotal Voter R1R1..R m-1 RmRm R m+1...RnRn R social order b...baa aa a ab b b b.

The Pivotal Voter (m) R1R1..R m-1 RmRm R m+1...RnRn R social order b...bba ab a aa..a b b.

After the Pivotal Voter R1R1..R m-1 RmRm R m+1...RnRn R social order b...bbb ab a aaa.a b.

Unanimity again R1R1..R m-1 RmRm R m+1...RnRn R social order b...bbb bb a aaa aa

Table 1 (top) before and Table 2 (bottom) after the pivotal voter R1R1..R m-1 RmRm R m+1...RnRn R b..baa...aa a..ab b b...b. R1R1..R m-1 RmRm R m+1...RnRn R b bba ab a aa..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 R1R1..R m-1 RmRm R m+1...RnRn R b..baa...aa a..ab b b...b. R1R1..R m-1 RmRm R m+1...RnRn R b bba ab a aa..a b b.

Table 1’ (top) before and Table 2’ (bottom) after the pivotal voter- see Gaertner pages 26/7. R1R1..R m-1 RmRm R m+1...RnRn R b..ba.....a....b.....b a...a. a..a.b...b. R1R1..R m-1 RmRm R m+1...RnRn R b bb..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. R1R1..R m-1 RmRm R m+1...RnRn R b..ba.....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) R1R1..R m-1 RmRm R m+1...RnRn 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) R1R1..R m-1 RmRm R m+1...RnRn R....a.....a....c c..cbc...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. R1R1..R m-1 RmRm R m+1...RnRn R....a.....a....c c..cbc...cc b..b.b...b. a..a.a...ab

Penultimately consider this (Step 5 first part) Pivotal Voter m is dictatorial. (Note that c cannot effect the social ranking between a and b) R1R1..R m-1 RmRm R m+1...RnRn R c..cac...ca....c b.....c b..b.b...b. a..a.a...ab

... and finally (Step 5 second part)... Pivotal Voter m is dictatorial wrt a versus any other option. R1R1..R m-1 RmRm R m+1...RnRn R b..bab...ba....b c.....b c..c.c...c. a..a.a...ac

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 c i (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.

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 x 5 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?