Social Choice Lecture 16

1. Social ChoiceLecture 16 John Hey

2. More Possibility/Impossibility Theorems • Sen’s Impossibility of a Paretian Liberal • Gibbard’s theory of alienable rights • Manipulability • Gibbard-Satterthwaite theorem

3. Sen’s Impossibility of a Paretian Liberal • The idea is that each individual has the right to determine things ‘locally’ – that is, those things that concern only him or her. So individuals are decisive over local issues. • For example, I should be free to choose whether or not I read Lady Chatterley’s Lover. • He gives a nice example. Three alternatives, a, b and c, and two people A and B. • a: Mr A (the prude) reads the book; • b: Mr B (the lascivious) reads the book; • c: Neither reads the book.

4. Lady C. • Mr A (the Prude): c > a > b • Mr B (the Lascivious): a > b > c • Now assume that Mr A is decisive over (a,c) and that Mr B is decisive over (b,c). • So from A’s preferences c > a and from B’spreferences b > c. From unanimity a > b. • Hence we have b > c (B) and c > a (A) and a > b (unanimity)! • WEIRD! (Intransitive).

5. Sen’s Theorem • Condition U (Unrestricted domain): The domain of the collective choice rule includes all possible individual orderings. • Condition P( Weak Pareto): For any x, y in X, if every member of society strictly prefers x to y, then xPy. • Condition L* (Liberalism): For each individual i, there is at least one pair of personal alternatives (x,y) in X such that individual i is decisive both ways in the social choice process. • Theorem: There is no social decision function that satisfies conditions U, P and L*.

6. Proof • P indicates Society’s preference and Pithat of individual i. • Suppose i is decisive over (x,y) and that j is decisive over (z,w). Assume that these two pairs have no element in common. • Let us suppose that xPiy, zPjw, and, for both k=i,j that wPkx and yPkz. • From Condition L* we obtain xPy and zPw. • From Condition P we obtain wPx and yPz. • Hence it follows that • xPy yPz zPw and wPx. • Cyclical.

7. Gibbard’s Theory of Alienable Rights • Background... • Going back to the Lady C example, Mr A may realise that maintaining his right to decisiveness over (a,c) leads to an impasse/intransitivity. • He cannot get c (his preferred option) because Mr B has rights over that and renouncing his right to decisiveness over (a,c), society will end up with a (which is preferred by Mr A to b – his least preferred). • (Might Mr B think similarly (mutatis mutandis) and give up his right to decisiveness?)

8. Gibbard’s own example • Three persons: Angelina, Edwin and the ‘judge’. • Angelina prefers marrying Edwin but would marry the judge. • Edwin prefers to remain single, but would prefer to marry Angelina rather than see her marry the judge. • Judge is happy with whatever Angelina wants. • Three alternatives: • x: Edwin and Angelina get married • y: Angelina and the judge marry (Edwin stays single) • z: All three remain single • Angelina has preference: x PA y PA z • Edwin has preference: z PE x PE y

9. The problem and its solution • Angelina has a libertarian claim over the pair (y,z). • Edwin has a claim over (z,x). • Edwin and Angelina are unanimous in preferring x to y. • So we have a preference cycle: yPz, zPx, xPy. • If Edwin exercises his right to remain single, then Angelina might end up married to the judge, which is Edwin’s least preferred option. • ‘Therefore’ it will be in Edwin’s own advantage to waive his right over (z,x) in favour of the Pareto preference xPy.

10. Gibbard’s Theory of Alienable Rights • Condition GL: Individuals have the right to waive their rights. • Gibbard’s rights-waiving solution: There exists a collective choice rule that satisfies conditions U, P and GL. • The central role of the waiver is to break a cycle whenever there is one ... • ... but the informational demands are high.

11. Manipulation • Suppose there are three people in society A, B and C and three propositions a, b and c. • A‘s preferences: a > b > c • B‘s preferences: b > c > a • C‘s preferences: c > a >b • There is clearly a problem with choosing by majority rule: a majority (A and C) prefer a to b, a majority (A and B) prefer b to c and a majority (B and C) prefer c to a. • Suppose however that we organise the voting in stages: first between two alternatives and then between the winner of the first stage and the third alternative.

12. Depends who chooses the order • A proposes a first vote between b and c; and then between the winner of that and a. Which will win? If no strategic voting, clearly a - A’s preferred option. • B proposes a first vote between a and c; and then between the winner of that and b. Which will win? If no strategic voting, clearly b - B’s preferred option. • C proposes a first vote between a and b; and then between the winner of that and c. Which will win? If no strategic voting, clearly c - C’s preferred option. • So the person who chooses the order can manipulate the voting to get what he/she wants. • But what happens if people vote strategically...

13. Strategic Voting • A proposes a first vote between b and c; and then between the winner of that and a. Which will win? If no strategic voting, clearly a - A’s preferred option. • But suppose B realises this and hence knows that his least preferred option is going to win, then at the first stage B will vote for c thus ensuring that c will win at the second stage. • C is very happy to go along with this, but A is clearly not (c is A’s least preferred). Can A do anything about it? • A can propose that voting is first over (a,c) and then over the winner of that and b. With strategic voting by C then a will win. But A relies on C to vote strategically!

14. The Gibbard-Satterthwaite Theorem • Result (a): If there are at least three alternatives and if the social choice function h is Pareto Efficient and monotonic, then it is dictatorial. (Very similar proof to that of Arrow presented in the lectures.) • Result (b): If h is strategy-proof and onto, then h is Pareto efficient and monotonic. • Theorem: Let h is a social choice function on an unrestricted domain of strict linear preferences. It the range of h contains at least three alternatives and h is onto and strategy proof, then h is dictatorial. • (onto: every element of choice set is chosen for some profile.)

15. Conclusions • It seems difficult to avoid dictatorship... • ... even with individuals protected by their own rights. • Manipulation is a serious problem, but can be self-defeating. (Also requires information about motives/intentions/behaviour.) • Giving people the right to waive their rights simplifies in some senses and complicates in others, but does not remove the fundamental problem of information. • But if preferences differ, it seems inevitable that conflicts exist, and that politicians do also.