Social Choice Lecture 16 John Hey

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

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.

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’s preferences b > c. From unanimity a > b. Hence we have b > c (B) and c > a (A) and a > b (unanimity)! WEIRD! (Intransitive).

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*.

Proof P indicates Society’s preference and P i that 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 xP i y, zP j w, and, for both k=i,j that wP k x and yP k z. 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.

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?)

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 P A y P A z Edwin has preference: z P E x P E y

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.

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.

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.

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...

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!

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.)

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.