Social Choice Session 3 Carmen Pasca and John

Social Choice Session 3 Carmen Pasca and John Hey

Plan for this session • We start with a review of Arrow’s theorem – using now Jen’s preferences to give completeness and showing that this leads to her being the dictator. • Then, within the same framework (ordinal preferences) we explore how we may weaken Arrow’s assumptions and possibly get some possibility. • After a short break we will invite comments from you as to other ways to get possibility. • (Note: in session 6 we explore the implications of cardinality/measurability of preferences – so we do not do this today. )

Arrow’s Impossibility Theorem • In social choice theory, Arrow’s impossibility theorem. . . states that, when voters have three or more discrete alternatives (options), no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. (Wikipedia) • We remind you of Arrow’s famous result and explore more its implications. • We again draw inspiration from John Bone at York and borrow some of his slides.

Arrow • Arrow’s conditions are these: • Universal Domain: is applicable to any profile. • Consistency: produces a complete, transitive ordering of available alternatives. • Pareto: if everyone prefers x to y then so should society. • Independence of Irrelevant Alternatives: orders x and y on the basis only of individual preferences on x and y. • Can we replace/eliminate/weaken any of these? • This is the question we address today.

Back to session 2 briefly • We go back to where we had to assume society’s preferences in order to get completeness. . . • . . . and we take an alternative way. • We go to slide number 12 of the Power. Point presentation of session 2. . . • . . . after which we once again shamelessly use John Bone’s thoughts and slides.

Can we plausibly relax Arrow’s conditions? Universal Domain: is applicable to any profile Perhaps there are some profiles at which we might not require a social ordering at all? Especially for profiles of (e. g. malicious) preferences. But perhaps also for profiles of welfare rankings. For example if x and y are different allocations of economic goods, and J is allocated more of every good under x than under y, then it might be reasonable to ignore any hypothetical profile in which J ranks y above x. Might this be enough to escape the problem?

Can we plausibly relax Arrow’s conditions? Pareto: if everyone prefers x to y then so should society Perhaps for similar reasons as with Universal Domain A Sen, “The impossibility of a Paretian Liberal”, Journal of Political Economy, vol 78 (1970), 152 -7 A single copy of Lady Chatterley’s Lover a: Ken (Prude) reads it b: Jen (Liberal) reads it c: neither reads it Jen Ken Pareto → abc cab Liberalism → Ignore “meddlesome” preferences? a. Pb b. Pc c. Pa (We will return to Sen shortly. )

Can we plausibly relax Arrow’s conditions? Independence of Irrelevant Alternatives Whatever determines the relative merits of x and y should not include individual welfare data regarding z …. . . but should instead be intrinsic to x and y. So at any two profiles in which the individual welfare data regarding (only) x and y is the same, the social ordering of x and y should also be the same. This becomes problematic when the individual welfare data comprises only rankings that is, comprises only ordinal welfare data

Can we plausibly relax Arrow’s conditions? Consistency: complete transitive ordering We may only have to decide which of the available alternatives should be (socially) chosen. So why require a complete, transitive social ordering of all available alternatives ? This is equivalent to requiring the social choice to be consistent across different (hypothetical) sets of alternatives Whatever determines the relative merits of x and y should not include the availability or otherwise of z …. . . but should instead be intrinsic to x and y.

A single copy of Lady Chatterley’s Lover a: Ken reads it Jen abc Ken cab b: Jen reads it choose: c: neither reads it a from {a, b} b from {b, c} c from {a, c} ? from {a, b, c} Whatever is the choice from {a, b, c}, there is inconsistency. But so what?

Majoritarianism Choose x from {x, y} iff more people prefer x to y than prefer y to x. Jen abc Ken cab Len bca choose: a from {a, b} b from {b, c} c from {a, c} ? from {a, b, c} Whatever is the choice from {a, b, c}, there is inconsistency. But so what?

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, which we have already seen. 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/liberal) 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 x. Py. • 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 Pi 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 x. Piy, z. Pjw, and, for both k=i, j that w. Pkx and y. Pkz. • From Condition L* we obtain x. Py and z. Pw. • From Condition P we obtain w. Px and y. Pz. • Hence it follows that • x. Py y. Pz z. Pw and w. Px. • 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 PA y PA z • Edwin has preference: z PE x PE 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: y. Pz, z. Px, x. Py. 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 x. Py. • • •

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.

Now a short break • After the break we invite you to suggest other ways of breaking Arrow’s impossibility. . . • . . . within the ordinal preferences framework. • Note that we could dispense with the notion of trying to find society’s preferences. . . • . . . we could immediately think about a decision rule of society which depends upon individual preferences. . • . . . Like doing something if a majority of the population want to do it. • Problems with this? Intransitivity?

Conclusions • So what do we conclude from all of this? • Essentially that, whatever way we do the aggregation, there is no method, that does not have objections, to aggregate individual preferences into a set of social preferences. • Is this surprising? • What are its implications? • That we cannot leave a computer in charge of society. • We need decision-making bodies with flexibility and some autonomy but also somehow under the control of the members of society.