Social Welfare Functions

• The UP (Utilitarian Principle) crucially refers to individual well-being.
• The structure of individual well-being was studied previously
• Throughout the course, it is presupposed that well-being is at stake for all currently living members of a given group or society N of individuals.
• UP compares outcomes by adding up all individual utilities per outcome, giving the "social welfare" level of that outcome.
• The conditions under which summing individual well-being levels appropriately reflect how good a given outcome is are important.

Social Welfare Functions

• Given a society of individuals N and outcomes A, a utility profile U maps (N × A) to R, U(A, x) or uA(x) denotes individual A's well-being in outcome x.
• The question is: how should a social planner rank outcomes given some utility profile?
• A social welfare function (SWF) is needed
• A SWF is a function that takes any utility profile on some (N, A) and outputs a complete weak order ≥ on A.
• SWFs can only determine the rightness of actions if consequentialism (P1) and welfarism (P2) are presupposed: actions' moral rightness is determined by individual well-being at the resulting outcomes.
• P1 and P2 are presupposed throughout lectures 5 and 6 but will be questioned in L7 and L8.

Informational Assumptions

• A key assumption made is that individual well-being levels are comparable to some extent
• I.e., individual i's well-being level n can be compared to individual j's well-being level m.
• Different senses of comparability lead to different appropriate social welfare functions.
• The choice for a SWF depends on the informational assumptions made about the problem
• It is assumed what type of information about individual well-being in society is available to the social planner.
• The assumption of comparability will be dropped and Social Choice Theory will be applied in the next lecture

Lecture Structure

• The topics covered in this lecture include introduction, utilitarianism & cardinal comparability, egalitarianism, prioritarianism, Rawlsian SWFs & ordinal comparability, summary and reflection, and literature.

Utilitarian SWF

• The Utilitarian SWF is determined by Sum-ranking Welfarism.
• x is better than y if and only if the sum-total of well-being in x is greater than in y

Utilitarian Principle

• P1: Consequentialism
• P2: Welfarism
• P3: Weak Pareto
• P4: Unit Cardinal Comparability
• P5: Transitional Equity
• Lectures focus on P4 and P5, their entailments, and how they could be changed, while the other principles are not questioned

Comparability and Equity

• P4 (Unit Cardinal Comparability):
  • Individual well-being gains and losses can be compared across outcomes.
  • People's well-being gains and losses can be compared with one another.
• P5 (Transitional Equity): If outcome x is obtained from outcome y by increasing the well-being of person i by amount k and decreasing the well-being of person j by k, then x and y are equally good.
• P5 meaningfully relies on P4, making P4 fundamental.
• P5 is not necessarily a principle that follows from P4.
• Part (i) of P4 requires individual well-being to be represented cardinally, which means being invariant under positive linear transformations.

Cardinality

• Cardinality by itself is not sufficient for a utilitarian approach

Unit Cardinal Comparability

• A statement about Ann and Bob's well-being is unit cardinal comparable if, when the statement is true for one utility profile (uA, uB), it remains true for every utility profile (u'A, u'B) obtained by applying positive linear transformations τA, τB to (uA, uB) as long as TA and TB are identical up to a constant
• Multiplying each utility u(x) with the same positive factor k and adding a constant kA for each individual A should not make a true statement false.

Full Cardinal Comparability

• A statement about Ann and Bob's well-being is fully cardinal comparable if, when true for one utility profile (uA, uB), it remains true for every utility profile (u'A, u'B) obtained by applying a single positive linear transformation τ to (uA, uB).
• Unit cardinal comparability suffices for utilitarianism, so full cardinal comparability is not required.

P4 Revisited

• P4: Well-being is cardinal and unit comparable between individuals
• Utilitarianism requires a unit cardinal comparable representation of Ann's and Bob's well-being, which is a utility profile (uA, uB) representing all unit cardinal comparable facts.
• To obtain a unit cardinal comparable representation, Ann's and Bob's well-being must be "measurable on the same scale" (up to a constant).
• VNM functions arguably provide a "same scale measurement".
• Utilitarianism makes non-trivial assumptions concerning the structure of well-being through P4: comparability is essential, and cardinality alone is insufficient.
• Problems arise even when granting P4.

Comparability & Equality

• P5 (Transitional Equity): If outcome x is obtained from outcome y by increasing the well-being of person i by amount k and decreasing the well-being of person j by k, then x and y are equally good.
• P5 implies that the Utilitarian SWF does not prioritize equality, because one outcome is just as good as another.
• Some disagree with utilitarianism because it does not value equality of well-being
• Intuition suggests that a society in which everyone is moderately well-off may be preferable to one in which an elite group thrives while the masses suffer.

Equality

• Four alternatives to P5 and utilitarianism more generally include:
  • Egalitarianism
  • Prioritarianism
  • Maximin
  • Leximin
• Focus on 2-person utility profiles, which can be generalized.
• Each alternative values "equality of well-being" in some sense but presupposes some form of full comparability.
• Knowing when two persons' well-being is "equal" is essential when caring about equality

Equality: Well-being vs Resources

• Making everyone's well-being equal does not mean giving everyone the same resources.
• For example, people with impaired mobility need more mobility aids in order to have the same well-being.
• Utilitarians still believe resources (money) should be distributed equally as doing so maximizes the sum-total of well-being
• This means that one's amount of well-being is a concave function of one's amount of resources, then an equal distribution of given resources can obtain a maximal sum of well-being..
• SWFs talk about how best to distribute well-being, not resources (discussed in Lecture 8).

Egalitarianism

• Egalitarianism is a political philosophy that favors equality of some sort.
• Egalitarianism can focus on equality of well-being, presupposing we have individual well-being levels at each outcome
• A simple way is to take inequality into account in the sum of utilities, for a two-person society

Simple Egalitarianism

• Simple Egalitarian SWF

Levelling Down

• Simple Egalitarian SWF illustrated
• Egalitarianism and Levelling Down
• Simple Egalitarianism states: L(eft) is better than M(iddle), as 15 > 14 and R(right) is better than M(iddle), as 14.7 > 14
• We can make Bob worse off which will make the situation better
• We can make the situation better (realize R) by making everyone worse off!
• We can improve a situation by "levelling down"

Levelling Down: Definition

• SWF allows for levelling down: situations x and y such that: x is obtained from y by making one or more persons worse off than in y, while the SWF states that x is better than y.
• A SWF that allows for levelling down is incorrect

Levelling Down: Utilitarian SWF

• The Utilitarian SWF does not allow for levelling down:
• If Ann is worse off then the situation gets worse.

Levelling Down: Strong Pareto

• If everyone is at least as well-off in y as in x and at least someone is better off in y than in x, then y is better than x.
• If an SWF allows for levelling down, then there are situations x and y.

Levelling Down illustrated

• Levelling Down and the Pareto's
• Weak Pareto: everyone is better off in y than in x, then y is better than x.
• Strong Pareto implies Weak Pareto.
• Strong Pareto implies No Levelling Down.
• No Levelling Down does not imply Strong Pareto.

Levelling Down illustrated

• Consider a situation in which well-being is very unequally distributed: a small fraction (the elite) has much more well-being than others (the masses).
• There is some policy that would take away some of the well-being of the elite.
• If this policy increase well being this is not subject to the levelling down objection
• If policy is subject to the levelling down objection and SWF says that this policy is better

Equality Respecting SWFs

• Simple Egalitarianism is equality respecting

Equality Respecting

• Consider possible distributions of a given value of well-being (k)
• Every individual gets a benefit from well-being
• This is equality respecting
• If equality is respected the objections threaten all Egalitarian SWFs
• It depends on if whether not such an SWF faces the objection depends on it detail. Also whether Simple Egalitarianism is equality respecting

Three Desiderata

• A SWF should avoid leveling down, satisfies Strong Pareto, and is equality respecting.
• In what follows, consider two alternative approaches and concrete SWFs for each of these, checking which of (i), (ii), and (iii) they satisfy.
• Alternatives rely on distinct informational assumptions: full cardinal comparability vs. full ordinal comparability.

Prioritarianism

• Prioritarianism means increases in well-being count, well-being for those who are worse off should count more than increases of well-being for those who are better off
• A broad class of SWFs can apply to this

Simple Prioritarian SWF

• Give priority to the worse-off
• The square root function is concave
• If transfer well-being from a better-off to a worse-off person, well being increases
• When new unit available, benefit worse off person overall

Concavity & Priority

• Each additional unit of u leads to an ever-smaller increase.
• Transfer 1 unit of well-being from Bob to Ann yields an increase in overall goodness. -An additional unit of well-being does more good when given to Ann than when given to Bob.
• Observe looking at the shape of the graph of the square root
• Or from the graph of any strictly increasing, concave function.

Prioritarian SWFs

• Prioitarian SWFs are and only if the function that it uses to evaluate alternatives is of the form f(u_A) + f(u_B) where f is some strictly increasing concave function.

Prioitarian SWFs: Equality

• Prioritarian SWFs satisfy Strong Pareto (and hence do not allow for leveling-down).
• Prioitarian SWFs are equality respecting.

Concave function

• Any unequal distribution of well-being can be improved upon by transferring well-being from a better-off to a worse-off person.
• Prioritarian SWFs require full cardinal comparable well-being information, making the informational assumptions strong

Rawls' Difference Principle

• Social and economic inequalities are to be arranged to the greatest benefit of the least advantaged
• DP is a principle that concerns the distribution of basic goods, and is thus social justice
• The "least advantaged" is cashed out in terms of well-being
• Rawls argues for the DP using his famous "Original Position" argument
• Reinterpret the DP as applying to the distribution of well-being as such

Basic Rawlsian SWF

• The difference principle can be interpreted as a recipe for a simple SWF

Basic Rawlsian SWF

• Basic Rawlsian SWF, BR
• If only if the worst-off person in x is better off than the worst-off person in y.

BR illustrated

• The worst-off person's well-being is given by min(uĄ, UB), so according to BR
• The best overall situation is the one for which is maximized.
• Example show that BR violates Strong Pareto

Levelling Down & Equality

• BR violates Strong Pareto, however
• BR does not allow for levelling-down
• Make the situation making someone worse-off
• BR is equality respecting
• This means any unequal distribution of well-being can be improved upon by transferring well-being from person who is not worst-off to a person that is worst off.

BR

• BR presupposes less information about persons' well-being than Prioritarianism

Full Ordinal Comparability

• Full ordinal comparable well-being statements: when the statement is true for one utility profile (uA, uB), the statement is true for every utility profile (u', u'B) that is obtained by applying a single strictly positive transformation τ to (uĄ, uB).
• The ranking of outcomes under BR (only) requires a reprentation of well-being that is full ordinal comparable:

Lexical Rawlsian SWF

• First compare outcomes in terms of the utility of the worst off.
• When these are equally good, compare the utility of the second worst off.
• Lexicographic ranking of tuples: (nk)1≤k≤m is better than (nk)1≤k≤m iff there.
• Re-order utilities(Ua(x))&en from "worst off given x" to "best off given x".
• LR requires ordinal comparable representation of well-being.
• Social welfare ranking of outcomes by LR is invariant under any strictly positive transformation of a utility profile.
• LR satisfies Strong Pareto (thus, also No Levelling Down) and is Equality Respecting.