Aggregating Well-being and SWFs

• Lecture 5 is about aggregating well-being and comparable well-being.
• The lecture will cover different social welfare functions (SWFs) and how they relate to individual well-being.
• The lecture will also discuss the informational assumptions that underlie different SWFs.
• The utilitarian principle (UP) compares outcomes by adding up individual utilities per outcome, resulting in a "social welfare" level.
• Lecture 4 discussed different types of well-being information: ordinal, cardinal, and ratio scale.
• Ordinal statements only capture that one thing is better than another.
• Cardinal statements capture the differences in well-being.
• Ratio scale statements capture relative magnitudes of well-being.
• Ordinal statements are invariant under strict positive transformations.
• Cardinal statements are invariant under positive linear transformations.
• Ratio scale statements are invariant under scalar transformations.
• The type of information needed depends on one's theory of what is morally right.
• The constitutive principles of the UP are: P4) Cardinal comparability and P5) Comparison.
• P4(i) The well-being of an individual has a cardinal structure.
• P4(ii) Well-being gains and losses of different people can be compared with one another.
• A utility profile U is a mapping from (N × X) to R, where N is a society of individuals and X is a set of outcomes.
• U(A,x) represents the well-being of individual A in outcome x.
• A social welfare function (SWF) takes as input a utility profile and outputs a complete weak order.
• SWFs determine the rightness of actions only if we presuppose consequentialism (P1) and welfarism (P2).
• The choice of a SWF depends on the informational assumptions about the problem at hand.
• The lecture focuses on the assumption that well-being is "comparable" across individuals.
• The structure of the lecture is as follows:
  • Introduction
  • Utilitarianism & Unit Cardinal Comparability
  • Egalitarianism & Full Cardinal Comparability
  • Prioritarianism & Full Ratio Scale Comparability
  • Rawlsian SWFs & Ordinal Comparability
• Utilitarianism uses a sum-ranking welfarism system: x is better than y if the sum-total of well-being in x is greater than in y.
• Utilitarian principle (UP):
  • P1 Consequentialism
  • P2 Welfarism
  • P3 Weak Pareto
  • P4 Unit Cardinal Comparability
  • P5 Transitional Equity

Comparability and Equity

• Unit Cardinal Comparability: individual well-being gains and losses can be compared across outcomes.
• It also dictates well-being gains and losses of different people can be compared (aka "unit cardinal comparability").
• The outcome x is obtained from the outcome y by increasing the well-being value of person i by amount k, and decreasing the well-being value of person j with k, then x and y are equally good.
• P5 can only meaningfully apply if P4 holds and P4 is fundamental.
• P5 does not follow logically from P4.

Cardinality

• Cardinal representation of each individual's well-being isn't enough.
• The representation should respect information to meaningfully and correctly apply the UP.
• A statement about the well-being of Ann and Bob is unit cardinal comparable iff when the statement is true for one utility profile (uA, uB), the statement is true for every utility profile (u'A, u'B) that is obtained by applying positive linear transformations TA, TB to uA resp. uB, where TĄ and TB are identical up to a constant.
• Comparisons of sum-totals of well-being are unit cardinal comparable.
• Recommendations by the UP are unit cardinal comparable.
• a statement is true for every utility profile (u'A, u's) that is obtained by applying a single positive linear transformation T to (uA, uB).
• Utilitarianism requires a unit cardinal comparable representation
• Ann's & Bob's well-being need to be "measurable on the same scale" to be unit cardinal comparable representation
• Utilitarianism makes assumptions concerning the structure of well-being.
• The Utilitarian SWF does not "care about equality".

Respecting Equality

• Discusses three alternatives to P5 and Utilitarianism more generally:
  • Egalitarianism
  • Prioritarianism
  • Maximin
• Each alternative cares about "equality of well-being" and presupposes full comparability, which is unavoidable.
• Making everyone's well-being equal does not entail giving everyone same resources.
• People with impaired mobility need mobility aids for the same level of well-being as people who can walk unaided.
• Utilitarians may argue resources have to be distributed equally as doing so maximizes sum-total of well-being.
• SWFs talk about how best to distribute well-being, rather than how best to distribute resources.

Egalitarianism

• Trend of thought in political philosophy that favors equality: People should get the same, be treated the same, or as equal.
• There is a focus on equality of well-being and at disposal are individual well-being levels at each outcome.

Simple Egalitarianism SWF.

• I(x) = |uĄ(x) – UB(x)| is the (in)equality at outcome x.
• E(x) = Σienui(x) – 2 · I(x) is the egalitarian welfare level of x.
• x > y iff E(x) > E(y).
• Obtained by changing the negative weight of inequality k (here: k = 2).
• For |N| > 2, one may consider the difference between the "worst of" and the "best of" person, or use more complex measures of inequality.
• Utilitarianism: x is just as good as y as 10 = 10
• Simple Egalitarianism: x is better than y as 10 > -6
• One can make the situation better (realize L) by making another worse off.
• One can make the situation better (realize R) by making everyone worse off.
• One can improve situation M by "levelling down”.
• A SWF that allows for levelling down is incorrect, hence Simple Egalitarianism is incorrect.

Levelling Down

• If there are situations x and y such that outcome x is obtained from outcome y by making one or more persons worse off than in y and making no one better off than in y, while the SWF says that outcome x is better than outcome y.
• The Utilitarian SWF doesn't allow for levelling down: making one worse off makes the situation worse.
• If a policy increases the well-being of at least one person that is not subject to the levelling down objection.
• If it does not, its subject to the levelling down objection.
• Policies face the levelling down objection, as does any SWF that says this policy makes things better.
• Simple Egalitarianism is equality respecting:

Equality Respecting SWFs

• Given a given "amount" k of well-being over n persons, an SWF is equality respecting if it says that the best distribution is the one where well-being is equally distributed
• A SWF is egalitarian if and only if the function that it uses to evaluate alternatives is of the form f(ua)+f(ив) + EQ(ид, иß) where f is some increasing function and where EQ specifies how equality of well-being contributes to the evaluation.
• The levelling down objection threatens all Egalitarian SWFs
• Specification decides how an SWF faces the objection of the exact way f and EQ are specified.

Prioritarianism

• It gives priority to those who are worse-off.
• If we have to choose between benefitting people that are worse off and benefitting people that are better off, we should give priority to the former.
• A Square root functions main idea is illustrated here:
• SP(x) = √ UA(x) + √ UB(x).
• If x > y then SP(x) > SP(y).
• Transferring well-being from a better-off to a worse-off person is better overall.
• Concavity of √: each additional unit of utility leads to an ever-smaller increase Prioritarian SWFs do not allow for levelling-down.
• Because f(ua) + f (up) is strictly increasing in both us and uв.
• It is both equality respecting and requires full ratio scale comparable well-being information.
• Stronger assumptions are required than for utilitarianism.
• Full ratio scale comparable well-being statements A statement about the well-being of Ann and Bob is full ratio scale comparable iff: when the statement is true for one utility profile, the statement is true for every utility profile that is obtained by applying a single scalar transformation T

Scalar Transformations

• Give two utility functions: u and v, you can apply a scalar transformation when there is an α > 0 in which if for every outcome x, v(x) = α·u(x).
• Scalar truth isn't effective if we multiply every well-being level with constant constant λϵ R
• affected if linear or other transformations are used.

Rawls’s Difference Principle

• States that social and economic inequalities are to be arranged so that they are to the greatest benefit of the least advantaged.
• The DP concerns the distribution of socio-economic goods, thus social justice cf. Lecture 8.
• The DP applies to the distribution of well-being as such.
• It is interpreted as a a recipe for a simple SWF:

Basic Rawlsian SWF, BR

• (x, y ∈ A)
  • x > y if and only if the worst-off person(s) in x better off than in y.
  • iff min(uĄ(x), UB(x)) > тіп(ид(у), ив(у)).
• worst-off person that is given by min(uA,uB), so it that the best overall situation is in which min(uA,uB) is maximized.
• Violates Strong Pareto: BR does not allow for levelling-down.
• Is equality respecting + transfers well-being from those that are more-off to those less-off.
• Presupposes one persons’ well-being is more than Prioritarianism. This needs to be precise.
• States that the ranking outcomes require only a representation of well-being as full ordinal (not changing the ranking of outcomes according to BR) to be comparable.
• if a single strictly positive transformation to a utility profile this changes nothing to the ranking of outcomes according to BR.
• If they apply different strictly positive transformations to the individual utility functions, this may affect the ranking of outcomes.
• Utilitarianism (Ut): uĄ + UB
• Simple Egalitarianism (SE): uĄ + ив - 2· | иа - ив|
• Simple Prioritarianism (SP): √ UA + √ UB
• Basic Rawls (BR): min(ид, ив)

SWFs

• have the following properties:

• SE and SP moreover belong to broad families of SWFs with similar properties.

Informational Assumptions

• Different SWFs makes different assumptions about the information in our numeric representation of well-being:

• SP presupposes more information than SE, and SE has more information than Ut and BR.
• BR and Ut are incomparable in terms of the amount of information they require.

Caring about Equality

• E, SP, and BR are all equality respecting.
• SWFs that care about equality.
• It is intrinsically important to them in EQ(uA, uB) term in the SWF.
• The other two rules are equality respecting is in virtue of the virtue of something else:
  • Relative priority for the worse-off, compared to the better-off (SP)
  • Absolute priority to the worst-off (BR)