COLLECTIVE DECISION MAKING

1. COLLECTIVE DECISION MAKING Pierre Dehez CORE University of Louvain pierre.dehez@uclouvain.be

2. Outline 1. Preferences, utility and choices 2. Cardinal welfarism distributive justice utilitarism vs egalitarism Nash bargaining social welfare orderings transferable utility games 3. Ordinal welfarism the case of two alternatives social choice procedures impossibility theorems possibility theorems

3. References Austen-Smith D. and J. Banks, Positive political theory I: Collective preferences, University of Michigan Press, 1999. Austen-Smith D. and J. Banks, Positive political theoryII: Strategy and structure, University of Michigan Press, 2005. Brams S., Game theory and politics, Dover, 2004. Brams S., Mathematics and democracy, Princeton University Press, 2008. Moulin H., Axioms of cooperative decision making, Cambridge University Press, 1998.* Moulin H., Fair division and collective welfare, MIT Press, 2003.* Taylor A., Mathematics and politics, Springer-Verlag, 1995. Peyton Young H., Equity. In theory and Practice, Princeton University Press, 1995. Handbook of social choice and welfare, Elsevier, 2002. * Moulin's monographies have inspired some of the material presented here.

4. 1. Preferences, utility and choices

5. Preferences Preferences over a set A of alternatives are defined by a (binary) relation over A: bis not preferred to a from which the strict preference and indifference relations are deduced: ais preferred to b  indifference between a andb

6. Preferences arerational if they verify the following properties: - completeness: - reflexivity: - transitivity: Completeness is by far the most demanding assumption! A relation satisfying reflexivity and transitivity is a preorder. We denote by L(A) the set of preorders on a set A.

7. Ordinal utilities A preference preorder carries no information on the intensity of preferences: if a is preferred to b and c is preferred to d, we don't know whether a is "more preferred" to b than c is preferred to d Under minimal assumptions, preferences can be represented by a utility function which associates a real number to each alternative, such that:

8. As such, this is an ordinal representation of preferences: only the sign of the difference provides an information on the preferences between a and b As a consequence, u and where is an arbitrary increasing transformation, both represent the same preferences. T(u) = u3 is the simplest nonlinear transformation with range

9. Choices Given a set of alternatives A and preferences a choice is a best element in A: or There may be several best elements. The set of solutions is called the choice set.

10. Let denote the choice set associated to a set A of alternatives and a preference relation Then: There is indifference between the elements of a choice set. In the multivalued case, a neutral mechanism is necessary to eventually retain a unique alternative. For instance a random mechanism.

11. Cardinal utilities Utilities are cardinal if utility difference have a meaning: means that a is preferred to b more intensely than c is preferred to d. Cardinal utility function are defined up to an increasing affine transformation: are utility functions representing the same preferences.

12. Collectivity: preference profiles Consider a set A of alternatives and n individuals indexed by i running from 1 to n, each having a preference relation A preference profileP specifies a preference relation for each member of the group: A utility profile can be associated to any alternative a A:

13. One of the questions addressed by social choice is the determination of a collective preference ordering for comparing utility profiles. There are several levels of independence that collective preferences may satisfy: 1. Ordinal, non-comparable: full independence 2. Ordinal, comparable: independence of common utility space 3. Cardinal, non-comparable: independence of utility scales 4. Cardinal, partially comparable: independence of zero utilities 5. Cardinal, comparable: independence of utility scales and zero utilities

14. Given a set of alternative A and a preference profile on A represented by utility functions u1,…,un we define the attainable utility set The problem is then to pick up a point in this set, possibly given the specification of a disagreement point d in U(A).

15. 1. Ordinal, non-comparable: full independence This is the situation where each individual utility level is defined up to an arbitrary increasing transformation: where the Ti's are arbitrary increasing transformation

16. 2. Ordinal, comparable: independence of common utility space This is the situation where individual utility levels are defined up to an arbitrary and common increasing transformation: where T is an arbitrary increasing transformation

17. 3. Cardinal, non-comparable: independence of utility scales This is the situation where each individual utility level is defined up to an increasing and affine transformation:

18. 4. Cardinal, partially comparable: independence of zero utilities This is the situation where each individual utility level is defined up to an increasing and affine transformation: Alternatively:

19. 5. Cardinal, comparable: independence of utility scales and zero utilities This is the situation where individual utility levels are defined up to an increasing and affine common transformation:

20. 2. Cardinal welfarism

21. 2.1 Distributive justice

22. "Equal treatment of equals" is the basic principle of distributive justice. It is a minimal and clear requirement of fairness. "Unequal treatment of unequals" instead is a vague principle. "Equals should be treated equally and unequals unequally, in proportion to the relevant similarities and differences" (Aristoste)

23. Liberalism: the social order emerges from the interaction of free wills. Methodological individualism is at the root of liberalism. Individuals are characterized by values, rights and obligations. Distributive justice has two sides: - procedural justice: is the distribution of rights fair ? - end-state justice: is the outcome fair ? We start with a simple problem of sharing a resource.

24. We assume that a utility index can be associated to each individual: where xi denotes the share of individual i in the resource. It is a cardinal utility andutility levels can be compared. Its definition depends upon the context. It is an "objective" index and the individual is not responsible for its shape. It is the information that a benevolent dictator needs to decide on the allocation of resources in a particular context.

25. Principle 1: ex ante equality There are basic rights like freedom of speech, access to education, freedom of religion, equal political rights (one person, one vote),… They induce ex ante equality: equal claim to the basic resources. Private ownership or differences in status (for instance seniority) are instances of unequal exogenous rights which justify unequal treatment.

26. Principle 2: ex post equality … justifies unequal shares of resources to compensate for involuntary differencesin individuals' primary characteristics like nutritional needs, health,… If ui is an objective utility index for individual i resulting from his/her primary characteristics, this principle allows for or equivalently This principle amounts to equalization of utilities.

27. Principle 3: reward or penalize … justifies unequal shares yi's of resources to compensate for voluntary differencesin individuals' characteristics: - past sacrifies justify a larger share - past abuses justify a lesser share How to reward individual contributions ? The answer if difficult when there are externalities (extraction of exhaustible resources, division of joint costs or surpluses).

28. Principle 4: best use of the resources (fitness) …resources must go to those that can make the best use them. Fitness justifies unequal treatment by differences in talent, independently of basic rights, needs or merits. Two definitions: sum-fitness: maximization of the sum of the individual utilities efficiency-fitness: Pareto optimality Sum-fitness implies efficiency fitness.

29. How should the benevolent dictator use these four partiallyconflictingprinciples very much depends upon the context. Examples: - access to the lifeboat, - allocation of organs for transplant, - seat rationing, - political rights.

30. Lifeboat exogenous rights: strict equality (lottery) or priority ranking based on social status or wealth compensation: priority to the weak ones (equality of ex-post survival chances) reward: exclude those responsible for the sinking ship… fitness: keep the crew, the women, the children,…

31. Transplants exogeneous rights: strict equality (lottery) or priority ranking based on social status or wealth compensation: priority to those suffering most or whose life expectancy is the shortest reward: priority to seniority on the waiting list fitness: maximization of the chances of success

32. Seats: auctioning or queuing exogenous rights: only a lottery would induce a strict equality reward: queuing reward efforts while auctioning does not fitness: queuing meets sum-fitness but involves a waste of time auctioning is better if individuals are comparable, because otherwise it favors the rich

33. Political rights fitness and reward: justify unequal voting rights which were commonplace in the past exogenous rights: justifies equal rights (beyond some obvious limitations justified by fitness) compensation: there are many examples of situations where voting rights are not equal (EU distribution of votes among countries supposed to take into differences in population sizes)

34. Allocation methods A given amount of some commodity has to be divided between a given number of individuals and each individual has a claim. The commodity could be a "good" or a "bad": - for a good, individuals express demands - for a bad, individuals have liabilities There may be an excess or a deficit.

35. Data: a set N = {1,…n} of "players" an amount E > 0 to be allocated players' claims: d1,…,dn > 0 Problem: find an allocation x = (x1,…,xn) sucth that x(N) = E. Two cases:

36. Examples: joint venture: E is the revenue generated by the cooperation and the di's are the stand-alone revenues (surplus) bankcruptcy: E is the firm's liquidation value and the di's are the creditors' claims (deficit) inheritance: E is the value of the deceased's estate and the di's are the heirs' deeds (deficit or surplus) taxation: E is the tax to be levied and the di's are the taxable incomes (deficit)

37. Assumption: equal exogenous rights  the allocation depends only on the distribution of claims or liabilities An allocation method is a rule  that associates an allocation to any given allocation problem such that

38. Proportional rule (in the case of a surplus or a deficit) satisfies: and

39. x2 PROP d2 x1 d1 0

40. Equal surplus rule (in case of a surplus) satisfies

41. x2 SURPLUS: d1+d2 < E and d1 > d2 ES d2 x1 d1 0 d1–d2

42. x2 SURPLUS: d1+d2 < E and d1 < d2 ES d2 d2–d1 x1 d1 0

43. Uniform gain rule (in case of a surplus) satisfies Here z can be interpreted as the common gain. This rule is also called "constrained" egalitarian.

44. y = f(z) 2d1 d1+d2 d1 z 0 d1 d2

45. x2 UG d2 x1 0 d1

46. x2 SURPLUS: d1+d2 < E UG ES PROP d2 x1 0 d1 d1–d2

47. Uniform gain rule (in case of a deficit) satisfies Here z can be interpreted as the common gain. This rule is also called "constrained" egalitarian.

48. y = f(z) d1+d2 2d2 d2 z d1 0 d2

49. x2 DEFICIT: d1+d2 > E d2 UG x1 0 d1

50. Uniform loss rule (in case of a deficit) satisfies The idea is to substract the same amount from the claims subject to the non-negativity constraint: z is the common loss (ex post deficits are equalized). This rule is also called levelling.