Problems for Part I.1
Except where mentioned otherwise all concepts and notation have been explained in class.
Please refer to your notes or the slides in case of doubt.
Exercise 1 This is an exercise on scoring rules.
(i) Show that a scoring rule is always rational and non-dictatorial.
(ii) Show by means of a simple example that scoring rules may not be weakly Paretian.
(iii) Show that a scoring rule is weakly Pareto optimal if s1 > s2 > . . . > sm.
(iv) Show by means of a simple example that scoring rules may fail to be independent of irrelevant alternatives.
Exercise 2 Consider a social choice problem with three alternativesx, y, z and three individ- uals 1,2,3. Consider the following two preference profiles:
R R1 R2 R3
x y z
y x y
z z x
and
R0 R01 R02 R03
z z x
y x y
x y z.
Show thatany social welfare function satisfies independence of irrelevant alternatives if only these preference profiles are possible.
Exercise 3 Let F be a given social welfare function.
Given two alternatives x, y ∈X,
• a set of individuals J ⊆ I is decisive for x against y, if for every preference profile (R1, . . . , Rn)∈ Rn such thatxPiy for all i∈J, xFp(R1, . . . , Rn)y,
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• individuali∈I has a veto forx against y, if for every preference profile (R1, . . . , Rn)∈ Rn such that xPiy,xF(R1, . . . , Rn)y.
A set of individuals J ⊂I is decisive, J is decisive for x against y for all ordered pairs of alternatives (x, y)∈X2. An individual i∈I has veto power, if i has a vetor for x against y for all ordered pairs (x, y)∈X2.
(i) Characterize the set of decisive coalitions for the majority and the plurality rule.
(ii) Is there an individual who has veto power when either the majority or the plurality rule is used?
Exercise 4 Assume for this exercise that individual preferences are rational and strict (i.e.
no individual is ever indifferent between two distinct alternatives) and that X is finite with
|X| ≥3.
A social welfare function F is anoligarchy with respect toJ ⊆I if (i)J is decisive, and (ii) every individuali∈J has veto power. A social welfare function F isoligarchicif there is some J ⊆I such that F is an oligarchy with respect to J.
A social welfare function F is quasi-transitive, if for all preference profiles (R1, . . . , Rn) and all triples of distinct alternativesx, y, z ∈X,xFp(R1, . . . , Rn)yandyFp(R1, . . . , Rn)zimply xFp(R1, . . . , Rn)z.
(i) Give an example of an oligarchic SWF that is not transitive.
(ii) (Harder) Let F be a quasi-transitive social welfare function that is weakly Paretian and independent of irrelevant alternatives. Show thatF is oligarchic.
You may benefit from structuring your proof along the following steps (of course, any other proof that you manage to come up with is fine as well).
Step 1: Given F and two alternatives x, y ∈ X, let I(x, y) ⊆ 2I be the set of decisive coalitions for xagainst y.
Show thatI(x, y) =I(x0, y0) for all x, x0, y, y0 ∈X.
Step 2: Show that I(x, y) is stable under intersection, that is, J, J0 ∈ I(x, y) implies J∩J0 ∈I(x, y).
Step 3: Show thatF is an oligarchy with respect to the intersection of all sets inI(x, y).
(iii) In light of the previous results, would you say that relaxing transitivity is a way out of the dilemma identified by Arrow’s Theorem?
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Exercise 5
(i) Consider a social choice problem with 11 individuals,I ={1, . . . ,11}, and an unrestricted set of alternativesX. Individual preferences are assumed to be rational but are otherwise unrestricted. Consider the social welfare functionF which for any two alternativesx, y ∈ X and any preference profile (R1, . . . , Rn), setsxFp(R1, . . . , Rn)y if and only if
|{i∈I :xPiy}| ≥7 andxRiy for all i∈ {1, . . . ,5}.
Show that this social welfare function is acyclic and weakly Paretian. Is it oligarchic and/or quasi-transitive?
(ii) (Harder) Now consider a social choice problem with 15 individuals, I0 = {1, . . . ,15}, and an unrestricted set of alternatives X. Individual preferences are again assumed to be rational but are otherwise unrestricted. Consider the social welfare function F which for any two alternatives x, y ∈ X and any preference profile (R1, . . . , Rn), sets xFp(R1, . . . , Rn)y if and only if
|{i∈I0 :xPiy}| ≥9 andxRiy for all i∈ {1, . . . ,5}.
Show that this social welfare function is not necessarily acyclic.
[Hint: Use a counterexample with 10 alternatives.]
Exercise 6 Show that the axioms of May’s Theorem are independent: For each of the three axioms used in the characterization give an example of a social welfare function that does not satisfy this particular axiom but satisfies the other two. Restrict attention to social choice problems with only two alternatives.
Exercise 7 Show that the example of the Condorcet paradox discussed in class is impossible if all individuals’ preferences are single-peaked with respect to the same strict ordering of alternatives.
Exercise 8 Exercise 21.D.10 from Chapter 21 of Mas-Colell, Whinston, Green (excluding (i)).
If you have any questions or corrections, please send them to awest (at) uni-bonn.de.
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