Problems for Part I.2

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Problems for Part I.2

All exercises on this problem set will be covered in class May 4th 2010. 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 Letf be a social choice function defined on the domain of all rational preference profiles. We say that f is

• anonymous, if for every permutation π : I → I we have that f(R^π) = f(R) where for alli∈I, R_π(i)^π =R_i, and

• neutral, if for any permutationσ :X →X we have thatf(R^σ) =σ(f(R)), where for all i∈I, σ(x)R_i^σσ(y) if and only ifxR_iy.

Show that a neutral and anonymous social choice function may fail to exist. [Hint: Consider the case where |N|=|X|= 2.]

Exercise 2 Suppose the SCF f is defined on the domain of all rational preference profiles.

LetD ⊆ Rbe some subset of the set of all rational preference relations. Let f be the restriction of f to the domain Dⁿ⊆ Rⁿ.

Show that if f is strategyproof on Rⁿ, then f is strategyproof on Dⁿ.

Exercise 3 Let f be a SCF that is defined on the set of all rational preference profiles. Is it always true that f is strategyproof if it is dictatorial? Either prove that this is the case or provide a counterexample.

Exercise 4 In this exercise you will show that if all individuals have strict rational preferences, strong monotonicity implies strategyproofness.

For the proof suppose that contrary to what is to be shown there is a strongly monotonic but not strategyproof SCF f. This means that f is strongly monotonic but there exist a strict preference profile R, an individual i ∈ I, and a strict preference relation R⁰_i, such that f(R⁰_i, R−i)P_if(R) =: x.

Using the following construction derive a contradiction to the assumed strong monotonicity of f:

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• Use the two preference relations Ri and R⁰_i to partition X into Y⁺ := {y ∈ X : yPix}, Y :={y ∈X : (xP_iy and yP_i⁰x) or y=x}, and Y⁻:={y∈X :xP_iy and xP_i⁰y}.

• Using this partition define a strict preference relation ˜R_i that “mixes” R_i and R⁰_i by setting

1. yP˜_iz if either (y∈Y⁺ and z ∈X\Y⁺) or (y∈Y and z ∈Y⁻) 2. ˜R_i|_{y,z} =R_i|_{y,z} if either {y, z} ⊆Y⁺ or {y, z} ⊆Y⁻

3. ˜R_i|{y,z} =R⁰_i|{y,z} if {y, z} ⊆Y

Exercise 5 In this exercise we consider a setting with two alternatives x and y and assume that all individuals have rational preferences. Consider the social choice functionf defined by f(R) =x if and only if xF^pl(R)y.

Show that f is strategyproof.

Exercise 6 Suppose there is an odd number n of individuals. Let Q be a strict preference relation on X and R_Q ⊆ P be the set of all strict rational preference relations that are single- peaked with respect to Q. Let f^C denote the social choice function that is defined for all preference profiles in Rⁿ_Q and selects the Condorcet winner for all profiles in Rⁿ_Q.

Show that f^C is strategyproof.

If you have any questions or corrections, please send them to awest (at) uni-bonn.de.

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