Problems for Part I.2
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(Rπ) = f(R) where for alli∈I, Rπ(i)π =Ri, and
• neutral, if for any permutationσ :X →X we have thatf(Rσ) =σ(f(R)), where for all i∈I, σ(x)Riσσ(y) if and only ifxRiy.
Show that a neutral and anonymous social choice function may fail to exist. [Hint: Consider the case where |I|=|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 Show that if all individuals have strict rational preferences, strong monotonicity implies strategyproofness.
[Hint: Consider some strongly monotone SCF f. Suppose there is a preference profile R such that x0 :=f(R0i, R−i)Pif(R) =:xfor some i∈I and R0i ∈ R. Use a preference relation ˜Ri that ranksx0 first and x second to derive a contradiction to the assumed monotonicity.]
Exercise 5
(i) Construct one simple example which shows that the Borda count is neither strongly monotone nor strategyproof.
(ii) Let f be an arbitrary Condorcet consistent SCF. Construct one simple example which shows thatf is neither strongly monotone nor strategyproof.
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Exercise 6 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 xFpl(R)y.
Show that f is strategyproof.
Exercise 7 Suppose there is an odd number n of individuals. Let Q be a strict preference relation on X and RQ ⊆ P be the set of all strict rational preference relations that are single- peaked with respect to Q. Let fC denote the social choice function that is defined for all preference profiles in RnQ and selects the Condorcet winner for all profiles in RnQ.
Show that fC is strategyproof.
If you have any questions or corrections, please send them to awest (at) uni-bonn.de.
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