Phil 321 Week 10 Tutorial Session: November 13
Main topic: Social choice theory
1. Do these qualify as SWFs?
a) Constant function. Fix a particular preference ordering k. Set W(P) = k for every set of individual
preference orderings P. Is W a SWF?
b) Randomization. Suppose that there are 6 individuals and three alternatives a, b, c. For each set
of individual preference orderings P, determine how W(P) ranks a pair of alternatives by rolling a
die and using the ranking by that individual as the social ranking of that pair of alternatives. Is W
a SWF?
2. Let k be one individual and let D be everyone else (a nonempty set). Define the social ordering
as follows: to figure out whether a ≻ b, b ≻ a or a ~ b, use k’s individual preference ordering
unless everyone in D has the same preference for a vs. b, in which case use the ranking in D.
a) Does this define a SWF?
b) Which of the Arrow conditions (O, D, I, P) is satisfied by this social ordering?
3. Proofs for some of the missing cases in Arrow’s Theorem.
4. Prove that Harsanyi’s utilitarianism (either version) violates condition I (Independence of
Irrelevant Alternatives).