ECOS3035: Economics of Political Institutions
Homework I
1. Consider a society of three individuals whose preferences over the four possible outcomes are:
Person 1: a ≻ c ≻ b ≻ d
Person 2: b ≻ a ≻ d ≻ c
Person 3: d ≻ c ≻ b ≻ a
Consider a series of pairwise comparisons between the alternatives and assume that individuals
vote sincerely.
(a) For the profile of preferences above, are social preferences intransitive?
(b) Now consider changing person 3’s preferences to d ≻ b ≻ c ≻ a. Are social preferences
transitive for the new preference profile?
(c) Relate your answers in parts (a) and (b) to Arrow’s theorem? Be brief here.
2. Consider a society of three individuals whose preferences over the three possible outcomes are:
Person 1: a ≻ b ≻ c
Person 2: b ≻ c ≻ a
Person 3: c ≻ a ≻ b
Consider a series of pairwise comparisons between the alternatives and assume that individuals
vote sincerely.
(a) For the profile of preferences above, are social preferences intransitive?
(b) Now consider changing person 3’s preferences to c ≻ b ≻ a. Are social preferences
transitive for the new preference profile?
(c) Order the alternatives a, b, and c from left to right and relate your answer in part (b) to
single peaked preferences.
3. Consider a society of three individuals and where the set of alternatives is {a, b, c, d}. For each
of the voting rules below, specify which of the following conditions from Arrow’s theorem
it satisfies or violates: Pareto Principle, Non-dictatorship, and Independence of Irrelevant
Alternatives. If a condition is satisfied, provide a short (one or two sentences) reason why. If
it is violated, provide a counter example.
1
• Borda Count: if there are m candidates, a candidate gets m points for every voter who
ranks her first, m − 1 points for a 2nd-place ranking, and so on. Candidates are then
ranked according to their vote totals.
• Unanimity Rule: an alternative x is strictly preferred to an alternative y if and only if
at least one member of society strictly prefers x to y and no members strictly prefer y
to x.
• Alternatives are always socially ranked in alphabetical order.
4. Prove that the axiom of anonymity implies non-existence of a dictator. This is a very short
proof.
5. Consider the Hotelling/Downs model of political competition with a unit mass of consumers
uniformly distributed over the interval [0, 1]. There are two candidates (candidate 1 and
candidate 2) who simultaneously choose a policy on the interval [0, 1], and then voters vote
for one of the two candidates. Let sj denote candidate j′s policy, j = 1, 2. A voter’s utility
depends on the policy s that wins and his location i ∈ [0, 1] and is given by ui(s) = −|i− s|.
A candidate gets a utility of 1 from winning and -1 from losing. Ties are broken with a fair
coin toss.
(a) Suppose candidate 1’s strategy s1 < 0.5. What is candidate 2’s best response (that is,
specify all the s2’s which are optimal for candidate 2, given s1).
(b) Suppose candidate 1’s strategy s1 = 0.5. What is candidate 2’s best response?
(c) Suppose candidate 1’s strategy s1 > 0.5. What is candidate 2’s best response?
(d) In a graph with s1 on the horizontal axis and s2 on the vertical axis, plot these best
responses for candidate 2.
(e) On the same graph plot candidate 1’s best responses for each s2.
(f) Depict the Nash equilibrium on this graph where both players are playing a best response
to each other.
6. Consider the same setting as in the previous question, but now assume that each candidate
maximizes their vote share.
(a) Suppose candidate 1’s strategy is either s1 < 0.5 or s1 > 0.5. Does candidate 2 have a
best response? If not, explain (in one sentence) why.
(b) Suppose candidate 1’s strategy s1 = 0.5. What is candidate 2’s best response?
(c) What is the Nash equilibrium of the game?