Problem Set 3: Micro 1
Nov 2020 - Feb 2021
(1) Assume that two players are competing over a monopoly rent R. Each player
i = 1, 2 simultaneously invests an amout xi ≥ 0 with the purpose of acquiring
this rent. Given that player i, j have invested xi , xj respectively, the probability of
success of player i is given by

xri


with xi , xj > 0
 r
xi + xrj
pi =
,
1 with xi > 0, xj = 0



1/2 with xi = xj = 0
where r > 0 determines the impact of differences in expenditures on probability
of winning the prize.
(a) Solve for the Nash equilibrium of this game (x∗i , x∗j ) with r = 1.
(b) Solve for the pure-strategy Nash equilibrium with r = 3 (if any).
(c) How does x∗i compare with R for the two cases r = 1 and r = 3? (5+2+3=10)
(2) In many markets consumers have switching costs. Consider the following
simple model of such a market: Two firms A and B simultaneously and noncooperatively set prices in a single period for a commodity that they can each
produce at zero cost. There are n + s customers, where n > 0 and s > 0 all with
reservation price R. Because of switching costs, s/2 customers can only buy from
A and s/2 can only buy from B. The n “new” customers buy from the cheapest
firm, if at all.
(a) Show that there are no pure strategy equilibria.
(b) Find a mixed-strategy Nash equilibrium in which each firm chooses price p
according to the distribution F(p). (10+10=20)
(3) Consider the following game for n players. Each of the players selects a
number from the set {1, 2, ..., 100} and a cash prize is split evenly among the
players who’s numbers are closest to two-thirds the average of the n numbers
chosen.
Show that the problem is solvable by iterated elimination of weakly dominated
strategies, meaning the method can be used to eliminate all but one strategy for
each player, which necessarily gives a Nash equilibrium. (A strategy µi of a player
0
i is called weakly dominated if there is another strategy µi that always does at least
as well as µi and is strictly better than µi for some vector of strategies of the other
players.) (10)
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(4) Five ferocious pirates are dividing their plunder (100 gold coins). They are
ordered from 1 to 5, and suggests ways of sharing the coins as follows. Pirate 1
suggests a way of sharing the coins (where no single coin can be subdivided). For
example, he might suggest {80, 5, 5, 5, 5} meaning that he gets 80 coins and the
others get 5 each. All five pirates then vote on the proposal. If a majority accept,
the division is carried out. If the majority decline, then Pirate 1 is thrown overboard. Ties are broken in favour of the proposer. If there is agreement, the game
ends. If Pirate 1 has been thrown overboard, we return to the Örst step, where Pirate 2 suggests a division, and so on. The process continues until an agreement is
reached. Explain carefully what happens on the path of the (unique) subgame perfect equilibrium (assuming any voter who is indi§erent (at equilibrium) between
accepting or rejecting an offer, votes to reject). (10)
(5) Platform Selection Problem. Two candidates 1 and 2, select platforms in an
election fought over a single issue/dimension. These platforms/positions are quantifiable as the variable Q. All voters are assumed to vote. Each voter’s preferences
can be described by a symmetric, single-peaked utility function over Q. Different
voters have different preferred platforms i.e. values of Q which maximize their
utility. All candidate platforms are believed with certainty by all voters. Thus each
voter votes for the candidate whose announced position is nearest on the Q axis
to the preferred position of the voter. The election is decided by simple majority.
The values of Q which are advocated by candidates 1 and 2 are denoted x and y
respectively.
The utility functions of the two candidates are given by
U1 (x, y) = α1 P1 (x, y) − β1 (x − x
b)2
U2 (x, y) = α2 P2 (x, y) − β2 (y − yb)2
where P1 (x, y) denotes the probability that candidate 1 wins the election and P2 =
1 − P1 the corresponding probability that 2 wins the election. The ideal point of
candidate 1 is x
b ∈ Q, while that of 2 is yb ∈ Q, with x
b > yb. Thus candidates care
not only about winning but also about how far they choose to locate from their
ideology (ideal/bliss point). The parameters αi , βi > 0 for i = 1, 2.
Denote the location of the median voter as Qm . Assume that Qm is not known
with certainty by the candidates, but the subjective probability density for Qm
(which both candidates agree on) is f (Qm ), with a corresponding c.d.f. F (Qm ).
Assuming that
1/2e for Qm − e < Qm < Qm + e
f (Qm ) =
0 elsewhere
2
solve for the Nash equilibrium of the game (x∗ , y ∗ ). (20)
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