Solution to exercise on the Downs model
a) Differentiating and solving the FOC gives p∗ = α. As the utility function is concave
everywhere, preferences are single peaked.
b) Single peaked prefs.⇒we can use the median voter theorem. Median agent is median
α = 1/2, so implemented policy is p = 1/2.
1
0,8
0,6
0,4
0,2
0
0
0,2
0,4
0,6
0,8
1
x
c) The figure shows party A’s vote share as a function of pA for the case of pB = 0.4.
Similarly, we see that A will get a majority (xA > 1/2) if pB < pA < 1 − pB when pB < 1/2,
pB > pA > 1 − pB when pB > 1/2 and win with probability 1/2 if either pA = pB or
pA = 1 − pB . Hence we get
uA (pA , pB ) =





1



if pB < pA < 1 − pB and pB < 1/2
or if 1 − pB < pA < pB and pB > 1/2


1/2 pA = pB or pA = 1 − pB




 0 otherwise
d) If pA = pB = 1/2 no party can gain by unilaterally deviate. If A chose either
pA < 1/2 or pA > 1/2 it will get less than 1/2 of the votes and hence loose for sure. Hence
pA = pB = 1/2 must be a Nash equilibrium.
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e) There are two typos here: The utility function should be
wA (p, xA , xB ) = − (p − ᾱA )2 + QvA (xA , xB )
(i.e. a minus and an over bar missing).
Q is weight put on being in office relative to implementing favorite policy. pA = pB = 1/2
is still a Nash equilibrium. To see why, assume A deviates. Then they loose for sure, and
implement policy is still 1/2. Hence for Q > 0 they are wore off.
f ) To see pA = pB = pC = 1/2 is not an equilibrium, just notice that A can win almost
1/2 of the votes and win by choosing pA = 1/2 + ε for ε a tiny positive (or negative) number.
g) If the utility is 1 for winning for sure etc., a possible Nash equilibrium is pA = 1/3,
pB = pC = 2/3. Then A wins for sure. To see that this is a Nash equilibrium, notice first
that A can’t do better by deviating. If B selects pB = 2/3 − ε for ε a small positive number,
they loose all voters to the right of 2/3, and competes with A for the voters in [1/3, 2/3].
They will then get the voters in [1/2, 2/3], which is 1/6 of votes, and hence still lose for sure.
By moving towards A, at some point C will take over as the winner, but nothing happens
to Bs utility. By choosing pB = 2/3 + ε, they will also get a share slightly below 1/3 and
still loose. Hence B cannot change its probability of winning by a unilateral deviation. The
same holds for C, hence this is a Nash equilibrium.
Notice that 1/3,2/3 is not the only combination where this holds.
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