The name is absent

two parties in government always have the same vote share in equilibrium : equa-
tion (3.1) implies that _Nv_C¹ = _Nv_C² . Moreover, since all electoral districts are
homogenous, either the two parties in government win the whole legislature, or
the opposition wins the whole legislature, depending on the realization of δ.Ifthe
two parties in government win, a coin is tossed to award the seat to one of them
district by district. With a continuum of districts, in equilibrium each winning
party in government ends up with half the seats in the legislature.¹⁴

This argument implies that the equilibrium expected seat share of party P =
1, 2 in a coalition government, is:

E(N^sp) = ¹ P^rob[N^vp ≥ NvC] ^,                   (4.1)

2

where Nv_c denotes the minimum threshold needed to win the election in any one
of the identical districts given the number of parties. As before, the expectation
and the probability refers to the uncertainty regarding the realization of δ. If the
opposition is also split in two parties (i.e., N = IV), we have IVv_c = 1. To win,
the coalition parties in government thus need to carry at least half the votes in any
district; since the votes are split equally between them (as they are between the
opposition parties), any one of the parties in government needs to win a quarter
of the votes in any district. If instead the opposition consists of a single party
(i.e., N = III), IIIv_c = 3. As the vote for the government is split in half between
the coalition parties, the government parties win the elections only if each of them
has a vote share at least as large as that of the single opposition party.

Recall that the random variable δ has a uniform distribution with mean 0 and
density ψ. Using (3.1) and simplifying, we then obtain the expected seat share
in the next legislature, at the policy formation stage, for a party in a coalition
government:

E(N^sC) = 4 + (4 -N vc)φ ⁺                           (4.2)

ψ              1⁴

+4[(VP - V*^c) + ₂∑(vJ - VJ)].

J=3

This expression is similar to that under proportional elections, equation (3.2).
The differences are that: (i) the density φ of the idiosyncratic reservation utility

¹⁴ The reader may wonder why, then, the coalition parties do not strategically agree to split
the districts among themselves running only a single coalition candidate in each district. But in
our simple model, these agreements would not be self-enforcing. To satisfactorily address this
issue, a richer model is needed.

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