The name is absent

b) a type of government (coalition or single-party majority) selected by nature
at the government formation stage, given the equilibrium party system.

c) a policy optimally selected by the parties in each possible government, given
the equilibrium party system and the type of government, and taking into account
the electoral rule and the expected equilibrium outcome at stage d).

d) an election outcome, given the electoral rule, the equilibrium economic
policy, type of government, and party system.

3. Proportional elections

Under proportional elections, the seat share of each party, Ns^P_G , is identical to its
nation-wide vote share, Nv_G^P . In the first subsection, we compute the equilibrium
economic policies under coalition governments and single-party governments, fo-
cusing on the last two stages of the game, elections and policy formation. In the
next subsection, we characterize the equilibrium party system and the equilibrium
type of government.

3.1. Equilibrium policy

3.1.1. Coalition government

Consider a four-party system (N = IV). We assume (without loss of generality
given the symmetry of the model) that parties 1 and 2 form a coalition government,
while parties 3 and 4 make up the opposition. We first describe equilibrium
vote shares and seat shares as a function of economic policy and then compute
equilibrium policies.

Equilibrium seat shares Consider the voters in group J = P, where P =1 or
2 is one of the two parties in government. Pick a voter in this group, with a value
of ωⁱ exactly equal to VJ — V*J — δ. By (2.6), this “swing voter” in group I is just
indifferent between voting for P and voting (randomly) for an opposition party.
All voters of the group with a lower value of ωⁱ vote for party P. Let F(∙) denote
the cumulative distribution function of ωⁱ in (2.6). The fraction of voters in group
J = P voting for party P is thus F (V P — V *P — δ), while the complementary
fraction 1 — F (V P — V *P — δ) votes (randomly) for the opposition.

The vote share, and hence also the seat share, for each of the parties in the
coalition government (P =1, 2) is:

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