ωi is replaced in the third term (4.2) by the density ψ of the popularity shock
δ; (ii) the constant in (4.2) depends on the number of parties in the opposition,
through the second term in NVC .15
Whatever the equilibrium policies at the policy formation stage, we have V J =
V*J. As explained above, in a four party system we have IVVC = 1/4. Hence, the
equilibrium seat share in the next legislature expected by each coalition partner at
the party formation stage is E(IVsC) = 1. By the symmetry of the model, this is
also the expected seat share of each of the two parties in the opposition, E(IVsPO) =
1. In a three party system, IIIVC = 1/3. Here, the expected equilibrium seat share
of each small party (P =1, 2) when in government is:
E (III sP )=1 -⅛ φ, (4.3)
while the single party in the opposition has an expected seat share of 2(1 -
E(IIIsPC)), namely:
E(J11 so) = 1 + 1 ψ . (4.4)
2 6φ
These two expressions illustrate an important difference between majoritarian and
proportional elections, where we had E (III sP ) = 1 and E (III sp ) = 2 .In terms of
expected seats, plurality rule implies an extra gain for a large party, and an extra
loss for a small party. This disproportionality effect raises the incentives to merge
into large parties under majoritarian elections.16
Finally, note that the term NVVC enters additively in the right hand side of (4.2).
Hence, the partial derivatives of E(sP) with respect to the policy instruments
do not depend on VV. This implies that the optimal policy choices of a coalition
government do not depend on the number of parties in the opposition, even though
its equilibrium expected seat share does.
15 Note that (4.2) gives us the equilibrium expected seat share: it was derived from (4.1), under
the assumption that the vote shares of the two parties in government are equal, so that the seats
gained by the whole coalition are split in half between them. Out of equilibrium, the coalition
partner with more votes gets all the seats accruing to the coalition, and his expected seat share is
given by the right hand side of (4.2) pre-multiplied by 2. We return to this point when deriving
the equilibrium policies in the next subsection, since there we also evaluate out-of-equilibrium
payoffs.
16 Since the expected seat shares must always lie between 0 and 1, (4.3) and (4.4) imply that
1 ≥ ψ-
1 ≥ 3φ ∙
22