large party is larger than 1/2, and given by (4.6). Using (2.5) and (4.5), his
expected payoff simplifies to:
IIIW P = 2(1 + γ) + 2- , P = 12 . (4.9)
γψ 3φ
Next, consider the two smaller parties, P =3, 4. Again, their expected payoffs
at the start of the government formation stage are the same as in (3.9). But their
expected seat share in the next legislature is now given by (4.3). Using (2.5) and
(4.5), the equilibrium expected payoffs of the small parties simplify to:
III W P = 2(1 + γ ) — ɪ, P = 3,4 . (4.10)
γψ 3φ
Of course, this difference between the expected welfare of the small and large
parties in a three-party system reflects the extra electoral bonus for a large party
facing two small parties that we discussed in Section 4.1.
4.2.2. Party formation
We now turn to the party formation stage. As we shall see, there is no universally
dominant strategy for the four existing groups. Because of this, we can find
conditions for two-party equilibria as well as four-party equilibria, while three-
party equilibria are ruled out by symmetry.
Specifically, we have a four-party equilibrium if all groups of legislators prefer
to remain split rather than to merge, given two group-specific parties on the
opposition side. More precisely, using the above notation, a four-party system is
an equilibrium if, say,
IV W1 ≥ - IIIW 12 . (4.11)
2
The left-hand side of (4.11) is the expected payoff of party(=group) 1 in a four-
party system. The right-hand side of (4.11) is the expected payoff accruing to
group 1 if it merges with party 2, given that the opponents have remained split:
the term IIIW12 is the expected payoffs of the large party P =12in a three-party
system, divided equally between the merging groups. Given the symmetry of the
model, if condition (4.11) holds for party P =1, it also holds for all the other
parties.
Given the results stated above, condition (4.11) can be re-written as:
' ≥ ψ. . (4.12)
γ ≥ 3φ ( )
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