which is exactly the same expression as in the four-party case. This should not
come as a surprise. As shown in the previous section, the rents appropriated
in equilibrium by each party in government are exactly the same, rCP = rSP.
Moreover, the probability of being included in government is the same ( 1 ), as is
the expected value of future rents. Future expected rents are the same because the
higher expected seat share of a single party, E(II-sSS) = 1 = 2E(IVsP), is exactly
compensated by the lower value of total future rents accruing to a single party
government, r*S = 1 r*C .
Three party system Finally, consider a three-party legislature, say P =12,
3 and 4. Then both a single-party government and a coalition government are
possible, with equal probabilities, 1/2. Here the parties differ, and we have to
keep track of their identity. The large party, P =12, can only be in a single-party
government. As argued in Section 3.1, however, equilibrium rents for a single-
party government, r*SP, do not depend on the number of parties in the opposition,
and neither does the value of future seats, RP = r*S. Moreover, the expected seat
share of a large party is always equal to 1/2, irrespective of the number of parties
in the legislature. Thus it follows that the expected payoff of a large party is
the same as in the two-party system,IIIWP =II WP, for P =12, as given by
the expression in (3.12). By similar reasoning, the smaller parties, P =3, 4, can
only be in a coalition government, and their expected payoff is the same as in a
four-party system IIIWP =IV WP as given by the expression in (3.10). All in
all, the number of parties represented in the legislature does not matter for the
parties’ expected payoffs.
3.2.2. Party formation
It is now straightforward to compute the equilibrium of the party formation stage.
As we have just seen, the expected equilibrium payoffs for parties are the same
independently of the number of parties in the legislature: each party formed during
the course of the party formation stage always has a continuation utility given
by both (3.12) and (3.10). But this payoff accrues entirely to the members of
legislative group J if it forms a group-specific party, while it is divided in half
between two groups merging into a large party. Therefore, no party will ever want
to merge. Indeed, remaining split is a dominant strategy at the party formation
stage. Applying the equilibrium conditions in Section 2.2.1, a four-party system
is thus the unique equilibrium of the game under proportional elections.
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