If this condition is met, the four-party system is an equilibrium under majoritarian
elections. But this condition is always the case, since as noted in Footnote 16 the
right-hand side of (4.12) is smaller than 1, while the left-hand side is above 1.
Hence, under majoritarian elections too, a four party equilibrium always exists.
But now, it is not the only equilibrium. A two-party system is also an equilib-
rium if all groups prefer to merge rather than to remain split, given that the two
opposition groups have also merged, namely if :
- IIW12 ≥ IIIW1 . (4.13)
2
The right-hand side of (4.13) is the expected payoff to group 1 of remaining a
group-specific party while the opposing groups have merged. The left-hand side
of (4.13) is the expected payoff accruing to group 1 if it merges with group 2:
the term IIW12 is divided in half because each group gets half the party payoff
resulting from the merger.
Given the results stated above, condition (4.13) for a two-party equilibrium
can be re-written as:
1+2 ≤ 2Ψ . (4.14)
γ ^ 3φ ( )
Note that the left-hand sides of (4.14) and (4.12) are the same, but the right-hand
side of (4.14) is twice as large as the right-hand side of (4.12). Hence, for some
parameter values both a two-party and a four-party equilibrium exist (since the
left hand side of (4.14) cannot exceed 2, this requires that γ ≥ 1).
A two-party equilibrium is more likely to exist if ψ∕φ = Std(ω)∕Std(δ) is
large. That is, if aggregate voter mobility is large relative to within-group voter
mobility. As the reader may recall, the same conditions make it more likely that
equilibrium rents are higher under proportional elections. This makes intuitive
sense. If, say, aggregate voter mobility is large (Std(δ) small), the election outcome
is very uncertain. Then, the electoral advantage of a large party facing two small
parties is very significant, cf. (4.4) and (4.3), which naturally raises the incentive
to merge. At the same time, rent extraction is punished much more strongly by
the voters under majoritarian elections: recall that voter utilities are multiplied
by ψ rather than φ in the expression for expected seat shares. Similarly, small
within-group voter mobility (Std(ω) large) raises the electoral advantage ofa large
party.
We summarize the results of this section as follows:
Proposition 4
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