The probability pA that party A will win the election equals the probability that
πA is at least one half. Using (A.3) through (A.5), this condition may be written
as
pA ≡ Pωr πA ≥ 1/2
(A.6)
= Pr '' (UA - UB) + 1 φo (Uoa - UB) + αih (Za - Zb) ≥ ω
ωφ φ o o
where φ ≡ αiφi +(1- αi) φo is the average dispersion of ideological preferences
across the two groups of voters. As the general ideological preference ω is uni-
formly distributed on the interval
1 1
2ψ , 2ψ
, the probability in (A.6) is
pA
α. ■ ' ■ B ■ 1 ι ■ ■ A ■ B ■ ∙ ■ Z ■ Z- ../
1∕ψ
1 + ψ [α<≠, (UA - UB) + (1 - αi) φo (Uaa - UB) + φαih (Za - Zb)]
2φ
(A.7)
Lobby officials choose their campaign efforts to maximise the objective function
(2.18), subject to the constraint that efforts cannot be negative. Since (A.7)
implies ∂pa∕∂Za = αihψ and ∂pa∕∂Zb = -αihψ, the first-order conditions for
the solution to the lobby problem may be written as follows, where μA and μB are
the Kuhn-Tucker multipliers associated with the non-negativity constraints on ZA
and ZB, respectively:
αihψ (UiA - UiB) - ZA - μA =0, -αi hψ (UiA - UiB) - ZB - μB =0, (A.8a)
ZA ≥ 0,ZB ≥ 0,μA ≥ 0,μB ≥ 0, μAZA =0, μBZB =0.
(A.8b)
From (A.8) we get the results stated in (2.19) which in turn imply that
ZA- ZB = αihψ (UiA - UiB) . (A.9)
Substituting (A.9) into (A.7), we find
PA = 1 + ψ [α< (Φi + φαiψh2) (UA - UB) + (1 - “i) Φo (UA - UB)] . (A.10)
2φ
32
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