1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

198 D. Kselman

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xS=

(^3) / 2 −{GP·( 1 −xP)}
1 +GP

xP if and only if xP<^3 / 2 −GP.

In turn, for anyGP>^1 / 2 Sub-lemma1 allows to expressxSas follows:
xS=
{
∅ if^1 / 2 <xP<^3 / 2 −GP,
(^3) / 2 −{GP·( 1 −xP)}
1 +GP ifxP>^3 /^2 −GP.
(A.4)
We now move to identifying ideological swing votersxSin the range[xP, 1 ].Given
our specification of programmatic utilityui,P(prog), for anyGP>^1 / 2 the following
expression implicitly definesxSwhen∼PchoosesvmandPchoosesxP>^1 / 2 :
1 −(xS−^1 / 2 )=GP·
{
1 −(xS−xP)
}

. (A.5)

This can be rewritten as:

xS=

(^3) / 2 −{GP·( 1 +xP)}
1 −GP

. (A.6)

Basedon(A.6) we can establish the following Sub-lemmas:

Sub-lemma 2Fo r a n yGP>^1 / 2 ,when∼PchoosesvmandPchoosesxP>^1 / 2 ,

there is no swing voter ideological voterxSin the range[xP, 1 ]for values ofxP<

(^1) / (^2) GP.
Sub-lemma 3Fo r a n yGP>^1 / 2 ,when∼PchoosesvmandPchoosesxP>^1 / 2 ,
there is no swing voter ideological voterxSin the range[xP, 1 ]for values ofxP>
(^3) / 2 −GP.
Proof of Sub-lemma 2 We are looking for swing ideological voters in the range
[xP, 1 ]. By definition, if (A.6) generates a valuexS>1, then there is no swing
ideological voterxSin the range[xP, 1 ]: no voters in the applicable range satisfy the
indifference condition in (A.6). It is then straightforward to establish that (algebra
omitted):
xS=
(^3) / 2 −{GP·( 1 +xP)}
1 −GP

1 if and only if xP<^1 / (^2) GP. 
Proof of Sub-lemma 3 We are looking for swing ideological voters in the range
[xP, 1 ]. By definition, if (A.6) generates a valuexS<xP, then there is no swing
ideological voterxSin the range[xP, 1 ]: no voters in the applicable range satisfy the
indifference condition in (A.6). It is then straightforward to establish that (algebra
omitted),
xS=
(^3) / 2 −{GP·( 1 +xP)}
1 −GP

^3 / 2 −GP.