1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

196 D. Kselman

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Theoretical Appendix

6.1 Proof of Lemma2 for the CaseGP≤^1 / 2

IfGP≤^1 / 2 andP’s opponent∼P choosesvm, it will be impossible to forP to

persuade any voters on programmatic grounds. To see this note that, whenGP≤

(^1) / 2 , no voter will have a purely programmatic utility forP greater than (^1) / 2 (i.e.
ui,P(prog)≤^1 / 2 for all voters). As well, note that all voters have a programmatic
utility of at least^1 / 2 for any candidate∼Pwho choosesvm: the voters least satisfied
with this platform are those with ideal pointsxi=1 andxi=0, and for these voters
ui,∼P(prog)=^1 / 2 for any party∼Pwhich chooses the median voter programmatic
vectorvm.
As a result, whenGP≤^1 / 2 andP’s opponent∼Pchoosesvm,Pwill only gain
the support of voters who are in its target set. In turn, any deviation from the outcome
v 1 =v 2 =vmwill need to involve a target set of at least half the electorate in order to
givePa chance of winning. Furthermore, any target set greater than a bare plurality
contains more voters than necessary to win the election, and thus will not represent
the necessary condition choicesxˆP(GP), andxˆP(GP)(recall above definition of
necessity).
By Assumption1 above, this bare plurality target set will include the median
voter. The median voter will be the voter from this target set whose allegiance will
be most difficult to gain, since the opposing party∼Pchooses the median voter’s
ideal point atvm. It follows thatxˆP(GP)=xm.
6.2 Lemma3 and the Ideological Swing Voter
WhenGP>^1 / 2 andP’s opponent∼Pchoosesvm, it may bepossibleto forPto
persuade some voters on programmatic grounds. In turn, there may exist payoff-
enhancing deviations forP which do not involve choosing a bare plurality target
set. Lemma3 establishes the necessary condition strategy for a payoff-enhancing
deviation which does not involve a bare plurality target set. Put otherwise, if the
strategy identified in Lemma3 leads does not lead toπP>^1 / 2 , then no deviation
without a bare plurality target set is payoff-enhancing. Lemma3 establishes the
necessary condition strategy for a payoff-enhancing deviation on the political right;
a symmetric condition applies on the political right.
Lemma 3Fo r a n yGP>^1 / 2 ,the necessary condition strategy without a bare
plurality target set on the political right isxˆP(GP)=^3 / 2 −GPandΘˆP(GP)=
[xm,(^3 / 2 −GP)].
This lemma, tells us that for anyGP>^1 / 2 the necessary condition strategy for
payoff-enhancing deviation on the political right involves the platformxˆP(GP)=