1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

200 D. Kselman

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which targets all voters between the median ideal point and the swing voterxS=

xˆP(GP)=^3 / 2 −GP. 

6.3 Proof of Lemma2 for the CaseGP>^1 / 2

The median voter receives a utility of ‘1’ from the set of actionsvm. On the other

hand, Lemma2 tells us that, whenη=1, the median voter’s utility for necessary

condition deviations whenGP<^1 / 2 will be:

um,P

(

x(Gˆ P,),ΘˆP(GP)

)

=GP+

(

1 −GP

δ+^1 / 2

)

. (A.9)

WhenGP>^1 / 2 , partyPcan consider both locally optimal deviations with a bare

majority is target set and the median policy stance (Lemma2), or deviations to the

political right or left (Lemma3). If the former, the median voter’s utility whenη= 1

will be (A.9). If the latter, the median voter’s utility for locally optimal deviations

whenη=1 will be:

um,P

(

x(Gˆ P,),ΘˆP(GP)

)

=(GP)^2 +

(

1 −GP

δ+ 1 −GP

)

. (A.10)

To prove Lemma2, I first establish that, for anyGP>^1 / 2 , the median voter will

always receive a higher utility from the deviation stipulated in Lemma2 than that

stipulated in Lemma3:(A.9)>(A.10) (algebra omitted). This in turn implies that

the strategy identified Lemma2 is more likely to yield payoff-enhancing deviations

than is that identified in Lemma3, i.e. if the strategy from Lemma2 yields a payoff-

enhancing deviation then so does the strategy in Lemma3, but not vice versa. This

establishes Lemma2 in the text, i.e. that for any value ofGP<1 Lemma2 identifies

the necessary condition strategy for payoff-enhancing deviations.

6.4 Proof of Proposition 1

Whenη=1, as long asδ>^1 / 2 theredoes notexist a payoff-improving deviation

fromvmto a valueGP<1, and conversely as longδ<^1 / 2 theredoesexist a payoff-

improving deviation fromvmto a valueGP<1.

Given a deviation fromvmto the necessary condition strategy, it is straightfor-

ward to see that, as long as the median voter prefers the deviating candidatePto

the her opponent∼P, then do all other voters inP’s target set. The median voter

receives a utility of ‘1’ from the set of actionsvm. On the other hand, whenη=1,

the median voter’s utility for the necessary condition strategy whenGP<1 will be:

um,P

(

x(Gˆ P,),ΘˆP

)

=GP+

(

1 −GP

δ+^1 / 2

)

. (A.11)