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APPENDIX B Supplement to Chapter 3 Proofs of Propositions Presented in Chapter 3 PROPOSITION 1: Ifa ~ =a ~,e =a~Ij1 a~Ij1 =0, prob((Ik )/probt(Im) > v2, and Y~k "* Y.~k for some voters r,s, then the unique Stackelberg equilibrium party announcements are rel = (argmaxvki EUrk)/aki' where Mk is the set ofvoters with the median ideal point in k.lfprobtClk)/probtClm) v2 IEU{(yw) - EU{(Yklj1 )1, which becomes probt(Ik)/probtC1m) > v2 by way of equations (3.8) and (3.10) and some algebraic manipulation. Votersj for whomYki = yj%, i = O,'1' flip a coin between ideologies by virtue of the condition placed on equation (3.8) in case of equality. Thus if the parties adopt positions that translate into distinct hi' all voters use ideology Ik, and, by standard arguments, there is a unique median ideological position and ° wins by adopting it. If both parties adopt the relevant rel, they both stand an equal chance of winning voters with and without ideal points ytk' An analogous argument proves the sufficiency of the strict inequality in the opposite direction. • COROLLARY IA: If the assumptions ofproposition 1 obtain except that probtC1k)/probllm) =v2, then there are equilibrium position announcements rei =(argmaxy'i EU/y")/aki' PROOF: When the assumed equality holds, voters not only are indifferent between ideologies but, again by equations (3.7) and (3.8), find that ideological translation does not affect their evaluation of the relative merits of party positions . This means that positions in the two ideologies can be measured on a common dimension. Specifically, we define the line y' such that Y' =Yk =Ym/v along with measures J.l(Yd and J.l(Y",) such that J.l(Yi) = 112 x the number of 191 192 Appendixes voters with ideal points atYi' This induces a measure /ley') = /l(Yk) + /l(Y,'/v). Since Ii /levi) =4N + 2, there will, in general, be a closed interval of median points (see Enelow and Hinich 1984, 8-12). In any event, given median points M,., candidates will take equilibrium positions nf = (argmax,,'ki EU/y')/aki .• COROLLARY 1C: For any mix of voters satisfYing the assumptions of proposition I, corollary IA, and corol/my IB, there are equilibrium position announcements nf =(argmax"'ki EU/y'·)/aki . PROOF: For those voters satisfying the assumptions ofcorollary 1A or 1B, the construction ofy' and !ley') follows directly. For those voters satisfying the assumptions of proposition 1, we apply 2!l(yd or 2!l(Ym) as appropriate.• PROPOSITION 2: The optimal candidate strategies are determinedfor any /inite number o{ideologies. PROOF: Since preferences over ideologies are determined by expected utility numbers, from proposition 1 and its corollaries each voter has a preference ranking-technically, a weak ordering-over ideologies that is independent of particular party position announcements. For each voter who has a uniquely preferred ideology Ik' we apply 2!l(Yk) as in corollary 1C. For each voter who has a preferred set of n ideologies Ik for which the assumptions of corollary IA hold, we apply !ley') =Ii !l(Yi )/n. The results of corollary IC follow. . • PROPOSITION 3: Ifcr ~8 =cr ~'8 =cr ~1jI =cr ~'1jI =0, y,* '*Y~kfor some voters r,s, and prob(t_I)CId/prob(t_I)(lm ) > v2 , then Stackelberg equilibrium candidate announcements are (n;'", k), where nJk = (argmax"ki EU/yk)/aki' PROOF: Clearly, 8 is not ill served by making ideological and position announcements that are consistent. If8 makes announcement (n~k' k), then \f' can announce (n~k' k) or (n~m' m). In the former case, the algebra of Bayesian updating confirms that \f' merely reinforces 8's choice; in the latter case the algebra confirms that prob(t_1 Pk)/prob(t_1 Pill) = prob,(Ik)/prob,(lIIl)' also confirming 8's choice. On the other hand, with an announcement of (n~m' m), 8 can at best equal its results with (n~k' k). • COROLLARY 3A: If the assumptions ofproposition 3 obtain except that probu_1)(Ik )/probu_1)(Im) =v2 and (1) there does not exist a n~k =(argmaxl'k8 EU/yk)/ak8 such that n~k E (argmaxl" EU/y")/ak8 or (2) there does not exist a n~1I1 =(argmax\'m8 EU~m)/am8 such that n~m E (argmax\,o EU~")/aI1l8' then the incumbent 8 loses. . Appendixes 193 PROOF: In response to announcement (1t~k> k), k =1,2, \f' announces (1t~n" m), 1t~m E (argmaxym", EU~")lam",' m...

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