Flip-Floppin, Primary Visibility and the Selection of Candidates Marina Aranov ONLINE APPENDIX Appendix A: Basic Election Model Proof of Claim. Consider the challener who is believed to be moderate with probability sch 0, at the culmination of the eneral election stae and who competes with the incumbent with known type t Inc R. The challener s chances of winnin eneral election are determined by the preferences of the eneral-election median voter W s Ch Pr Challener with s Ch wins Pr Eum, s Ch > um, R Pr s Ch m M s Ch m L > m R R F sch M sch L R sch M pch sch L It is straiht-forward to check that the arument of F is strictly increasin in sch since L < M < R. Moreover, if sch 0 then W 0 F RL > 0 and if p Ch sch then > W F RM > F RL > 0, QED. Proof of Claim. Suppose that 0, and voters conjecture that, dependin on her type, the challener exerts efforts ê L and êm in the eneral election stae. Then, expected payoffs of liberal and moderate challeners who exert efforts e L and em, respectively, denoted by EΠ tch L e L and EΠtCh M e M, can be written as EΠ tch L e L e L W λ he L, M, n W µ W λ where EΠ tch M e M e M W µ he M, M, n W µ W λ µ and pch λ are specified in equations a and b. Assume that voters beliefs after observin liberal and moderate sinals durin the eneral election campain are the same, that is, µ pch λ and W pch µ W pch λ 0. In this case, both types of challeners would choose zero effort, which means that sinals are perfectly informative and we must have µ and pch λ 0. Thus, pch µ pch λ. Assume next that µ < pch λ then usin Claim we obtain W pch µ W pch λ < 0. In this case, liberal challener will exert zero effort, in which case moderate sinal would fully reveal a moderate challener µ. In that case, we obtain W pch µ W pch λ > 0, which contradicts our assumption above. Thus, the only beliefs ê L, êm that miht be consistent with equilibrium are µ > λ W µ W λ > 0
Given these beliefs, the moderate challener would choose to exert no effort, e M 0, and liberal sinal becomes perfectly informative of a liberal type, λ 0. For any pair of beliefs, êl define the best-response function of liberal challener el êl, pch as the one that maximizes her expected payoff deπ tch L e L de L h e e L, L, n W µ W 0 0 This best-response function is decreasin in ê L and satisfies 0 < el, pch < el 0, pch <. Therefore, there exists a unique fixed point e L such that e L êl, pch êl el. This optimal effort for the liberal challener is determined by equation 3b specified in Theorem, QED. Proof of Claim 3. We will use the Implicit Function theorem to prove this claim. Define where µ S, e L h e e L, L, n W µ W 0 0. pch hel,l,n S, e L h ee L, L, n dw pch µ d µ S, e L h e L ee e L, L, n W dw µ W 0 h ee L pch µ, L, n d el S,eL S,eL e L he L, L, n > 0 hel, L, n µ > 0 QED. he L, L, n < 0 hel, L, n Proof of Claim. Recall that Euz j, p k sk denotes expected utility of voter who has ideal point z j when the winner of the primary stae is candidate k who enerated sinal s k and, thus, believed to be moderate with probability p k sk : Euz j, p k s k p k s k W µ uz j, M W µ uz j, R he L p k s k, L, n W µ he L, L, n W 0 uz j, L he L, L, n W µ he L, L, n W 0 uz j, R p k s k W 0 pk s k µ p k s k W µ uz j, M µ W W 0 pk s k µ W 0 µ W pch uz i, L µ W 0 uz j, R On the other hand, a moderate politician who was revealed to be moderate in the primary stae and advances to the eneral election stae brins voter with ideal point z j expected utility of Euz j, where Euz j, W uz j, M W uz j, R The voter with ideal point z j will vote in the primary for a candidate k with belief p k sk 0, over candidate l with p l sl if and only if Euz j, p k sk Euz j,. Define function Sz j which captures the utility difference for a voter z j between supportin an uncertain type
p k sk 0, and a moderate type pl sl. This function can be re-written as Sz j Euz j, p k s k Euz j, γ uz j, L γ uz j, R uz j, M where γ p k sk W 0 pk sk pch µ W µ W 0 W p k sk W pch µ 0, µ First, we consider under which conditions Sz j 0 for z j 0, d. Sz j 0 can be written as W 0 uz j, L uz j, M W 0 uz j, M uz j, R > W 0 pch µ > p k s k µ W µ W 0 uz j, L W µ uz i, M W pch µ W 0 µ uz j, R If riht-hand side of the inequality is neative, then we only need to make sure that the left-hand side is positive. If, however, the riht-hand side is positive, then p k s k W 0 pch µ µ W µ W 0 uz j, L W µ uz j, M W pch µ W 0 µ uz j, R < < W 0 pch µ µ W µ W 0 uz j, L W µ uz j, M W pch µ W 0 µ uz j, R W 0 uz j, L uz j, M W pch µ W 0 µ µ uz j, L µ uz j, M uz j, R Concavity of function W sch uarantees that W µ W 0 µ µ uz j, L µ uz j, M uz j, R is decreasin in µ. Collectin all the terms, we obtain that the riht-hand side of inequality is bounded above by W 0uz j, L uz j, M W W 0uz j, M uz j, R, which is exactly the left-hand side of inequality. Thus, inequality is satisfied as lon as function W µ is weakly concave and the left-hand side of the inequality is non-neative, which simply means that voter z j prefers to nominate a liberal over a moderate type because the ains from nominatin a politician who is ideoloically closer to him outweihs the risk expressed in the utility terms from this candidate losin the election: W 0 uz j, L uz j, M W W 0 uz j, M uz j, R 0 Euz j, 0 Euz j, To see this notice that for all z j 0, d we have assumed that uz j, L > uz j, M > uz j, R, which ensures that µ uz j, L µ uz j, M uz j, R is decreasin in µ. Moreover, d W pch µ W 0 µ d µ dw d W W 0 p 0 as lon as dw pch µ W p d µ Ch µ W 0 µ, which is uaranteed for all µ 0, since d W p 0. d µ 3
Second, consider dsz j γ duz j, L γ duz j, R duz j, M M γl γr dz j dz j dz j dz j Therefore, if M γl γr then dsz j dz j 0 and it is enouh to make sure that S d 0 to uarantee that all voters with z j 0, d vote for the uncertain over moderate type in the primary election. If, however, M < γl γr then dsz j dz j > 0 and it is enouh to make sure that S0 0 to uarantee that all voters with z j 0, d vote for the uncertain over moderate type in the primary election, QED. Proof of Claim 5. Assume that voters believe that ê M 0,, ê L 0 and candidate A follows this stratey. We will show that candidate B wants to follows this stratey as well. First consider what liberal candidate B would do: where deπ tb L e L de L because EΠ tb L e L e L W hel, L, n hê M, M, n W ˆp µ e L ˆp λ he L, L, n 3 hêm, M, n ˆp µ ˆpλ ˆp λ ˆp λ he L M hê and ˆpλ ˆpλ, L, n hê M, M, n, M, n hê h ee L M, L, n, M, n W 3 hêm, M, n W ˆp µ e L ˆp λ < 0 hê M, M, n W 3 hêm, M, n W ˆp µ e L ˆp λ < < hêm, M, n W 3 hêm, M, n W 0 < hêm, M, n W 0 < 0 The first inequality follows from the fact that W ˆp µ e L ˆp λ 0 since liberal challener could choose zero effort in the eneral election stae and preferred not to. The second inequality follows from the condition W 0 > W. Therefore, liberal candidate B prefers to put no effort in the primary campain. Consider now incentives of moderate candidate B: EΠ tb M e M e M W hem, M, n hê M, M, n deπ tb M e M de M W ˆp µ he M, L, n 3 hêm, M, n 3 hê h ee M M, M, n, M, n W ˆp µ hêm, M, n W Define best-response function of moderate candidate B, e M moderate candidate B, e M 0,, that solves deπt exists and it is unique for all ê M 0,. ê M B M e M de M e M. This is the effort level of 0. Notice that best-response We are left to show that there exists a unique fixed point such that e M ê M êm, which is determined by the equation 3a. This follows from three observations: e M 0 > 0, e M <, and 3 dem ê M > 0 and d e M ê M < 0 by the Implicit Function theorem and dê M dê M
assumptions imposed on the scrutiny function h. We, therefore, conclude that there exists a unique fixed point e M ê M êm that constitutes part of the equilibrium stratey, QED. Proofs of Claims 6 and 8. As Theorem asserts, the optimal efforts of candidates e M, e L are determined by equations 3a and 3b, and depend upon n and n, which are exoenous parameters that capture prominence levels of the primary and the eneral election staes, respectively. To simplify exposition, we abuse notation and use the followin shortcuts e M Xn, n x and he M, M, n x, n e L Y n, n y and he L, L, n hy, n h Then equations 3a and 3b can be re-written as 3 V x, y, n, n x Ux, y, n, n h y W W W 0 0 W 0 Findin n, n, n n n n and n n is a straiht-forward application of Cramer s rule: n n n n n n n n where h hy x W > 0 hyy W W 0 h y W n W W 0 W n h < 0 h y h yn n 3 xx W 3 W x W 3 xn n x n W x hy 3 W W h y < 0 h n W W < 0 W 3 x hn 3 W n h x x W W n h W < 0 > 0 5
If scrutiny function satisfies condition C3, which asserts that xx x for all x 0,, then the sin of is neative. To see why this is the case, notice that G 3 xx W W W 0 x h xx 3 W W x W W W x h3 W h W 0 h3 First inequality follows from concavity of function W. Further, W 0 > W uarantees that h3 3 W W > W W 0 Once, the sin of is determined, so is the sins of the denominators < 0 and > 0 Thus, both liberal challeners in the eneral election and moderate candidates in the primaries exert less effort when primaries are more visible: e L n e M n n n n n n n neative positive < 0 positive neative < 0 In addition, we obtain by substitutin n and n d h h d dh < 0 into the derivative above and performin alebraic manipulations iven the assumptions imposed on scrutiny and winnin eneral election functions. Thus, voters believe that the likelihood that moderate type enerates sinal λ in the primary and sinal µ in the eneral election is lower when the primary race is more prominent. This completes the proof of Claim 6. Further, hiher prominence level of the eneral election incentivizes moderate candidates in the primary to exert hiher effort and results in more likely revelations of the liberal challeners in the eneral election stae: e M n n n n neative neative > 0 6
dhe L, L, n dhy, n h n h y dn dn n which completes the proof of Claim 8, QED. h n hn n h y < 0 hn hy n hy n Proof of Claim 7. We will prove that the likelihood of the candidate from the Democratic party winnin the election is decreasin in the prominence level of the primary selection process. We will continue usin the notation introduced in Proofs of Claims 6 and 8 to save trees: PrDemocrat wins eneral election d dh 3 d h d 3 dh 3 h d W W 0 h W W dh h3 W 0 d W 0 h W h W 0 W 0 d W W W 3 h h W 0 W W 3 W 3 W W W W d h dh h3 W First, note that W is weakly concave, and, therefore, W x Thus, 3 W W 0 W 0 W x W 0 x x 0,. If h W 0 W 3 h W h h3 h W h > 0 then the proof is complete, since both dh h y n < 0 as well as d n x n < 0, which uarantees that the whole derivative is neative. Assume that h W 0 W 3 h W h h3 h W h < 0. Then, we will use the fact that d dh h < and we will re-write the whole derivative as 3 d < dh W 3 W 0 3 W h W 0 h 3 W 3 h W 3 h h W dh 3h 3 h 3 h W W 0 W h h h 7
We will show now that the last bracket is positive: 3 3h h h W 3 3h h W 3 h W 0 W h h 3 h W 0 W This last inequality can be re-written as W xab W 0aW b where a 3h h, b and x h. Notice that x > b a b 3 3h h 3 3h 3 h > h h 3 h > 0 TRUE! Thus, since function W is strictly increasin and weakly concave we et W x > W b a b a a b W 0 b a b W QED. 8
Appendix B: Properties of Scrutiny Function In this section we discuss the role of the assumption Ac which states that a candidate that exerts no effort enerates sinal which coincides with her type for sure, that is, h0, t, n 0. This assumption plays an important part in Claim 3. In particular, when h0, t, n 0, we show in Claim 3, that the liberal challener exerts more effort in pretendin to be moderate durin the eneral election campain when prior on her bein moderate at the beinnin of the eneral election campain is hiher, del d > 0. Consider the basic election model in which candidate that exerts no effort has a very small but positive chance of eneratin sinal opposite from her true type. To capture this, we will modify assumption Ac: { } Ac h0, t, n ɛ > 0 where ɛ h0, t, n < min, h, t, n In the remainder of this section, we characterize optimal behavior of challener in the eneral election stae dependin on her type Claim * and then study how it varies with prior belief about challener s type, Claim 3*. Claim *. Assume that the parameters of the election ame satisfy assumptions Aa, Ab, Ac, Ad-Af, A-A6. If the belief about challener s type at the beinnin of the eneral election stae is deenerate, 0 or, then she exerts no effort irrespectively of her type. If, however, voters are uncertain about challener s type after the primary race, i.e. 0,, then the unique equilibrium in the eneral election subame prescribes the moderate challener to exert no effort and the liberal challener to put positive effort in mimickin the moderate type, where the amount of mimickin e L is determined by equation * below where µ h e e L, L, n W µ W ɛ ɛ hel λ, L, n and pch λ... ɛ ɛ hel, L, n Proof of Claim *. First note that Claim and its proof remain unchaned with modified assumption Ac*, as they do not depend on the properties of the scrutiny function h. Suppose that 0, and voters conjecture that, dependin on her type, the challener exerts efforts ê L and êm in the eneral election stae. Then, expected payoffs of liberal and moderate challeners who exert efforts e L and em, respectively, denoted by EΠtCh L e L and EΠ tch M e M, can be written as EΠ tch L e L e L W λ he L, M, n W µ W λ EΠ tch M e M e M W µ he M, M, n W µ W λ 9
where µ λ hê M, M, n hê M, M, n hêl, L, n hê M, M, n hê M, M, n hêl, L, n Assume that voters beliefs after observin liberal and moderate sinals durin the eneral election campain are the same, that is, µ pch λ and W pch µ W pch λ 0. In this case, both types of challeners would choose zero effort since deπtch L de L 0. Thus, we must have ɛ ɛ ɛ pch ɛ ɛ ɛ deπtch M de M < which is satisfied only for ɛ. In other words, as lon as ɛ <, W pch µ W pch λ is not part of the equilibrium system of beliefs. Assume next that µ < pch In this case, liberal challener will exert zero effort since λ then usin Claim we obtain W pch µ W pch λ < 0. deπ tch L de L h e e L, L, n W µ W λ < 0 and we must have he M, M, n he M, M, n ɛ < pch he M, M, n he M, M, n ɛ The inequality above is false as lon as ɛ h0, t, n < h, t, n he, t, n for all e 0, Thus, the only beliefs ê L, êm that miht be consistent with equilibrium are µ > λ W µ W λ > 0 Given these beliefs, the moderate challener would choose to exert no effort, since deπ tch M de M h e e M, M, n W µ W λ < 0 For any pair of beliefs, êl define the best-response function of liberal challener ēl e L êl, pch as the one that maximizes her expected payoff where µ deπ tch L e L de L ɛ ɛ hêl h eē L, L, n W µ W λ 0, L, n and pch λ ɛ ɛ hêl, L, n 0
We will show that this best-response function is decreasin in ê L. Define function S ē L, êl 0 and use Implicit Function Theorem to obtain the required derivative. For the simplicity of exposition, we will use the followin shortcuts in this part: p, hēl, L, n hē L and hê L, L, n hê L. S ē L, ê L h eē L W µ W λ 0 S ê L S ē L h eeē L W µ W λ < 0 h eē L W p p ɛ h eê L µ p ɛ p W p p ɛ h eê L hê L λ < 0 p ɛ p hê L ēl ê L < 0 Therefore, there exists a unique fixed point e L such that ē L el êl, pch êl el. This optimal effort for the liberal challener is determined by equation specified above, QED. Claim 3*. Assume that the parameters of the election ame satisfy assumptions Aa, Ab, Ac, Ad-Af, A-A6 and 0,. Then for all δ <, there exists ɛ > 0 such that ɛ < ɛ we have del > 0 on the domain 0, δ. d Proof of Claim 3*. We will use the Implicit Function theorem to prove this claim. Use the followin shortcuts to simplify the exposition: p, p µ µ, p λ λ and he L hel, L, n. Define Sp, e L h e e L W p µ W p λ 0 where p µ and p λ are described in equation * above. Sp, e L h eee L e L W p µ W p λ h ee L W p dpµ µ de L because W p µ dpµ de L where W p λ dpλ de L dp µ dp Sp, e L p W p p ɛh ee L p µ p ɛ p W p λ he L h ee L W p dpµ µ W p dpλ λ dp dp ɛhe L p ɛ p he L and dpλ dp W p dpλ λ < 0 de L ɛ he L p ɛ p he L p p ɛ h ee L p ɛ p he L < 0 Under what conditions Sp,e L p > 0? Sp, e L > 0 W ɛhe L p µ p p ɛ p he L > W p λ ɛ he L p ɛ p he L Recall that the winnin function W p is strictly increasin in p Claim. Thus, W p > 0
for all p 0,. In particular, W p µ W p λ W W 0 > 0. Now, consider the followin inequality W W 0 ɛ hel > ɛhe L p ɛ p he L p ɛ p he L The left-hand side of this inequality is a positive constant, while the riht-hand side approaches zero from above when ɛ approaches zero. Therefore, for any p, e L there exists ɛ p, e L > 0 such that for all ɛ < ɛ p, e L the inequality above is satisfied. This means that for all ɛ < ɛ p, e L Sp we have,e L p > 0, which is enouh to uarantee that del > 0, QED. d Example. To intuit Claim 3* and appreciate the role of assumption Ac* consider the basic election ame with the scrutiny function he, t, n ɛ n e probability winnin function W 8 9 0 and the prominence level of the eneral election is n 0. First note that these functions satisfy assumptions Aa, Ab, Ad-Af and A-A6. Moreover, if ɛ 0, then assumption Ac is satisfied, while if ɛ > 0 then assumption Ac is satisfied. Fiure depicts optimal effort of the liberal challener in the eneral election stae as a function of prior belief about her type,, for various values of ɛ: 0.0 and 0.00 and 0. Symbol X marks optimum when the peak is interior. Fiure : Optimal effort of liberal challener in the eneral election as a function of prior belief,. 0.00 epsilon 0.0 0.00 epsilon 0.00 0.00 epsilon 0 0.003 0.003 0.003 0.00 0.00 0.00 0.00 0.00 0.00 0 0 0 0 0. 0. 0.6 0.8 0 0. 0. 0.6 0.8 0 0. 0. 0.6 0.8 p p p As Fiure illustrates, for any positive value of ɛ, the peak of the function is interior. However, as ɛ approaches zero, the peak shifts to the riht, that is that maximizes e L pch approaches. When ɛ 0, the optimal effort of liberal challener is strictly increasin in on the domain 0, but drops discontinuously at to zero.
Appendix C: Election Game with Partially Informative Primaries Proof of Theorem. Behavior of the challener in the eneral election stae. Similar to the basic election model, the challener s chances of winnin eneral election are determined by the preferences of the eneral-election median voter. Since all reistered Democrats vote for the challener irrespectively of his type, the median voter in the eneral election stae has an ideal point z j z, and, consequently, believes that any Democratic nominee is equally likely to be a liberal or a moderate type. The arument presented in Claim and its proof holds here as well. That is, the unique equilibrium in the eneral election stae is for moderate challener to exert no effort, ẽ M 0, and for liberal challener to exert effort level ẽ L 0, which is determined by the equation h e ẽ L, L, n W µ W 0 where µ Behavior of reistered Democrats in the primary election. hẽ L, L, n At the end of the primary campain, reistered Democrats contemplate candidate l who is moderate for sure, p l, and candidate k whose type is uncertain and who is believed to be moderate with probability p k 0, : Euz j, p k p k Euz j, p k Euz j, W µ uz j, M W µ uz j, R hẽ L, L, n W µ hẽ L, L, n W 0 hẽ L, L, n W µ hẽ L, L, n W 0 uz j, L uz j, R Euz j, p k Euz j, W 0 hẽ L, L, n W µ W 0 uz j, L uz j, R W µuz j, M uz j, R The last inequality holds true for the majority of the reistered Democrats, because it is implied by Euz j, 0 Euz j, which is uaranteed by conditions C and C. Behavior of candidates in the primary stae. Assume that voters believe that ê M 0,, êl 0 and candidate A follows this stratey. We will show that candidate B wants to follows this stratey as well. First consider what liberal candidate B would do: EΠ tb L e L e L 3 hêm, M, n he L, L, n ẽ L hẽ L, L, n W µ hẽ L, L, n W 0 Notice that the expected payoff of the liberal challener last brackets does not depend on behavior in the primary stae. Therefore, deπ tb L e L de L h ee L, L, n ẽ L hẽ L, L, n W µ hẽ L, L, n W 0 < 0 3
Thus, liberal candidate B prefers to exert no effort in the primary campain. Consider now the incentives of the moderate candidate B: EΠ tb M e M e M hem, M, n hê M, M, n W µ deπtb M e M de M h ee M, M, n W µ Define best-response function of moderate candidate B, ẽ M ê M B M e M de M ẽm. This is the effort level of moderate candidate B, ẽ M 0,, that solves deπt 0. Notice that bestresponse exists and it is unique for all ê M 0,. It is easy to see that there exists a unique fixed point such that ẽ M ê M êm, which is determined by the equation 7a. This completes the proof of Theorem, QED. Proof of Claim 9. We will start by showin that conditional on observin a moderate sinal µ in the eneral election campain, belief about the challener in the election ame with partially informative primaries is hiher than the one in the basic election ame in which the challener won the nomination after eneratin liberal sinal, µ > pch µ. Assume that it is not the case, and, in fact, µ pch µ. Thus, we must have hẽ L, L, n pch hel, L, n pch he L, L, n hẽ L, L, n But at the same time we know that if µ pch µ then ẽl e L. This follows from the properties of the scrutiny function h ee < 0 and the equations 3a and 7a. Thus, hẽ L, L, n he L, L, n, and coupled with condition that < above. Therefore, it must be that µ > pch µ, which implies that ẽl > e L. To show that ẽ M > e M we consider equations 3b and 7b and notice that W pch µ > W pch µ > W pch µ hem, M, n contradicts inequality W W µ which coupled with the properties of the scrutiny function uarantees that ẽ M > e M, Q.E.D. Proof of Claim 0. To simplify the exposition, we will use the followin notation h hẽ L, L, n and hẽ M, M, n. Then, the probability that a Democrat wins the election in the model with partially informative primaries is PrDemocrat wins eneral election h W h3 µ h W 0 W h W µ W 0 µ h W 0 W 3 h W µ µ We compare this expression to the probability of a Democrat winnin the eneral election in the basic model, which is specified in Proof of Claim 7. To show that Democrats enjoy hiher
chance of winnin the election in the ame with partially informative primaries compared with the basic ame, we use properties of the winnin function W as well as Claim 9 which ranks effort levels of candidates in these two versions of the ame. In particular, the proof follows from alebraic manipulations usin the followin observations: W > W µ W h > W W µ > W 0 W 0 > W W x > x W 0 x W > h > h > 0 > > > 0 > h > h Q.E.D. 5
Appendix D: Election Game with Endoenous Primary Prominence Proof of Theorem 3. The election ame with endoenous primary prominence as defined in Section 5. is the same as the basic election model studied in Sections - except for the investment decisions of primary candidates. Therefore, to prove Theorem 3, it suffices to show that expected payoff of a moderate candidate is decreasin and expected payoff of a liberal candidate is increasin in the primary prominence. This would uarantee that liberal candidates are happy to invest in boostin primary visibility, while moderate ones refrain from doin so. Ex-ante expected payoff of candidate k who has type t k L can be written as EΠ tk L 3 hem, M, n he M e L W 0 he L, M, n, L, n W he L, L, n hem, M, n deπ tk L dhem, M, n 3 hem, M, n el n 3 hem, M, n he L e L W 0 he L, L, n h ee L, L, n el n, L, n W he L W he L he M, M, n, L, n hem, M, n W 0 he M, M, n W he L W 0, L, n hem, M, n he M, M, n W 0, L, n hem, M, n d he M,M,n he M,M,n he L,L,n To show that this derivative is positive, deπtk L > 0, we use equilibrium condition 3b, the fact that if liberal challener chose to exert positive effort in the eneral election stae this means he prefers this action to exertin no effort at all he M e L W 0 he L, M, n, L, n W he L W 0 > W 0, L, n hem, M, n as well as the comparative static results obtained in Claim 6 after substitutin derivatives e M n and el n into the expression above, and, finally, the fact that sequence of sinals λ in the primary and µ in the eneral election is less likely to come from the moderate candidates when primaries are more visible, d he M he M,M,n,M,n he L,L,n < 0, which is the last part of Claim 6. Ex-ante expected payoff of candidate k who has type t k M can be written as EΠ tk M e M hem, M, n 3 he M, M, n he M, M, n hem, M, n W he L W, L, n hem, M, n deπ tk M em n 3 hem, M, n hem, M, n 3 he M, M, n hem, M, n dhe M, M, n W W dhem, M, n W he L he L he M, M, n, L, n hem, M, n he M, M, n, L, n hem, M, n d he M,M,n he M,M,n he L,L,n Similarly, to show that this derivative is neative, deπtk M < 0, we use equilibrium conditions as well as comparative statics results obtained in Claim 6, QED. 6