In this part of the proof we establish a number of properties that a Nash equilibrium, x ˆ , should satisfy, employing a series of Lemmas All statements are presented assuming the existence of at least one equilibrium.

Lemma 1: In any Nash equilibrium,xˆ,B gets a strictly positive expected vote share in the general election.

Proof: Assume that in some equilibrium, xˆ, B gets an expected vote share equal to zero in the general election. Then, if B deviates toxB = l, she wins her party’s primary with certainty. This is so because: a)

vB > vA and b) for everyxA > l−vB +vAit is true that ui(xA, vA) < ui(l, vB)for every i < l+vB −vA and for every xA < l + vB −vA it is true that ui(xA, vA) < ui(l, vB) for every i > l − vB +vA. That is, for every xA ∈ [0, m] a majority of the primary voters of the leftist party votes for B. Moreover, since

vB >max{vA, vC, vD},Bis voted in the general election at least by all voters with an ideal policy in(l−ε, l+ε) for someε >0. That is, her general election vote share is at least equal toΦ(l+ε)−Φ(l−ε), which is strictly positive by the fact thatΦ is strictly increasing. In other words,B has incentives to deviate from the posited strategy and, hence, a strategy profile such that B gets an expected vote share equal to zero in the general election, cannot be an equilibrium.

Lemma 2: In any Nash equilibrium,xˆ,B wins the primary of the leftist party with certainty.

Proof: Since by Lemma 1 in any Nash equilibriumBgets a strictly positive expected vote share in the general election, it must be the case thatB advances to the general election with a strictly positive probability. Given that candidates are allowed to use only pure strategies, there are two cases: eitherB advances to the general election with certainty or with probability 1₂. CandidateB advances to the general election with probability 1₂ if and only if she receives exactly the same share of votes in the primaries of the leftist party asA. This may happen if and only if|xA−xB|> vB−vAand either xA+vA+₂xB−vB =lor xA−vA+₂xB+vB =l. In both cases the

candidate located to the left oflexpects a positive vote share conditional on qualifying to the general election (by the fact that valence differences are not very large) and can deviate marginally towards l, securing a sure win in the primaries and practically doubling her expected general election vote share. Hence, it cannot be that in equilibriumBadvances to the general election with any probability smaller than1.

The proof of the next lemma is similar to the proof of Lemma 1 –but not precisely identical. To make the proof as easy to follow as possible, we preferred to provide the whole line of reasoning in its support (which is, to a great extent, a repetition of the arguments in the proof of Lemma 1), rather than to just point out where one should make the minor modifications.

Lemma 3: In any Nash equilibrium,xˆ,Cgets a strictly positive expected vote share in the general election.

Proof: Assume that in some equilibrium,xˆ,Cgets a zero expected vote share in the general election. Then, ifC deviates to xC = r, she wins her party’s primary with certainty. This is so because: a)vC > vD and b) for everyxD > r−vC +vD it is true that ui(xD, vD) < ui(r, vC)for everyi < r +vC −vD and for every

xD < r+vC−vDit is true thatui(xD, vD)< ui(r, vC)for everyi > r−vC+vD. That is, for everyxD ∈[m,1] a majority of the primary voters of the rightist party votes forC. Moreover, since valence differences are not very large,Cis voted in the general election at least by all voters with an ideal policy in(r−ε, r+ε)for some

ε >0. That is, her general election vote share is at least equal toΦ(r+ε)−Φ(r−ε), which is strictly positive by the fact thatΦis strictly increasing. In other words,Chas incentives to deviate from the posited strategy and, hence, a strategy profile such thatC gets an expected vote share equal to zero in the general election, cannot be

an equilibrium.

The arguments supporting Lemma 4 are symmetric to the ones supporting Lemma 2 and are, hence, skipped.

Lemma 4: In any Nash equilibrium,xˆ,C wins the primary of the rightist party with certainty.

Lemma 5: In any Nash equilibrium,xˆ, it is the case thatxˆA= ˆxB−vB+vA.

Proof: Consider first that, in equilibrium,xˆA<xˆB−vB+vA. Since by Lemma 2 it is the case thatBwins with certainty the leftist party’s primaries, it should be the case that xˆA+vA+ˆxB−vB

2 < l. By Lemmas 3 and 4 we

know thatC advances to the general election with certainty and that she receives there a strictly positive vote share. Hence, there existsε >0such that ifBdeviates toxˆB+εit will still be the case thatxˆA+vA+ˆx₂B+ε−vB < l (Bwins the leftist party’s primary with certainty) and moreoverBwill secure a strictly positive increase in her general election vote share.10 _{So in equilibrium it cannot be the case thatxˆ}

A < xˆB −vB +vA. Now consider 10_{Since in any equilibrium,xˆ,Cqualifies to the general election with certainty (Lemma 4) and expects a positive general election}

that, in equilibrium,xˆA>xˆB−vB+vA. IfxˆA>xˆB+vB−vAthen, given thatBwins with certainty the leftist party’s primaries (by Lemma 2), it should be the case that xˆA−vA+ˆxB+vB

2 > l. Again, there existsε > 0such

that ifBdeviates toxˆB+εit will still be the case that ˆxA−vA+ˆx₂B+ε+vB > l(Bwins the leftist party’s primaries with certainty) and moreoverBwill secure a strictly positive increase in her general election vote share. So in equilibrium it cannot be the case thatxˆA >xˆB−vB+vAeither.

Lemma 6: In any Nash equilibrium,xˆ, it is the case thatxˆA=l.

Proof: Consider first that xˆA < l. Then by Lemma 5 it follows thatxˆB = ˆxA+vB−vA < l+vB −vA. This suggests that there existsε >0such that ifBdeviates toxˆB+εit will be the case that xˆA+vA+ˆx₂B+ε−vB < l (Bwins the leftist party’s primary with certainty) and moreoverBwill secure a strictly positive increase in her general election vote share (applying the same reasoning as footnote 10). Hence, in equilibrium, it cannot be thatxˆA < l. Now consider thatxˆA > l. By Lemma 5 it follows thatxˆB = ˆxA+vB−vA> l+vB−vA. This suggests thatAcan deviate toxˆA =l, win her party’s primary with certainty and secure a strictly positive vote share in the general election (by the fact that valence differences are not very large). Hence, in equilibrium, it cannot be thatxˆA> leither.

Lemma 7: In any Nash equilibrium,xˆ, it is the case thatxˆB =l+vB−vA. Proof: This is a trivial implication of Lemmas 5 and 6.

Lemma 8: In any Nash equilibrium,xˆ, it is the case thatxˆC =r−vC +vD andxˆD =r.

Proof: Consider a Nash equilibrium, xˆ. Since: a) by Lemma 4,C wins with certainty the primary of the rightist party, b) by Lemmas 6 and 7xˆA = l andxˆB = l +vB −vA, and c) valence differences are not very large; it must be the case that xˆC ∈ [r −vC +vD, r+vC −vD], because whenC makes such a choice, D does not qualify to the general election with a positive probability for anyxD ∈ [m,1]. Otherwise –that is, if

ˆ

xC ∈/ [r−vC +vD, r +vC −vD]–D could locate atr, win the primary of the rightist party with certainty and secure a positive vote share in the general election. Moreover, by the fact thatxˆB < mand that valence differences are not very large, the general election vote share ofC, conditional onC qualifying to the general election, is strictly decreasing on [r− vC +vD, r +vC − vD]. Hence, in any equilibrium, xˆ, it is the case thatxˆC = r−vC +vD. Finally, if in equilibrium xˆC = r−vC +vD and xˆD 6= r, then C can deviate to

ˆ

xC =r−vC+vD−εfor someε >0, and thus, qualify to the general election with certainty and increase her vote share (Lemma 3), we must have eitherxˆB+vB −vC <xˆCor ˆxB+vB −vC = ˆxC. In both cases a marginal move ofB

towards the right induces an increase in her general election vote share: in the first case, only a marginal increase, and in the latter a substantial one.

general election vote share (sincexˆB < mand valence differences are not very large). So, if in equilibrium we havexˆC =r−vC +vD, we should also havexˆD =r.

Hence, by Lemmas 6, 7 and 8, it follows that there exists a unique strategy profile that is a candidate for an equilibrium in our game;xˆ= (l, l+vB−vA, r−vC+vD, r).

Step 2 In this part of the proof we have to show that no candidate has incentives to deviate from xˆ = (l, l+vB − vA, r −vC +vD, r). First of all, note that for this profile, in each primary the party’s median voter is indifferent between the high and low valence candidate. This is also true for all more extreme voters than the median that split their support between the two primary candidates equally. All more more moderate voters than each party’s median strictly prefer the high valence candidate over the low valence candidate and hence candidatesB andC are the ones competing in the general election. Given that valence differences are not very large, in this strategy profile vote shares in the general election are hencePA(ˆxA,xˆ−A : v,Φ, l, r) =

PD(ˆxD,xˆ−D :v,Φ, l, r) = 0andPB(ˆxB,xˆ−B :v,Φ, l, r) = 1−PC(ˆxC,xˆ−C :v,Φ, l, r) = Φ(xˆB+vB+ˆ₂xC−vC). CandidateAwill have incentives to deviate if there existsx´A∈[0, m]such thatPA(´xA,xˆ−A:v,Φ, l, r)>0. But ifA deviates to x´A < l then ´xA+vA+ˆ₂xB−vB < l (B wins the primary of the leftist party with certainty) and hencePA(´xA,xˆ−A : v,Φ, l, r) = 0. IfA deviates to x´A > l then at least all voters with ideal policies in [0, l +vB −vA) will vote for candidateB in the primary of the leftist party and, hence, B will win the primary of the leftist party with certainty, inducingPA(´xA,xˆ−A:v,Φ, l, r) = 0. Therefore, candidateAhas no incentives to deviate away fromxˆA=l. Similar arguments rule out incentives for deviation away fromxˆD =r for candidateD.

If candidateB deviates tox´B < l+vB−vAthen, even if she wins in the primary of the leftist party, she gets a vote share ofΦ(x´B+vB+ˆxC−vC

2 )that is strictly smaller thanΦ( ˆ

xB+vB+ˆxC−vC

2 ). That is, her payoff is strictly

smaller thanPB(ˆxB,xˆ−B :v,Φ, l, r).If candidateB deviates tox´B > l+vB−vAthen xˆA+vA+´₂xB−vB > l and henceAwins the primary of the leftist party with certainty. That is,PB(´xB,xˆ−B : v,Φ, l, r) = 0. Therefore, candidateB has no incentives to deviate fromxˆB = l+vB −vA. Similar arguments rule out incentives for deviation away fromxˆC =r−vC+vD for candidateC.

Proof of Corollary 1. In the unique equilibrium characterized in Proposition 1Bis the winner ifΦ(xˆB+vB+ˆxC−vC

2 )> 1 2 ⇐⇒ l+ 2vB −vA+r−2vC +vD > 2m,C is the winner ifΦ( ˆ xB+vB+ˆxC−vC 2 ) < 1 2 ⇐⇒ l+ 2vB−vA+

l+ 2vB−vA+r−2vC +vD = 2m.

Proof of Proposition 2. As in the Proof of Proposition 1 in Step 1 we identify a unique strategy profile that satisfies the equilibrium properties and then show in Step 2 that this profile is indeed an equilibrium. The proof is similar to Proposition 1, however further arguments are needed since by permitting open primaries each deviation implies a change in the primary electorates. Recall that in this case valence differences not being very large means thatvB∈(0,v˜B)for some˜vB >0.

Step 1Additional to similar arguments as in the case of closed primaries we now start the proof by showing