Proof of Proposition 1. Denote the beliefs of consumer T 2 fL; Cg, given messagem2 fL;C;Sg, by
T(m) P(t=ljm). Then let
pL(m) v+ 2 L[ L(m) 1=2] (11) and
pC(m) v 2 C[ C(m) 1=2] (12)
denote the willingness to pay of liberals and conservatives, given message m, where UL(m; pL(m)) = 0 from (1) andUC(m; pC(m)) = 0 from (2). Given that pricing does not a¤ect beliefs, and the price is set at the pro…t maximizing level, we can restrict attention top2 fpL(m); pC(m)g, form2 fL;C;Sg.
Consider a candidate uninformative equilibrium, where both CEO types choose (mC; mL) = (S;S) with probability one. Beliefs on the equilibrium path are L(S) = L(S) = 1=2 by Bayes’Rule. Recall our assumption that beliefs o¤ the equilibrium path are given by the prior: for any m 6= S, beliefs are also T(m) = 1=2 forT 2 fL; Cg. It then follows from (11) and (12) that pL(m) = pC(m) = v, for any messagem: pricep=vis pro…t maximizing, with pro…ts = (nL+nC)v. Thus, deviating to any message pair (mC; mL)6= (S;S) is not pro…table.
To show that an informative equilibrium does not exist, we proceed by contradiction. Consider a candidate equilibrium where liberals do not always maintain their prior beliefs, i.e. where the CEO sets mL =C and mL=L with positive probability, with L(C)<1=2 and L(L)>1=2. Equilibrium pro…ts, when messagemL=C is sent, are bounded above bymaxfpL(C)(nL+nC); pC(C)nCg, where (11) implies
pL(C)< v and wherepC(C) v
Now consider a deviation where the CEO chooses message pair (mC =C; mL =L) with probability 1. Deviation pro…ts are then bounded below by minfpL(L); pC(C)g(nL+nC). These strictly exceed equilibrium pro…ts given message mL = C, since (11) implies pL(L) > v; and since nL > 0, nC > 0. Hence, this is not an equilibrium. The proof for the case when liberals always maintain their prior beliefs, but conservatives do not, is entirely analogous.
Proof of Proposition 2. As argued in the proof of Proposition 1, in any candidate equilibrium, we
can restrict attention to prices pL(m) and pC(m), de…ned by (11) and (12), for messagesm 2 fL;C;Sg. Denote consumer beliefs given message m by (m) = P(t = ljm), which must now the same for both consumer types.
For part (i) of Proposition 2, this follows immediately from the proof of Proposition 1.
For part (ii), …rst consider a fully revealing candidate equilibrium, where a liberal CEO sendsm=L
with probability 1, and a conservative CEO sendsm=Cwith probability 1. We then have (L) = 1 and
(C) = 0, by Bayes’ Rule, and recall that we assume out-of-equilibrium beliefs (S) = 1=2. Thus, (11) implies pL(L) > pL(S)> pL(C), wherepL(L) =v+ L , pL(S) =v, and pL(C) = v L . Moreover, (12) impliespC(L)< pC(S)< pC(C), withpC(L) =v C ,pC(S) =v, and pC(C) =v+ C .
Suppose that, in this fully revealing equilibrium, given any message m 2 fL;Cg, only one consumer type buys. That is, the price givenm=Lis p=pL(L), giving pro…ts =nL(v+ L) = L for a liberal CEO; and the price given m = C is p = pC(C), giving pro…ts =nC(v+ C) = C for a conservative CEO. A conservative CEO can deviate to m = L and earn L, whereas a liberal CEO can deviate to
m=C and earn C. By C 6= L, one of these deviations must be pro…table.
Suppose instead that, in this fully revealing equilibrium, there is at least one messagem2 fL;Cgsuch that both consumer types buy. The price, given m, must satisfy p < v, by pL(C) < v and pC(L) < v,
giving pro…ts < v(nL+nC). The CEO sendingm can instead deviate to message S, which along with pricep=v, results in strictly higher pro…ts: =v(nL+nC) = LC. Thus, a fully revealing equilibrium cannot exist.
We now consider candidate equilibria that are partially revealing. Suppose a liberal CEO sends message m =S with probability , messagem =L with probability (1 ), and message m=C with probability (1 )(1 ), with 2[0;1)and 2[0;1]; and a conservative CEO sends message m=S
with probability , messagem=Lwith probability 2[0;1], and messagem=Cwith probability1 . We will show that, for any belief pair( 1; 2), with 1 > 12 and 2< 12, we can …nd( ; )2[0;1] [0;1]
such that (L) = 1 and (C) = 2, i.e. so that these beliefs follow from Bayes’Rule and CEO strategies
for any <1.
This amounts to showing, that we can …nd and such that
1 = + and 2 = 1 (1 ) + (1 )
Expressing and in terms of 1 and 2 gives: ( 1; 2) = 1(1 2 2) 1 2 ; (13) and ( 1; 2) = (1 1)(1 2 2) 1 2 : (14)
The denominator of (13) and (14) is strictly positive, by 1 > 12 and 2 < 12; the numerator of (13)
is strictly positive, by 2 < 12; and the numerator of (14) is non-negative, by 2 < 12. To establish ( 1; 2)2[0;1], it remains to show that 1(1 2 2) 1 2, which indeed holds by 2 1 2 > 2. To
establish ( 1; 2) 2[0;1], it remains to show that(1 1)(1 2 2) 1 2, which indeed holds by (1 1)(1 2 2) < 1(1 2 2) 1 2. Thus, for any 1 > 12 and 2 < 12, beliefs ( 1; 2) can be
induced through an appropriately chosen CEO (mixed) strategy, with = ( 1; 2) and = ( 1; 2).
Given ( ( 1; 2); ( 1; 2)), liberal consumers are willing to pay pL(L) > v conditional on receiving message L, and pL(C) < v conditional on receiving message C, where pL(L) = v+ 2 L( 1 12) and
pL(C) =v+ 2 L( 2 12) . Conservative consumers are willing to pay pC(L)< v conditional on receiving message L and pC(C) > v conditional on receiving message C, where pC(L) = v 2 C( 1 12) and
pC(C) =v 2 C( 2 12) .
Hence, the optimal price given message L, if the …rm only wants to serve only one consumer type, is pL(L), such that only liberals buy. Denote the corresponding pro…ts by ( 1) = nLpL(L). The optimal price given message C, if the …rm wants to serve only one consumer type, is p = pC(C), such that only conservatives buy. Denote the corresponding pro…ts by ( 2) =nCpC(C).
We can rule out a candidate equilibrium with ( = 1; = 0), since messages Land C would then be perfectly revealing. For mixed strategy equilibria, note that by inducing the appropriate beliefs 1 2(12;1],
we can obtain any pro…ts ( 1) =nL[v+ 2 L( 1 12) ]2(nLv; L]. By inducing the appropriate beliefs 2 2[0;12), we can obtain any pro…ts ( 2) =nC[v 2 C( 2 12) ]2(nCv; C]. It follows that, for any
2(maxfnLv; nCvg; minf L; Cg], there is( ; ), such that both messagesm=L and m=C are sent with strictly positive probability, and the CEO is indi¤erent between these messages (and thus willing to mix).
Consider such a candidate equilibrium, with given pro…ts 2 (maxfnLv; nCvg; minf L; Cg]. If
2 (0;1) in this candidate equilibrium, so that the message m = S is sent with positive probability, then we need to check that the CEO is indi¤erent between any m 2 fL;C;Sg. Message m =S induces prior beliefs (S) = 1=2; the pro…t maximizing price isp =v, so that all consumers buy, and pro…ts are LC = (nL+nC)v. Thus, equilibrium pro…ts , given m 2 fC;Lg, must equal LC. Notice that LC >
maxfnLv; nCvg. Hence, an informative equilibrium with pro…ts = LC, where the …rm sometimes serves both consumers types, exists if and only if LC minf L; Cg. In any equilibrium with > LC, then we must have = 0, so that a niche strategy is indeed used with probability 1.
If = 0 in the candidate equilibrium, we must still rule out that the CEO might deviate by sending message m = S, and earning LC = (nL+nC)v. Again LC > maxfnLv; nCvg, so we have the result stated in the proposition: if LC minf L; Cg, then there exists an informative equilibrium where both CEO types earn if and only if 2 [ LC; minf L; Cg]. If LC > minf L; Cg, then an informative equilibrium does not exist.
Proof of Proposition 3. Consumer beliefs L =P(t =ljL) = 1+1 , and C = P(t= ljC) = 0, follow
immediately from Bayes’ Rule and the CEO’s equilibrium strategy. Given these beliefs, (11) and (12) implypL(L) =v+ 2(1+1 12) L =v+11+ L > v 11+ C =pC(L)along withpC(C) =v+ C >
v L =pL(C). Thus, the consumer behavior as stated in the proposition is optimal, given pricing and consumer beliefs. Pro…ts givenm =L are nL(v+11+ L ), and pro…ts given m=C arenC(v+ C ). Thus, by the de…nition of , the …rm is indi¤erent between sending the two messages; it earns pro…ts of nC(v+ C ) = C with probability one. Notice, moreover, that > 0, by L = nL(v+ L ) >
nC(v+ C ) = C, and that <1, by C > LC > nLv. It remains to check that there is no pro…table deviation involving a price for which all consumers buy. The most pro…table such deviation would be m=S, i.e. deviate to the message of staying silent, so that consumers maintain their prior beliefs, with p=vequal to the willingness to pay of both types. Doing so yields pro…ts = LC. Hence, this deviation is not pro…table, by LC < C.
Proof of Proposition 4. Consider a candidate uninformative equilibrium, where both CEO types earn
LC = (nL+nC)v. A necessary condition for a deviation to be pro…table is that it leads consumers to update their beliefs away from the prior. Notice from (3) and (4) that consumer willingness to pay is linear in beliefs P(t =ljm). Moreover, the non-pecuniary payo¤ for the …rm, 2k[P(t = ljm) 1=2] is also linear in these same beliefs.
In particular, this linearity means that when considering a deviation for a liberal CEO, we can restrict attention to messagem=L and beliefs (L) =P(t=ljL) = 1. The payo¤ for a liberal CEO, given these beliefs and the resulting optimal pricep=v+ L, is L+k. The corresponding payo¤ for a conservative …rm is L k. Thus, the deviation is equilibrium dominated for a conservative CEO, but not a liberal CEO, if and only if both L+k LC and L k < LC hold. This is equivalent to
k > maxf LC L; L LCg (15)
Thus, applying the our restriction on out-of-equilibrium beliefs, the deviation tom=L reveals the CEO to be liberal, i.e. (L) = 1, making the deviation pro…table for a liberal CEO, if and only if (15) holds. Similarly, a deviation to m=C reveals the CEO to be conservative, i.e. (C) = 0, making the deviation pro…table for a conservative CEO, if and only if
k > maxf LC C; C LCg (16)
i. LC < C: then both (15) and (16) hold for allk < C LC, but (16) is violated for allk > C LC. ii. C < LC<( L+ C)=2: then both (15) and (16) hold for allk < LC C, but (16) is violated for
all k > LC C.
iii. ( L+ C)=2< LC < L: then both (15) and (16) hold for allk < L LC, but (15) is violated for all k > L LC.
iv. L< LC: then both (15) and (16) hold for allk < LC L, but (15) is violated for allk > LC L. Taking together, i. - iv. implies the result as stated in the Proposition.
Proof of Proposition 5. First consider a partially revealing candidate equilibrium as described in
Proposition 3, but with replaced by k, de…ned by (9). Given the CEO’s equilibrium strategy, (11) and (12) implypL(L) =v+1+1 kk L > v 11+ kk C =pC(L)along withpC(C) =v+ C > v L =pL(C). By the de…nition of k, a conservative CEO is indi¤erent betweenm=Landm=C. Moreover, comparing condition (9) with Proposition 3 (i) shows that k = whenk= 0. Holding k constant, the right-hand side of (9) is increasing ink, whereas the left-hand side of (9) is decreasing ink. The left-hand side of (9) is decreasing in k, for allk < L2 C given that Case i or ii of Proposition 5 applies (i.e. C >2 LC L), since 2 LC L > nL(v L ). Thus, it must be that k is decreasing in for allk < L2 C. Direct substitution shows that k = 0 ifk = L2 C. Hence, k 2(0; ) for all 0 < k < L2 C. Moreover, for
k L C
2 , there is no k2(0;1)that satis…es (9), so the partially revealing equilibrium does not exists
for such largek.
To check when the partially revealing equilibrium will exist, given 0 < k < L C
2 , consider the
incentives of a liberal CEO. A liberal CEO prefers m=Ltom=C if (10) holds. The right-hand side of (10) is smaller than the right-hand side of (9), and the left-hand-side of (10) is larger than the left-hand side of (9), for allk >0. Thus, since (9) holds whenever 0< k < L C
2 , (10) must hold as well.
Moreover, in this candidate equilibrium, the payo¤ of a conservative CEO is uc = C+k, given by the right-hand side of (9). The payo¤ of a liberal CEO is ul =nL(v+11+ k
k L ) +k 1 k
1+ k, given by the
left-hand-side of (10). Comparing conditions (10) and (9) shows that ul> uc.
Now consider a deviation from the partially revealing candidate equilibrium that involves an unex- pected message, that of staying silent,m=S. If consumers believe the deviating CEO is liberal for sure,
(S) = 1, then a liberal CEO can earn L+k > ul, given price p = v+ L , which is pro…table. If consumers believe the deviating CEO is liberal for sure, (S) = 1, then a conservative CEO can earn L k, given pricep=v+ L , which is pro…table if L k > C+k. If consumers maintain their prior beliefs following the deviation, (S) = 1=2, then a conservative CEO can earn LC. This is pro…table if LC > C+k. Thus, our restriction on out-of-equilibrium beliefs will imply that the deviation reveals the CEO as liberal (and thus makes the deviation pro…table for a liberal CEO), if k <( L C)=2 and
k < LC C. We showed above thatk < L2 C must hold in the partially revealing equilibrium, so the relevant condition for a deviation to be pro…table isk < LC C.
If the deviation does not reveal the CEO as liberal, and instead induces prior beliefs (L) = 1=2, then the resulting pro…ts are uc = ul = (nL+nC)v = LC. Thus, given 0 < k < L2 C, the relevant condition for a deviation from the partially revealing equilibrium to be pro…table, for a conservative CEO, is LC > C +k, i.e. again k < LC C, as stated in the Proposition.
Now suppose that k L C
2 , so a partially revealing equilibrium does not exist. Consider a fully
this equilibrium, for a liberal CEO, we have m = L i.e. selling to liberal consumers; for a conservative CEO, we havem=C, i.e. selling to conservative consumers. Letk0 k L C
2 0. Then the incentive
constraint for a conservative CEO can be written as nL(v+ L ) k0 L C
2 nC(v+ C ) +k
0+ L C
2 ;
which reduces tok0 0. Similarly, the incentive constraint for a liberal CEO can be written as nL(v+ L ) +k0+
L C
2 nC(v+ C ) k
0 L C
2 :
which reduces to k0 > C L, where C L < 0. Thus, for both incentive constraints to hold, the relevant condition isk0 0, i.e. k L C
2 .
Now consider a deviation from the fully revealing candidate equilibrium, where a CEO sendsm=S. A conservative CEO earnsuc= C+kin this candidate equilibrium, whereas a liberal CEO earnsul= L+k. Sinceul > uc, a liberal CEO can only possibly pro…t from a deviation if LC > L+k. Since L k < C+k fork L C
2 , a conservative CEO can only possibly pro…t from a deviation if LC > C+k. Thus, if LC < C+2 L, then no CEO type could possibly pro…t from a deviation, for allk L2 C. Consumers will therefore maintain their prior beliefs (S) = 1=2. As a result, given the deviation to m = S, all consumers buy at pricep=v, resulting in payo¤uc =ul= (nL+nC)v= LC < C+k < L+k. Hence, if LC < C+2 L, then no CEO type has a pro…table deviation from the fully revealing equilibrium, given out of equilibrium beliefs (S) = 1=2. This establishes the result in parts (i) and (ii).
Now suppose C+ L
2 < LC <
3 L C
2 . Then, a liberal CEO cannot possibly pro…t from a deviation,
because LC < L+kfor allk L2 C, and because the deviation payo¤ is bounded above bymaxf L+
K; LCg. For given k L2 C, it is either the case that a conservative CEO can possibly pro…t from a deviation, or cannot do so. In the former case, consumers infer the CEO is conservative for sure, (S) = 0, when observing an unexpected message. As a result, the highest payo¤ a conservative CEO can earn by deviating to an unexpected message is C +k, which is equal to his equilibrium payo¤. In the latter case, by de…nition, a deviation will not be pro…table. It follows that, for C+ L
2 < LC <
3 L L
2 , a fully
revealing equilibrium exists for all k L C
2 , as stated in part (iii).
Now suppose instead 3 L C
2 < LC. Then, a liberal CEO can possibly bene…t from a deviation if L C
2 < k < LC L, because LC > L+kholds for these values ofk, but not for anyk > LC L.
A conservative CEO can also possibly bene…t from a deviation if L C
2 < k < LC L, because LC > L+k > C+k. As a result, for L2 C < k < LC L, consumers who observe an unexpected message will maintain their prior beliefs, (S) = 1=2. Thus, after the deviation, all consumers buy, yielding a payo¤ of LC. This deviation is pro…table for both CEO types, LC > L+k > C+k. Thus, the fully revealing equilibrium does not exist, for L C
2 < k < LC L.
For k > LC L, it is either the case that a conservative CEO can possibly pro…t from a deviation, or cannot do so. In the former case, consumers infer the CEO is conservative for sure, (S) = 0, when observing an unexpected message. As a result, the highest payo¤ a conservative CEO can earn by deviating to an unexpected message is C +k, which is equal to his equilibrium payo¤. In the latter case, by de…nition, a deviation will not be pro…table. It follows that, for 3 L C
2 < LC, a fully revealing
equilibrium exists for all k > LC L.
We now show that no other informative equilibrium exists, other than that identi…ed in the Proposition. Consider a partially informative candidate equilibrium where both CEO types mix, with message-price pairs given by (m; p) = (L; pL(L)) and (m; p) = (C; pC(C)). Denote consumer beliefs, conditional on
receiving these messages, by (L) P(t =ljL) and (C) P(t=ljC). Then the indi¤erence condition for a liberal CEO is
nLpL(L) + 2k (L) 1
2 =nCpC(C) + 2k (C) 1 2 ;
and the indi¤erence condition for a conservative CEO is nLpL(L) 2k (L)
1
2 =nCpC(C) 2k (C) 1 2 :
These conditions cannot hold simultaneously, for k >0, unless 1 = 2 = 1=2, which would contradict
the fact that the equilibrium is informative.
Now consider a partially informative candidate equilibrium, where a liberal CEO mixes, with message- price pairs (m; p) = (L; pL(L))and (m; p) = (C; pC(C)), and where a conservative CEO chooses (m; p) = (C; pC(C)) for sure. Then Bayes’ rule implies (L) P(t = ljL) = 1, so that pL(L) = v+ L , and
(C)2(0;1=2). Then we can write the indi¤erence condition for a liberal CEO as
nL(v+ L ) +k=nCpC(C) + 2k (C) 1 2 :
The left-hand side is strictly larger than L=nL(v+ L ), byk >0. The right-hand side is strictly less than C = nC(v+ C ), by pC(C) v+ C , k > 0 and (C) 2 (0;1=2). It follows that the indi¤erence condition for a liberal CEO must be violated. Thus, there exists no informative equilibrium where both CEO types send both m= L and m =C with positive probability; or where a liberal CEO sends bothm=Landm=C with positive probability, and a conservative sendsm=C (but notm=L) with positive probability. Notice that this is the case regardless of whether both CEO types send message m=S with probability 2(0;1)in equilibrium, or with probability zero.
As a result, the only informative equilibrium has a liberal CEO send m = L (but not m = C) with positive probability, and a conservative CEO send both m = L and m = C with positive probability. It cannot be that both CEO types also send message m =S with probability 2 (0;1)in equilibrium,