Internet Appendix for Proxy Advisory Firms: The Economics of Selling Information to Voters

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1 Internet Appendix for Proxy Advisory Firms: The Eonomis of Selling Information to Voters Andrey Malenko and Nadya Malenko The first part of the Internet Appendix presents the supplementary analysis for the proofs in the main Appendix. The seond part presents a detailed analysis of litigation pressure and poliy proposals. A. Supplementary analysis for the proofs. Supplementary analysis for the proof of Proposition : Proof that for any q, the equilibrium w s 0 = 0, w s =, and w 0 = exists. Consider the deision of shareholder i with signal s i when other informed shareholders i.e., shareholders that aquired private signals vote aording to strategy w s s j, and uninformed shareholders i.e., shareholders that did not aquire private signals vote aording to strategy w 0 =. Given q, the probability that eah shareholder votes for in state θ {0, } equals Pr [v j = θ = ] = q w s p + w s 0 p + q = qp + q, Pr [v j = θ = 0] = q w s p + w s 0 p + q = q p + q. Shareholder i s vote affets the deision if other shareholders vote for and vote against. The expeted value of the proposal to shareholder i in this ase is ũ s i = E [u, θ s i, P IV i ] = Pr [θ = s i, P IV i ] Pr [θ = 0 s i, P IV i ], where P IV i denotes the event in whih shareholder i s vote determines the outome i.e., if i j v j =. Applying the Bayes rule, ũ s i = Pr[s i θ=] Pr[ j i v j= θ=] Pr[s i θ=0] Pr[ j i v j= θ=0] Pr[s i θ=] Pr[ j i v j= θ=]+pr[s i θ=0] Pr[ j i v j= θ=0] = D s i Pr [s i θ = ] Pr [s i θ = 0] + q p q p where D s i > 0. The best response of shareholder i is to vote for v i = if ũ s i 0 and vote against v i = 0 if ũ s i 0. When s i =, Pr [s i θ = ] Pr [s i θ = 0] = p > 0. When s i = 0, Pr [s i θ = ] Pr [s i θ = 0] = p < 0. Therefore, the optimal strategy of shareholder i is indeed v i = s i. Hene, w s s = s is an equilibrium. Similarly, for an uninformed shareholder, the expeted value of the proposal onditional on Citation format: Malenko, Andrey, and Nadya Malenko, Internet Appendix to Proxy Advisory Firms: The Eonomis of Selling Information to Voters, Journal of Finane [DOI STRING]. Please note: Wiley- Blakwell is not responsible for the ontent or funtionality of any supporting information supplied by the authors. Any queries other than missing material should be direted to the authors of the artile.,

2 being pivotal is ũ 0 = D 0 qp + q qp q q p + q q p q = 0, for some D 0, and hene it is indeed optimal to mix between voting for and against.. Proof that and derease in N and approah zero as N. Denote both as funtions of N via N and N, respetively. Using their expressions in 7 and the fat that C N+ N+ = 4N N+ C, we get N N+ = N N+ p N < N and N+ = N N+ < N. Furthermore, lim N N = 0 and lim N N = 0, beause lim N P q, N, = 0 for any q 0,. 3. Value of signals. We derive the value of the private signal V s q r, q s and the value of the advisor s reommendation V r q r, q s to shareholder i for given q r, q s. 3.. Value of a private signal. Shareholder i s vote only makes a differene only if j i v j =. Conditional on s i = and on being pivotal, his utility from being informed is E [u, θ s i =, P IV i ]. Similarly, onditional on being pivotal and his private signal being s i = 0, the shareholder s utility from being informed is E [u, θ s i = 0, P IV i ]. Overall, the shareholder s value of aquiring a private signal is V s q r, q s = Pr s i = Pr P IV i s i = E [u, θ s i =, P IV i ] Pr s i = 0 Pr P IV i s i = 0 E [u, θ s i = 0, P IV i ]. By the symmetry of the model, E [u, θ s i =, P IV i ] = E [u, θ s i = 0, P IV i ] and Pr P IV i s i = = Pr P IV i s i = 0, so we get V s q r, q s = Pr P IV i s i = E [u, θ s i =, P IV i ] = Pr P IV i s i = Pr θ = s i =, P IV i Pr θ = 0 s i =, P IV i = p Pr P IVi, where Pr P IV i = Pr P IV i θ = = π Pr P IV i r =, θ = + π Pr P IV i r = 0, θ = = πp q n + q r + q s p, N, + π P q n q r + q s p, N,. Hene, V s q r, q s is given by Value of the advisor s signal. As before, shareholder i s vote makes a differene only if j i v j =. Conditional on r = and on being pivotal, his utility from being informed is E [u, θ r =, P IV i]. Similarly, onditional on r = 0 and on being pivotal, shareholder i s utility from being informed is E [u, θ r = 0, P IV i]. Overall, the shareholder s value of aquiring the advisor s signal is V r q r, q s = Pr r = Pr P IV i r = E [u, θ r =, P IV i] Pr r = 0 Pr P IV i r = 0 E [u, θ r = 0, P IV i].

3 By the symmetry of the model, E [u, θ r =, P IV i ] = E [u, θ r = 0, P IV i ] and Pr P IV i r = = Pr P IV i r = 0, so we get V r q r, q s = Pr P IV i r = E [u, θ r =, P IV i ] = Pr P IV i r = Pr θ = r =, P IV i Pr θ = 0 r =, P IV i = Pr P IV i r =, θ = Pr r = θ = Pr P IV i r =, θ = 0 Pr r = θ = 0 = Pr P IV i r =, θ = π Pr P IV i r =, θ = 0 π. Note that Pr P IV i r =, θ = = P q r + q s p + q n, N, and Pr P IVi r =, θ = 0 = P q r q s p + q n, N,. Hene, Vr q r, q s is given by Supplementary analysis for the proof of Proposition : Derivation of the ondition under whih equilibrium w s s i = s i, w r r = r, and w 0 = exists. Aording to Proposition, we an restrit attention to subgames that follow the information aquisition stage at whih eah shareholder i aquires r with probability q r, aquires s i with probability q s, and stays uninformed with probability q n = q r q s. Suh an equilibrium only exists if given q r, q s, it is optimal for a shareholder who aquired a signal to follow it. It will be useful to ompute the probabilities that a random shareholder j votes for the proposal, onditional on the advisor s reommendation r and the true state θ: Pr [v j = r =, θ = ] = q r + q s p + q n, IA Pr [v j = r = 0, θ = ] = q s p + q n, IA Pr [v j = r =, θ = 0] = q r + q s p + q n, IA3 Pr [v j = r = 0, θ = 0] = q s p + q n. IA4 First, onsider a shareholder with private signal s i. Sine his vote affets the deision only when he is pivotal, he ompares E [u, θ s i, P IV i ] with zero or, equivalently, Pr θ = s i, P IV i with, and votes for if and only if the former is higher. By Bayes rule, Pr θ = s i s i, P IV i = Pr P IV i θ = s i p Pr P IV i θ = s i p + Pr P IV i θ s i p = p >, where we used the independene of s j and r from s i onditional on θ: beause of independene, v j is independent from θ = s i or θ s i i.e., from whether shareholder i s private signal is orret or not. Therefore, it is always optimal for a shareholder who aquired a private signal to follow it. Seond, onsider a shareholder that aquired r. A shareholder ompares E [u, θ r, P IV i ] with zero and votes for if and only if the former is higher. Using Bayes rule and Pr θ = = Pr r, we get E [u, θ r, P IV i ] Pr P IV i r = Pr θ = r, P IV i Pr P IV i r Pr θ = 0 r, P IV i Pr P IV i r = Pr P IV i r, θ = Pr r θ = Pr P IV i r, θ = 0 Pr r θ = 0. IA5 It is suffi ient to onsider r = : sine the model is symmetri, voting against is optimal for r = 0 whenever voting for is optimal for r =. When r =, the shareholder finds it optimal to vote 3

4 for if and only if By independene of s i, s j, j i, and r onditional on θ, v j = N r, θ = P Pr P IV i r, θ = Pr j i Plugging this into IA6 gives Pr P IV i r = θ = π. IA6 Pr P IV i r =, θ = 0 π π P + qr + q sp π P + qr q sp Pr [v j = r, θ], N, N., N,, N,. IA7 The intuition for IA7 is as follows. Consider a shareholder with the advisor s reommendation deiding whether to follow it. If q s > 0, a split vote is a signal that the advisor s reommendation is more likely to be inorret r θ, sine a split vote is more likely when private signals of shareholders disagree with the advisor s reommendation than when they agree with it. Therefore, as long as q r > 0 and q s > 0, the information ontent from being pivotal lowers the shareholder s assessment of the preision of the advisor s reommendation. This logi is refleted in the left-hand side of IA7, whih gives the ratio of probabilities that the advisor is orret and inorret: the first term π π is the prior, while the seond term reflets additional information from the fat that the vote is split. Finally, onsider an uninformed shareholder. Sine the event of being pivotal is uninformative about state θ, suh a shareholder is indifferent between voting for and against the proposal, so it is optimal for him to mix between the two options. Therefore, if q r and q s satisfy IA7, then voting in the diretion of the signal that a shareholder has private or advisor s is an equilibrium. If IA7 is violated, there is no equilibrium with a positive value of the advisor s reommendation. However, sine all these sub-games imply zero value of reommendation of the advisor, they are not reahed on equilibrium path if q r > 0. In partiular, whenever V r q r, q s f 0, whih is implied by any equilibrium with q r > 0 where V r q r, q s is the value of the advisor s reommendation to a shareholder, this ondition is satisfied. Therefore, we do not verify IA7 in subsequent derivations. 5. Supplementary analysis for the proof of Lemma. 4

5 The solutions to A7, if they exist, are given by qr ψ = 4 f+ p + p p ψ πc qs ψ = p 4 f+ p + p p ψ qr ψ = qs ψ = p 4 4 πc f+ p + p p ψ πc f+ p + p p ψ πc p p f+ p ψ πc p p, p f+ p ψ πc p f+ p ψ πc p, p f+ p ψ πc,. IA8 IA9 Note that qj 0 = qa j and qj 0 = qb j for j {r, s}. Sine p,, it is easy to see that qr ψ + qs ψ qr ψ + qs ψ and that qr ψ + qs ψ is stritly dereasing in ψ. Eah solution satisfies q r, q s > 0 if and only if f + ψ < π p + ψ f < f + π pψ p. Proof of Claim : If f f, then there is no equilibrium q r, q s > 0. First, sine stritly positive solutions A5-A6 do not exist for f f, there is no equilibrium q r, q s > 0 satisfying q r + q s <. Seond, by ontradition, suppose there is an equilibrium p f+ p p ψ π q r, q s > 0 with q r + q s =. Then, f+ p + p p ψ π and qr i ψ + qs i ψ = for some ψ 0 and some i {, }. Sine qr ψ+qs ψ qr ψ+qs ψ, we have qr ψ+qs ψ. This, together with the inequality above, implies p p 4 f + p + p p ψ πc p 4 f + p + p p ψ = p Λ = q 0, πc whih ontradits Assumption that q0 0,. p Proof of Claim : If f + p if and only if f [ f, f p 4 πc p + p 4 p f + p π C p 4 f + p, where f is given by A4. πc p ψ, there is an equilibrium q r, q s > 0 5

6 Note that q b r + q b s = p p 4 f + p πc p p 4 p f π C IA0 is stritly dereasing in f. Also, when f = f, the seond term is zero and hene, given the [ inequality assumed by the laim, qr b + qs b for f = f. Hene, qr b + qs b for any f f, f with strit inequality for f f. As shown above, qr, b qs b > 0 for f < f. Hene, there is an equilibrium qr, b qs b [ > 0 if f f, f. By Claim, there is no equilibrium q r, q s > 0 if f f. If f < f, system A3 has no solution, so there is no equilibrium q r, q s > 0 with q r + q s <. Finally, A3 with + ψ and f + ψ instead of and f does not have a solution if f < p + p p ψ C N π = f + p p ψ. Sine ψ 0, it does not have a solution if f < f, so there is no equilibrium q r, q s > 0 with q r + q s = in this ase either. p Proof of Claim 3: If f + p >, there exists f f suh that p 4 πc there is an equilibrium q r, q s > 0 if and only if f [ f, f. By Claim, there is no equilibrium q r, q s > 0 if f f. Note that when f = f, the two roots in IA0 are equal, and hene qr b + qs, b given by IA0, is below one. Also, when f = f, the seond term in IA0 is zero and hene, given the inequality assumed by the laim, qr b + qs b > for f = f. Sine qr b + qs b is stritly dereasing in f, there is a unique ˆf f, f at whih IA0 equals one and sine f < f, f both qr b and qs b are stritly positive. Hene, if f ˆf,, there is an equilibrium q r, q s > 0 with q r + q s <. If f = ˆf, there is an equilibrium q r, q s > 0 with q r + q s =. Finally, if f < ˆf, then q a r + q a s q b r + q b s >, so there is no equilibrium of type q a r, q a s or q b r, q b s. Next, onsider f ˆf. Consider equilibria with q r + q s =. Define ˆf + C N π p p 3p p p π. IA We next show that if ˆf ˆf, then the neessary and suffi ient onditions for equilibrium of the type q r ψ, q s ψ > 0 with q r ψ + q s ψ = to exist for some ψ 0 is f [ ˆf, ˆf ]. To prove this, note that suh an equilibrium exists if and only if f is suh that equation q r ψ + q s ψ = has a solution ψ 0 with q r ψ > 0 ondition q s ψ > 0 is implied by it from IA8. Hene, ψ 6

7 must satisfy p p 4 f + p + πc p p ψ 4 p f + p π C p ψ ψ ψ l, IA 0 ψ ψ h, IA3 where the first inequality follows from q r ψ + q s ψ = and ψ l = p p ψ h = p p p3p 4p πc f N π C + f p p., IA4 Hene, this system is equivalent to ψ l ψ ψ h. IA5 Note that ψ h ψ l f ˆf, given by IA. Therefore, if f < ˆf, there is no equilibrium q r ψ, qs ψ > 0. We next show that if f [ ˆf, ˆf ], so that IA5 is non-empty, suh an equilibrium exists. When ψ = ψ h, IA3 binds and sine ψ l ψ h, then IA is satisfied and hene qr ψ h + qs ψ h sine it equals the left-hand side of IA when IA3 binds. When ψ = ψ l, IA binds and hene qr ψ l + qs ψ l sine it equals the left-hand side of IA plus a non-negative number. As shown above, qr 0 + qs 0 = qr a + qs a qr b + qs b for f ˆf. Hene, when ψ = max {0, ψ l }, we have qr ψ + qs ψ for f ˆf. Sine qr ψ + qs ψ is stritly dereasing in ψ, it must be that ψ h 0 otherwise, qr ψ h + qs ψ h > qr 0 + qs 0. Thus, the interval [max {0, ψ l }, ψ h ] is non-empty and by the intermediate value theorem there exists a unique ψ [max {0, ψ l }, ψ h ] at whih qr ψ + qs ψ =. Note also that for this ψ, qr ψ > 0 and qs ψ > 0 follows from IA8. Indeed, suppose by ontradition that qr ψ 0. Sine qr 0 = qr a > 0 for f < f, then by the intermediate value theorem, there exists ψ 0, ψ ] suh that qr ψ = 0. Sine ψ ψ and qr ψ + qs ψ is stritly dereasing in ψ, qr ψ + qs ψ qr ψ + qs ψ =, and hene qs ψ. Sine qs ψ and qr ψ satisfy V s q r, q s = V r q r, q s f = ψ and qr ψ = 0, we have V s 0, q s ψ = + ψ, and hene qs ψ is the equilibrium of the benhmark ase with no advisor but with a higher ost, = + ψ. Sine the ost if higher, it must be that qs ψ qs 0 = q0, but then q 0, whih ontradits Assumption. Hene, indeed, qr ψ > 0. Therefore, there exists an equilibrium q r ψ, qs ψ > 0 with qr ψ + qs ψ = for some ψ 0 if and only if f [ ˆf, ˆf ]. Sine ψ l = ψ h for f = ˆf, then IA and IA3 bind for ψ = ψ h. Thus, qr ψ h + qs ψ h =. By IA8, qr ψ h 0,, and hene qs ψ h = qr ψ h 0,. Hene, equilibrium of the type q r ψ, qs ψ > 0 with qr ψ + qs ψ = for some ψ 0 exists for f = ˆf. We next prove that there exists a utoff level ˆf3 ˆf suh that equilibrium of the type qr ψ, qs ψ [ > 0 with qr ψ + qs ψ = for some ψ 0 exists for f ˆf3, ˆf ] and does not exist for f < ˆf 3. To see 7

8 this, define V f min ψ [0,ψ h f] {q r ψ, f + qs ψ, f} = max ψ [0,ψ h f] { q r ψ, f qs ψ, f}, where ψ h f is given by IA4. Define Φ = {f [0, ˆf ] : V f } and note that equilibrium of the type qr ψ, qs ψ > 0 with qr ψ+qs ψ = exists if and only if f Φ. Indeed, if V f >, then qr ψ, f + qs ψ, f > for any ψ 0 sine for ψ > ψ h f, this funtion is not well defined and hene no suh equilibrium exists. On the other hand, suppose that V f and is ahieved at ψ f. Then qr ψ f, f + qs ψ f, f. In addition, sine ψ h f < ψ l f for f < ˆf, then IA is violated and IA3 binds for ψ = ψ h f, and hene qr ψ h f, f + qs ψ h f, f > for f < ˆf. By the intermediate value theorem, there then exists ψ [ψ f, ψ h f] suh that qr ψ, f + qs ψ, f =. Sine IA9 implies that qr ψ 0,, and hene qs ψ = qr ψ 0, as well, this onstitues an equilibrium. Next, note that V f is dereasing in f. Indeed, define the Lagrangian L f, ψ, λ, µ qr ψ, f qs ψ, f + λψ + µ ψ h f ψ and note that V f = max ψ,λ,µ L f, ψ, λ, µ. By Envelope theorem, V f = L f f, ψ, λ, µ = qr ψ, f f q s ψ, f f + µ ψ h f. Note that µ 0 aording to the Kuhn-Tuker onditions. Beause qr ψ, f + qs ψ, f dereases in f for [ a given ψ and ψ h f 0, it follows that V f 0. The fat that V f 0 implies that Φ = ˆf3, ˆf ] for some ˆf 3 ˆf. Hene, equilibrium qr ψ, qs ψ [ > 0 with qr ψ+qs ψ = exists for f ˆf3, ˆf ] and does not exist for f < ˆf 3, as required. Moreover, note that ˆf 3 f. This is beause, IA9 does not have a solution if ψ > ψ h f < f + p p ψ and hene does not have a solution if f < f. Consider two ases. First, if ˆf ˆf, then ombining f the results above, equilibrium q r, q s > 0 with q r + q s < exists if and only if f ˆf,, equilibrium qr ψ, qs ψ > 0 with qr ψ + qs ψ = exists if and only if f [ ˆf, ˆf ], and equilibrium qr ψ, qs ψ [ > 0 with qr ψ + qs ψ = exists for f ˆf3, ˆf ] and does not exist for f < ˆf 3. Combined with Claim, this implies that equilibrium q r, q s > 0 exists if and only if f [ ˆf 3, f. Seond, if ˆf > ˆf, then equilibrium q r, q s > 0 with q r +q s < exists if and only if f ˆf, f, there exists an equilibrium q r, q s > 0 with q r +q s = for f = ˆf, and equilibrium qr ψ, qs ψ [ > 0 with qr ψ+qs ψ = exists for f ˆf3, ˆf ] and does not exist for f < ˆf 3. Combined with Claim, this implies that equilibrium q r, q s > 0 exists if and only if f [min ˆf 3, ˆf, f. Combining the two ases, we onlude that equilibrium q r, q s > 0 exists if and only if f [f, f, where f min ˆf 3, ˆf. Sine, as shown above, ˆf > f and ˆf 3 f, then f f, whih proves Claim Supplementary analysis for the proof of Proposition 3: Properties of A6. Let us fix fee f and vary π. Reall from Lemma that equilibrium with omplete rowding out exists if and only if f < f = π p, i.e., π > + f p. The derivative of the left-hand side of A6 in π is: k= N+ P p a, N, k + π dp a dπ 8 k= N+ P q p a, N, k > 0,

9 sine N k= N+ beause N k= N+ P p a, N, k >, N k= N+ P q p a, N, k > 0, and dpa dπ P p a, N, k + N k= N+ > 0. Indeed, N k= N+ P p a, N, k > P p a, N, k = and P p a, N, k > P p a, N, k for p a > and k > N. Seond, N k= N+ P q p a, N, k > 0 for p a >, as shown in the proof of Part. Finally, dpa dπ > 0 follows diretly from A4. Therefore, the left-hand side of A6 is stritly inreasing in π. Note also that the advisor s presene stritly dereases firm value for π + f p. Indeed, in this ase, p a p 0, so we obtain π k= N+ P p 0, N, k π < k= N+ P p 0, N, k < k= N+ P p 0, N, k, whih is true, as just shown above, sine p 0 >. Finally, when π, the advisor s presene stritly inreases firm value. Indeed, lim p a = π + 4 f C so the left-hand side of A6 onverges to k= N+ P lim p a, N, k > π > + 4 k= N+ C p P p 0, N, k beause N k= N+ P q q, N, k > 0 for q >, as shown in the proof of Part. = p 0, 7. Supplementary analysis for the proof of Proposition Proof that when f = f, firm value is stritly higher in equilibrium with inomplete rowding out than in equilibrium with omplete rowding out. To see this, onsider any equilibrium with p a > and p d <. Sine Ω q r, q s = P p a, N,, Ω q r, q s = P p d, N, and sine p a > and p d <, we have p a = ϕ Ω and p d = ϕ Ω, where ϕ is given by A. Aording to A0, firm value is Û Ω, Ω = N k= N+ πp ϕ Ω, N, k + π P ϕ Ω, N, k = N k= N+ πp ϕ Ω, N, k π P ϕ Ω, N, k + π = πf Ω π f Ω + π, where f x N k= N+ P ϕ x, N, k. In equilibrium with omplete rowding out and f = f, we have p a = + q r >, p d = q r <, and aording to A Ω q r, 0 = Ω q r, 0 = f π Ω r. Consider Ω Ω r + π π ε and Ω Ω r + π π π ε with ε p f π π > 0. Note that πω π Ω = f and πω + π Ω = p 0.5, i.e., Ω and Ω satisfy A. Hene, for f = f, equilibrium with inomplete rowding out is haraterized by probabilities of being pivotal 9

10 Ω and Ω. We next prove that Û Ω, Ω = Û funtion Ũ x Û Ũ x = π π π Ω r + π π x, Ω r + Ω r + π π ε, Ω r + π π x for x 0 is inreasing beause f Ω r + π π x f Ω r + π π x π π = π π π ε > Û Ω r, Ω r. Indeed, Ωr+ π π x Ω r+ π π x f y dy > 0 by Auxiliary Lemma A. Hene, indeed, when f = f, firm value is stritly higher in equilibrium with inomplete rowding out than in equilibrium with omplete rowding out. 7.. Comparison of ˆπ and π. Simplifying, P and hene + N, N, ˆπ + = C + P P Plugging in p 0 = p q 0 + = Λ + N, N,, N, C N = C N, N p P + N = C N N,., N, into A0, we get ˆπ π if and only if P P, N,, N, P + N p, N, N k= N+ N k= N+ P + Λ, N, k P + N, N, k. Furthermore, from the indifferene ondition in the benhmark ase, P and hene, ˆπ π if and only if + Λ, N, = p, P N k= N+, N, P +, N, k N P + N, N, P N k= N+, N, P + Λ, N, k P + Λ, N,. Consider funtion g x = P, N, L x P x, N, where L x = N k= N+ P x, N, k is the same as defined in the proof of Lemma A3. Then, the above inequality is equivalent to g + g N + Λ. Differentiating, g x = L x P, N, P x, N, + Px x, N, L x P, N, P x, N, Using the expressions for L x and P x x, N, in IA3 and IA3 in the proof of Lemma, 0

11 A3, it follows that the sign of g x oinides with the sign of Note that g x = N P, N, N P x, N, N g x = NP x x, N, [ x N x x = N x x+x x x L x < 0 N x x x ] x L x x x L x L x. Sine g = 0, g x < 0 for x,. Therefore, g x is stritly dereasing in x,. Hene, ˆπ π g + g N + Λ is satisfied if and only if + N + Λ Λ. Note N also that Λ π, as follows from A0. Hene, if Λ N, then ˆπ π and π, so N the advisor improves the quality of deision-making ompared to the benhmark ase if and only if π > π. If Λ >, then ˆπ > π and π, so the advisor never improves the quality of deisionmaking. Hene, in both ases, the advisor improves the quality of deision-making ompared to N the benhmark ase if and only if π > π. 8. Proof that the advisor does not benefit from adding i.i.d. noise to its reommendations. Consider the following extension of the model. Given the advisor s signal r, the signal that shareholder i observes is: { r, with prob. ε, r i = r, with prob. ε, for ε [0, ]. Signals r i are independent aross shareholders, onditional on r. If ε = 0, the model redues to the basi model in whih all shareholders observe the same reommendation. By the same logi as in the basi model, at the voting stage, shareholders that aquired a ertain signal must vote aording to the signal. Consider the information aquisition stage. The values of signals are: V s q r, q s, ε = p πω q r, q s, ε + π Ω q r, q s, ε V r q r, q s, ε = ε πω IA6 q r, q s, ε π Ω q r, q s, ε where the probabilities of a split vote onditional on the advisor being orret and inorret are, respetively, Ω q r, q s, ε = P qr ε + q s p + q n = P + qr ε + q s p IA7 = Ω q r ε, q s, 0 Ω q r, q s, ε = P qr ε + q s p + q n = P qr ε + q s = Ω q r ε, q s, 0 where P x P x, N, and qn = q r q s. Hene, p V s q r, q s, ε = V s q r ε, q s, 0, V r q r, q s, ε = ε V r q r ε, q s, 0, IA8

12 where V i q r, q s, 0 = V i q r, q s are the values of the signals in the basi model with ε = 0. Denote qr ε, f and qs ε, f the equilibrium frations of shareholders aquiring the advisor s and private signal, respetively, given ε and fee f. For simpliity, we assume that is suffi iently large, so that at least some shareholders will remain uninformed, i.e., qr ε, f + qs ε, f < this restrition is analogous to restrition > ĉ in Assumption of the basi model, whih guarantees, aording to the proof of Lemma A3, that there is no equilibrium with q r + q s =. Then, the following three ases are possible: qr ε, f > 0, qs ε, f > 0, requiring V s qr ε, f, qs ε, f, ε =, V r qr ε, f, qs ε, f, ε = f; qr ε, f = 0, qs ε, f > 0, requiring V s qr ε, f, qs ε, f, ε =, V r qr ε, f, qs ε, f, ε f; 3 qr ε, f > 0, qs ε, f = 0, requiring V s qr ε, f, qs ε, f, ε, V r qr ε, f, qs ε, f, ε = f. Hene, IA8 implies that, respetively, either V s qr ε, f ε, qs ε, f =, V r qr ε, f ε, qs ε, f = f ε or V s qr ε, f ε, qs ε, f =, qr ε, f q r ε, qs ε, f f ε, or 3 V s qr ε, f ε, qs ε, f, V r qr ε, f ε, qs ε, f = f ε. This, in turn, implies that qr ε qr ε, f and qs = qs ε, f will form an equilibrium of f the same type in the model with ε = 0 and fee ε q r diretly follows from qr ε, f sine ε. Hene, the advisor an ahieve exatly the same profits by setting ε = 0 and fee f ε and perhaps even higher profits by setting a different fee.therefore, the advisor an never stritly benefit from adding i.i.d. noise to the signals distributed to shareholders. Intuitively, when the advisor adds noise to its signal, two effets at in opposite diretions. The first, negative, effet is that given the same probabilities of being pivotal, noise ε > 0 dereases the shareholder s value from buying the advisor s reommendation beause the signal now has a lower preision. This is aptured in the term ε of the seond equation of IA6 and requires the advisor to derease the fee by a fator ε. The seond, positive, effet is that given the same fration q r of shareholders subsribing to the advisor, the probability of eah shareholder being pivotal is higher when ε > 0, beause shareholders subsribing to the advisor will not always vote with eah other. This allows the advisor to apture a higher fration of shareholders while keeping their probability of being pivotal the same. This effet is aptured by terms ε in eah equation of IA7. The two effets fully offset eah other. 9. Auxiliary Lemma A. Funtion f x N k= N+ P ϕ x, N, k, where ϕ x is defined by A, is stritly dereasing and stritly onave. Proof of Auxiliary Lemma A. It will be useful to ompute the derivative: ϕ x =, C N ψ x IA9 where ψ x x C N 3 4 x C. IA0

13 The first derivative of f x is f x = k= N+ P q ϕ x, N, k ϕ x < 0, sine ϕ x < 0 and N k= N+ P q q, N, k > 0 for any q >, inluding q = ϕ x. The former follows from IA9. The latter follows from N k= N+ P q, N, k k Nq q q < 0 for any k < N seond derivative of f x is f x = dϕ dx k= N+ P q q, N, k = k=0 P q q, N, k and P q q, N, k = beause q >. Therefore, f x is stritly dereasing. The P qq ϕ x, N, k + d ϕ dx Sine N k=0 P q q, N, k = 0 and N k=0 P qq q, N, k = 0, f x = dϕ dx k=0 P qq ϕ x, N, k = k=0 P qq ϕ x, N, k ψx C Plugging in P q, P qq and simplifying, C d ϕ k= N+ P q ϕ x, N, k dx k=0 P q ϕ x, N, k ψ x C ψx N ψ x f x = [ k=0 P ϕ x, N, k k Nϕx ϕx ϕx k ϕx N k + C ϕx k=0 P q ϕ x, N, k N ψ x k Nϕx ϕx ϕx ]. Next, we an alulate ψ x: C N ψ x = N 3 4 = ϕx x C N 3 4 N + 4 ϕx ϕx N +. x C IA Thus, C [ k=0 P ϕ x, N, k k Nϕx ϕx ϕx k ϕx N ψ x f x = N k + ϕx ϕx N 3 4 ϕx ϕx N + k Nϕx ϕx ϕx ]. 3

14 Multiplying by ϕ x ϕ x : = k=0 C N ψ x ϕ x ϕ x f x k Nq k q N k q P q, N, k + k Nq N 3 q 4 N q q L q, IA where we denote ϕ x by q,. It follows that f x < 0 if L q > 0 for any q,. To prove it, denote ζ q, k k Nq k q N k q + C k Nq = k k N q C k + N N q CNq, where C N 3 q 4 N q q. Then, L q = k=0 P q, N, k k k N q C k=0 P q, N, k k+ N N q CNq k=0 P q, N, k. Consider the first two terms:. Term : k=0 k k Ck N qk q N k = k= k k N! k!n k! qk q N k = N N q m=0 P q, N, m = N N q Pr [ k k B N, q ].. Term : = qn k=0 kck N qk q N k = k=0 P q, N, k k= k N! k!n k! qk q N k = qn Pr [ k k B N, q ]. IA3 Hene, Note that Lq qn = N q Pr [ k k B N, q ] N q C Pr [ k k B N, q ] + N q C Pr [ k k B N, q]. Pr [ k k B N, q] = I N+ q, N+, Pr [ k k B N, q ] = I N+ q,, Pr [ k k B N, q ] = I N+ q, N 3 IA4, where I q is the regularized inomplete beta funtion. Aording to the property of the regularized inomplete beta funtion, I x a, b + = I x a, b + xa x b a!b! bba,b, where B a, b = a+b! is 4

15 the beta funtion. Hene, I N+ q, N+ = N+ I q, I N+ q, Plugging into the expression for Lq qn : Lq qn = N q I N+ q, + N q C = N q = N+ I q, N 3 q N+ q N 3 N 3 N+ q q N 3 N 3 N+ B, N 3 Dividing by q N+ q N 3 and simplifying, B N+, N 3 I N+ q, + q N+ q N+ B, + q N+ q N 3 N 3 B N+, N 3. IA5 N q C I N+ q + q N+ q B N+, + N q C q N+ q, N+ B., L q q N+ q N = q N!! N 3 q C! q N!! N 3!. Hene, Lq N 3!!q q N+ q N+ N! = 4 q 4 q + = q N 3 N q q Lq N 3!!q q N+ q N+ N! = q q +. IA6 Sine q q + > 0, we onlude that L q > 0 for any q,. Therefore, f x < 0, whih ompletes the proof. 0. Auxiliary Lemma A. Funtion f x, defined by IA34, is stritly onave. Proof of Auxiliary Lemma A. Differentiating f x and using the definition of f x, f x = f x ϕ x xϕ x. Using f x from the proof of Auxiliary Lemma A above, in partiular, expressions IA, IA6, IA9, and the derivative of IA9, we an write f x = x ϕ x ϕ x + N ϕ x ϕ x ϕ x C N ψ x + xψ x. N ψ x C N ψ x C Multiplying both sides by C N ψ x, using IA, IA0, and A and simplifying gives N ψ x f x = N x ϕ x ϕ x ϕ x < 0 C 5

16 sine ϕ x,. Therefore, f x is stritly onave.. Lemma A3 omparison of shareholder welfare aross equilibria. Suppose that ĉ,, where ĉ is defined in the proof. Then, in the range f [ f, f, all equilibria an be ranked in their shareholder welfare expeted value of the proposal minus information aquisition osts. Speifially, there exist three equilibria, with equilibrium a having the highest and having the lowest shareholder welfare: a equilibrium with inomplete rowding out of private information aquisition and q r p q s, given by A5 in the Appendix; b equilibrium with inomplete rowding out of private information aquisition and q r p q s, given by A6 in the Appendix; equilibrium with omplete rowding out of private information aquisition: q s = 0 and q r given by 3. Equilibria a and b oinide when f = f. When f = f, there exist two equilibria: a equilibrium that is equivalent to the benhmark ase, q s = q 0, q r = 0, and b equilibrium with omplete rowding out: q s = 0, q r 0,, and equilibrium a has higher shareholder welfare than b. Proof of Lemma A3. We start by defining ĉ in the statement of the lemma. Consider qr a, qs a given by A5 and define S f, qr a + qs a = p p 4 f + Consider the following funtion of : p πc + p p 4 p f π C. S max S f,, f [f, f] IA7 where f = p N π C and f = π p, as defined before. We show that S is stritly dereasing in. Indeed, S f, = p 4 p N πc, p S f, = p 4 πc p C. IA8 For any, one of three ases must hold: S = S f, ; S = S f, for some f f, f ; 3 S = S f,. As lear from IA8, S f, and S f,, orresponding to ases and 3, are stritly dereasing in. In ase, i.e., when IA7 reahes the maximum at an interior point f, we an apply the envelope theorem: S = Sf, < 0. Together, this implies that S is stritly dereasing in. Note also that when =, defined in 7, then S f, =. Hene, S. In addition, when =, defined in 7, then f = f, and hene S = S f, = 0. Sine S is stritly dereasing in, there exists a unique ĉ [, at whih S ĉ =, and S < for any 6

17 ĉ,. To sum up, we define where S is given by IA7. Suppose that ĉ,. Then, ĉ S, p p 4 f + p πc = S f, S <, and hene f = f aording to A8. Aording to the proof of Lemma, there exists an equilibrium q r, q s > 0 if and only if f [ f, f. Let us find all suh equilibria. Sine S <, then q a r +q a s < for any f [ f, f. Therefore, q b r + q b s q a r + q a s <. In addition, q a r, q a s > 0 and q b r, q b s > 0 beause f < f. Thus, both equilibria A5 and A6 exist. Sine q r ψ+q s ψ is stritly dereasing in ψ and q r 0 + q s 0 = q a r + q a s <, we have q r ψ + q s ψ q r ψ + q s ψ < for any ψ 0, where q i r ψ, q i s ψ, i =,, represent potential solutions for q r + q s = and are given by IA8 and IA9 in the Internet Appendix. Therefore, there is no equilibrium with q s + q r = when f [ f, f. Thus, in addition to equilibrium with omplete rowding out, there exist exatly two other equilibria when f [ f,, f, and these equilibria feature inomplete rowding out with q r + q s < : A5 with q a r p q a s and A6 with q b r p q b s. The expeted welfare of a shareholder is the expeted per-share value of the proposal, U q r, q s, given by A0, minus the expeted information aquisition ost: W q r, q s = k= N+ πp p a, N, k + π P p d, N, k q rf q s. IA9 We next rank these three equilibria in shareholder welfare for f [ f, f and show that the equilibrium with inomplete rowding out of private information and q r < p q s has the highest shareholder welfare, followed by the equilibrium with inomplete rowding out of private information and q r > p q s, whih is followed by the equilibrium with omplete rowding out of private information. First, we show that the equilibrium with inomplete rowding out of private information and q r > p q s, denoted qs, b qr, b has lower shareholder welfare than the equilibrium with inomplete rowding out and q r < p q s, denoted qs a, qr a. As shown above, q r + q s <. Using A9, we get q r = p a p d and q s = pa+p d p, and plugging these into IA9, W q r, q s an be rewritten as k= N+ Using A, πp p a, N, k + π P p d, N, k f + W q r, q s = π k= N+ P p a, N, k Ω p a + π p a p p f p d + k= N+ P p d, N, k Ω p d + p. p. Aording to A, A3, and A9, p a, Ω, and Ω are idential in both equilibria and p d q b s, q b r = p d q a s, q a r <. Therefore, to show that W qa r, q a s > W q b r, q b s, it is neessary and suffi ient to 7

18 show that for p d > k= N+ P p d, N, k Ω p d > k= N+ P p d, N, k Ω p d. Using N k= N+ P q, N, k = N k= N+ P q, N, N k = N k= N+ P q, N, k and Ω = P p d, N,, this is equivalent to k= N+ P p d, N, k > p d P p d, N, N. IA30 Denote the left-hand side and the right-hand side by L p d and R p d, respetively. Differentiating the left-hand side of IA30, L x = N k= N+ P x x, N, k = k=0 P x x, N, k = x x k=0 P x, N, k k Nx = x x k=0 kp x, N, k Nx k=0 P x, N, k Note that k=0 P x, N, k = I N+ x, N+, where Ix a, b is the regularized inomplete beta funtion. In addition, aording to IA3 and IA4 and IA5 in the proof of Auxiliary Lemma A, k=0 N + kp x, N, k = NxI x, N [ = Nx where B a, b is the beta funtion. Hene, x x L x = Nx N+ x x B N+, = N + I x, N + N+ x x N!! = Nx x P! N+ ] x x B N+,, x, N, N IA3 Differentiating the right-hand side of IA30, R x = P x x, N, x + P x, N, = P x, N, x x x x + = P x, N, x x x < P x, N, N = L x. IA3 Sine L = R = 0, it follows that L x > R x for any x >. Hene, indeed, W qa r, qs a > W q b r, q b s. Seond, we show that the equilibrium with inomplete rowding out of private information and q r > p q s, denoted q b r, q b s, has higher shareholder welfare than the equilibrium with omplete rowding out of private information, denoted q r, 0, whenever the two o-exist, i.e., f [f, f.. 8

19 Consider funtion ϕ x, defined as the higher root of x = P ϕ x, N, and given by A. Sine Ω = P p a, N,, Ω = P p d, N,, and sine pa > and p d < in both of these equilibria, we have p a = ϕ Ω and p d = ϕ Ω. Plugging these expressions for p a and p d, we an re-write IA9 as N k= N+ = N k= N+ πp ϕ Ω, N, k + π P ϕ Ω, N, k q rf q s πp ϕ Ω, N, k π P ϕ Ω, N, k + π q rf q s, IA33 where we used N k= N+ P x, N, k = k=0 P x, N, k = N k= N+ P x, N, k to get to the seond line. For equilibrium q b r, q b s, let us plug q r = p a p d, q s = pa+p d p, p a = ϕ Ω, and p d = ϕ Ω into IA33. Then, using A and simplifying, we an write shareholder welfare W q b r, q b s as the following funtion of Ω and Ω : where Ŵ Ω, Ω = π f Ω π f Ω + π, f x k= N+ P ϕ x, N, k x ϕ x IA34 and Ω and Ω are given by A. Similarly, for equilibrium qr, 0, let us plug q r = p a p d, q s = 0, p a = ϕ Ω, and p d = ϕ Ω into IA33. Using the fat that in this equilibrium, Ω = Ω = Ω r = and simplifying, we an f f π write shareholder welfare W qr, 0 as Ŵ Ω r, Ω r, where Ω r = π. Note next that Ω = Ω r + π π ε and Ω = Ω r + π π π ε, where ε p f π π > 0 sine f < f. Thus, to prove that W qr, b qs b > W qr, 0, it is neessary and suffi ient to prove that Ŵ Ω r + π π ε, Ω r + π π ε > Ŵ Ω r, Ω r. Define funtion W x Ŵ Ω r + π π x, Ω r + π π x for x 0. Differentiating, W x = π π π f Ω r + π π x f Ω r + π π x π π = π Ωr+ π π x Ω r+ π π x f y dy. Auxiliary Lemma A shows that funtion f is stritly onave, and hene W x > 0 for any x > 0. Thus, Ŵ Ω r + π π ε, Ω r + π π ε = W ε > W 0 = Ŵ Ω r, Ω r, whih proves the statement. Combing the two results above, we an onlude that when ĉ, and when f [f, f, multiple existing equilibria rank in shareholder welfare in the following way: The equilibrium with inomplete rowding out of private information and q r < p q s has the highest shareholder welfare, followed by the equilibrium with inomplete rowding out of private information and q r > p q s, whih is followed by the equilibrium with omplete rowding out of private information. Finally, when f f, the equilibrium qr a, qs a onverges to equilibrium q s = q0, q r = 0, and the equilibrium qr, b qs b onverges to equilibrium with omplete rowding out, qs = 0, q r 0,. By monotoniity and the welfare omparison above, it follows that when f = f, the first equilibrium 9

20 has higher shareholder welfare than the seond.. Proof that the model is equivalent to a more general setup with u, u 0, = u 0, 0 u, 0 Suppose that u, u 0, = u 0, 0 u, 0 =. We show that shareholders make exatly the same information aquisition and voting deisions as in the urrent setup.. Same voting deisions. Whenever a shareholder deides how to vote, he votes for if and only if his utility from voting for is greater than his utility from voting against onditional on the event of being pivotal and whatever information he knows. Denote this information set I i. The shareholder s utility from voting for minus his utility from voting against onditional on this information set is Pr θ = I piv u, u 0, + Pr θ = 0 I piv u, 0 u 0, 0 = Pr θ = I piv Pr θ = 0 I piv, whih is the same as in the basi model. Hene, given the same information aquisition deisions, shareholders make the same voting deisions.. Same information aquisition deisions. Shareholder i s vote only makes a differene only if the votes of other shareholders are split. Denote this set of events by P IV i. Let us find the value of any signal to the shareholder. Denote the signal aquired by the shareholder private or advisor s by S i. If S i =, then by aquiring the signal, the shareholder votes for for sure, instead of randomizing between voting for and against. Hene, onditional on S i = and on being pivotal, his utility from being informed is E [u, θ S i =, P IV i ] E [u, θ + u 0, θ S i =, P IV i ] = E [u, θ u 0, θ S i =, P IV i ] Similarly, onditional on being pivotal and the signal being S i = 0, the shareholder s utility from being informed is E [u 0, θ S i =, P IV i ] E [u, θ + u 0, θ S i =, P IV i ] Overall, the shareholder s value of aquiring the signal is = E [u 0, θ u, θ S i =, P IV i ] Pr S i = Pr P IV i S i = E[u,θ u0,θ S i=,p IV i ] Pr S i = 0 Pr P IV i S i = 0 E[u,θ u0,θ S i=,p IV i ], Sine, by assumption, u, u 0, = and u, 0 u 0, 0 = and are the same as in the basi model, the value of any signal to the shareholder is the same as in the basi model, and hene the shareholders make the same information aquisition deisions. 3. Supplementary analysis for the proof of Proposition 8. To prove that lim t 0 π t =, we prove the auxiliary result that R π is bounded away from zero for π in the neighborhood of. To prove this property, we first prove that R π is stritly inreasing in π. The envelope theorem 0

21 implies that at any point at whih R π is differentiable, R π = Nf π q r π f π, π, IA35 where f π denotes the fee hosen by the advisor when the preision of the signal equals π. If, given π, the equilibrium features omplete rowding out of private information aquisition, then q r f π, π is given by 3, so q r π f π, π = 4 N π qr f π, π f π C π. IA36 If the equilibrium features inomplete rowding out of private information aquisition, then q r f π, π is given by A5, so N q r π f π, π = π 4 f π+ p πc f π+ p πc + π p f π πc 4 p f π πc. IA37 Sine f π > 0 for any π,, IA36 IA37 imply that qr π f π, π > 0 at any π, at whih R π is differentiable. Thus, R π > 0 at any π, at whih R π is differentiable. In addition, at any point π at whih R π is not differentiable, it annot be dereasing, sine for any suh π the seller an hoose the fee that is optimal for π ε for infinitesimal positive ε and ahieve higher revenues. Therefore, R π is stritly inreasing in π,. Next, we prove that R π is bounded away from zero for π in the neighborhood of. When π is lose enough to, A3 has no solution, and hene equilibrium annot feature inomplete rowding out. In equilibrium with omplete rowding out, R π is given by IA35 IA36, so to prove that it is bounded away from zero, it is suffi ient to prove that f π is bounded away from zero. Suppose this is not the ase, that is, lim π f π = 0. Sine q r <, this implies lim π R π = 0. This, however, is not possible sine R π is stritly inreasing, as shown above. This ompletes the proof of this statement. B. Analysis of regulation B.. Litigation pressure Suppose that a shareholder gets an additional payoff > 0 if it subsribes to and follows the advisor s reommendation. Given q r and q s, the gross value to a shareholder from aquiring a private signal and the reommendation of the advisor is V s q r, q s and V r q r, q s +, respetively. As before, the value from staying uninformed is zero. Therefore, for a fixed fee f, the game is idential to the subgame of the basi model with fee f. The equilibrium probability that a shareholder buys and follows the advisor is therefore given by q r f, where q r is given by 4. Speifially, if f < f +, the equilibrium features omplete rowding out of private

22 information aquisition q r = qr H f and q s = 0, while if f [f +, f +, it features inomplete rowding out q r = qr L f and q s > 0. Sine q r is dereasing in fee f, for any fee f, the demand for the advisor s reommendation is higher than in the basi model. The advisor responds to the inreased demand by inreasing its fee. The next proposition summarizes the effet of an inrease in regulatory pressure on the informativeness of deision-making: Proposition B. litigation pressure. A marginal inrease in :. dereases firm value if the equilibrium features inomplete rowding out of private information aquisition i.e., equilibrium fee exeeds f + ;. does not aff et firm value if the equilibrium features omplete rowding out of private information aquisition and limit priing i.e., equilibrium fee equals f + ; 3. inreases firm value if the equilibrium features omplete rowding out of private information aquisition and unonstrained maximization i.e., equilibrium fee is below f +. Proposition B. suggests that greater litigation pressure is a deliate issue. It inreases the demand for the advisor s reommendation for any quality of the advisor s reommendation, whih has two effets. On the one hand, it inreases the inentives to vote informatively. On the other hand, it shifts the inentives from doing proprietary researh to following the advisor s reommendations. As a onsequene, the total effet on the quality of deision-making depends on the quality of the advisor s information. As the basi model shows, if the quality is low, there is overreliane on the advisor s reommendation and ineffi ient rowding out of private information prodution. In this ase, higher litigation pressure leads to even more ineffi ient rowding out of private information prodution, whih redues the quality of deision-making. In ontrast, if the quality of the advisor s reommendation is high, there is underreliane on the advisor s reommendation, beause the profit-maximizing advisor pries information so as not to sell it to all shareholders. In this ase, greater litigation pressure inreases the quality of deision-making by inreasing the fration of shareholders who follow the advisor instead of voting uninformatively. B.. Reduing proxy advisory fees Consider the effet of a marginal redution in the fee harged by the advisor from the equilibrium f to a lower level. As the next proposition shows, whether suh a redution in market power is benefiial depends on the equilibrium information aquisition deisions by shareholders, and in partiular, on how muh private information they aquire. To see this, suppose, first, that given the equilibrium fee f, shareholders do not aquire any private information. In this ase, it is optimal for the quality of deision-making that more shareholders rely on the advisor, sine following the advisor dominates uninformed voting. Therefore, if omplete rowding out of private information aquisition ours in equilibrium, a marginal redution of the advisor s fee inreases the informativeness of voting. In ontrast, if the equilibrium features inomplete rowding out of private information aquisition, a redution in the advisor s fee has a negative effet of rowding out some of this private information aquisition. By the same logi as in Proposition 3, this is ineffi ient and lowers the quality of deision-making. The following result formalizes these arguments:

23 Proposition B. restriting market power. A marginal redution in the advisor s fee inreases firm value if equilibrium features omplete rowding out of private information aquisition, but dereases firm value if equilibrium features inomplete rowding out of private information aquisition. Proposition B. implies that restriting the advisor s market power will lead to more informative voting only if the advisor s information is suffi iently preise. In ontrast, if the advisor s information is impreise, dereasing its market power will lower the quality of deision-making beause it will lead to even greater overreliane on the advisor s reommendations. B.3 Dislosing the quality of reommendations In this setion, we examine how dislosing the quality of the advisor s reommendations affets the informativeness of deision-making. Speifially, onsider the following modifiation of our baseline setting. The atual preision of the advisor s signal an be high or low, π {π l, π h }, π l < π h, with probabilities µ l and µ h, µ h + µ l =. Let π µ l π l + µ h π h denote the expeted preision of the signal. Let us ompare the quality of deision-making in two regimes when the preision of the advisor s signal is publily dislosed and when it remains unknown to the shareholders. If the preision of the advisor s signal is dislosed, the timing of the game is as follows. First, preision π {π l, π h } is realized and learned by all parties. Then, the advisor deides on the fee it harges for its reommendation. After that, shareholders non-ooperatively deide what signals to aquire and how to vote. If the preision of the advisor s signal is not dislosed, the timing of the game is idential to that in the previous setions: The advisor sets the fee it harges, shareholders deide what signal to aquire, not knowing whether π = π l or π = π h, and then deide how to vote. The proof of the proposition below shows that the equilibrium in this game oinides with the equilibrium of the basi model for π = π. We make a simplifying assumption that unertainty about the preision of the advisor s signal is rather high: Assumption high preision unertainty. π l = and π h is suh that omplete rowding out of private information aquisition ours in equilibrium of the basi model with π = π h. This assumption implies that if the quality of the advisor s information is low, its signal is ompletely uninformative. Clearly, if shareholders know that the advisor s signal is pure noise, no shareholder buys it, and the equilibrium is idential to the benhmark model without the advisor. In ontrast, if the quality of the advisor s information is high and shareholders know about it, no shareholder aquires private information. The next proposition gives suffi ient onditions under whih dislosure improves the quality of deision-making: Proposition B.3 dislosure of preision. Firm value is stritly higher when the preision of the advisor s signal is dislosed if at least one of the following onditions is satisfied:. V π h > V 0, i.e., firm value is higher with the advisor than without when π = π h ; or. Complete rowding out of private information aquisition ours when π = π. 3

24 The intuition is as follows. Dislosing the preision of the advisor s reommendations allows shareholders to tailor their information aquisition deisions to the quality of the reommendations: shareholders do not aquire the advisor s reommendations if π = and do not aquire private information if π = π h. Under the first ondition in Proposition B.3, suh tailored information aquisition deisions are rather effi ient: they ensure that the advisor s reommendations do not affet the vote when they are uninformative, and that they have a relatively large effet on the vote when they are suffi iently informative V π h > V 0. Hene, dislosure leads to more informed voting deisions than if shareholders made their deisions based on the average preision π and sometimes relied on the advisor s reommendations when they are ompletely uninformative. A similar argument applies under the seond ondition in Proposition B.3: without dislosure, shareholders do not aquire private information and ompletely rely on the advisor s reommendations, even though they are sometimes uninformative. In ontrast, with dislosure, shareholders perform independent researh when the advisor s reommendations are uninformative, leading to more informed voting deisions. Interestingly, however, dislosing the preision of the advisor s reommendations does not always improve the quality of deision-making: Dislosure may enourage even stronger rowding out of private information aquisition and derease firm value. To see this, onsider the numerial example of Figure 3 and suppose that π l =, π h = 0.7, and µ l = µ h =, so that π = 0.6. Without dislosure, expeted firm value is given by V 0.6, whih, as Figure 3 demonstrates, is very lose to value V 0 in the benhmark ase without the advisor. This is beause the expeted preision of the advisor s signal is suffi iently low, so that there is relatively little rowding out of private information aquisition. In ontrast, with dislosure, expeted firm value is the average of V 0 and V 0.7, and this average is lower than V 0.6. Thus, in this example, dislosure makes voting deisions less informed and dereases firm value. The reason is that when π = π h, the advisor s reommendations are not preise enough to improve deision-making but are suffi iently preise to ompletely rowd out private information aquisition. This ineffi ient rowding out of private information when π = π h is detrimental for firm value, and even the more effi ient deision-making when π = π l is not suffi ient to ounterat its negative effet. Proofs for the setion Analysis of regulation Proof of Proposition B.. Let f denote the equilibrium fee that the advisor harges. Consider part of the proposition. Sine q r +q s < by Assumption, then f = arg max f fqr L f, where qr L is given by A5. Using a hange of variable φ f, we have: f = + arg max φ φ + ql r φ. We first prove that the maximizer φ, denoted φ, is dereasing in. Consider any >. Denoting φ i = φ i, i {, }, we have φ + q L r φ φ + q L r φ, φ + q L r φ φ + q L r φ, 4

25 or equivalently, implying φ q L r φ φ q L r φ [ q L r φ q L r φ ], φ q L r φ φ q L r φ [ q L r φ q L r φ ], [ q L r φ q L r φ ] [ q L r φ q L r φ ]. IA38 Suppose, by ontradition, that φ > φ. Sine ql r φ φ < 0 by A5, then qr L φ < qr L φ, and hene IA38 implies, giving a ontradition. Hene, φ φ, i.e., φ is dereasing in. Sine ql r φ φ < 0 and φ is dereasing in, then the equilibrium probability that a shareholder aquires information from the advisor, qr L φ, inreases in. Hene, aording to A9, p a p d = q r inreases in. By the argument similar to that in the proof of Proposition 3, firm value dereases in. Indeed, sine q r + q s <, a marginal inrease in inreases the distane between x a = P p a, N, and x d = P p d, N,, while keeping the total probability of being pivotal, πx a + π x d, unhanged at. Aording to A0, firm value equals πf x a + π f x d, where f x = N k= N+ p P ϕ x, N, k and ϕ x is defined by A. Sine, aording to Auxiliary Lemma A, funtion f x is onave, firm value dereases with the distane between x a and x d when πx a + π x d remains unhanged, and hene dereases with. Consider part of the proposition. In this ase, f = f+, and hene q r = qr H f = qr H f. Thus, both qr and q s = 0 are unaffeted by a marginal hange in, and hene firm value is unaffeted by as well. Finally, onsider part 3 of the proposition. In this ase, f = arg max f fqr H f. Using a hange of variable φ f, we have: f = + arg max φ φ + qh r φ. Sine the ross-partial derivative of the maximized funtion qh r φ φ is negative, the maximizer φ, denoted φ, is dereasing in. Therefore, the equilibrium probability that a shareholder aquires information from the advisor, qr H φ, inreases in. As shown in the proof of Proposition 3, N k= N+ P q q, N, k > 0 for q > and hene, aording to A8, firm value inreases in. Proof of Proposition B.. First, suppose that omplete rowding out of private information aquisition ours in equilibrium. Then q r = qr H f is given by 3, and a marginal derease in f inreases q r. The expeted value of the proposal is given by A8. As shown in the proof of Proposition 3, N k= N+ P q q, N, k > 0 for q > and hene expeted value inreases when f dereases. Next, onsider the ase of inomplete rowding out of private information aquisition. Sine q r + q s < by Assumption, q r, q s are given by A5. A marginal derease in f inreases q r and hene, aording to A9, p a p d = q r inreases. By the argument similar to that in the proof of Proposition 3, firm value inreases in f. Indeed, a marginal derease in f inreases the distane between x a = P p a, N, and x d = P p d, N,. while keeping the total 5