Internet Appendix for "Nonbinding Voting for Shareholder Proposals"

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1 Internet Appendx for "Nonbndng Votng for Shareholder Proposals" DORON LEVIT and NADYA MALENKO The Internet Appendx contans the omtted proofs from the man appendx (Secton I) and the analyss correspondng to two extensons of the model (Secton II) - endogenous proposal submsson and the presence of manageral retalaton. A. Supplemental Proofs for Secton I I. Supplemental Proofs Proof of Lemma A.: The shareholder s bene t from votng for the proposal depends on hs belefs about the state of the world. Snce! g >! b, other shareholders votes are nformatve about ther sgnals and could be used to update the shareholder s belefs. However, when votng, the shareholder does not observe the votes of other shareholders and therefore has to consder N possble stuatons, correspondng to the number of a rmatve votes among the other N shareholders, T N, beng 0; ; ::: or N. Let us denote by s (T N ) the shareholder s expected utlty from votng "for" relatve to hs expected utlty from votng "aganst," gven T N a rmatve votes among other shareholders and hs own sgnal s. Then the shareholder s expected bene t from votng for the proposal relatve to votng aganst the proposal,!;t (s), s gven by!;t (s) T 0 s (T ) Pr(T N T js), and the shareholder prefers to vote a rmatvely f and only f!;t (s) 0. When the shareholder votes strategcally, he takes nto account the fact that hs decson a ects hs utlty only n certan events, namely, those when the shareholder s vote s pvotal, Ctaton format: Levt, Doron, and Nadya Malenko, 0, Internet Appendx for "Nonbndng Votng for Shareholder Proposals," Journal of Fnance 66, , Please note: Wley-Blackwell s not responsble for the content or functonalty of any supportng nformaton suppled by the authors. Any queres (other than mssng materal) should be drected to the authors of the artcle.

2 that s, changes the nal outcome. Therefore, each shareholder condtons hs decson not only on hs prvate sgnal, but also on the nformaton that must be true when he s pvotal. In the current settng, the shareholder s only pvotal f he changes the manager s decson to accept the proposal. The shareholder antcpates that the manager follows the decson rule T and understands that hs vote changes the manager s decson f and only f the number of a rmatve votes among the other N shareholders, T N ; s exactly T. Thus, whenever T N 6 T, the shareholder s vote does not change the manager s decson, and hence the shareholder s nd erent wth respect to hs vote, whch mples s (T N ) 0. If T N T ; then the shareholder s vote becomes pvotal for the manager because the proposal s accepted f and only f the shareholder votes a rmatvely. Let Pr (jt N T ; s) be the posteror belef that the state s condtonal on beng pvotal for the manager and the sgnal s. Then the expected value that a shareholder wth sgnal s gans by votng for the proposal n ths event s gven by s (T ) Pr (GjT N T ; s) Pr (BjT N T ; s) : (IA.) Combnng these arguments, the expected relatve bene t from votng for the proposal s gven by!;t (s) s (T ) Pr(T N T js): (IA.) We argue that!;t (g) >!;T (b). Note that Pr(T N T js) > 0 snce T [0; N ]. Then pluggng (IA.) nto (IA.) and usng Bayes rule, we can rewrte!;t (s) as!;t (s) Pr(G; T N T js) Pr(B; T N T js) Pr (T N T ; sjg) Pr (T N T ; sjb) : Wrtng out the probabltes Pr (T N!;T (s) C N T T ; sj) explctly, we get ( T G ( G) N T T B ( B) N T ( ) f s g T G ( G) N T ( ) T B ( B) N T f s b (IA.3) and thus!;t (g) >!;T (b), T G ( G ) N T ( ) > T B ( B ) N T ( ); whch s true snce > 0:5. Last, from (IA.3) t mmedately follows that!;t (g) > 0, T B ( B) N T < T G ( G) N T!;T (b) > 0, T B ( B) N T T B G < B G ( G) N T G ;

3 and notng that Pr[T N;!T jb] Pr[T N;! T T B ( B) N T jg] T G ( G) N T completes the proof. Proof of Theorem, Case - bad type mxng equlbra: Ths equlbrum exsts and s responsve f and only f () T! [0; N ] and ()!;T! (b) 0. Note that n ths case, and accordng to (A3), B G!;T! (b) 0, Pr [T N;! T!jB] Pr[T N;! T!jG], T! N log log G B + log Snce T! s an nteger, ths s only possble f the rght-hand sde s an nteger. Note that when! b spans the nterval (0; ) ; G B spans the nterval (; ). Therefore, for any nteger I N ; N there exsts some! b (0; ) such that the rght-hand sde of the above equaton equals I. Moreover, f T! N ; N ; then the condton T! [0; N ] (condton () above) s sats ed. Thus, ths type of mxed strategy equlbra exsts f and only f there exsts! b (0; ) such that both the left-hand sde, T!; and the rght-hand sde are equal to some nteger I N ; N. Recall that T! b! ( )c and! ( ) s gven by (A). Thus, when! g, Snce N log log G B +log Expressng log G B some nteger I T! $ log G B N log + log + log G B log + log I T! and I s an nteger, t must be that % : log : log G B +log [0; ). through I; ths s equvalent to the requrement that I [0; ) for N log N ; N. Hence, a responsve equlbrum exsts f and only f there exsts log I N ; N such that I [0; ); whch s equvalent to log N log N [0; log ). Clearly, I the equvalent requrement s that the last nequalty s sats ed for the lowest nteger I n the N nterval ; N ; whch s N N+ + when N s even and when N s odd. Pluggng these values nto the last nequalty, we conclude that responsve equlbra wth a bad type mxng exst f and only f (A5) holds. Proof of Theorem, Case 3 - good type mxng equlbra: Ths equlbrum exsts and s responsve f and only f () T! [0; N ] and ()!;T! (g) 0. Note that n ths case, and accordng to (A3), G B log!;t! (g) 0, Pr [T N;! T!jB] Pr[T N;! T!jG] B, T! G N log B G log + log B G : Snce T! s an nteger, ths s only possble f the rght-hand sde s an nteger. Note that 3

4 when! g spans the nterval (0; ) ; B G spans the nterval (; ). Therefore, for any nteger I 0; N there exsts some! g (0; ) such that the rght-hand sde of the above equaton equals I. Moreover, f T! 0; N, then the condton T! [0; N ] (condton () above) s sats ed. Thus, ths type of mxed strategy equlbra exsts f and only f there exsts! g (0; ) such that both the left-hand sde T!; and the rght-hand sde are equal to some nteger I [0; N ). Recall that T! b! ( )c and! ( ) s gven by (A). Thus, when! b 0, $ % N log B T! G log log + log + B G log + log : B G Snce N log B G log +log B G I T! and I s an nteger, t must be that log log +log B G + [0; ). Expressng log B G through I; ths s equvalent to the requrement that N I + N log [0; ) for some nteger I 0; N. Hence, a responsve equlbrum exsts f and only f log there exsts I 0; N such that N I [ ; 0); whch s equvalent to log N log h N log. ; 0 Clearly, the equvalent requrement s that the last nequalty s sats ed for N I the hghest nteger I n the nterval 0; N ; whch s N when N s even and N ; when N s odd. Pluggng these values nto the last nequalty, we conclude that responsve equlbra wth a good type mxng exst f and only f (A6) holds. B. Supplemental De ntons and Proofs for Secton II Full Descrpton of Preferences. We rst formalze the preferences of the shareholders, the manager, and the actvst descrbed n Secton II of the man text for a gven pro le of strateges (!; d M ; e) of the shareholders, manager, and actvst, respectvely. Note that the number of a rmatve votes T, gven that shareholders follow votng strateges! and the state of the world s ; follows a bnomal dstrbuton wth parameters (N; ) ; where s de ned by (A). Let E! T [j] be the condtonal expectaton operator over the number of a rmatve votes under ths dstrbuton. Denote by U (d; ) each agent s utlty from the ultmate decson d fa; Rg when the state of the world s. As de ned n the setup, U S (d; ) v (d; ) U M (d; ) k M v(d; ) ( k M ) fdag U A (d; ) k A v(d; ) + ( k A ) fdag ; where v (d; ) s the value to the rm from acceptng (rejectng) the proposal for d A (R). Last, f d s the acton taken by the manager, denote the opposte acton by ^d, that s, b d fa; Rg nd. log 4

5 Then shareholders expected utlty gven state of the world and the pro le of strateges (!; d M ; e) s gven by " # ( e(t; d M (T ))) U S (d M (T ) ; )+ W S (!; d M ; e; ) E! T e(t; d M (T )) U S ( d b M (T ) ; ) + ( ) U S (d M (T ) ; ) : Smlarly, but wth addtonal costs c M ncurred by the manager f the proxy ght s successful, the manager s expected utlty s gven by " # ( e(t; d M (T ))) U M (d M (T ) ; )+ W M (!; d M ; e; ) E! T e(t; d M (T )) hu M ( d b M (T ) ; ) c M + ( ) U M (d M (T ) ; ) : Last, takng nto account solctaton costs ncurred by the actvst, the actvst s expected utlty s gven by " # ( e(t; d M (T ))) U A (d M (T ) ; )+ W A (!; d M ; e; ) E! T e(t; d M (T )) U A ( d b M (T ) ; ) + ( ) U A (d M (T ) ; ) c A : Proof of Lemma B.: Gven! and T, the bene t of the actvst from acceptng the proposal relatve to rejectng t s ( k A ) k A (! (T )). Thus, f the manager accepts the proposal, e (T; A), [ ( k A ) + k A (! (T ))] c A,! (T ) A : Smlarly, f the manager rejects the proposal, e (T; R), [( k A ) k A (! (T ))] c A,! (T ) R : The last nequalty of the statement follows from an explct comparson of A, R ; and. Proof of Lemma B.: Gven posteror belef! (T ), the manager s bene t from acceptng the proposal relatve to rejectng t when no proxy ght s organzed s gven by X (! (T )) k M (! (T ) ) ( k M ); (IA.4) where X (! (T )) > 0,! (T ) > NP. The superscrpt NP shows that ths threshold belef corresponds to the case of no proxy ght. We start wth the argument that the manager does not accept the proposal f hs posteror belefs are below A. Intutvely, f! (T ) < A ; then the actvst prefers that the proposal be rejected and hence the manager, who s more based aganst the proposal than the actvst, also prefers that the proposal be rejected. To show ths formally, note that accordng to Lemma B., when! (T ) A the actvst ntates a proxy ght f and only f the manager accepts 5

6 the proposal. Hence, n ths range, d M (T ) A, ( ) X (! (T )) > c M : Note that NP A and hence the nequalty! (T ) A mples that! (T ) NP ; whch mples that X(! (T )) 0 accordng to (IA.4). Thus, the left-hand sde of the nequalty above s nonpostve, whle the rght-hand sde s postve. Hence, d M (T ) R n that range, whch mples that > A. Next, we consder two cases: Case I - R < NP : Because P s also smaller than NP ; then accordng to condton (A8) we need to show that max P ; R. Frst, consder T such that! (T ) ( A ; R ]. Accordng to Lemma B., the actvst never ntates a proxy ght n ths range and thus d M (T ) A, X (! (T )) > 0,! (T ) > NP : Therefore, snce R < NP, the manager rejects the proposal f! (T ) R. Next, consder T such that! (T ) > R. Then the actvst ntates a proxy ght only when the manager rejects the proposal and hence d M (T ) A, X (! (T )) > [X (! (T )) c M ],! (T ) > P : The superscrpt P shows that ths threshold belef corresponds to the case of a proxy ght. Combnng the two ranges, we conclude that the manager accepts the proposal f and only f! (T ) > maxf P ; R g; that s, max P ; R ; as requred. Case II - R NP (and hence NP ( A ; R ]): Because NP P ; then accordng to (A8) we need to show that NP. Note that smlarly to Case I, f! (T ) ( A ; R ]; then d M (T ) A,! (T ) > NP. Hence, t remans to prove that the manager also accepts the proposal f! (T ) > R. Smlarly to Case I, f! (T ) > R ; then d M (T ) A,! (T ) > P, whch holds n ths range because P NP R <! (T ). Thus, NP ; as requred. Proof of Theorem, Case II - R < P : Suppose by way of contradcton that there exsts a responsve equlbrum (! b ;! g ) wth! g >! b. Snce R < P, then accordng to Lemma B. P ; and accordng to Lemma B. a proxy ght to approve the proposal s organzed n equlbrum when! (T ) ( R ; P ]. Note that n ths case, a shareholder s vote has the potental to change not only the manager s decson (f the posteror belefs are around P ), but also the actvst s decson to ntate a proxy ght (f the posteror belefs are around R ). We wll show that under the condtons of the theorem, the posteror belefs n both of these events are su cently optmstc, nducng shareholders to vote n favor of the proposal. Let! (N) and! (0) be the hghest and lowest posteror belefs, respectvely, that can be acheved for any T [0; N] gven the strateges!. There are four cases to consder: II. If P >! (N) > R >! (0) ; then the manager always rejects the proposal and 6

7 shareholders are only pvotal for the actvst s decson to launch a proxy ght. II. If! (N) > P >! (0) > R ; then the actvst always launches a proxy ght and shareholders are only pvotal for the manager s decson. II.3 If! (N) > P > R >! (0) ; then shareholders are pvotal for both the manager s and the actvst s decsons. II.4 In all other cases, the outcome of the vote does not change ether the actvst s or the manager s decson and hence the equlbrum s non-responsve by de nton. We prove the theorem for Case II.3, but t wll become clear that Cases II. and II. follow by the same logc. Suppose that shareholders are pvotal for both decsons. Note that by votng for the proposal when there are T a rmatve votes among other shareholders, the shareholder changes the probablty that the proposal s ultmately accepted by T ; where T takes the followng values: f T T! R! R ; then T ; f T T! P! P ; then T ; otherwse, T 0. Ths s because for T T! R the proposal s never accepted, for T (T! R ; T! P ] the proposal s accepted f and only f the proxy ght succeeds, whch happens wth probablty ; and for T > T! P the proposal s accepted wth probablty one. Therefore, the expected value that a shareholder wth sgnal s gans by votng for the proposal when the number of a rmatve votes among the other N shareholders, T N ; equals T s gven by T [ Pr (GjT N T; s) ]. It follows that a shareholder wth sgnal s wll vote a rmatvely f and only f hs expected relatve bene t from dong so s postve:! (s) T R! Pr GjTN T! R ; s + T P! Pr GjTN T! P ; s!;t R! (s) + ( )!;T P! (s) 0; where!;t (s) s gven by (IA.3). Note that by the same logc, the relatve bene t of votng a rmatvely s gven by! (s)!;t R! (s) n Case II. and by! (s) ( )!;T P! (s) n Case II.. Before provng that no responsve equlbrum exsts n ths case, we pont out two propertes. Frst, recall that the proof of Lemma A. dd not use any propertes of T. Hence, t follows from the same proof that!;t (g) >!;T (b) for T T! R ; T! P and hence! (g) >! (b). In other words, the expected bene t from votng for the proposal relatve to votng aganst t s strctly hgher for the shareholder wth a good sgnal than for the shareholder wth a bad sgnal. Smlarly to the proof of Theorem, t follows that there can only be three types of responsve equlbra: the equlbrum n pure strateges (! b 0;! g ), and two types of mxed strategy equlbra (! b (0; ) ;! g and! b 0;! g (0; )). Second, accordng to (A3),!;T (g) > 0, Pr[T N;!T jb] Pr[T N;! < T jg]!;t (b) > 0, Pr[T N;!T jb] B G Pr[T N;! < B T jg] G : 7

8 Snce Pr[T N;!T jb] Pr[T N;! s decreasng n T and T R T jg]! < T! P ; then!;t R! (s) > 0 mples!;t P! (s) > 0. Let us now prove that no responsve equlbrum exsts under the condtons of the theorem f Case II.3 s realzed. Recall that k A c A, R ( c A R ). Frst, consder equlbra wth! g and! b [0; ). Repeatng the argument n the proof of Theorem and usng (A4) and (A5), we conclude that snce R ( R ) ; then!;t R! (b) > 0 for any! (! b ; ) and any N. Therefore,!;T P! (b) > 0 as well. Intutvely, when R s su cently hgh, the shareholder s belefs condtonal on beng pvotal for the actvst s decson are su cently optmstc to nduce hm to vote a rmatvely even f hs sgnal s bad. Because the manager s threshold s even hgher than the actvst s, the shareholder s belefs condtonal on beng pvotal for the manager s decson are even more optmstc. It follows that! (b) > 0 and thus a shareholder wth a bad sgnal strctly prefers to vote for the proposal, whch contradcts! b <. Hence, a responsve equlbrum of ths type cannot exst. Second, consder equlbra wth! b 0 and! g (0; ). Repeatng the argument n the proof of Theorem and usng (A6), we conclude that snce R ( R ) >, t follows that!;t R! (g) > 0 and hence!;t P! (g) > 0; mplyng that! (g) > 0. Thus, a shareholder wth a good sgnal strctly prefers to vote a rmatvely, whch contradcts! g <. Hence, a responsve equlbrum of ths type cannot exst ether. Proof of Theorem 3, Case I - R P : Accordng to Lemmas B. and B., when a responsve equlbrum exsts, R P mples that the manager s threshold belefs are gven by mn R ; NP ; correspondng to the threshold T! b! ( )c. Moreover, each shareholder s only pvotal for the manager s decson and hence hs expected bene t from votng for the proposal relatve to votng aganst t s!;t! (s); whch sats es (A3). Recall that expresson (A7) n the proof of Theorem was derved for a general. Therefore, we can replcate the proof of Theorem and conclude that a responsve equlbrum exsts f and only f (A7) holds for mn R ; NP. Note, however, that k M + ( ), NP ( NP ) and hence (A7) s volated for NP k M under the condtons of the theorem. Thus, a responsve equlbrum exsts f and only f ) R and ) R sats es (A7). Recallng that R + c A k A ; t s straghtforward to show that for even N the condton h (A7) that R ( R ) ; ( ) s equvalent to condton (4) n the statement of the theorem. The condton c A < ensures that the nterval n (4) s non-empty. Hence, t remans to ensure that R. Accordng to (A8), R f and only f P R NP ; or equvalently, f and only f P R NP. The rst nequalty s sats ed n Case I by assumpton. P R NP The second nequalty s sats ed because R < ( R ) NP for k NP M + ( ) and under condton (4). We conclude that f R P ; k M + ( ) ; and c A (c A ; ), then condton (4) s necessary and su cent for the exstence of a responsve equlbrum. In such equlbrum a 8

9 proxy ght s never organzed. Proof of Theorem 3, Case II - R < P : There are two possbltes: h c II. Suppose k M > M cm, + ( ) and note that ths condton s equvalent to P < ( P ). We show that n ths case, condton (4) s su cent for the exstence of a pure strategy responsve equlbrum. Recall that (4) s equvalent to P > ( P ). We conclude that < R R h ( ) ; ( ) R < P R < P and hence R < P mples that : (IA.5) Consder a pure strategy responsve equlbrum: (! g ;! b ) (; 0). In ths equlbrum,! (0) (! (0) )N < ( ) and ( ) < ( )N! (N) ; whch mples! (N)! (0) < R < P <! (N) and hence that shareholders are pvotal wth strctly postve probablty for both the manager s and the actvst s decsons. In partcular, for T T R!! R ; the manager rejects the proposal and there s no proxy ght. For T! R < T T! P! P ; the manager rejects the proposal and the actvst ntates a proxy ght. Hence, wth probablty the proposal s accepted. Last, for T > T! P the manager accepts the proposal. Therefore, the expected bene t from votng for the proposal relatve to votng aganst t for a shareholder wth sgnal s s gven by! (s)!;t R! (s) + ( )!;T P! (s). Shareholders wll optmally vote accordng to ther sgnals f and only f! (b) 0! (b). It follows that su cent condtons for exstence of a pure strategy equlbrum are!;t R! (b) 0!;T R! (g) and!;t P! (b) 0!;T P! (g). Accordng to the proof of Theorem, condton (A4) for a general threshold belef and T! b! ( )c mples that!;t! (b) 0!;T! (g). Snce both R and P satsfy condton (A4) accordng to (IA.5), we conclude that a pure strategy responsve equlbrum ndeed exsts. h c M + ( cm ). Note that ths condton s equvalent to P P ( ). We show that n ths case, condton (5) n the statement of the theorem s su cent for the exstence of a responsve equlbrum n whch a proxy ght occurs wth postve probablty. There are two possble cases. II. Suppose k M (a) Suppose P ( P )N. Ths mples that P! (N) for any!. Hence, the manager always rejects the proposal and shareholders are never pvotal for hs decson. The only threshold at whch shareholders are pvotal s the actvst s threshold T! R : for T T! R ; the manager rejects the proposal and there s no proxy ght. 9

10 For T > T! R ; the manager rejects the proposal but the actvst organzes a proxy ght, and wth probablty the proposal s eventually accepted. Hence, the expected relatve bene t from votng for the proposal for a shareholder wth sgnal s s!;t R! (s). A su cent condton for the exstence of a pure strategy responsve equlbrum s therefore!;t R! (b) 0!;T R! (g). Note that condton (5) s equvalent to < R < and mples that condton (A4) s sats ed for R R. Moreover, because! (T ) (! (T ) )T N when shareholders play pure strateges, (A4) also mples that! (0) < R <! (N). Hence, by repeatng the arguments n the proof of Theorem, there exsts a responsve equlbrum n pure strateges, and n ths equlbrum a proxy ght s organzed wth strctly postve probablty. (b) Suppose ( ) P P < ( )N. We show that n ths case, condton (5) s su cent for the exstence of a bad type mxng responsve equlbrum n whch the manager always rejects the proposal and a proxy ght s ntated wth strctly postve probablty. Consder bad type mxng equlbra:! b (0; ) ;! g. Note that the maxmal and mnmal posteror belefs followng the nonbndng vote,! (N) and! (0) ; satsfy! (N)! (N)! (0)! (0) and hence as! b spans (0; ), N G ( )!b + B! b + ( ) N < ; N! (N) spans ; (.! (N) )N Note also that f condton (5) holds, then < R < < ( R ) P for any P and R n ths range. P Therefore, there exsts a range of! b (0; ) over whch spans the nterval ( R ; R P ), or equvalently, over whch log P G B spans N! (N)! (N) R log ; P log R N P. For these values of! b ;! (N) < P and hence the manager always rejects the proposal and shareholders are only pvotal for the actvst s decson to organze a proxy ght. We next demonstrate that there exsts a responsve equlbrum for some! b wthn ths range. Consder a mod caton of the proof of Theorem. Because shareholders are only pvotal at the threshold T! R! R, such an equlbrum exsts and s responsve f and only f () T! R [0; N ] and ()!;T R! (b) 0. Note that when! b (0; ), B G and accordng to (A3)!;T R! (b) 0, Pr T N;! T R! jb Pr[T N;! T R! jg], T R! N log log G B + log : 0

11 Snce T! R s an nteger, ths s only possble f the rght-hand sde s an nteger. Snce log G R B spans the nterval log ; P log, the rght-hand sde can be equal N R N P N log N log to any nteger I ; (I ; I ) N ; N for N log P P +log N log R R +log some! b n ths range. Moreover, f T! R (I ; I ) ; then the condton T! R [0; N ] (condton () above) s sats ed. Thus, ths type of mxed strategy equlbra exsts f and only f there exsts a! b wthn ths range such that both the left-hand sde, T! R ; and the rght-hand sde are equal to some nteger I (I ; I ). T! R! R and! R s gven by (A). Thus, when! g, 6 N log log R T! R 4 7 log G B + log + R 5 log G B + log : Recall that Snce N log log G B +log [0; ). Expressng log G B log R I R N log I T R! and I s an nteger, t must be that log R R log G B +log through I; ths s equvalent to the requrement that [0; ) for some nteger I (I ; I ). Hence, a responsve equlbrum exsts f and only f there exsts I (I ; I ) such that I N log R R log [0; ); whch s equvalent to log R N [0; log ). Clearly, the equvalent requrement s R I that the last nequalty s sats ed for the lowest nteger I n the nterval (I ; I ) ; whch s bi c +. Because R > ; t follows that log R > 0. Because R R R R < and bi c I ; to show that log R that log < N log I +, I < N ; where I ( ) P P ; t s su cent to show that < N log R bi c+ N log N log P P +log N log log( N ) +log t s su cent to show. Fnally, because < N, N > ; whch s always true n Case II.(b). We conclude that there ndeed exsts a bad type mxng responsve equlbrum n whch the manager always rejects the proposal and a proxy ght s ntated wth strctly postve probablty. C. Supplemental Proofs for Secton III Proof of Proposton : Let P (k A ; k M ; c M ; c A ;!) be the ex-ante probablty that the proxy ght s organzed n equlbrum. Recall from Lemmas B. and B. that P > 0,! R

12 T R! < T P!! P. Frst, suppose that P > 0. Then P s gven by P Pr T (T R! ; T P! ] 0:5 T P!X T T R! + [Pr(T jg) + Pr(T jb)] : Note that gven the votng strateges (! b ;! g ) (0; ), each of the terms Pr(T jg) + Pr(T jb) CT h N TG ( G ) N T + TB ( B ) N T only depends on N; T; and. As shown n the proof of Proposton, T! R weakly ncreases wth k A ; whle T! P does not depend on k A. Therefore, the terms Pr(T jg) + Pr(T jb) n the sum reman constant but the number of terms weakly decreases wth k A, whch mples that P weakly decreases wth k A. For the same reason, snce T! R weakly ncreases wth c A and T! P does not depend on c A ; P weakly declnes wth c A. Fnally, snce T! P weakly declnes wth both k M and c M and T! R does not depend on these parameters, P weakly declnes wth k M and c M. Hence, whenever P > 0; t weakly decreases wth all four parameters. It remans to show that P cannot ncrease from zero to postve probablty as one of the parameters ncreases. Suppose that P 0; whch s sats ed f and only f T! R T! P. As k A or c A ncrease, the left-hand sde of the last nequalty weakly ncreases and the rght-hand sde does not change, and hence the nequalty contnues to hold. Smlarly, as k M or c M ncrease, the left-hand sde does not change whle the rght-hand sde weakly decreases, and hence P contnues to be zero. Combnng all the results, the probablty of the proxy ght s weakly decreasng n k M ; k A ; c A ; and c M. D. Supplemental Proofs for Secton IV Proof of Lemma D.: A shareholder wth sgnal s votes a rmatvely f and only f hs posteror belef condtonal on beng pvotal, Pr (Gjs; pv), sats es Pr (Gjs; pv), + Pr(s;pvjB) Pr(s;pvjG), ( B ) N ( B ) N G G Pr(sjG) Pr(sjB) ; snce condtonal on hs beng pvotal, there are N a rmatve votes out of N total votes among the other shareholders. Note also that snce > ; we have that Pr (Gjg; pv) > Pr (Gjb; pv). Usng ths nequalty, a responsve equlbrum n pure strateges exsts f and only f Pr (Gjb; pv) Pr (Gjg; pv), ( ). Smlarly, an equlbrum wth! b 0;! g (0; ] exsts f and only f Pr (Gjg; pv),! g H ; where H H+ h ( ) N N H. It can be shown that (0; ], ( ) N <. Fnally, an H+

13 equlbrum wth! b [0; );! g exsts f and only f Pr (Gjb; pv),! b L+ ; L+ L h where L ( ) N + N. It can be shown that L+ [0; ), ( L+ L ) < ( ) N +. Combnng the three cases, a responsve equlbrum exsts f and only f (( ) N ; ( ) N + ). A. Proposal Submsson II. Extensons of the Model The analyss n the man text gnores shareholders decson of whether to submt a proposal to a vote. In ths secton we endogenze the process of proposal submsson and demonstrate that t s the large shareholders who are more lkely to submt proposals. Intutvely, only large shareholders have su cent ncentves to ncur the varous costs assocated wth the proposal submsson. A.. Setup Suppose there are two knds of shareholders. There are N small shareholders, each of whch holds one share, and there s one large shareholder who holds M shares, where M s an nteger. Each shareholder, regardless of hs sze, observes a prvate bnary sgnal s fb; gg, as descrbed n the basc model n Secton I. All sgnals have the same precson. We augment the basc model wth an ntal stage of proposal submsson. In partcular, once shareholders observe ther prvate sgnals, each shareholder, large or small, can submt a proposal on whch a nonbndng vote s held. Shareholders make ther decsons to submt the proposal smultaneously, and the vote takes place f and only f at least one shareholder submts the proposal. Submttng a proposal, however, s costly. It requres tme, e ort, and legal advce. Therefore, submsson of a proposal requres a personal cost c > 0. The cost s dentcal across all shareholders, large or small. If the proposal s submtted by at least one shareholder, a nonbndng vote s held as descrbed n Secton I. We assume that the manager can observe the vote of the large shareholder and of whoever submts the proposal. In addton, we focus attenton on cases n whch a responsve equlbrum at the votng stage exsts; otherwse, not submttng a proposal s a strctly domnant strategy snce the vote does not a ect the nal outcome. More spec cally, we assume that k M > + ( ) ; whch ensures that there always exsts a responsve equlbrum n pure strateges (each shareholder votes a rmatvely f and only f hs sgnal s good). Shareholders ncur the cost c by submttng a proposal even f the proposal s submtted by other shareholders as well. 3

14 Note that under ths condton, there may also exst responsve equlbra n mxed strateges. However, we select the unque equlbrum n pure strateges, whch s also the most nformatve of all responsve equlbra. A.. Analyss When shareholders consder whether to submt a proposal, they already observe ther prvate nformaton. Therefore, potentally, nformaton could be revealed by the shareholder s decson of whether to submt a proposal. Due to sgnalng consderatons, the game descrbed above has multple equlbra. We consder both symmetrc and asymmetrc equlbra at the submsson stage, but to keep the analyss smple we focus our attenton on pure strategy equlbra. In other words, for a gven sgnal, each shareholder submts the proposal n equlbrum ether wth probablty zero or wth probablty one. Fnally, we focus on equlbra n whch every shareholder whose prvate sgnal was revealed through hs submsson decson votes unnformatvely at the votng stage (e.g., votes n favor of the proposal regardless of hs prvate sgnal). Gven unnformatve votng, the manager optmally gnores the vote of such a shareholder, and gven that the manager dsregards hs vote, t s optmal for ths shareholder to vote unnformatvely. Hence, these are ndeed equlbrum strateges. We denote by x s (m g ; m b ) the bene t per share from the proposal submsson gven the shareholder s own sgnal s and the nformaton that m g f0; ; :::; N g other shareholders have good sgnals whle m b f0; ; :::; N g other shareholders have bad sgnals, where m g + m b f0; ; :::; N g. More spec cally, x s (m g ; m b ) s the bene t n the absence of sgnalng consderatons on the part of the shareholder, that s, hs bene t from submsson when the manager correctly nfers hs true sgnal s. In the proofs we show that the bene t from submttng the proposal ncreases wth m g, decreases wth m b ; and s hgher for shareholders wth a good sgnal than for those wth a bad sgnal. These results are ntutve gven that the proposal adds value f and only f the state s good. Nevertheless, t s worth mentonng that the bene t from proposal submsson stems partly from aggregatng the dspersed nformaton held by all shareholders of the company. Therefore, a pror, the bene t from proposal submsson could be postve even f the shareholder observes a bad sgnal. To rule out the stuaton n whch even a completely unnformed small shareholder mght have an ncentve to submt the proposal to a vote, we make the followng assumpton. We assume that f no nformaton on the value of the proposal s avalable, then the bene t per share from proposal submsson does not outwegh the cost of submsson. ASSUMPTION IA.: The ex-ante bene t per share from the nonbndng vote s smaller than the cost of submsson: x X Pr [s] x s (0; 0) < c: sg;b As n the man paper, for smplcty, we focus on the case when N s even. 4

15 We demonstrate n the proofs that f Assumpton IA. holds, there are only three knds of pure strategy equlbra at the submsson stage: equlbra n whch the proposal s never submtted, equlbra n whch a sngle (ether small or large) shareholder submts the proposal wth postve probablty, and equlbra n whch one small shareholder submts the proposal f and only f hs sgnal s good and the blockholder submts the proposal f and only f hs sgnal s bad. Next we nd condtons on M (the sze of the large shareholder) under whch each of these equlbra exst. The followng proposton provdes the characterzaton of the extended game. PROPOSITION IA.: Suppose Assumpton IA. holds. There exst M 3 M > M > 0 such that: () If M < M ; there exsts a unque equlbrum and n ths equlbrum the proposal s never submtted by any shareholder. () If M M ; there exts an equlbrum n whch the blockholder submts the proposal wth strctly postve probablty. () If M > M ; then n any equlbrum the blockholder submts the proposal wth strctly postve probablty. (v) If M > M 3 ; there s no equlbrum n whch at least one small shareholder submts the proposal wth strctly postve probablty. An mmedate corollary follows from the above proposton. COROLLARY IA.: If M > M 3 ; then n any equlbrum the large shareholder submts the proposal wth strctly postve probablty and small shareholders never submt the proposal. Intutvely, when the large shareholder has su cent holdngs n the rm, he has enough ncentves to submt the proposal and bene t from the nformaton aggregaton durng the nonbndng vote. At the same tme, small shareholders free rde on the large shareholder and thereby save the cost of submsson. Note, however, that we need the sze of the blockholder to be su cently large to rule out equlbra whereby small shareholders submt the proposal wth postve probablty. Indeed, as follows from the proof of Proposton IA., f M < M and x b (0; 0) > c; there exsts an equlbrum n whch a sngle small shareholder submts the proposal f and only f hs sgnal s good, and all other shareholders never submt t. The analyss of the model also suggests that the larger the sze of the blockholder s stake, the less nformaton s revealed through hs decson to submt the proposal to a vote. Indeed, when the blockholder s stake s not very large, he nds submsson bene cal only f hs prvate sgnal about the value of the proposal s postve. However, when the blockholder s stake s 5

16 su cently large, then submttng the proposal s pro table even f hs prvate sgnal s negatve. Ths s because the nonbndng vote e cently aggregates other shareholders nformaton and thereby helps more nformatve decson makng by the manager. As a result, submsson of the proposal conveys postve news about the blockholder s sgnal when hs stake s small, and does not convey any nformaton when hs stake s large. Formally, COROLLARY IA.: There exst M and M wth M M 3 M > M such that f M > M ; then there exsts a unque equlbrum and n ths equlbrum the blockholder submts the proposal wth probablty one. If M > M > M ; then there s no equlbrum n whch the blockholder submts the proposal wth probablty one, but there exsts an equlbrum n whch the blockholder submts the proposal wth strctly postve probablty smaller than one. Fnally, note that when M < M ; the proposal s never submtted by any shareholder even though there may be sgn cant bene t from nformaton aggregaton durng the vote f the number of shareholders s large. Thus, the collectve acton problem among shareholders may result n ne cency when the proposal s not submtted although t s optmal for the vote to take place from the shareholders aggregate pont of vew. A.3. Proofs To prove Proposton IA., we rst prove the followng clams: IA. For any (m g ; m b ) as de ned above, () x g (m g ; m b ) > x b (m g ; m b ), and () x s (m g + ; m b ) > x s (m g ; m b ) > x s (m g ; m b + ). IA. Suppose that n a pure strategy equlbrum more than one shareholder submts a proposal wth postve probablty. Then no shareholder submts the proposal wth probablty one. IA.3 There s no pure strategy equlbrum n whch there s a small shareholder who submts the proposal f and only f hs sgnal s bad. IA.4 There s no equlbrum n whch at least two shareholders submt the proposal f and only f ther sgnal s good and all other shareholders do not submt the proposal. IA.5 There s no equlbrum n whch the blockholder submts the proposal f and only f hs sgnal s bad, at least two small shareholders submt the proposal f and only f ther sgnal s good, and all other small shareholders do not submt the proposal. If x g (0; 0) < c, then there s also no equlbrum of ths type n whch only one small shareholder submts the 6

17 proposal f and only f hs sgnal s good and all other small shareholders do not submt the proposal. As follows from these clams, there are only three knds of pure strategy equlbra at the submsson stage: equlbra n whch one small shareholder submts the proposal f and only f hs sgnal s good and the blockholder submts the proposal f and only f hs sgnal s bad (whch exsts only f x g (0; 0) c), equlbra n whch a sngle (small or large) shareholder submts the proposal wth postve probablty, and equlbra n whch the proposal s never submtted. Proposton IA. then demonstrates under whch condtons on the sze of the large shareholder each of these equlbra exst. CLAIM IA.: For any (m g ; m b ) as de ned above, () x g (m g ; m b ) > x b (m g ; m b ) and () x s (m g + ; m b ) > x s (m g ; m b ) > x s (m g ; m b + ). Proof of Clam IA.: Frst, note that under the condtons of Theorem, there always exsts a responsve equlbrum at the votng stage n whch nformaton s fully revealed. We focus on ths equlbrum, whch mples that shareholders whose nformaton was not revealed by ther submsson decson wll vote accordng to ther sgnals. Moreover, n ths equlbrum the endogenous majorty rule T equals N (so the proposal s accepted f and only f the number of good sgnals s N + or more). 3 Second, once m g and m b are revealed through the submsson decsons, the posteror belef that the state s good of a shareholder wth sgnal s s gven by 8 < f s g; +( s Pr [Gjm g ; m b ; s] )mg m b + (IA.6) : f s b: +( )mg m b Moreover, f the proposal s submtted, the manager wll accept t f and only f the number of good sgnals out of the remanng unknown N m g m b sgnals together wth the shareholder s own sgnal s at least T + m g. Let T N mg m b be the number of good sgnals held by shareholders whose nformaton the shareholder does not know. Then, 8 g Pr T N mg m b + m g T jg >< x s (m g ; m b ) g Pr TN mg m b + m g T jb f s g; b Pr T N mg m b + m g T >: + jg ( b ) Pr T N mg m b + m g T + jb f s b: (IA.7) Ths s because a shareholder wth a good sgnal knows that f hs sgnal s not revealed by hs submsson decson, then he wll vote for the proposal, so that the manager wll correctly nfer hs good sgnal n any case. Hence, only T a rmatve votes among the other N shareholders 3 It can be ver ed from the proof of Theorem for the pure strategy responsve equlbrum (we rule out T! N snce ). 7

18 wll be needed to nduce the manager to accept the proposal. In contrast, a shareholder wth a bad sgnal knows that the manager wll always correctly nfer hs bad sgnal, and hence T + a rmatve votes wll be needed. To prove that x g (m g ; m b ) > x b (m g ; m b ), note that x g (m g ; m b ) x b (m g ; m b ) g Pr T N mg m b + m g T + jg g Pr TN mg m b + m g T + jb b Pr T N mg m b + m g T + jg + ( b ) Pr T N mg m b + m g T + jb + g Pr T N mg m b + m g T jg g Pr TN mg m b + m g T jb : Because g + ( b ) g b, by rearrangng terms we get Note that Pr T x g (m g ; m b ) x b (m g ; m b ) g N mg m b + m g T + jg b + Pr T N mg m b + m g T + jb + g Pr T N mg m b + m g T jg g Pr TN mg m b + m g T jb : g Pr T N mg m b + m g T jg > g Pr TN mg m b + m g T jb, g > Pr TN mg m b + m g T jb g Pr T N mg m b + m g T jg, ( )mg m b+ > ( )T m g () N mg m b (T m g) T m g ( ) N m g m b (T m g), T > N : Thus, snce g > b ; we have that x g (m g ; m b ) > x b (m g ; m b ) ; whch proves part () of Clam IA.. To see part () of Clam IA., note that the bene t from submsson gven m m g + m b sgnals s the expected bene t from submsson gven m + sgnals condtonal on m g ; m b. Hence, by solatng the unknown sgnal ^s of another shareholder, we get x s (m g ; m b ) Pr (^s gjs; m g ; m b ) x s (m g + ; m b ) + Pr (^s bjs; m g ; m b ) x s (m g ; m b + ) : In other words, x s (m g ; m b ) s a weghted average of x s (m g ; m b + ) and x s (m g + ; m b ) wth weghts Pr (^s gjs; m g ; m b ) and Pr (^s gjs; m g ; m b ) strctly wthn the unt nterval. Thus, t remans to show that x s (m g + ; m b ) > x s (m g ; m b + ). Note that x b (m g + ; m b ) 8

19 x g (m g ; m b + ). Ths s because n both sdes of the equalty, the shareholder knows that there are overall m b + bad sgnals and m g + good sgnals. Gven the rst part of the clam, we get x g (m g + ; m b ) > x b (m g + ; m b ) x g (m g ; m b + ) > x b (m g ; m b + ) ; and hence for s fb; gg we have x s (m g + ; m b ) > x s (m g ; m b + ) as requred. CLAIM IA.: Suppose that n a pure strategy equlbrum more than one shareholder submts a proposal wth postve probablty. Then no shareholder submts the proposal wth probablty one. Proof of Clam IA.: Suppose by way of contradcton that there s an equlbrum n whch one shareholder submts the proposal wth probablty one and at least one other shareholder submts the proposal wth postve probablty. There are four cases to consder: Case : The second shareholder submts the proposal f and only f hs sgnal s bad. Let us show that ths shareholder has a strctly pro table devaton of not submttng the proposal when hs sgnal s bad. The bene t from devatng s c. Snce the proposal s always submtted n equlbrum by the rst shareholder, the only cost from devatng s that hs sgnal s msnterpreted by the manager, whch may lead to ne cent decson makng. In partcular, f he devates from hs equlbrum strategy, hs bad sgnal s nterpreted as a good sgnal, and hence when (and only when) there are T N good sgnals among the other N shareholders, the proposal s accepted even though t should be rejected. However, combned wth the shareholder s bad sgnal, n ths scenaro there are exactly N good sgnals out of N sgnals n the economy and hence the shareholder s nd erent between approval and rejecton of the proposal. Therefore, the cost of msnterpretaton for a shareholder wth a bad sgnal s exactly zero and devaton s always bene cal. Case : The second shareholder s small and submts the proposal f and only f hs sgnal s good. We show that ths shareholder has a strctly pro table devaton of not submttng the proposal f hs sgnal s good. The bene t of devatng s c. Snce the proposal s always submtted n equlbrum by the rst shareholder, the only cost from devatng s that hs good sgnal s msnterpreted by the manager as bad and the proposal can be overrejected. Note that ths msnterpretaton only matters for the nal decson when there are exactly N good sgnals among the other N shareholders and hence the proposal s rejected though t should be accepted. The followng two steps demonstrate that devaton s optmal: Step - calculatng the shareholder s expected loss when hs good sgnal s msnterpreted to be bad: The loss occurs only when there are exactly N good sgnals among the other N shareholders. Gven that the shareholder has a good sgnal, the probablty of ths 9

20 event s gven by Pr [Gjs g] C N N N ( ) N N + Pr [Bjs g] C N N ( )N N N C N N [ ( )]N + ( ) : The loss from rejectng the proposal when there are exactly N + good sgnals out of N s gven by Pr [GjN + ] Pr [BjN + ] Pr [GjN + ] + ( Overall, the expected loss from devaton s ) + ( ) > 0: L C N N [ ( )]N + ( ) + ( ) C N N [ ( )]N ( ) > 0: Hence, to show that devaton s bene cal, we need to show that c L > 0, C N N [ ( )]N ( ) < c: Step - showng that L < x: Recall that accordng to Assumpton IA., x < c, where x s the expected bene t from submttng the proposal wthout any prvate nformaton. Hence, t s su cent to show that L < x. Because decson makng s based on e cent aggregaton of sgnals (T N), the proposal s accepted upon submsson when the number of good sgnals s at least N +. For any number of good sgnals above N +, the bene t from the proposal approval s hgher than the bene t when there are exactly N + good sgnals, whch equals as calculated above. Hence, +( ) 0

21 x P N +( ) tn+ Pr [t], where t s the number of good sgnals. Note that tn+ Pr [t] tn+ C N t tn+ tn+ C N t C N t h t ( ) N t + N t ( ) t h t ( ) N t + h t ( ) N t + C N N [ ( )] N and hence t s su cent to show that L < C N N [ ( )]N X N Ct N t0 tn+ C N t N t ( ) t C N N [ ( )]N ; + ( ), > CN N [ ( )]N [ ( )] : Lettng y ( ) 0; 4, t s su cent to show that N t ( ) t A AA > CN N max y N y[0; 4] y N : One can show that for any N 3; the maxmum s acheved at y 4. Hence, we need to show that N > 3C N N. Note that N P N 0 CN and C N N C N N (N )! (N )! (N)! C N N C N N+ (N )! (N + )! (N )! : Snce N > C N N + CN N + CN N + CN N+ CN N + CN N, t s su cent to show that C N N < CN N ; whch always holds for N > 6. Case 3: The second shareholder submts the proposal regardless of hs sgnal. Snce there s only one large shareholder, there s at least one small shareholder who submts the proposal wth probablty one. By devatng and not submttng the proposal, the shareholder gans c. Devaton from submsson, however, becomes an o -equlbrum event. Because the shareholder

22 wants e cent nformaton aggregaton, he ncurs the maxmum possble costs of devaton f hs sgnal s msnterpreted. However, as was argued n the rst and second cases, the cost from msnterpretaton of the sgnal s strctly smaller than c for both a good and a bad sgnal. Hence, t s always optmal to devate under any o -equlbrum belefs. Case 4: The second shareholder s large and submts the proposal f and only f hs sgnal s good. Snce the second shareholder s the large shareholder, then t must be the case that a small shareholder (the rst one) submts the proposal wth probablty one. We wll show that the small shareholder has ncentves to devate and not submt the proposal f he has a bad sgnal. Frst, note that there s no other small shareholder who submts the proposal wth postve probablty. Ths follows from the above three cases. Suppose that the small shareholder has a bad sgnal. If the small shareholder submts the proposal, he gets x b (0; 0) c < x c; whch by Assumpton IA. s negatve. If the shareholder devates and does not submt the proposal, then the followng two scenaros are possble. If the large shareholder has a bad sgnal as well, the proposal s not submtted and the payo s zero. If the large shareholder has a good sgnal, the proposal s submtted. The worst-case scenaro n terms of o -equlbrum belefs s that the small shareholder s sgnal s nterpreted as good. But, repeatng the argument n Case, the loss from the resultng msnterpretaton s zero. Hence, the payo n ths scenaro s nonnegatve. Combnng the two scenaros, devaton gves a nonnegatve payo and hence devaton s optmal. CLAIM IA.3: There s no pure strategy equlbrum n whch there s a small shareholder who submts the proposal f and only f hs sgnal s bad. Proof of Clam IA.3: Let m s 0 be the number of shareholders who submt the proposal f and only f they observe sgnal s n equlbrum. Suppose by way of contradcton that at least one of the m b shareholders s small. The relatve bene t of the small shareholder wth a bad sgnal from submttng the proposal relatve to not submttng t s gven by Pr T mb +m g m b jb x b (m b ; m g ) c; (IA.8) where Pr T mb +m g m b jb s the probablty that the other m b +m g shareholders who are supposed to submt the proposal wth postve probablty n equlbrum do not submt the proposal (ths s because smlar to Case n the proof of Clam IA., shareholders wth a bad sgnal have no cost of msnterpretaton of ther sgnal f they decde not to submt the proposal as they are supposed to). For m b we get Pr T mg 0jb x b (0; m g ) x b (0; m g ) x b (0; 0) < x < c: Hence, f we show that (IA.8) decreases n m b, the clam holds for any m b ; m g because the small shareholder has ncentves to devate and not submt the proposal. To see that (IA.8) decreases

23 n m b note that Let Pr T mb +m g m b jb ( ) m b ( ) mg + ( ) m b mg m b ( ) mg+ + ( ) m b mg+ [( ) ] mg+ m b m g + ( ) m b m g : f (m b ) Pr T mb +m g m b jb x b (m b ; m g ) [( ) ] mg+ : Gven (IA.6) and (IA.7), t can be shown that f (m b ) m b m g Pr T N mb m g N m b + jg ( ) m b m g Pr T N mb m g N m b + jb m b m g N m b m X g N m b + ( ) m b m g N m b m X g N m b + C N m b m g C N m b m g ( ) N m b m g N m b m X g N m b + C N m b m g ( ) N m b m g +m b m g ( ) N m b m g ( ) +m b m g N m b m g It remans to prove that f (m b ) f (m b + ). Note that f m b N + ; then f (m b ) m b m g ( ) m b m g and f (m b ) f (m b + ), m b m g 3, whch holds f m b N +. If m b < N + ; then f (m b ) f (m b + ) f and only f : N m b m X g N m b + N m b m X g N m b + C N m b m g C N m b m g +m b m g ( ) N m b m g ( ) +m b m g N m b m g ++m b m g ( ) N m b m g ( ) ++m b m g N m b m g : Note that N m b m X g N m b + N m b m X g N m b + C N m b m g C N m b m g ++m b m g ( ) N m b m g ( ) ++m b m g N m b m g +m b m g ( ) N m b m g ( ) +m b m g N m b m g 3 :

24 Hence, f (m b ) f (m b + ) f and only f N m b m X g N m b + h C N m b m g C N m b m g +m b m g ( ) N m b m g ( ) +m b m g N m b m g 0: Snce C N m b m g C N m b m g + C N m b m g ; then f (m b ) f (m b + ) f and only f N m b m g X N m b + C N m b m g +m b m g ( ) N m b m g ( ) +m b m g N m b m g Note that each component n the summaton s postve because 0: +m b m g ( ) N m b m g ( ) +m b m g N m b m g > 0, > N m b + : Therefore, f (m b ) f (m b + ) ndeed holds. CLAIM IA.4: There s no equlbrum n whch at least two shareholders submt the proposal f and only f ther sgnal s good and all other shareholders do not submt the proposal. Proof of Clam IA.4: Consder an equlbrum n whch there are m shareholders who submt the proposal f and only f ther sgnal s good and all other shareholders do not submt the proposal. There s at least one small shareholder among m. Consder hs decson to submt the proposal gven a good sgnal. Let (m) be the probablty that the other m shareholders have a bad sgnal. The bene t of a small shareholder from submsson s X (m) (m) x g (0; m ) + ( (m)) L (m) c; where L (m) > 0 s the cost that the manager msnterprets the shareholder s good sgnal as bad, condtonal on the shareholder havng a good sgnal and knowng that out of the m shareholders, at least one of them has a good sgnal as well. Recall that the cost of msnterpretaton s ncurred f and only f there are exactly N good sgnals among the other N shareholders. A drect calculaton mples that Pr [T X (m) Pr [Gjg] m 0jG] Pr [T N m NjG] + P m Pr [T m jg] Pr [T N m N jg] Pr [T Pr [Bjg] m 0jB] Pr [T N m NjB] + P m Pr [T : m jb] Pr [T N m N jb] 4

25 Rearrangng, X (m) ( ) m P N m N CN m ( ) N m + P h m C m ( ) m h C N m N N ( ) ( ) N m (N ) P m N m N CN m ( ) N m + P h h m C m ( ) m C N m N ( )N N m (N ) mx m C N m + ( ) N + N+ ( ) N N X N N m C N N + m C N m m ( ) + N + ( ) N+ N h + ( ) N ( ) + N mx h N+ ( ) N ( ) N+ N X N m C N N+ m h + ( ) N ( ) + N mx + ( ) [ ( )] N C N m N + C m C N m N C m C N m mx C m C N m N! ; N A C m C N m N! A! where P the dervaton above holds for any m N. Note that by Vandermonde s dentty, N 0 Cm C N m N CN N, and snce Cm 0 for > m ; P m Cm C N m N CN N C N m N. Therefore, X (m) X N m C N N+ m h + ( ) N ( ) + N +( ) [ ( )] N C N N : Note that snce + ( ) N ( ) + N 0 for N + and C N m C N m ; X (m) s decreasng n m. Therefore, to prove that the small shareholder wants to devate and not submt the proposal, t s su cent to show that X () < c. Note that X () N+ C N h + ( ) N ( ) + N + ( ) [ ( )] N C N N : 5