Online Appendix for Corporate Control Activism

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1 Online Appendix for Corporate Control Activim B Limited veto power and tender offer In thi ection we extend the baeline model by allowing the bidder to make a tender offer directly to target hareholder. We aume that if no acquiition agreement i reached at the econd round of negotiation then with probability λ [0, 1] the target remain independent and whoever control the target board can conume hi private benefit. With probability 1 λ the bidder can make a tender offer. In thi cae, whoever control the target cannot conume hi private benefit. Whether a tender offer i poible i revealed at the beginning of the econd round of negotiation. For implicity, we focu on conditional offer for all target hare. The poibility of making a tender offer affect the analyi of the baeline model only if the bidder can at leat partly overcome the free-riding problem of Groman and Hart (1980). That i, the bidder mut make ome profit, otherwie, the option of making a tender offer i never exercied. Target hareholder mut alo make a profit from the tender offer, otherwie, they are indifferent between keeping the firm independent and elling it via tender offer. For implicity, we aume there are no additional cot that are aociated with the tender, m > 0, and the free-rider problem exit. 33 We prove the following reult. Propoition 5 Suppoe the firt round of negotiation failed. Then: (i) The bidder never run a proxy fight. (ii) The activit run a proxy fight if and only if π A π I (κ/α) /λ. (19) where π A and π I are given by (1). If the activit run a proxy fight, he win the control of the target board and then reache an acquiition agreement with an expected takeover premium of (1 λ) + λπ A. 33 The free-rider problem in takeover effectively give target hareholder a bargaining power. If b then the target board might have incentive to leverage thi a bargaining power by allowing a tender offer. For implicity, we aume thi poibility away. Thi aumption would not change qualitatively the main reult ince the credibility of the activit arie only when < b. 1

2 (iii) If the activit doe not run a proxy fight, the incumbent retain control and the target remain independent if and only if a tender offer i not feaible and b >. Proof. Suppoe the econd round of negotiation failed. If the bidder cannot make a tender offer, the target remain independent. If the bidder can make a tender offer, becaue of the free-rider problem, hareholder tender their hare if and only if the offer i higher than q + (hareholder are not playing weakly dominated trategie). Therefore, the bidder make a tender offer of q + per hare, target hareholder tender their hare, the bidder take over the target and make a profit of m. Conider the econd round of negotiation. All partie involved rationally expect that if the econd round fail, the above dynamic would unfold. Therefore, if the bidder cannot make a tender offer, the outcome of the negotiation in thi tage i identical to the baeline model. Suppoe that the bidder can make a tender offer. Since the bidder can buy the firm with a tender offer of q + if the econd round of negotiation fail, the highet premium the bidder would be willing to pay i. Similarly, the incumbent will not agree to ell the firm for a premium lower than. Therefore, the bidder and the incumbent will reach an agreement in the econd round with a premium of. Thi conclude part (iii) of the propoition. The negotiation between the bidder and the activit in the econd round (if the latter control the board) are the ame a above. Next, uppoe the bidder control the target board. When tender offer i poible he cannot conume hi private benefit, and hence he will offer q + to the hareholder, ince thi i the lowet price that the hareholder accept. However, if tender offer i not poible, then he will conume hi private benefit (i.e., extract firm value) of η and offer hareholder the lowet price that i acceptable to them, which i q η. 34 For all of thee reaon, π 2 in Lemma 1 can be rewritten a (1 λ) + λπ 2 under the incumbent and the activit control, while it can be written a (1 λ) + λ ( η) under the bidder control. Therefore, in the econd round of negotiation, the expected hareholder value under the incumbent control i q + 1 {b }[(λ 1 m + (1 λ) ) + (1 ) (λb + (1 λ) )] + 1 {b> } (1 λ) = q + (1 λ) + λπ I. 34 Implicitly, we aume that the bidder private benefit η are larger than m, and therefore, he ha no incentive to make a tender offer to hareholder. Knowing thi, hareholder will agree to any price higher than q η if the bidder i already controlling their board. Alternatively, if the bidder could completely freeze out target hareholder and olve the free-rider problem, hareholder cannot expect any poitive premium once the bidder take control of their board. 2

3 Similarly, the expected hareholder value under the activit and the bidder control i q + (1 λ) + λπ A and q + (1 λ) + λ ( η), repectively. Thee obervation imply that the bidder can never win a proxy fight, which prove part (i). Moreover, they imply that the activit will run a proxy fight if and only if (1 λ) + λπ A (1 λ) + λπ I κ/α π A π I (κ/α) /λ, which conclude part (ii). C Optimal level of b The intitutional and legal environment often leave corporate inider with opportunitie to extract private benefit which are central to our analyi. In principle, hareholder can limit the extent of thee private benefit. Below we how that the reitance of the board can play a poitive role for target hareholder in our framework. While higher b could reduce the probability that the bidder initiate takeover negotiation, target hareholder might till prefer an incumbent with b > 0 over b = 0, ince larger b increae the target bargaining power during the takeover negotiation. Propoition 6 Conider the etup of Section 4 and uppoe that target hareholder can chooe b at the outet, before the activit receive hi ignal and trade. Let b be the level of b that maximize the expected target hareholder value. Then b > 0 if ( G (1 ) ˆ ) ( /g (1 ) ˆ ) > ˆ. (20) where ˆ E[ > 0]. Proof. Suppoe b < δ. Note that 0 < δ δ/α for all α [0, 1] > 0. According to Corollary 1 and Propoition 4, the expected hareholder value i given by q + µh(0) where Note that ( h(0) = G (1 ) b ) ( ) ( b) df ( ) [ + (1 )b] df ( ). b h(0) b = G (w(0) v(0)) [ bf(b) + (1 )(1 F (b))] g(w(0) v(0))v(0)(1 )(1 F (b)) 3

4 and h(0) lim b 0 b [ ( = (1 )(1 F (0)) G (1 ) ˆ ) ] g((1 ) ˆ ) ˆ h(0) Therefore, lim b 0 > 0 if and only if (20) hold. Thi implie that b > 0. Note that if b G (c) = 1 e λc (exponential ditribution with parameter λ > 0) and < 1 then (20) hold. 2 D The value of commitment In thi ection we extend the baeline model by auming that before the firt round of negotiation tart, the bidder can fully commit to act in the bet interet of target hareholder if they elect him to their board. Under commitment, hareholder expect to receive the fair price (which i q + ) if they elect the bidder to their board. In thi ection we aume B A = 0. The next reult characterize the equilibrium of the game. Propoition 7 Suppoe the bidder i committed to act in the bet interet of target hareholder once elected to the target board. A unique equilibrium exit. In equilibrium, the target i acquired if and only if { min b, 1 κ α, 1 1 m } κ 1 1 m. (21) If thi condition hold then the bidder reache an agreement in the firt round of negotiation with the incumbent board in which the bidder pay an expected takeover premium per hare of π 1 = + 1 {b } (1 )b +κ 1 κ if 1 κ/ if min{b, } α < min{b, κ/ α } (22) and acquire full control of the target. If condition (6) doe not hold, no proxy fight i initiated and the target remain independent under the incumbent control. Suppoe i drawn from ditribution F. The next reult follow directly from a comparion between Propoition 2 (when B A = 0) and Propoition 7. 4

5 Corollary 3 The net expected value that the bidder obtain from a commitment to act in the bet interet of target hareholder i L = () min{b, κ/ α } () min{b, κ/ α, 1 [(1 ) κ] df ( ), (23) κ 1 } which i decreaing in α and increaing in b. Moreover, if b 1 κ 1 then L = 0. D.1 Proof Proof of Propoition 7. promie an expected premium of Under the aumption above, both the bidder and the activit can. Therefore, target hareholder reelect the incumbent whenever b, and are indifferent between electing the bidder or the activit when < b. Therefore, the bidder and the activit will run a proxy fight only if the other party i not expected to do o. Subject to thi contraint, the incentive of the activit to run a proxy fight are the ame a in Propoition 1 part (ii), when B A = 0. However, unlike part (i) of Propoition 1, here the bidder can win a proxy fight. The bidder expected profit from running a proxy fight i (1 m) ( ) = (1 ) κ, and therefore, the bidder will run (and win) a proxy fight if and only if the activit doe not run a proxy fight and 1 κ 1 m 1 1 m < b. (24) κ/ We proceed in everal tep. Firt, uppoe < min{b,, 1 κ }. We prove that α 1 the target remain independent under the incumbent board control. Baed on the dicuion above, neither the bidder nor the activit run a proxy fight. Since < b, the incumbent board and the bidder will not reach an agreement in the econd round of negotiation. Therefore, in the firt round of negotiation the incumbent board reject any offer lower than q + b and the bidder reject any offer higher than q +. agreement in the firt round a well, and the target remain independent. Thu, the partie will not reach an Second, we prove that if b then the bidder pay q + the target after the firt round of negotiation. Baed on the dicuion above, if b + (1 )b and acquire then neither the bidder nor the activit run a proxy fight, and both the bidder and the incumbent expect to reach an agreement in the econd round in which the bidder pay + (1 ) b. Therefore, the bidder will not agree to pay more than thi amount and the incumbent board will not accept le than thi amount. They will reach an agreement in the firt round of 5

6 negotiation in which the bidder pay a premium of + (1 ) b. Third, uppoe max{ κ/, 1 κ } < b. In principle, there i an equilibrium of the α 1 ubgame (that follow the failure of the firt round) in which the bidder the run a proxy fight and an equilibrium in which the activit run a proxy fight. Conider the former equilibrium. We prove that the bidder pay an expected price of q + +κ and acquire the target in the firt round of negotiation. If the firt round of negotiation fail, the bidder will run a proxy fight and win. In the econd round, the expected premium i q +, and the bidder expected profit i (1 ) κ > 0. In the firt round of negotiation, hareholder would reject any offer lower than q +, and accept any offer higher than that amount. If the bidder i the propoer, he will offer q +, and both the board and the hareholder will accept it. If the board i the propoer, he will offer q + +κ, which leave the bidder with a profit of (1 ) κ 0. Indeed, q + (1 m) p qm = (1 ) κ p = q + + κ 1 m. The bidder will accept thi deal. Overall, the expected takeover premium i q + +κ, a required. Conider the latter equilibrium. All player expect that once the activit obtain control of the board, he will reach a ale agreement in which the bidder pay in expectation q + per hare. The bidder realize that any lower offer will be rejected by hareholder, who expect the activit to negotiate a higher offer at the econd round. The bidder can afford to pay q +, but he will not pay more than q +, ince he alway ha the option to pay that much in the econd round when he negotiate with the activit. The incumbent board undertand the bidder incentive and that the takeover of the target i inevitable, and therefore, he will loe hi private benefit of control. However, by accepting the offer q + the board can avoid the cotly proxy fight. Therefore, the incumbent and the bidder reach an agreement in the firt round of negotiation where the offer i q + Fourth, uppoe κ/ < min{b, 1 κ }. If the firt round fail only the activit run α 1 a proxy fight. Therefore, a in the third tep above, the bidder reache an agreement in the firt round of negotiation with the incumbent board in which he pay q + per hare and acquire the target. 1 κ Fifth, uppoe κ/ < min{b, }. If the firt round fail only the bidder run 1 α a proxy fight. Therefore, a in the third argument above, the bidder reache an agreement in the firt round of negotiation with the incumbent board in which he pay an expected price of q + +κ per hare and acquire the target.. 6

7 Finally, the tatement of the propoition i the union of the argument above ubject to the aumption that if max{ κ/, 1 κ } < b and the firt round of negotiation fail, α 1 then the equilibrium in which the activit run a proxy fight i elected. 35 E Comparative tatic of b Below we give uffi cient condition under which the probability of a takeover when b i high and the equilibrium exhibit treatment i higher than when b i low and the equilibrium exhibit election. Propoition 8 Suppoe α R and B q+µe[max{0, }] A < κ. There exit b < δ/α < b < uch that if b L [b, δ/α] and b H [b, ) then the probability of a takeover when b = b L i trictly lower than when b = b H. Moreover, the equilibrium exhibit election when b = b L and treatment when b = b H. Proof. Suppoe α R. According to Propoition 4, q+µe[max{0, }] α R = min{α, q+µh(α ) R R implie α q+µe[max{0, }] q+µh(α ) h(α ) E [max {0, }] for all α and b 0, α and b 0. Therefore, α = α for all b 0. Suppoe that b H b L δ/α. Note that by (17), θ (b H ) > θ (b L ) Since α (b H ) = α (b L ) = ᾱ, it i reduced to µg(w (b H ) v (b H )) (1 F (min {b H, δ/α (b H )})) > µg(w (b L ) v (b L )) (1 F (min {b L, δ/α (b L )})). }. Since for all α θ (b H ) > θ (b L ) w (b H ) v (b H ) > w (b L ) v (b L ) (25) Moreover, α (b H ) = α (b L ) = ᾱ further implie lim (w (b H) v (b H )) = b H w (b L ) v (b L ) = δ/ᾱ δ/ᾱ (1 ) (1 ) ( B A ᾱ ( δ ᾱ ) df ( ), ) df ( ), 35 Thi election i conervative in the ene that it give an upper bound on the bidder value from commitment, which i our main interet in thi ection. 7

8 where the econd line i atified ince b L = δ/α and by Corollary 2 the equilibrium exhibit election if b = b L (thi i becaue δ max{ q+µh(0 b=b L), 1 } = δ = b R α α L ince q+µh(0) 1 b [0, ) R α by the firt tep). Hence, lim b H θ (b H ) > θ (b L ) B A < δ B A < κ. The continuity of θ ( ) a a function of b and Corollary 2 conclude the proof. F Activit propoal Activit invetor often have the capacity to propoe way to increae the tandalone value of the target. Doe it complement or ubtitute the activit ability to preure the incumbent to ell the firm? To anwer thi quetion, uppoe that the bidder can increae the value of the target by only through it acquiition, while the activit can make a propoal that increae the value of the target by ε 0, but only if it remain independent. The incumbent loe hi private benefit if the target i acquired or the propoal i implemented. The propoal can be implemented by the incumbent or the activit, but without the activit, the incumbent i either unaware or doe not have the expertie to implement thi propoal. For implicity, we aume m = 0 and B A = 0. Suppoe ε < min {b, κ/α}. Since ε < b, the incumbent would not voluntarily implement the activit propoal. The activit intervention can be interpreted a the removal of ineffi ciencie caued by the incumbent conumption of private benefit. However, ince ε < κ/α, the activit doe not have enough incentive to run a proxy fight if the ole purpoe i implementing the propoal. Neverthele, the analyi in the baeline model continue to hold with the exception that the activit threat of running a proxy fight i credible if and only if κ/α 1 ε < < b, and in thi region, the takeover premium i + (1 ) ε. Intuitively, the upide from the takeover increae the incentive of the activit to run a proxy fight. Since the propoal increae the tandalone value of the firm once the activit obtain control of the target board, it alo increae the takeover premium that the activit can negotiate with the bidder. Similarly, the ability to increae the tandalone value of firm increae the credibility of the activit threat to run a proxy fight when the incumbent reit elling the firm. In thi repect, corporate control activim and non-control activim are complement. Moreover, ince the activit relaxe the reitance of the incumbent to the takeover, the bidder expected profit i higher when the activit i preent. In fact, it can increae with ε even conditional on the 8

9 activit preence, if increae in the likelihood of a takeover i firt order relative to increae in premium once the takeover take place. That aid, if ε i uffi ciently large, the bidder expected profit would decreae with ε and the activit take, and a takeover i le likely when the activit i preent than when he i not. In thoe cae, corporate control activim and non-control activim are ubtitute. The formal reult i given below. Propoition 9 Suppoe the activit can make a propoal, then: (i) If the firt round of negotiation fail, then the bidder never run a proxy fight, while the activit run a proxy fight if and only if κ/α ε < b and < b, or ε < κ/α and ε + κ/α ε < b. If the activit run a proxy fight, he win. (ii) Let Π B (α, ε) be the bidder expected profit. Then: (a) If ε < min {b, κ/α} then α > 0 Π B (α, ε) Π B (0, ε) and lim κ/α ε b ε+ 0. Π B (0,ε) ε > (b) If ε > b then for all α > 0, Π B (α, ε) i trictly decreaing in ε, Π B (α, ε) < Π B (0, ε), and takeover i le likely when the activit i preent than when he i not. Proof. Conider the following three cae: 1. Firt, uppoe max {ε, b}. If the incumbent retain control of the board and the firm remain independent, the incumbent implement the activit propoal if and only if ε b. Therefore, the reervation value of the incumbent in thi cae i q + max {ε, b} per hare. Since max {ε, b}, an agreement in which the bidder pay an expected premium of + (1 ) max {ε, b} i alway reached under the control of the incumbent board. On the other hand, if the activit obtain control of the board, he will reach an agreement with the bidder in which the expected takeover premium i + (1 ) ε. Therefore, the activit ha no incentive to run a proxy fight. Overall, the expected firm value i q + + (1 ) max {ε, b}. 2. Second, uppoe < ε and b ε. Since < ε and b ε, if the incumbent retain control of the board, the incumbent i willing to implement the activit propoal but refue the ell the firm. Since < ε, a takeover cannot increae the value of the firm even if hareholder extract all the urplu. Therefore, the activit ha no incentive to run a proxy fight, and the value of the firm under the incumbent control i q + ε. 9

10 3. Third, uppoe max {ε, } < b. Since max {ε, } < b, if the incumbent retain control of the board, the incumbent refue the ell the firm or implement the activit propoal. Therefore, under the incumbent control the firm value i q. Suppoe the activit control the target board. If ε > then he would implement the propoal, and if ε then he would reach an acquiition agreement in which the bidder pay an expected premium of +(1 ) ε. Therefore, under the activit control firm value i q+ε+ max {0, ε}, and hareholder alway elect the activit if he decide to run a proxy fight. The activit ha incentive to run a proxy fight if and only if α [q + ε + max {0, ε}] κ αq, which hold if and only if ε κ/α or, ε < κ/α and ε + κ/α ε. Part (i) follow from the interection of thi condition with max {ε, } < b. Conider the firt round of negotiation. All partie involved anticipate the dynamic above if the firt round fail. Therefore, if max {ε, b} then the bidder pay q+ +(1 ) max {ε, b} and take over the target after the firt round of negotiation. If < ε and b ε then the target remain independent and the activit propoal i implemented. If max {ε, } < b then the bidder pay q + ε + max {0, ε} if ε κ/α or, ε < κ/α and ε + κ/α ε, and otherwie, the target remain independent but the activit propoal i not implemented. Integrating over all value of, which i drawn from cdf F ( ), firm value i q + v (α, ε) where ε + ( ε) df ( ) if b ε ε v (α, ε) = v (0) + b [ε + max {0, ε}] df ( ) if κ/α ε < b v (0) + b [ε + ( ε)] df ( ) if ε < min {b, κ/α}, min{b,ε+ κ/α ε } which can be rewritten a v (0) + b [ε + ( ε)] df ( ) if ε < min {b, κ/α} min{b,ε+ κ/α ε } v (α, ε) = ε + ( ε) df ( ) ε +(1 ) if ε min {b, κ/α}. max {0, b ε} df ( ) b (26) 10

11 Moreover, the expected value created by the takeover and the activit propoal i df ( ) if ε < min {b, κ/α} min{b,ε+ w (α, ε) = κ/α ε } ε εdf ( ) + df ( ) if ε min {b, κ/α}. ε (27) Conider part (ii) and note that Π B (α, ε) = w (α, ε) v (α, ε). If ε < min {b, κ/α} then [ Π B (α, ε) = (1 ) ( b) df ( ) + b b min{b,ε+ κ/α ε } ( ε) df ( ) Clearly, Π B (α, ε) Π B (0, ε) for α > 0 where the inequality i trict if b > ε + κ/α ε. Suppoe b > ε + κ/α ε, and note that Π B (α, ε) ε and lim κ/α ε b ε+ ε > b. Then, Since ( ) 2 ( 1 = (κ/α ε) f ε + κ/α ε ) b (1 ) df ( ) ε+ κ/α ε Π B (α,ε) ε > 0. Thi complete part (ii.a). To ee part (ii.b), uppoe that ( ε) df ( ) if α > 0 ε Π B (α, ε) = (1 ) ( b) df ( ) if α = 0. b ε > b ε ( ε) df ( ) < b ( b) df ( ) and ( ε) df ( ) i trictly decreaing in ε, Π ε B (α, ε) i trictly decreaing in ε and Π B (α, ε) < Π B (0, ε) for all α > 0 a required in the tatement. Moreover, takeover probability i 1 F (ε) if the activit i preent and 1 F (b) if the activit i not preent, where the former decreae in ε and i larger than the latter ince ε > b, concluding the proof. ]. G Hidden value In reality, corporate board often have private information about the tandalone value of the target q, and bidder often have private information about the expected ynergy. The baeline model abtract from thee information aymmetrie and the reulting advere election in order to focu on agency problem a the key friction. Thi ection how that the aymmetric 11

12 information can in fact exacerbate the commitment problem of bidder in takeover and ometime enhance the ability of the activit to reolve it. For implicity we aume m = 0 and B A = 0. G.1 Uncertainty about q Incumbent board often jutify their reitance to takeover by claiming that the fundamental value of the target under their control i higher than the propoed takeover offer, even if the offer repreent a ignificant premium relative to the unaffected tock price. Eentially, they claim that baed on their private information the target i undervalued by the market a a tandalone firm. In thi ection we olve the baeline model under the aumption that q {q L, q H } i uncertain, q H > q L 0, and q i privately oberved by whoever control the target board, including the activit and the bidder if they win a proxy fight. We denote the prior by τ = Pr[q = q H ]. We alo aume that the identity of the propoer, the value of the offer, and the counter-party repone (i.e., accept or reject) are made public in each round. We focu attention on Perfect Bayeian Equilibria in pure trategie. Therefore, any equilibrium i either pooling or fully eparating. Lemma 2 Suppoe no information about q i revealed in the firt round of negotiation, and conider the econd round of negotiation. (i) If the bidder control the target board then: (a) If E [q] q L then the bidder offer hareholder E [q] and take over the target with probability one. (b) If < E [q] q L then the bidder offer hareholder q H and take over the firm if and only if q = q H. (ii) If the target board ha private benefit of control per hare β {0, b} and the bidder make an offer to the target board then: (a) If β + (q τ H q L ) then the bidder offer q H + β and the board accept the offer with probability one. (b) If β < β + (q τ H q L ) then the bidder offer q L + β and the board accept the offer if and only if q = q L. 12

13 (c) If < β then the takeover alway fail. (iii) If the target board ha private benefit of control per hare β {0, b} and the target board make an offer to the bidder then: (a) If β + (1 τ) (q H q L ) then the target board ak for E [q] + regardle of hi type and the bidder accept the offer. (b) If β < β + (1 τ) (q H q L ) then the target board ak for q L + if q = q L and the bidder accept the offer. If q = q H the target remain independent. (c) If < β then the takeover alway fail. Proof. Suppoe information about q i not revealed in the firt round. The proxy fight tage doe not reveal any information about q, ince q i only oberved by thi tage by the incumbent. Conider part (i) and uppoe the bidder control the target board. There i no information aymmetry between the bidder and the target board (ince it i controlled by the bidder), but target hareholder till need to approve the deal. We proceed in four tep. Firt, we how that the takeover ucceed with a trictly poitive probability in any equilibrium. To ee why, uppoe on the contrary that the takeover alway fail. Therefore, no offer π [q H, q H + ] i on equilibrium path, becaue otherwie it would be accepted by hareholder. However, ince > 0, if q = q H then the bidder trictly prefer an off-equilibrium offer π H (q H, q H + ) over hi equilibrium offer ince the former would be accepted by hareholder and generate a profit, creating a contradiction. Second, conider a pooling equilibrium where the takeover alway take place. Shareholder accept the pooling offer only if it i higher than E [q]. The bidder ha incentive to make the pooling offer when q = q L only if it i maller than q L +. Therefore, a pooling equilibrium exit if and only if E [q] q L +. In thi cae, the target i taken over for ure. Notice that the only pooling equilibrium that urvive the Groman and Perry (1986) criterion i the one in which the pooling offer i E [q]. Third, conider a eparating equilibrium. There are three ub-cae to conider: 1. The bidder make different offer depending on q and the takeover alway take place. However, the bidder ha incentive to deviate to offering the lower offer even if q = q H. So thi equilibrium cannot exit. 13

14 2. The takeover take place if and only if q = q L. Suppoe the bidder offer π when q = q L. However, the bidder ha incentive to deviate by offering π alo when q = q H. So thi equilibrium cannot exit. 3. The takeover take place if and only if q = q H : if q = q L the bidder doe not take over the firm and if q = q H the bidder offer π H and the offer i accepted by hareholder. Thi i an equilibrium only if π H = q H, becaue if π H > q H then whenever q = q H the bidder i trictly better off by deviating to an offer π (q H, π H ) which would be alway accepted by the hareholder, and if π H < q H then hareholder would reject π H. Therefore, π H = q H. Moreover, thi can be an equilibrium only if hareholder reject any offer lower than q H. However, off-equilibrium belief that upport thi equilibrium and atify the Groman and Perry (1986) criterion exit if and only if E [q] > q L +. Fourth, overall, if the off-equilibrium belief are required to atify the Groman and Perry (1986) criterion, then the unique outcome i a decribed in part (i) of the propoition tatement. In thi cae, target hareholder expected value i E [q]. Conider part (ii). Suppoe the target board ha private benefit of control per hare β {0, b} and the bidder make an offer to the target board (β = 0 if the activit control the board and β = b if the incumbent retain control). Since β 0 hareholder approve any offer that i approved by the target board. If < β then a takeover can never ucceed, becaue otherwie either the bidder or the target board (or both) make negative profit on the equilibrium path. In thi cae, in equilibrium, the bidder make an offer trictly maller than q L + β, which i alway rejected by the target board. Suppoe that > β. 36 If the bidder offer q H +β then the takeover ucceed for ure. If the bidder offer q L + β the takeover ucceed with probability 1 τ, only when q = q L. The bidder prefer the higher offer if and only if E [q] + q H β (1 τ) (q L + q L β) β + 1 τ (q H q L ). τ Note that the bidder doe not have any incentive to make any other offer. Hence if β + (q τ H q L ) the offer i pooling and hareholder value i q H +β, if β < β + (q τ H q L ) the offer i eparating and hareholder value i E [q] + (1 τ) β, and if < β the takeover never take place and hareholder value i E[q]. 36 If = β then the equilibrium can have the propertie of part (ii.b) or (ii.c). 14

15 Conider part (iii). Suppoe the target board ha private benefit of control per hare β {0, b} and the target board make an offer to the bidder. Note that hareholder approve any offer aked by the target board ince β 0. If < β then a takeover can never ucceed, becaue otherwie either the bidder or the target board (or both) make negative profit on the equilibrium path. In thi cae, in equilibrium, the target board alway ak a uffi ciently high offer that i rejected by the bidder. Thi equilibrium can be upported by off-equilibrium belief that q = q L upon oberving any off-equilibrium path offer π q L + β, which atify the Groman and Perry (1986) criterion. Suppoe > β. 37 We proceed in three tep: 1. Firt, we how that in any equilibrium the takeover ucceed with a trictly poitive probability. Suppoe not. Then, no offer π [q L + β, q L + ] i on the equilibrium path, becaue otherwie it would be accepted by the bidder and the hareholder. However, in any uch equilibrium if q = q L then the target board trictly prefer any off-equilibrium offer π (q L + β, q L + ) over the equilibrium offer ince the former would be accepted by the bidder and hareholder, creating a contradiction. 2. Second, uppoe the target board make a pooling offer where the takeover alway take place. Then, he mut ak the bidder to pay no more than E [q] +. The board ha incentive to make thi offer when q = q H only if it i higher than q H + β. Therefore, the pooling equilibrium exit if and only if E [q] + q H β 0 β + (1 τ) (q H q L ). When it exit, the pooling equilibrium require that the off-equilibrium belief are uch that higher offer are rejected by the bidder. Notice, however, that the only pooling equilibrium that urvive the Groman and Perry (1986) criterion i the one in which the pooling offer i E [q] Third, uppoe the target board make a eparating offer. Then, the takeover cannot take place with probability one, becaue otherwie the target board making the lower equilibrium offer trictly prefer making the higher equilibrium offer. Moreover, the takeover take place if and only if q = q L, becaue otherwie if q = q L then the target board trictly prefer making the equilibrium offer that i made when q = q H. Therefore, it mut be that the target board i aking from the bidder no more than q L + when q = q L and thi offer i accepted, and when q = q H hi offer i rejected by the bidder. 37 If = β then the equilibrium can have the propertie of part (iii.b) or (iii.c). 15

16 Moreover, the target board ha no incentive to ak for the eparating offer when q = q H if and only if the eparating offer i maller than q H + β. In addition, ince the bidder accept any offer equal to or maller than q L + under any off-equilibrium belief, the eparating offer made when q = q L i at leat q L +, and ince the bidder ha to make nonzero profit in equilibrium, it cannot be trictly larger than q L +. Hence, the eparating offer i q L +. Therefore, the eparating equilibrium exit if and only if q L + q H + β. Thi equilibrium, however, urvive the Groman and Perry (1986) criterion if and only if E [q] + q H β < 0. We conclude, if β + (1 τ) (q H q L ) the offer i pooling and hareholder value i E [q] +, if β < β + (1 τ) (q H q L ) the offer i eparating and hareholder value i E [q] + (1 τ), and if < β the takeover never take place and hareholder value i E[q]. Lemma 3 Suppoe the firt round of negotiation fail and no information about q i revealed. Then: (i) The bidder never run a proxy fight. (ii) If the activit own α hare of the target, the activit run a proxy fight if and only if Γ (α) where Γ (α) = : 1 κ/α 1 τ τ [ (qh q L ) 1 { τ (q H q L ) } b 1 {b } τ 1 {0 <b} + 1 {()(qh q L ) <b+()(q H q L )} ] < b + 1 τ τ (q H q L ) Whenever the activit run a proxy fight, he win. (28) Proof. Suppoe no information about q i revealed in the firt tage. Baed on part (i) of Lemma 2, hareholder value under the bidder control i E [q]. Therefore, electing the bidder to the board i a weakly dominated trategy, and trictly dominated if extraction of value i poible. Baed on part (ii) and (iii) of Lemma 2, the expected hareholder value under the 16

17 incumbent control i [ E [q] + (1 τ) 1 { b} +τ 1 { b+()(qh q L )} ] + (1 ) E [q] + (1 τ) b 1 {b+ τ (q H q L )> b} + ((1 τ) (q H q L ) + b) 1 { b+ τ (q H q L )}, and the expected hareholder value under the activit control, if he chooe to run a proxy fight, i [ E [q] + (1 τ) 1 { 0} +τ 1 { ()(qh q L )} ] + (1 ) [ E [q] + (1 τ) (q H q L ) 1 { τ (q H q L )} The activit run a proxy fight if and only if the increae in value under her control i greater than κ/α, which hold if and only if Γ (α). Remark: Baed on Lemma 3, note that lim Γ (α) = b = : [ min [ min [ κ/α 1 κ/α 1 τ τ { κ/α τ (q H q L ) 1 { τ (q H q L ) } 1 {0 } + 1 {()(qh q L ) } } ), } ), ) (q H q L ), 1, (1 τ) (q H q L ) { κ/α, (q τ H q L ) (1 τ) 1 if if κ/α κ/α 1 q H q L τ + 1 if q H q L < κ/α ] q H q L < κ/α τ + 1 [ ) Thi demontrate that if q H q L i large then lim b Γ (α) κ/α, and if q H q L i [ ) mall then κ/α, lim b Γ (α). Thu, advere election can either increae or decreae the incentive of the activit to run a proxy a fight. Finally, note that lim Γ (α) = 0 [ (q τ H q L ), b + (q τ H q L ) ) if [ { min b, (q τ H q L ) }, b ) if κ/α + b q H q L κ/α q H q L < κ/α + b if q H q L < κ/α Thi demontrate that unlike the baeline model, here the activit may run a proxy fight even if > b. Intuitively, the activit who i le biaed againt the takeover can overcome the advere election problem while the incumbent cannot

18 To conclude, the exitence of private information reduce the bidder credibility even further ince it create advere election and additional opportunitie for the bidder to abue the power of the target board once elected. The exitence of private information, however, ha an ambiguou effect on the activit. On the one hand, private information increae the activit incentive to run a proxy fight ince the activit can extract information rent from the bidder once he get acce to the target private information. On the other hand, private information create advere election which decreae the probability of reaching an acquiition agreement with the bidder, thereby weakening the activit incentive to run a proxy fight. The latter effect dominate the former if q H q L i large. G.2 Uncertainty about In thi ection we olve the baeline model under the aumption that { L, H } i uncertain, H > L > 0, and i privately oberved by the bidder. We denote the prior by ψ = Pr[ = H ]. We alo aume that the identity of the propoer, the value of the offer, and the counter-party repone (i.e., accept or reject) are made public in each round. We focu attention on Perfect Bayeian Equilibria in pure trategie. Therefore, any equilibrium i either pooling or fully eparating. Lemma 4 Suppoe the firt round of negotiation fail and no information about i revealed in the firt tage or in the proxy fight tage. Conider the econd round of negotiation. Then: (i) If the bidder control the target board then the bidder offer hareholder q and take over the target with probability one. (ii) If the target board ha private benefit of control per hare β {0, b} and the bidder i the propoer then: (a) If β L then the bidder offer q + β and the board accept the offer. (b) If L < β H then the bidder offer q + β when = H and the board accept the offer, and when = L the takeover fail. (c) If H < β then the takeover alway fail. (iii) If the target board ha private benefit of control per hare β {0, b} and the target board i the propoer then: 18

19 (a) If β L ψ H then the target board ak for q + 1 ψ L and the bidder accept the offer with probability one. (b) If L ψ H < β 1 ψ H then the target board ak q + H and the bidder accept the offer if and only if = H. (c) If H < β then the takeover alway fail. Proof. Suppoe information about i not revealed in the firt round or in the proxy fight tage. There are three cae. Firt, uppoe the bidder control the target board. There i no information aymmetry between the bidder and the target board, but target hareholder till need to approve the deal. Shareholder will approve any offer higher than q regardle of their belief about. Since L 0, regardle of the realization of the bidder offer hareholder q and the offer i accepted. In thi cae target hareholder value i q. Notice that thi argument hold for any et of hareholder belief about. Second, uppoe the target board ha private benefit of control per hare β {0, b} and the bidder i the propoer. Notice that regardle of hi belief about, the target board reject any offer below q + β and accept any offer above q + β. Therefore, the bidder ha incentive to offer q + β, provided that her expected profit i non-negative. Thi complete part (ii). 38 Third, uppoe the target board ha private benefit of control per hare β {0, b} and the target board i the propoer. Since β 0 hareholder approve any offer that i aked by the target board. If H < β then a takeover can never ucceed, becaue otherwie either the bidder or the target board (or both) make negative profit on the equilibrium path. In thi cae, in equilibrium, the target board make an offer trictly larger than q + H, which i alway rejected by the bidder. Suppoe H > β. 39 If the board ak for q + L then the takeover ucceed for ure, and the board expected profit i L β. If the board ak for q + H then the takeover ucceed with probability ψ, only when = H, and the board expected profit i ψ ( H β). The board prefer the former over the latter if and only if L β ψ ( H β) β L ψ H, 1 ψ where L ψ H < 1 ψ L < H. Note that the target board doe not have any incentive to make any other offer. Thi complete part (iii). 38 Note that if H = β then the equilibrium can have the propertie of part (ii.b) or (ii.c). 39 If H = β then the equilibrium can have the propertie of part (iii.b) or (iii.c). 19

20 Lemma 5 Suppoe the firt round of negotiation failed and no information about wa revealed. Then: (i) The bidder never run a proxy fight, and no information about i revealed. (ii) If the activit own α hare of the target, the activit run a proxy fight if and only if Proof. b Λ (α) where max {ψ H, L } ψ H 1{ L ψ H Λ (α) = b : 0 b <b L 1 { 1 ψ H} b L ψ H 1 ψ [ 1{b L } + ψ 1 { L <b H }] 1 Whenever the activit run a proxy fight, he win. Baed on part (i) of Lemma 4, hareholder value under the bidder control i q regardle of their belief about. Therefore, electing the bidder to the board i a weakly dominated trategy, and trictly dominated if extraction of value i poible. Thi alo implie that the bidder deciion not to run a proxy fight i not informative about. Baed on part (ii) and (iii) of Lemma 4, the expected hareholder value when the target board private benefit are β {0, b} i Π SH [ (β) = q + ψ H 1 { L ψ H 1 ψ + (1 ) β [ 1 {β L } + ψ 1 { L <β H }]. Therefore, the activit run a proxy fight if and only if Π SH (0) Π SH (b) κ/α, + <β L 1 { } H} β L ψ H 1 ψ which hold if and only if b Λ (α). Since κ > 0, whenever the activit run a proxy fight, he i elected by hareholder. ] } κ/α Reference [1] Groman, Sanford J., and Motty Perry, 1986, Perfect equential equilibrium, Journal of Economic Theory 39,