Online Appendix for Communication and Decision-Making in Corporate Boards

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1 Online Appendix fo Communication and Decision-Making in Copoate Boads Nadya Malenko This online Appendix pesents seveal extensions of the basic model. Section I consides two extensions that captue diectos desie to suppot the CEO. Section II analyzes a geneal linea speci cation of the decision-making ule. Finally, Section III consides a model with voting. All poofs ae collected in Section IV. I. Confomity due to fea of the CEO Fea of CEO etaliation is one of the impotant easons fo pessue fo confomity in the boadoom. In this section, I analyze two extensions that incopoate fea of the CEO into diectos pefeences. I show that in both extensions, similaly to the basic model, pessue fo confomity at the decision-making stage can have an oveall positive e ect on m value because it encouages moe communication. I.A. Asymmetic confomity biases In this extension, I assume that it is paticulaly costly fo a diecto to deviate fom othe boad membes when he citicizes the CEO than when he suppots the CEO. I show that the esults of the basic model continue to hold in this setting as well. Speci cally, suppose that if a is the ultimate action chosen by the boad, the CEO favos a highe action ove a lowe action. Fo example, in the context of investment decisions, the CEO could have a bias towads moe investment (highe a) due to empie-building pefeences. Suppose that diecto i s utility, given the nal action a and othe diectos actions a j, j 6= i, is given by ( (a ) i (a i a i ), if a i > a i U i (a; a ; :::; a N ; ) = (a ), if a i < a i :

2 In othe wods, the diecto su es a loss if he is less suppotive of the manage than the aveage diecto (a i < a i ) but does not su e any loss if he is moe suppotive of the manage than othes. Othe assumptions of the basic model emain unchanged. Because this speci cation of diectos pefeences makes the model less tactable, I focus on the case of two symmetic diectos (c i = c, i = ) and a unifom distibution of signals. As in the basic model, diectos communication stategy takes a theshold fom: a signal is evealed if and only if it lies outside the inteval [t; T ] fo some t; T. The next esult deives the equilibium at the decision-making stage taking this communication stategy as given. Lemma B.. Suppose that at the communication stage, x i is evealed if and only if x i 6 [t; T ] : Then the following stategies constitute an equilibium at the decision-making stage.. If both signals wee communicated, then a = a = x + x :. If no signal was communicated, then a i = x i + t + T + T t (T x i) : (B.) 3. If x was communicated and x was not, then a = x + A, and 8 >< x + x, if x > A a = x + A, if x [A ; A] >: x + x +, if x < A ; (B.) whee A = t+t + T T t + T t t+t ; T. The intuition is the following. If both signals wee evealed, diectos coodinate on the optimal action x + x, which maximizes m value. Even if x + x is low and hence the boad s decision goes against the CEO s pefeences, diectos do not su e any loss because they jointly oppose the CEO. Howeve, if a diecto did not eveal his signal, the othe diecto does not know his view on the optimal decision and is afaid to be the one who is less suppotive of the CEO. Fea of the CEO then induces the othe diecto to bias his action upwads. In paticula, if diecto did not eveal his signal, then instead of the action E [ji ] = x + t+t, which maximizes m value given his infomation, diecto s action is eithe (x + t+t ) + (T x T t ) o x + A, which ae both geate than x + t+t. Next, conside diectos communication stategy. Because a diecto is punished fo being less suppotive of the manage than the othe diecto, a diecto with a negative signal has

3 paticulaly stong incentives to shae it with the othe diecto pio to voting, when the manage is not pesent. By pivately shaing his negative signal with the othe diecto, he can avoid the cost of etaliation because he ensues that the othe diecto will also oppose the manage at the decision-making stage. The following lemma con ms this intuition. Lemma B.. A theshold equilibium at the communication stage takes the following fom: diecto i communicates his signal if and only if x i t fo some t [ k; k]. As in the basic model, a stonge confomity bias has two opposite e ects on m value. Fist, diectos distot thei actions moe if they did not shae infomation with each othe. Second, they ae also moe likely to shae infomation. The intuition behind the positive e ect of confomity on communication is simila to the basic model. When pessue fo confomity is stong, a diecto with negative infomation about the CEO s pefeed decision will not vote against it unless he is con dent that othe diectos shae his concens. Thus, to be able to vote against the CEO without su eing etaliation, the diecto has incentives to convince othes of his position by shaing his infomation befoe the vote. Because diectos shae thei negative infomation befoehand, they can be moe e ective in jointly opposing the CEO than if each of them voted individually based on his pivate infomation. To see this fomally, note that (B.) and (B.) imply the following popeties, which ae simila to the basic model: st, if a signal was communicated, it is used e ciently by both diectos, and second, if a signal was not communicated, confomity induces the diecto to use it ine ciently. Stonge pessue fo confomity inceases ine ciency fom not communicating the signal and thus gives diectos stonge incentives to incu the costs of communication. The next esult shows that this positive e ect dominates when is su ciently small. Poposition B.. Fim value is maximized at a stictly positive value of. Thus, similaly to the basic model, some degee of confomity is bene cial fo m value due to its positive e ect on pe-vote communication. I.B. Confomity to the CEO In this extension, I assume that the CEO is one of the diectos and focus on diectos desie to confom to the CEO. Conside the following vaiation of the setup. Let the CEO 3

4 coespond to diecto in the model. Denote the CEO s bias by b and assume, without loss of geneality, that b > 0. Fo example, if a is the amount of investment in a poject, a positive bias coesponds to empie-building pefeences. Suppose that othe diectos ae identical and have no diectional biases: b i = 0, i =, c i = c > 0, k i = k fo i >. To emphasize the impotance of confoming to the CEO s position, conside the following pefeences: U (a; ) = (a (b + )) ; U i (a; a ; a i ; ) = (a ) (a i a ) ; i > : This speci cation implies that the CEO does not cae about confoming to othes ( = 0), while diectos only cae about confoming to the CEO. Suppose also that the CEO s costs of communication ae zeo. Finally, in the notations of Section 3, suppose that the boad s decision equals a i with pobability p i, whee p is the CEO s decision-making powe and p i = p = p is the decision-making powe of each of the othe diectos. N Repeating the aguments of Lemma A. in the Appendix, it can be shown that the equilibium stategies at the decision-making stage ae given by a i = g i + a i (p ; :::; p N ) ; whee g = b, g i = b fo i >, and a p+ i (p ; :::; p N ) is given by (0) fo signals that wee communicated and by (0) fo signals that wee not. Intuitively, the desie to confom to the CEO distots diectos actions: they not only put less than optimal weight ( p thei pivate signals, but also bias thei decisions (by p+ p+ < ) on b) towads moe investment, i.e., a highe a. Howeve, this distotion is smalle when the CEO is less in uential (p is lowe). The poof of Poposition B. shows that at the communication stage, diectos eveal thei signals if they ae su ciently low, x i < t. Intuitively, since the CEO is biased towads ove-investment, diectos do not eveal infomation that futhe suppots a high amount of investment. Impotantly, t inceases with, and hence pessue fo confomity at the decision-making stage impoves communication. The following lemma shows that Poposition 3 continues to hold. Poposition B.. Fim value is maximized at a stictly positive value of. The implication of the lemma is that even when diectos cae exclusively about confoming to the CEO, open ballot voting (coesponding to a highe ) could be moe e cient 4

5 than secet ballot voting due to its positive e ect on the pe-vote discussion. The intuition is simila to the basic model. Conside a diecto with negative pivate infomation about a poject favoed by the CEO. If voting is by secet ballot, the diecto will vote against the poject knowing that his vote will not be obseved. Howeve, since the poject is suppoted by the CEO, the diecto may be eluctant to voice his concens pio to the vote, leading to ine cient communication. In contast, if voting is by open ballot, the diecto will not vote against the poject if he knows that the CEO still suppots it. The diecto s concen about m value will then induce him to expess his esevations duing the discussion and do his best to convince the CEO and othe diectos that the poject should not be undetaken, even though voicing this opinion may be costly. II. Geneal linea model This section povides the analysis of the model fo the geneal linea speci cation of the function h (a ; :::; a N ), which aggegates individual diectos actions into the decision a taken by the boad. Speci cally, the geneal linea speci cation is chaacteized by m linea combinations ( (j) ; ::: N(j) ); i(j) 0; P N i= i(j) =, and pobabilities q j > 0; P m j= q j =, attached to these combinations, such that h (a ; :::a N ) is equal to P N i= i(j)a i with pobability q j : Then m value is given by V 0 X m q j j= N X k= k(j) a k! : Suppose that the utility of diecto i is U i (a; ) = mx j= q j N X! X k(j) a k b i i w k(i) (a i a k ) ' i (a i ) ; k= k6=i whee P k6=i w k(i) =. The st tem e ects the diectional bias b i, the second tem e ects the confomity bias, and the thid tem e ects the e ect of the diecto s actions on his eputation - the diecto bene ts fom his own action being close to the shaeholdes optimal action, even if the action taken by the boad is su ciently di eent fom this optimal action. Denote by I i the diecto s infomation set afte the communication stage. Taking the 5

6 st-ode condition, a i = P j q j i(j) ' i + i + P j q b j i + ' i + P j q j i(j) i(j) ' i + i + P j q E [ji j i ]+ X i(j) k6=i i w k(i) Pj q j i(j) k(j) ' i + i + P j q E [a j k ji i ] : i(j) Denote F i = P j q j i(j) ' i + i + P j q ; j i(j) ' i + P j Q i = q j i(j) ' i + i + P j q ; (B) j i(j) Z k(i) = P iw k(i) j q j i(j) k(j) ' i + i + P j q : j i(j) Using the condition P N i= i(j) =, it is staightfowad to check that Q i + X k6=i Z k(i) = : (B) Then, the st-ode condition can be witten as a i = F i b i + ( X k6=i Z k(i) )E [ji i ] + X k6=i Z k(i) E [a k ji i ] : The following esults epeat the esults of the basic model fo this geneal speci cation. Lemma C. (equilibium at the decision-making stage). Suppose that at the communication stage signals x i ; i J C wee communicated and that y i is the expected value of x i conditional on no communication. Denote J NC = f; :::; NgnJ C, X = P ij C x i, and Y = P ij NC y i. Then thee is a linea equilibium at the decision-making stage chaacteized by the following stategies:. If diecto i communicated his signal, i J C, his action is given by a i = g i + X + Y:. If diecto i did not communicate his signal, i J NC, his action is given by a i = g i + X + Y + Q i (x i y i ) ; 6

7 whee g i solves the linea system g i = F i b i + X k6=i Z k(i) g k ; (B3) and F i ; Q i ; Z k(i) ae given by (B) and satisfy (B). Lemma C. (equilibium at the communication stage). Suppose that conditional on diecto i not communicating his signal, othe diectos believe that the expected value of x i is y i : Then diecto i has incentives to communicate x i if and only if it satis es H i (x i y i ) > 0; whee H i () = A i B i c i ; whee A i = B i = mx q j i(j) Q i + i Q i + ' i ( Q i ) ; j= mx q j " N #! X X i(j) Q i k(j) g k b i i Q i g i w n(i) g n + ' i g i ( Q i ) ; j= k= n6=i Q i is given by (B), and g i solves (B3). Lemma C.3 (expected m value). Suppose that at the communication stage diecto i communicates his signal if and only if x i C i, and let y i = E [x i jx i = C i ]. Then expected m value is given by V 0 E" m X j= q j N X i= i(j) g i! N X " X m q j k(j) Q k # Z k= j= x k 6C k (x k y k ) f k (x k ) dx k ; whee Q i is given by (B) and g i solves (B3). Lemma C.4 (ine cient decision ules) Suppose that h (a ; :::; a N ) is non-deteministic, that is, thee exists i such that i(j) 6= i(j 0 ) fo some j; j 0, whee q j > 0, q j 0 > 0. Then, h (a ; :::; a N ) does not e ciently aggegate infomation accoding to De nition. Poposition C.. Suppose that h (a ; :::; a N ) is non-deteministic, that is, thee exists i such that i(j) 6= i(j 0 ) fo some j; j 0, whee q j > 0, q j 0 > 0. Suppose also that b k = ' k = 0 7

8 fo all k, and k = 0 fo k 6= i. Then m value is maximized at i > 0.. III. Model with voting This section povides the analysis of a model with voting. The boad, which consists of two diectos, is contemplating a poposal. The value of the m is equal to V (a; ) = (a ) ; whee a f; g is the decision made by the boad, and = x + x is the unknown state of the wold. Decision a = (a = ) coesponds to the boad accepting (ejecting) the poposal. At the decision-making stage, each diecto casts a vote a i f; g, whee a i = (a i = ) coesponds to voting in favo of (against) the poposal. The decision-making ule is as follows. If both diectos vote in favo of (against) the poposal, it is accepted (ejected) with pobability. If diectos disagee with each othe, the poposal is accepted with pobability 0:5. Signals x ; x ae independent and ae unifomly distibuted on [ ; ]. Diecto i pefectly obseves x i but has no infomation about the othe signal. The timeline is as follows. At the communication stage, diectos simultaneously decide whethe to eveal thei signals to each othe. Communicating the signal entails a cost c > 0. Then, at the decision-making stage, diectos simultaneously cast thei votes, and the boad s decision is detemined. Each diecto caes about shaeholde value. In addition, diectos have a confomity bias: they incu a loss if thei vote deviates fom the vote of the othe diecto. In paticula, a diecto s utility function is given by U i (a i ; a j ; a; ) = V (a; ) fa i 6= a j g: Given the symmety of the setup, we conjectue that the equilibium at the communication stage takes the following theshold fom: thee exists a theshold d such that signal x i is evealed if and only if jx i j > d. We late veify this conjectue. Fist, conside the equilibium at the decision-making stage. Thee ae thee possible scenaios depending on the outcome of the communication stage: ) no signal was evealed; ) only one signal was evealed; 3) both signals wee evealed. The following poposition chaacteizes the equilibium. 8

9 Poposition D.. If no signal was evealed at the communication stage, diecto i votes fo the poposal if and only if x i > 0.. If only signal x i was evealed at the communication stage, diecto i votes fo the poposal if and only if x i > 0. If x i > 0, diecto j votes fo the poposal if and only if x j > x i. If x i < 0, diecto j votes fo the poposal if and only if x j > x i. 3. If both signals wee evealed at the communication stage, both diectos vote fo the poposal if and only if x + x > 0. Given this equilibium at the decision-making stage, we next veify the conjectued equilibium at the communication stage and nd the communication theshold d. Poposition D. if jx i j > d, whee d = +p +8c. At the communication stage, diecto i eveals his signal x i if and only Since + p +8c deceases in, the esult of the basic model continues to hold: stonge confomity biases impove communication between diectos. IV. Poofs Poofs of Section I Poof of Lemma B.. ewitten as Given the andom dictato ule, the diecto s utility can be U i (a; a ; :::; a N ; ) = Conside each of the following thee cases sepaately. ( N P N j= (a j ) i (a i a i ), if a i > a i N P N j= (a j ), if a i < a i : () Suppose that both signals wee communicated. The actions a = a = x + x constitute an equilibium because the utility of both diectos is equal to zeo, which is the global maximum, and hence no po table deviation exists. Thee also exist othe equilibia. In unepoted esults, I pove that all possible equilibia take the fom a = a = a fo some a [x + x ; x + x + ]. I select the most 9

10 e cient of these equilibia, a = x + x, which maximizes m value, but the esults ae obust to the equilibium selection. () Suppose that signals x ; x [t; T ] and hence wee not communicated. t+t Denote = and = + T. We next pove that the best esponse of diecto T t T t to the stategy a = x + of diecto indeed takes the fom a = x + if x [t; T ]. Note that a > a, x > (a ). Conside the following thee egions fo a : Fist, if a > T + = T + t+t, then a < a fo all x [t; T ], and hence (up to a constant), U = (a ) ; which is an inveted paabola that has a maximum at E [] = x + T +t. Hence, the optimal action in this egion is max(x + T +t; T + t+t ). Second, if a < t + = + t + t+t, then a > a fo all x [t; T ], and hence E[U ] equals E (a ) T t Z T t (x + a ) dx = E (a ) + a T + t : Note that E[U ] 0 > 0, a < + x + T +t, and hence the optimal action in this egion is min + x + t+t ; + t + t+t. Thid, if t + < a < T +, then E[U ] equals E (a ) T t Z T (a ) (x + a ) dx = E (a ) (T + a ) ; (T t) and E[U ] 0 > 0, a + < x T t + T +t + (T + ), a T t < x +. Hence, T + the optimal action in this egion is [x + ] t+, whee [x]b a is x tuncated by a fom below and by b fom above. Summaizing these thee cases, it can be shown that the best esponse of diecto is T + a = [x + ] t+, which coincides with the conjectued stategy x + when x [t; T ]. (3) Suppose that x was communicated and x was not. Fist, let us nd the best esponse of diecto to a = x + A. Conside the following two egions fo a : If a > x + A, E[U ] = E (a ), which is an inveted paabola that has a maximum at a = x +x. Hence, the optimal action in this egion is max (x + x ; x + A). If a < x + A, E[U ] = E (a ) (a a ), which is an inveted paabola with a maximum at a = x + x +. Hence, the optimal action in this egion is min (x + x + ; x + A). Summaizing the two cases, it can be shown that the best esponse of diecto is (B.). Second, let us nd the best esponse of diecto to a given by (B.). Conside the following two egions fo a : 0

11 If a > x + A, then a > a if and only if () x > A and () x > a x (> A). Any a > x + T is dominated by a = x + T because a > a in both cases, but x + T is close to the optimal action E []. Fo a x + T, E[U ] = E (a ) T t (T + x a ) ; which is an inveted paabola that has a maximum at a = x + T +t Hence, the optimal action in this egion is x + A. + T T t + T t = x + A. If a < x + A, then E[U ] = E (a ) T t [(T + x a ) + (x + A a ) ]; which is an inveted paabola that has a maximum at a = x + A + > x T t+ + A. Hence, the optimal action in this egion is also x + A. Summaizing these two cases, the best esponse of diecto is indeed x + A. Poof of Lemma B.. Let [t; T ] be the equilibium non-communication egion. Conside the best esponse of diecto to this communication stategy of diecto. Let U C (x ; x ), U NC (x ; x ) be the utility of diecto when he communicates and does not communicate his signal, espectively, given signal x of the othe diecto. Also let U C (x ) = E x U C (x ; x ) and U NC (x ) = E x U NC (x ; x ) be the expected utility of diecto fom communicating and not communicating his signal, whee the expectation is taken ove all possible ealizations of x. Then diecto chooses to communicate his signal if and only if (x ) > 0, whee (x ) = U C (x ) U NC (x ) c: By continuity of the best esponse and utility functions, if t (o T ) ae inteio points, it must be that (t) = 0 ( (T ) = 0). Below I pove that (t) > (T ) : () The statement of the lemma follows fom this inequality. Indeed, t and T cannot both be inteio points because othewise, (t) = (T ) = 0, contadicting (). Similaly, if t = k and T ( k; k), then (T ) = 0 and hence (t) > 0, implying that the diecto pefes to communicate his signal at x = k. Thus, the only possible case is T = k and t [ k; k]. If c is vey lage, diectos neve communicate thei signals (t = k), and when c conveges to zeo, thee is full communication in the limit (t! k). Poof that (t) > (T ) : Fist, note that U C (x ) does not depend on x. Indeed, by Lemma B., U C (x ; x ) = 0 if x 6 [t; T ]. If x [t; T ], x is not evealed and hence a = x + A, a is given by (B.), and

12 since = x + x, then (a ), (a ), and (a a ) + do not depend on x. Thus, fo any x ; U C (x ; x ) does not depend on x, and hence U C (x ) does not depend on x eithe. Thus, it emains to pove that U NC (t) < U NC (T ). Possible values of x fall into two egions: x = [t; T ] and x [t; T ]. ) Fist, if x = [t; T ], diecto communicates his signal. Then a = x +A, and accoding to the poof of Lemma B., the optimal action of is given by (B.). Hence, 8 < U NC (x ; x ) = : (A x ), if x > A (A x ) (A x ), if x [A ; A] (A x ) (A x ), if x < A : This function inceases below A, eaches a maximum of zeo at x = A, and then deceases. Recall that A t+t ; T and that t; T ae equally distanced fom T +t. Hence, t < A < T and T A < A t. Theefoe, t < A T, and hence U NC (t; x ) < U NC (A T; x ). In tun, U NC (A T; x ) < U NC (T; x ) because the tem (A x ) takes the same value in the two points, and the tems (A x ) and (A x ) ae stictly negative. Theefoe, U NC (t; x ) < U NC (A T; x ) < U NC (T; x ). ) Second, if x [t; T ], diecto does not communicate his signal, and a = x + t+t + (T x t+t T t ) = x +, whee = and = + T. Accoding to the poof of T t T t Lemma B., a (T ) = T + a and a (t) = t + a. Hence: Then U NC (T; x ) = U NC (t; x ) = (T + x x ) (T + x T ) ; (t + x x ) (t + x t ) (x t) : H (T ) R T t = 3 U NC (T; x ) dx = R T [(T + x t ( )) + (T ( ) + x ) ]dx (T +T ( )) 3 (T +t( )) 3 ( ) + 3 (T +T ( )) 3 (t +T ( )) 3 and H (t) R T t U NC + (x t)]dx = 3 (t; x ) dx = R T [(t + x t ( )) + (t ( ) + x ) (t +T ( )) 3 (t +t( )) 3 ( ) + 3 (T +t( )) 3 (t +t( )) 3 + (T t) : Using the expessions fo and and denoting T t = d, we get T + T ( ) = d, T + t ( ) = d, t + T ( ) = d, and t + t ( ) = d. Hence, 3H (T ) 3H (t) = d3 (d ) 3 ( d) 3 +( d ) 3 + d3 ( d) 3 (d ) 3 +( d ) 3 3 (T t) ( ) = 6d + 3 (T t) < 0; and hence, R T U NC t (T; x ) dx > R T U NC t (t; x ) dx. Combining the esults fo the two anges of x, we get the inequality U NC (t) < U NC (T ).

13 Poof of Poposition B.. We show that the deivative of m value at = 0 is stictly positive, i.e., the positive e ect of on communication exceeds its negative e ect at the decision-making stage. Using the notations in the poof of Lemma B., let d T t and ecall fom Lemma B. that T = k. Conside m value V (; d) as a function of and d. It satis es: V (; d) = E x ;x (a x x ) + (a x x ) : Using Lemma B. and B., the following thee cases ae elevant:. If x i = [t; T ] fo i = ;, both signals ae evealed and a i = x + x.. If x i [t; T ] fo i = ;, neithe signal is evealed, and a i = x i +, so (a i x x ) = (( ) x i + x j ) 3. If x i [t; T ] and x j = [t; T ], a j = x j + A, and a i is given by (B.). Fo a vey small, t < A because A > T +t. whee R T I = R T t I = R x =[t;t ] V (; d) = 4k (I + I ) ; (( ) x t + x ) dx dx ; R T (A x t ) dx + R A t () dx + R A A (A x ) dx dx : Integating ove x ; x, using = of Lemma B., we get T t = d and the popeties deived in the poof I = [( )T +T ]4 [( )T +t ] 4 [( )t+t ] 4 +[( )t+t ] 4 = d Also, using the popeties T A = d + d I = (k d) 34( ) [d] 4 [ d] 4 [d ] 4 +[d+] 4 6 = 4 3 (d4 + d ) :, t A = d+ + d d 3 + (d + ) d d 3 + d d, and A t = d d, we get + d! + 3 : 3 Denote [d 0 ] =0 = d 0, which, as shown below, is stictly negative and nite. Then, d d I = 6 =0 3 d3 d 0 ; d I d Combining togethe, =0 = 3 (k d) [ d d d 3 +(d+) 3 4 ] (+ d) 3 =0 3 d3 d 0 = (k d) 3 [6d d 0 4 ] 3 d3 d 0 : d 4k V (; d) d = =0 4d3 d (k d) d d 0 = 4kd d 0 : Hence, to show that d d [V (; d)] =0 > 0, it is su cient to show that d0 = [d 0 ] =0 < 0. 3

14 Poof that [d 0 ] =0 < 0. Equivalently, we want to show that [t 0 ] =0 > 0, whee [ k; t] is the communication egion. Using the notations in the poof of Lemma B., t satis es U C (t) U NC (t) c = 0. Conside the function G (t; ) = Uh C (t) U NC i(t). By the implicit function theoem, h the inequality i [t 0 ] =0 > 0 < 0. We will show that, @t h i =0 =0 > 0. With a slight change of notations, let d = k t(d), t (d) = k d =0 i i G (d; ) G (t (d) ; ). Then, we need to show > 0 and > If x is evealed, then a = x + A, and using (B.) and the deivations above, U C (x ) k = R T x (x =t t A) dx + R A ( ) dx t + R A = (T A)3 (t A) 3 + (A t) + 3 (T A) = d3 +(d+) 3 3(+ =0 (x A A) dx + R T (x A A) dx d) 3 + d d + d d (+ d ) ; and if x = t is not evealed, then U NC (x ) k = R x =t x (t A) + + (A t ) dx =[t;t ] + + R x [t;t ] (( ) t + x ) + (( ) x + t ) dx + R T (x t t) dx (d+) = (k d) + + d (+ d + 8 d ) + 3 d3 + d d + d 4d : d Hence, G (d; ) k = (k d) (d+) (+ d ) + + d d 3 +(d+) 3 d 3(+ d) 3 d + d d + d d3 + d d + d 4d 3 3 d (+ d ) : Di eentiating with espect to d,, and setting = 0 : h i = d + (k d) d + 3 6d = 4d (k d) + 4d = 4dk > 0; h i=0 = (k d) 3d (d + ) d 3 3 d = 6 (k d) d + 5d > 0: =0 Hence, [d 0 ] =0 < 0, which completes the poof. Poof of Poposition B.. Conside the communication stage. Repeating the poof of Lemma A.3, it can be shown that diecto i > has incentives to eveal x i if and only if H(x i y i ) > 0, whee H () = pp + c b : p ( p) + Since the coe cient fo the linea tem is negative, the poof of Lemma 3 implies that diectos eveal thei signals if and only if x i < k +, whee < 0 is the smallest p p+ 4

15 oot of H (). Denoting B = pp + b and C = c p( p)+ p p+, we get = B p B + C and d C0. Calculating these deivatives, it is staightfowad to see that B 0 > 0, C 0 > 0, < 0, and hence d > 0. Thus, communication d Using Lemma A.4 fo a unifom distibution, expected m value is " X X # 3 E (V ) = V 0 E" p i gi Ti t i p + p ; p + 3k i i> whee [t i ; T i ] = k + ; k fo i >. Hence, T i =, and we can ewite E (V ) as E (V ) = V 0 E" b p + ( p ) ( p + ) + N p + p( 3k p + ) 3 : de(v ) To pove the poposition, we show that lim!0 d d( ) 3 cient condition is < 0, and hence lim!0 Poofs of Section II d t i > 0. Since lim!0 d d ( p+ ) = 0, a su - > 0. Note that lim!0 B 0 > 0, lim!0 C < 0, lim!0 > d( ) 3 = 3 C0 > 0. Poof of Lemma C.. Plugging these stategies into the st-ode condition deived above and using the fact that E [x k y k ji i ] fo k J NC ; i 6= k, we veify that these ae indeed equilibium stategies: ) Fo i J C : X a i = F i b i + ( Z k(i) (g i + X + Y ) = g i + X + Y: ) Fo i J NC : a i = F i b i +( X k6=i k6=i Z k(i) ) (X + Y ) + X k6=i Z k(i) ) (X + Y y i + x i )+ X k6=i Z k(i) (g i + X + Y ) = g i +X+Y +Q i (x i y i ) : Poof of Lemma C.. Suppose that the equilibium communication and non-communication egions of diecto i ae some sets C i and NC i, C i [ NC i = [ k i ; k i ]. That is, the diecto communicates his signal x i if and only if x i C i. Denote y i = E [x i jx i NC i ] and i = x i y i. Conside the decision of diecto with signal x whethe to pay c to communicate his signal. The diecto does not know the signals of othe diectos and thus conditions his decision on all possible values of x ; :::; x N : Suppose that among the emaining signals, signals x i ; i J C lie in thei espective egions C j and ae theefoe communicated, and signals x i ; i J NC lie in thei espective egions NC j and ae not communicated. It can be shown, using the equilibium at the decision-making stage, that if the diecto communicates 5

16 his signal, then his payo upon communication, U C, is equal to P h m P j= q j kjnc k k(j)q k PN i + i= i(j)g i b " P kj C w k() [g g k ] P kj NC w k() [g g k Q k k ] ' g PkJ NC k : If the diecto does not communicate his signal, then it can be similaly shown that his payo upon non-communication, U NC, is equal to P m j= q j h (j) Q + P kj NC k k(j) Q k + PN i= i(j)g i b P kj C w k() [g g k + Q ] P kj NC w k() [g g k + Q Q k k ] ' g PkJ NC k + (Q ) : The diecto aveages these payo s ove all possible values of x ; :::; x N ; " and chooses to communicate his signal if and only if Z Z U C f :::f N f " dx :::dx N d" > c + U NC f :::f N f " dx :::dx N d": (B4) If we open the backets in U C and U NC ; it is easy to see that the expessions inside the integals ae some linea combinations of quadatic tems i ; ", inteaction tems i j ; i ", linea tems i ; ", and a constant. Note also that the signal of diecto k; k 6= entes U C and U NC with a non-zeo coe cient only if x k NC k. Also, because i = x i E [x i jx i NC i ], then Z i f i (x i ) dx i = 0: NC i It follows that all linea tems fo i ; i, all inteaction tems i j ; i, and all tems including ", on both sides of (B4) integate to zeo. Hence, only quadatic tems " ; i, i J NC [ fg, the linea tem, and the constant emain. Note also that the constant tems and the coe cients fo tems " and i ; i J NC in both U C and U NC ae the same. Besides, the integal ove i is taken ove the same set NC i on both sides of (B4). Hence, the integals ove tems " and i ; i J NC on both sides of (B4) cancel out. Finally, and do not ente the expession fo U C and only ente U NC : The coe cient fo in the expession fo U NC is equal to A, and the coe cient fo is equal to B ; whee A > 0 and B ae given in the statement of the lemma. Hence, (B4) is equivalent to A B c > 0, which poves the lemma. Poof of Lemma C.3. Denote i = x i y i. Fo any given ealization of x ; :::; x N ; ", suppose that signals x i ; i J C ae communicated in equilibium and signals x i ; i J NC ae not communicated. Using the deivations in the poof of Lemma C., m value satis es i " V (x ; :::; x N ; ") = V 0 m X j= q j " X kj NC k k(j) Q k + NX i(j) g i "# ; i= 6

17 and expected m value is Z E (V ) = [ V (x ; :::; x N ; ")] f (x ) :::f N (x N ) f " (") dx :::dx N d": By the same agument as in the poof of Lemma C., the integal ove all linea tems i and inteaction tems i j is equal to 0. Also, because all quadatic tems i ente additively, the integal ove these tems is equal to the sum of the coesponding integals fo individual signals. The coe cient befoe i fo i J C is 0. Finally, note that i J NC if and only if x i 6 C i. Integating ove all possible ealizations of x ; :::; x N ; ", we get the expession in the statement of the lemma. P Poof of Lemma C.4. Since m value is V 0 (h (a ; :::; a N ) N i= x i "), then, accoding to De nition, a decision ule e ciently aggegates infomation if and only if expected m value is V 0 E". The absence of communication means that the communication inteval C i = ;, and hence y i = E [x i jx i = C i ] = 0. Accoding to Lemma C.3, when b i = i = ' i = 0, expected m value is given by N " X X m E [V ] = V 0 E" q j k(j) Q k # Z x kf k (x k ) dx k ; k= whee Q i = P j q j i(j) ( P j q j i(j) ). Hence, expected m value equals V 0 if mx q j k(j) Q k = 0 j= j= E" if and only fo all k. Hence, fo any j such that q j > 0, it must be that k(j) Q k =. It follows that if q j > 0 and q j > 0; then k(j ) = k(j ), i.e., the coe cient on a k is the same fo the two linea combinations. Hence, any such decision ule is equivalent to a deteministic decision ule, whee q = and q j = 0 fo j >. The poof fo othe paametes of diectos pefeences is simila. Poof of Poposition C.. () If b k = ' k = 0 fo all k, then B k = 0 in the statement of Lemma C. and hence thee exists an equilibium in which diecto k communicates his signal x k if and only if jx k j > d k, whee d k = ( c k ) = c k = ( P A m k j= q ) = : j k(j) Q k + k Q k Note that Q k, and hence d k, only depend on k. Hence, expected m value only depends on though the following tem V ( ) = " X m q j (j) Q # Z d j= d x f (x) dx = V ( ) V ( ) : 7

18 It is staightfowad to show that lim!0+ V 0 ( ) = 0 and thus, lim V 0 ( ) = lim V ( ) lim ( )!0+!0+ P mx m = lim 4 k= q j q! 3 k (j) P (k) m!0+ j= k= q 5 c 3= k lim (k)!0+ Q f (d ) lim (A ) 5=!0+ 3= = c 3= lim lim f (d ) lim (A ) :!0+!0+!0+ Q!0+ V 0 Finally, if (j) 6= (j 0 ), then lim!0+ A > 0. Indeed, lim A =!0+ " P mx k q j q k (k) (j) P k q k (k) j= Each tem is non-negative and hence the sum can only equal zeo if (j) P P k q k (k) # : k q k (k) equals fo all j. Howeve, this contadicts the fact that (j) 6= (j 0 ) fo some j; j 0. Thus, lim!0+ V 0 ( ) is stictly positive and hence expected m value is maximized at > 0. Poofs of Section III Poof of Poposition D.. Suppose no signal was evealed. We veify that voting fo the poposal if and only if x > 0 is indeed the best esponse stategy of diecto given that diecto follows this stategy. Since diecto did not eveal his signal, diecto infes that x is unifomly distibuted on [ d; d]. Then, the utility of diecto fom voting fo and against the poposal is, espectively: U (a = ) = R 0 (x d + x ) + (x + x + ) + U (a = ) = R 0 d (x + x + ) d dx R d 0 dx R d d (x + x 0 ) dx d ; (x + x ) + (x + x + ) + dx d : and hence the elative bene t of voting fo the poposal, U (a = ) U (a = ), is Z d 0 Z 0 d (x + x + ) (x + x ) d dx + (x + x ) + (x + x + ) + d dx = x ; which is positive if and only if x > 0. Hence, it is indeed optimal to vote fo the poposal if and only if x > 0.. Suppose only signal x was evealed. Fist, suppose x > 0 and, accoding to the conjectued equilibium, diecto votes fo the poposal. Conside the best esponse of 8

19 diecto. U (a = ) = U (a = ) = (x + x ) ; (x + x ) + (x + x + ) + ; and hence the elative bene t of voting fo the poposal, U (a = ) U (a = ), is positive if and only if (x + x ) + (x + x + ) + > 0, x > as conjectued. Next, conside the best esponse of diecto given that diecto follows a theshold stategy and votes fo the poposal if and only if x > T. Fist, if T ( d; d), then U (a = ) = R T (x d + x ) + (x + x + ) + d R d (x + x T ) d ; U (a = ) = R T d (x + x + ) d dx and hence U (a = ) U (a = ) is given by R d T x ; (x + x ) + (x + x + ) + d dx ; Z d T Z T d (x + x + ) (x + x ) d dx = x (x + x ) + (x + x + ) + d dx + T d : Similaly, if T < d, then U (a = ) U (a = ) = x +, and if T > d, then U (a = ) U (a = ) = x. Using these expessions, if diecto follows a theshold stategy with T = x, then the elative bene t fo diecto fom voting fo the poposal is eithe x + d o x +. Since it is stictly positive fo x > 0, this vei es the conjectued equilibium. Similaly, if x < 0 and hence diecto believes that diecto will vote against the poposal, his elative bene t of voting fo the poposal, U (a = ) U (a = ), is positive if and only if (x + x ) + (x + x + ) > 0, x > as conjectued. Given this, the elative bene t fo diecto fom voting fo the poposal is eithe x x d o x. Since it is stictly negative fo x < 0, this vei es the conjectued equilibium. 3. Finally, if both signals wee evealed and diecto believes that diecto votes fo the poposal if and only if x + x > 0, the best esponse fo diecto is to follow exactly the same stategy - it both leads to the highest m value and allows him to avoid the cost of non-confomity. x ; 9

20 Poof of Poposition D. Conside diecto with signal x > 0 and his expected utility fom evealing and not evealing his signal, given that diecto follows a theshold communication stategy with theshold d. We will show that it is optimal fo diecto to eveal his signal only if it exceeds some theshold. The poof fo the case x < 0 is simila, given the symmety of the setup. Denote U c (U nc ) diecto s utility given signals x ; x if he eveals (does not eveal) his signal, and U U c U nc. Conside the following fou anges of x : (0; d ), (d ; d), (d; d + ), and (d + ; ). If d < 0 o d + >, some of these anges may not exist. Case. 0 < x < d If x < d, diecto eveals his signal. If diecto eveals his signal, both diectos vote against since x + x < < 0 and hence U c = (x + x + ). If diecto does not eveal his signal, diecto votes against and diecto votes against as well because x + x < nc. Hence, U = (x + x + ). Thus, U = 0. Similaly, if x > d, diecto eveals his signal. If diecto eveals his signal, both diectos vote fo since x + x > 0 and hence U c = (x + x ). If diecto does not eveal his signal, diecto votes fo and diecto votes fo as well because x + x >. Hence, U nc = (x + x ) and U = 0. Finally, if d < x < d, diecto does not eveal his signal. If diecto eveals his signal, he votes fo the poposal, and diecto votes fo if x > x, whee 0 > x > d. If diecto does not eveal his signal, he votes fo the poposal, and diecto votes fo if x > 0. Combining these cases, diecto s expected elative bene t of evealing his signal is " R x d EU c = c (x + x ) + (x + x + ) + dx # + R d (x x + x ) dx " R 0 + (x d + x ) + (x + x + ) + # dx + R d (x + x 0 ) dx = c + + x : Case. d < x < d If x < d, diecto eveals his signal. If diecto eveals his signal, both diectos vote against since x + x < 0 and hence U c = (x + x + ). If diecto does not eveal his signal, diecto votes against and diecto votes fo if and only if x + x >, x > x +. Since x + > d + > d, diecto always votes against in this ange and hence U nc = (x + x + ) and U = 0. If x > d, U = 0 fo the same agument as in Case. Finally, if d < x < d, diecto does not eveal his signal. If diecto eveals his signal, he votes fo the poposal, and diecto votes fo if x > x. Since x < d < x, diecto always votes fo in this ange and hence U c = (x + x ). If diecto does 0

21 not eveal his signal, he votes fo the poposal, and diecto votes fo if x > 0. Hence, U nc (x + x ), U = 0 fo x > 0, and U nc = (x + x + ) (x + x ), U = (x + x + ) (x + x ) + fo x < 0. Combining these anges of x, diecto s expected elative bene t of evealing his signal is EU c = c + Z 0 d (x + x + ) (x + x ) + dx = c + (x + ) d d : = Case 3. d < x < d + If x < d, diecto eveals his signal. If diecto eveals his signal, diectos vote fo if and only if x > x, whee x < d. Hence, fo x ( ; x ), U c = (x + x + ) and fo x ( x ; d), U c = (x + x ). If diecto does not eveal his signal, diecto votes against and diecto votes fo if and only if x > x +. Since x + > d + = d, diecto always votes against in this ange and hence U nc = (x + x + ). Thus, U = 0 fo x ( ; x ) and U = (x + x + ) (x + x ) fo x ( x ; d). If x > d, U = 0 fo the same agument as in Case. Finally, epeating the agument in Case, if d < x < d, U is the same as in Case. Combining these anges of x, diecto s expected elative bene t of evealing his signal is Z d EU c = c + (x + x + ) (x + x ) x dx Z 0 + d (x + x + ) (x + x ) + dx = c + x + d x d + d : Case 4. d + < x < If x < d, diecto eveals his signal. If diecto eveals his signal, diectos vote fo if and only if x > x, whee x < d. Hence, fo x ( ; x ), U c = (x + x + ) and fo x ( x ; d), U c = (x + x ). If diecto does not eveal his signal, diecto votes against and diecto votes fo if and only if x > x +. Since x + < d, then U nc = (x + x + ) fo x ; x + and U nc = (x + x + ) (x + x ) fo x x + ; d. Thus, U = 0 fo x ( ; x ), U = (x + x + ) (x + x ) fo x x ; x +, and U = (x + x + ) (x + x ) + fo x x + ; d. If x > d, U = 0 fo the same agument as in Case. Finally, epeating the agument in Case, if d < x < d, U is the same as in Case.

22 Combining these cases, diecto s expected elative bene t of evealing his signal is Z x + EU c = c + (x + x + ) (x + x ) x dx Z d + x + (x + x + ) (x + x ) + dx Z 0 + d (x + x + ) (x + x ) + dx = c + x + x : 4 In all the fou anges of x, diecto s expected elative bene t of evealing his signal, EU c, is stictly inceasing in x. It is also easy to veify that EU c is continuous at the points d, d, and d + and hence is oveall continuous. It follows that the diecto s best esponse communication stategy is a theshold one: EU c > 0 if and only if x exceeds some theshold, which vei es the conjectued equilibium. Finally, to nd the equilibium theshold d, note that EU c must equal zeo fo x = d, which gives the following equation on d : c + (d + ) d d = 0, d + d c = 0: Since d > 0, d = +p +8c, which completes the poof.