Persuasion, Pandering, and Sequential Proposal

Transcription

1 Persuasion, Pandering, and Sequential Proposal Johannes Shneider November 5, 2015 Abstrat I study a model in whih an informed sender an propose a projet to an uninformed reeiver. The reeiver an aept or rejet the projet s implementation. If the reeiver rejets, the sender an propose a different projet to the reeiver, whih, in turn, may be aepted or rejeted. Overall, only one projet an be implemented. Both players share preferenes about the features of the projet. Aross projets, preferenes are not aligned. There are two lasses of equilibria. There always is a mixed-strategy equilibrium in whih the sender panders towards the reeiver-preferred projet and the sender-preferred projet is only implemented with a probability less than one. If the time horizon is suffiiently long a seond type of equilibrium exists in whih the sender persuades by waiting. In priniple, many waiting equilibria exist. The shortest waiting equilibrium is ex-ante Pareto-dominated by the mixed strategy equilibrium, but a waiting equilibrium with intermediate waiting time is preferred by the reeiver to all other equilibria. As an appliation, I onsider a firm that needs learane of a proposed merger by an anti-trust authority. Merger realizations are private to the firm. Both players prefer higher synergies. The authority prefers high ompetition, the firm prefers low. Keywords: Persuasion, Dynami Negotiation, Information Design, Merger Regulation JEL odes: C78, D21, D82, L44, L51 I would like to thank my advisor Volker Noke for guidane and support. I also benefitted from onversations with and omments reeived by Maro Ottaviano, Thomas Tröger, Martin Peitz, Li Hao, Aleksandra Khimih, Takakazu Honryo, Keiihi Kawai, Peter Vida and the partiipants of the MaCCI Annual Conferene 2014, EEA-ESEM 2014, Earie Conferene 2014 and various seminars in Mannheim and Toulouse. Center for Dotoral Studies in Eonomis, University of Mannheim. josndr@outlook.de 1

2 1 Introdution An agent who implements a ertain projet often imposes an externality on the rest of the eonomy through implementation. Examples inlude mergers that hange the market onentration, patent appliations that limit ompetition, and researh projets that require resoures that otherwise are available for different purposes. To mitigate potential welfare losses aused by the projet hoies, agents are often required to apply to an authority who deides whether the proposed projet should be implemented. In many ases, however, the authority is less informed than the agent about both, the projet s realized quality itself, and the quality or existene of a potential alternative projet. If private benefits of the projet do not oinide with soial benefits, the agent might have a strategi inentive to withhold a projet for private benefits. Seeing an appliation, the authority must therefore answer the following questions: How large is the likelihood that an even better projet exists, but is not proposed by the agent? To what extend is it worth denying the urrent proposal in hope for a better proposal tomorrow? The aim of this paper is to study dynami projet hoie in a sender-reeiver framework. The sender has multiple rounds to propose projets to the reeiver, but only one projet an be implemented in total. The reeiver, in turn, an only implement projets if they have been proposed. Thus, all the reeiver an use as a signal for the quality of both the proposed and the not proposed projets is the hoie of the sender s proposal. That means the reeiver might rejet a proposal in hope for a better proposal tomorrow, even if she expets the urrent proposal to be benefiial. I show that in priniple two types of equilibria arise: the first is a mixed-strategy equilibrium in whih the authority aepts the reeiver s less preferred projet only with some probability. Similar to a related, stati model by Che, Dessein, and Kartik (2013) the sender panders towards the reeiver-preferred projet as she expets this to be implemented with a larger probability. However, the dynami setting allows for a seond type of equilibria, if the time horizon is large enough. These are equilibria in whih the sender persuades by waiting to propose the sender-preferred projet. The set of waiting equilibria is potentially large. The waiting equilibrium with the shortest waiting time yields the same payoff for the sender, but a lower payoff for the reeiver than the mixed strategy equilibrium. The reeiver s profits are highest in a waiting equilibrium with intermediate waiting time. The reason is that in suh a ase good realizations of the sender-preferred projet are implemented, but all others are deterred by the waiting period. As the waiting time inreases also good realizations are deterred and as the waiting time shrinks worse realizations are implemented. Both is not benefiial to the reeiver. 2

3 I ontribute to the understanding of projet hoie problems in many environments in that I show that hoosing an equilibrium with onsiderable delay is to the benefit of the reeiver, while the sender never prefers any delay. This may be important in setting up institutions that might determine whih type of equilibrium is played in reality. Authorities prefer to pik equilibria that involve onsiderable delay while private parties always prefer a quik solution. As an appliation, I onsider a firm (firm 0) within an oligopoly that proposes a merger with another firm to an antitrust authority. Upon seeing only the proposed merger, but not the realized synergies, the authority needs to deide whether to allow the merger or to blok it. Even if the merger is preferred to the status quo, the authority faes the trade-off whether to allow the proposed merger or to blok it in hope that firm 0 proposes an alternative merger with another firm in the next period. Different from Noke and Whinston (2010), I onsider non-disjoint mergers whih is why a myopi poliy is not optimal in this setting. I show that the implemented merger depends on the hoie of equilibrium. The analysis is motivated by two observations: first, a merger proposed by a given firm has typially two omponents: the merging partner and a realization of synergies. Seond, if a merger is denied, a firm an offer an alternative merger in the subsequent period. The two dimensions of a given merger proposal are partiularly interesting as preferenes are aligned in one dimension, but orthogonal in the other. Preferenes about post-merger marginal ost are aligned: both the firm and the authority prefer less ostly projets. Aross projets, however, preferenes differ: while the firm favors less ompetition post-merger, the authority wants to keep the level of ompetition as high as possible. The dynami omponent beomes partiular interesting if synergies are non-verifiable. Then the merging firms an only ommit on their identities, but not on the partiular realization. The authority, however, observing the merging firms identities an only form beliefs about the merger s quality based on the merger hoie by the proposing firm. Without ommitment power this means that the authority updates, both her beliefs on the proposed merger as well as on the non-proposed mergers before making a deision whether to aept or rejet the merger. Although both, firm and authority, disount future payoffs in the same way, inreasing the time needed to implement the firm-preferred projet is to the advantage of the authority. If the equilibrium is hosen suh that deisions are always made early in the proess, the firm is willing to propose her preferred projet whenever synergies are not far worse than in the authority-preferred projet. Different to that, an equilibrium that involves a longer waiting time for the firm-preferred merger results in the firm proposing the authority-preferred merger more often. As the authority-preferred merger an be allowed right at the beginning, 3

4 the time loss is mitigated for the authority. However, any equilibrium with a very long waiting time is neither preferred by the authority: on the one hand, the firm would, of ourse, always propose the authority-preferred merger if available. On the other hand, she would also do so in ase the synergies in this merger are very low, but those in the firm-preferred merger are very high. As authority and firm would prefer the same implementation in suh a senario senario, both would prefer an equilibrium with intermediate waiting time. The remainder of this paper is strutured as follows: in Setion 2, I give an overview on the related literature. Setion 3 introdues the general model and the solution onept. In Setion 4, I derive equilibria for the ase of two possible projets, and analyze the differenes. Setion 5 onludes. 2 Literature Review The paper by Che, Dessein, and Kartik (2013) is probably the losest to mine. Similar to my model, they are interested in a game between an informed sender and an uniformed reeiver. In their model, however, the reeiver an in priniple deide freely what to do, even without the proposal of the sender. While this assumption is reasonable in ase a deision maker hires an expert to tell her what to do, my model is more onerned about an authority that regulates what firms are allowed to do. In many situations authorities have only the power to blok ertain ations, but annot enfore a partiular ation of the firm. The pandering equilibrium in their model has similar harateristis to the mixed strategy equilibrium in my model. I show, however, that adding a dynami omponent also allows for a seond lass of equilibria, whih are worthwhile to study, as the authority is atually better off in some equilibria within this lass. As proposals are needed, my model is not a model of heap talk per se. However, given the multidimensionality of the projet realizations, I relate to a model by Chakraborty and Harbaugh (2010) who introdue multidimensionality into the heap talk literature. They show, that multidimensional heap talk often leads to full revelation. The major differene to this model is that senders in Chakraborty and Harbaugh (2010) an hoose in whih dimension to ommuniate to the reeiver, while the dimension is fixed in my model. Moreover, they, too, rely on the fat that the reeiver has all the deision power and an implement whatever produt she wants. Different to the literature on sequential delegation, e.g. Ková and Krähmer (2013), who show that sequential delegation supersedes if the differenes in preferene is small, I do not have gradual information arrival in my model, but a fixed multidimensional type spae. This model also ontributes to the old literature on multidimensional signaling of Quinzii 4

5 and Rohet (1985), Wilson (1985), and Milgrom and Roberts (1986). While Wilson (1985) and Milgrom and Roberts (1986) onsider one dimensional types with multidimensional signals, my model has multidimensional types, as e.g. Quinzii and Rohet (1985), but only a one-dimensional signal. Moreover, I onsider a dynami environment whih is why, even without expliit ost of signaling, the firm is equipped with a signaling motive. Sine produing the signal itself is ostless, this paper also relates to dynami bargaining models with one-sided inomplete information going bak to Sobel and Takahashi (1983). In line with that Fudenberg and Tirole (1983) and Admati and Perry (1987) formulated a theory whih uses time as a strategi variable. While Sobel and Takahashi (1983) and Fudenberg and Tirole (1983) have a similar negotiation struture to mine, that is one party makes an offer, the other an only aept or rejet and the game ontinues upon rejetion, Admati and Perry (1987) show that strategi delay an be used to signal one s type in two sided-bargaining problems. The main differene to the bargaining literature is that the signal spae in this paper is redued to the identity of the projet and payments are not allowed. Firms want to signal that whatever they propose is without alternative whih, if redible, would lead to immediate implementation. Thus, the firm tries not so muh to signal anything about the value of the urrent projet, but tries to disourage the authority s hope for a better projet in the future. In the appliation part, the model losest to mine is that of Noke and Whinston (2013). They, model a delegated hoie problem to derive optimal merger ontrol. The differene is that is, despite their model being stati, that the firm an verify her synergies in the proposed merger. Thus, the only private information the firm has is about the harateristis of the not proposed mergers. Further, Noke and Whinston (2013) assume full ommitment of the authority, whih is not assumed in my model. Asymmetri information between firm and authority has been studied also by Besanko and Spulber (1993) who were probably the first to model merger ontrol expliitly as a game between firms and authorities. 1 Ottaviani and Wikelgren (2011) introdue a hoie of timing on the side of the authority and derive onditions when ex-post merger ontrol is better than ex-ante merger ontrol. Dynami merger review is studied by Noke and Whinston (2010) and Sørgard (2009). Both papers look at disjoint mergers and how the deision on proposing the merger depends on the approval rule of the authority. Sørgard (2009) derives a rule that determines an optimal investigation probability, while Noke and Whinston (2010) onsider sequential proposal. They show that a myopi poliy is optimal if mergers are disjoint. The 1 A small literature on merger remedies also onnets to this paper, as projets an also be seen as different types of remedies. Papers that asses remedies before are, e.g. Lyons and Medvedev (2007), Cosnita and Tropeano (2009), Vasonelos (2010), and Cosnita-Langlais and Tropeano (2012). 5

6 main differene to my model is, that mergers are not disjoint in my model. Muh to the ontrary, I look at the different merger options by on single firm and ask how this effets the authorities deision. I find, different to the setup of Noke and Whinston (2010) that a myopi poliy is not optimal in suh a ase. In fat I show, that the authority an atually use the time dimension to sreen merger realization. 3 The General Model This setion gives a formal desription of the game to be played. 3.1 Setup Consider a game of two players, a sender (the firm) and a reeiver (the AA). The firm an hoose one among a set of projets to reommend to the AA by sending message m, and the AA an aept or rejet the proposed projet and a proposed projet only. The number of projets is N. The realisation of eah projet an be one of the following 1. with probability 1 > λ i > 0 projet i is not available, 2 or 2. with probability 1 λ i projet i is available. If projet i is available, its realisation i follows a random variable. I assume eah i to be independently drawn from a distribution F i with a density f i that is ontinuous on its support [, ]. 3 The firm reeives a payoff of δ t 1 π i ( i ) if projet i is implemented in period t, the AA reeives a payoff of δ t 1 w i ( i ). I assume that the funtions are ordered suh that > π 1 () > π 2 () >... > π N () > 0 and w 1 () < w 2 () <... < w n () < for any [, ]. 4 differentiable on [, ] and dereasing in its argument. Eah π i, w i is twie ontinuously The game has a total number of T periods and both players disount with fator δ < 1. The value of the status quo is normalised to 0. 2 Note: The non-availability probability subsumes not only the ases in whih a ertain projet is not available in the usual sense (in a merger ontext this ould be due to personal problems between CEOs,...). Moreover non-availability may also our if the projet is neither profitable for the firm nor the AA (for example if synergies are too low). In suh a ase the projet never gets proposed and may therefore be treated as if it was not available at all. 3 I assume ommon bounds for the ease of notation. None of the results depend on this if there is at least some overlap between the two. 4 In the ase of the merger framework, this reflets the firms preferenes.p. towards larger mergers (funtions of lower ardinality) while the AA prefers smaller mergers (funtions of higher ardinality). In addition, to reverse the preferene order ompletely simplifies omputations, but the general results do not depend too muh on this partiular preferene order. All I need is that firm and AA do not share the exat same order of preferenes. 6

7 For the sake of simpliity and to make the deisions non-trivial assume the following: Assumption 1. Given it is the only projet that is available, eah projet has, in expetation, a positive payoff to both players. In other words: E[w i ] > 0 i. Assumption 2. The probability distributions are suh that E[w n+1 ] > E[w n ] n. Assumption 3. Suppose the firm naively proposes the projet that maximises its payoff under full aeptane. Then, the distributions are suh that there exists at least one ase in whih the AAs best response is to wait for a better projet in the next round. More formally, this means: j i {1, 2,..., N} : E[δw j π i max k N π k] > E[w i π i max k N π k]. Assumption 1 is only to failitate omputations. It serves to ensure that the threat of non-availability of other projets is present for any projet i. Assumption 2, too, is for simpliity. It ensures that the AA not only pointwise prefers projets with higher ardinality, but prefers them in expetation, too. Both assumptions an be relaxed to milder versions, in whih they only hold in onditional expetations. Finally, the last assumption is made to make the problem interesting. It is, thus, ruial to the problem. If assumption 3 was not fulfilled, preferenes would be ompletely aligned, and the solution to the problem would be trivial. All, but the realisations of eah projet is ommon knowledge. The timing of the game is as follows: 1. Before the beginning of the first period, the firm privately observes the vetor of realisations ψ = ( 1, 2,..., N ). I may refer to the vetor also as the firm s type. The firm s type remains onstant throughout the game. To deal with the non-availability of ertain projets, assume that if a projet was not available (whih as noted above happens with probability λ i ), the entry in ψ is some number >. 2. At the beginning of eah period, the firm an propose (a) nothing (by posting identity 0 ) 7

8 (b) exatly one projet identity that i. is available, i.e. where i and 5 ii. has not been proposed in any previous period At the end of eah period, after observing a non- 0 -proposal, the AA an (a) aept the proposal: in this ase we arrive at a terminal node, the proposed projet gets implemented and payoffs realize (b) deny the proposal: in this ase the game advanes to the next period. 4. After the firm has proposed 0, the game ontinues independently of AA s ation. 7 I am looking for perfet Bayesian Nash equilibria in whih no player plays a weakly dominated strategy Ation and Strategies The firm s hoie is a funtion that determines its proposal at eah time t. This funtion is a mapping from the set of possible types ψ defined as Ψ to some message m t. That is, m t : Ψ M t (Ψ, H t 1 ). By the rules of the game the set of available messages M t depends both on the type of the firm (sine non-available states are exluded) and the history (sine re-proposal is exluded). Thus, at time t and for a given state ψ the set of available messages, M t (ψ), is defined as M 1 (ψ) = {i : i } 0 M t+1 (ψ) = M t (ψ) \ m t 0 t 0. 5 Again this assumption is made to simplify the argument. An alternative way to model redued form unavailability would be to assume that an unavailable projet has a negative payoff to firm and authority. This way, if the AA aepts the proposal with positive probability, the firm would never propose it or withdraw her offer after it has been made, whih is essentially the same as proposing 0. To avoid unneessary notation, I simply assume suh a proposal is not possible. 6 This assumption redues the set of possible equilibria. Allowing re-proposal might ause equilibria in whih the firm persistently proposes the same projet. All equilibria I derive in this setting here, easily survive in a world were re-proposal is possible. Moreover, the set of equilibria I derive here without reproposal survives ertain refinements, whih they survive in the re-proposal setup as well. 7 A variant in whih this is not the ase is studied in setion Sine this is a two player game, this restrition is equivalent to using an strategi form trembling-hand refinement. 8

9 Thus, the firm an only post an available projet identity that has not been proposed before (i.e. a positive number) or 0 (the Null-Projet) whih means that it does not wish to implement anything. The strategy of the AA is slightly more omplex and involves a funtion that determines the aeptane probability and a funtion of beliefs over types. Before taking an ation, the AA forms a belief about the type of the firm. The belief funtion takes the following form β t : H t 1 (Ψ), where H t = {m 1,..., m t } is an ordered set of reeived messages 9 and (Ψ) is the set of probability distributions over the possible states. The AA hooses further a funtion ρ t in eah period to determine the aeptane probability after having reeived proposal m t. That is ρ t : M max 0 \ H t 1 [0, 1]. where M max 0, the maximal message spae at the beginning of the game, is simply the vetor {1,..., N} sine in priniple eah projet has a positive probability of being available and thus in the message set. After eah round, the AA an remove all elements out of the message set that already have been proposed sine we exluded re-proposal. 3.3 Equilibrium Desription A onfiguration { {m } T t=1, {ρ t } T t=1, {β t } T t=1} is a trembling-hand perfet Bayesian Nash equilibrium if the following onditions hold: m t is hosen out of a set of best responses to the equilibrium probability funtion of the AA. This set of best responses, M (ψ, ρ t, H t 1 ) is desribed as M (ψ, ρ t, H t 1 ) = {i M t : V (i, ψ, ρ t, H t 1 ) V (j, ψ, ρ t, H t 1 ) j M t } ( ) where V (i, ψ, ρ t, H t 1 ) is the value of proposing projet i in period t after having 9 Formally, to be preise we need to define the order H as m i H m j i > j and H t as the set of all messages send up to point t. H t ( H t, H ) is then the funtion that desribes this ordered set. For the ease of notation I am going to suppress the arguments H t and H. 9

10 proposed history H t 1 in the past and ating optimally thereafter. 10 To put some struture on V (i, ψ, ρ, H t 1 ) it is useful to distinguish between two types of value funtions: i. V (i, ) denotes the beginning of period value funtion, that is the value of proposing projet i at the beginning of a period and ating optimally thereafter. ii. Ṽ (i, ) is the end of period value funtion, that is, it desribes the value the hoie of i has on all future periods given the firm ats optimally after that partiular period. Note that Ṽ is only an auxiliary onstrution to properly desribe the V, sine for message m t it holds that Ṽ (m t, ρ t+1, ψ, H t 1 ) = δv (m t+1, ρ t+1, ψ, H t 1 m t ). Ṽ serves the simple purpose to spell out the beginning of period value funtion: 11 V (i, ) = ρ t (i)π i ( i ) + (1 ρ t (i)) Ṽ (i, ). The equilibrium probability funtion ρ t is hosen suh that, for all messages sent with a ex-ante positive probability, it holds that 0 if E t [w mt H t ] < δυ (H t ) ρ t (m t ) 1 if E t [w mt H t ] > δυ (H t ). ( ) [0, 1] else Where E t [ H t ] denotes the (rational) expetations of the AA in period t onditional on a history of messages H t, that is already inluding m t. Further, Υ (H t ) is the expeted value at the beginning of the next period onditional on rejeting this periods offer and ating optimally thereafter. Beliefs β t are onsistent with Bayes rule whenever possible. No player plays a weakly dominated strategy. 4 Equilibrium for N=2 For the sake of tratability, I am onsidering the ase of only two projets here. 10 Note that the time dependeny of the deision is already inorporated in the variables ρ t and H t 1 suh that the value funtion it self an be modelled as time independent. 11 For the ease of notation it is, without loss of generality, assumed that ρ t (0) := 0. 10

11 4.1 Equilibrium of the modified game Before looking at the general game, I want to fous on a slightly modified version for the ease of exposition. Definition 1. A game is alled modified if it follows the setup as it was laid out in setion 3 exept for the additional rule that the game ends diretly whenever the firm proposes 0, i.e. the null-projet. 12 This modifiation is primarily of didati use as it redues the number of equilibria. Later, I am going to show that the result also survives in the general game even under quite demanding refinement riteria and disuss other equilibria (and there relation to this one) that exist only in the general game. The modified game with only two possible projets has a simple type spae, namely a vetor ψ = ( 1, 2 ), and a message set in the first period that is at most {0, 1, 2}. Under assumption 1 and 3, proposing naively annot result in an equilibrium. The AA would never aept projet 1 and thus proposing 1 when 2 is also available is not optimal. If λ 2 > 0, rejeting 1 always and aepting 2 an also not be an equilibrium if we do not allow for weakly dominated strategies. Sine we ruled out weakly dominated strategies, the firm would always propose projet 1 if it is the only projet available and under assumption 1 the AA should aept it. Moreover, it is even possible to state the following. Lemma 1. For the ase of N = 2 and under assumption 1 and 3, no pure strategy tremblinghand perfet equilibrium exists in the modified game. The proof of this lemma an be found in appendix A, as an all others not in the text. The intuition behind this lemma is straight forward. Proposing 0 is always weakly dominated if any other proposal is available sine a proposal of 0 neessarily terminates the game and keeps the status quo, whih is never preferred by the firm if any projet is available. Thus the firm is always going to propose something. Assumption 3 rules out that the AA just waives whatever projet is proposed. Sine with some probability eah projet is not available, it annot be optimal to unonditionally blok a ertain projet either. The firm best responds to suh a bloking strategy by proposing the projet only if it is the only one available. With this strategy of the firm, however, unonditional bloking is not optimal sine eah projet is profitable in isolation (assumption 1). Thus, pure strategies played by the AA an never be optimal. 12 An equivalent formulation would be a setting in whih the authority an shut down negotiations by formally aepting the null projet and implementing it. Again, the equilibrium proposed in this setion would survive suh a game. 11

12 Sine every finite game has at least one trembling-hand-perfet Nash equilibrium, 13 the trembling-hand-perfet equilibria must involve at least one mixing party. In fat, sine projet realizations are ontinuously distributed, the AA is the only player that is going to mix on more than a probability zero set. To onstrut the equilibrium, first reall the equilibrium onditions from equation ( ) and equation ( ). For the equilibrium onstrution assume that the seond projet is the onditionally better looking one, i.e. Assumption 4. E[w 1 π 1 > π 2 ] < δe[w 2 π 1 > π 2 ]. Although it puts some restrition on the distributions, this assumption still allows for a wide range of distributions. Whenever both state variables 1, 2 are identially distributed or if 2 does not have too muh weight on high ost draws we always an find a δ < 1 suh that the assumption is fulfilled. Further assume that projet 2 is always aepted. 14 Under these assumptions, the probability q ρ (ψ) of proposing projet 1 for a given aeptane probability ρ 1 (1) = ρ is haraterised by 0 if ρπ 1 ( 1 ) ζπ 2 ( 2 ) q ρ (ψ) = 1 if ρπ 1 ( 1 ) > ζπ 2 ( 2 ) (1) with ζ = (1 δ + ρδ) [0, 1], whih aounts for both the alternative proposal and the seond round. In equation (1) we may assume that the firm does not mix sine indifferene only happens on a probability zero set of ψ. With the help of equation (1), it is possible to onstrut a ut-off funtion ρ ( 2 ) that assigns, given ρ and 2, a unique value suh that for all 1 < the firm prefers to propose projet 1 while she is going to propose projet 2 whenever 1 >. The funtion takes the following form: ρ ( 2 ) := max{ 1 : ρπ 1 ( 1 ) ζπ 2 ( 2 )} if it is not if 2 = else. ( ) Note that the funtion ( ) is defined for eah ρ > 0 over 2 [, ] and has the following properties: i. it is ontinuous in ρ and 2, sine π 2 is ontinuous; 13 The proof of this is due to Selten (1975) and an be found in various textbooks suh as Fudenberg and Tirole (1991) and Mas-Colell, Whinston, and Green (1995). 14 Later, I show that in equilibrium this is indeed the ase. 12

13 ii. it is weakly inreasing in 2 sine π 1 ( 1 ) and π 2 ( 2 ) are both dereasing in their arguments; iii. it is weakly inreasing in ρ sine the left-hand side of the inequality inreases in ρ and π 1 () dereases in ; iv. for fixed ρ it is weakly inreasing in δ sine the right-hand side dereases in δ, giving the inequality (weakly) more slak at all points. v. as ρ 0, ρ for all 2. Sine π 2 () > 0 for all 2 there exists an ε 2 > 0 suh that ρ < ε 2 ρ ( 2 ) =. Monotoniity of π 2 ( 2 ) ensures that ε 2 is dereasing in 2 ; vi. sine π 1 > π 2 by definition, it holds that there also exists an ρ < 1 suh that for all ρ > ρ we have ρ ( 2 ) 2 ; vii. as a onsequene of the monotoniity in both arguments, it also holds that the smallest 2 at whih the highest ρ ( 2 ) is reahed 15 (weakly) dereases in ρ. (), on the other hand, inreases in ρ. To define the equilibrium ation of the AA, reall equation ( ). By Lemma 1 we have exluded pure strategy equilibria, thus, 1 > ρ > 0. To re-write ondition ( ), define 1 2 to be the event in whih projet 1 is reommended although projet 2 is available, too, and define 1 0 to be the event at whih 1 is sent and it is the only message other than 0 that is feasible. Sine ondition ( ) defines the optimal behaviour of the firm for any ρ and therefore determines the probabilities and expetations of events 1 2 and 1 0, it is suffiient to rewrite ondition ( ) as follows E[w 1 m 1 = 1] = E[δw 2 m 1 = 1] (1 λ 2 )E[w ] + λ 2 E[w ] = (1 λ 2 )E[δw ] (1 λ 2 )E[w ] + λ 2 E[w 1 ] = (1 λ 2 )E[δw ] ( ) (1 λ 2 ) (E[w 1 δw ]) + λ 2 E[w 1 ] = 0, where λ 2 = λ 2 λ 2 +(1 λ 2 )P (m=1 1 2 ). The first step divides expetations into two subsets. The seond then makes use of the fat that, given that projet 1, but not projet 2, is available, the firm would always propose 15 Typially this highest ρ ( 2 ) =, but one may hoose a ρ small enough that even ρ () <. 13

14 1 in order to not play a weakly dominated strategy. Notiing that w 2 and w 1 in the first term are onditioning on the same event gives the third step of equation ( ). Lemma 2. If assumption 1 and 4 hold, there always exists a ρ suh that 1 > ρ > 0 and equation ( ) are fulfilled. Intuitively, this lemma says that it is always possible to hoose a ρ so low that whenever both projets are available, the firm hooses the seond projet. In turn, this hoie means that the firm only (weakly) prefers to offer projet 1 if nothing else is available. If it did so whenever only 1 was available, the AA would want to aept that proposal sine it is welfareinreasing in expetations. Sine gradual hanges in ρ lead at most to a gradual hange in both expetations, the differene of the two is ontinuous. If the differene is ontinuous, the indifferene ondition must, in line with the intermediate value theorem, eventually be satisfied for some 1 > ρ > 0. In fat, it is possible to derive an even stronger statement Lemma 3. If the onditions for Lemma 2 hold, then whatever ρ solves equation ( ) is unique. The intuition to this lemma is, again, quite simple. As ρ inreases, the firm inreases the number of states in whih it proposes projet 1. For any given 2, this inrease leads to additional states that are all worse than the worst state under the original regime. Therefore, the expeted payoff the AA earns from implementing projet 1 an only fall in ρ for proposals of 1. The expetations of postponing implementation, on the other hand, inrease for the same reasons, sine there is a larger set of desirable 2 that lead to a proposal of 1. Thus, inreasing ρ dereases the expeted payoff when implementing 1 and inreases the expeted payoff of waiting for any ρ. Consequently equation ( ) an at most hold for one ρ (0, 1). To desribe the equilibrium, it remains to show that it is optimal to always aept projet 2, i.e. E[w 2 m = 2] δe[w 1 m = 2] needs to hold under the equilibrium ρ. This is guaranteed by assumption 2, as the following lemma shows. Lemma 4. The AA prefers projet 2 over projet 1 whenever 2 is proposed. The intuition underlying this lemma is that, in expetation, whenever the firm proposes a ertain projet, this projet needs to be better than a ertain threshold. In turn, whenever the firm does not propose a projet it must have a worse realization than this threshold. Sine in equilibrium the AA is indifferent under the proposal of 1, the expeted payoff from projet 2 s realization must be higher for the authority if the firm proposes 2. Expetations of waiting after the proposal of projet 2 must be smaller by the same argument. Thus always aepting projet 2 is optimal in equilibrium. 14

15 Combining Lemma 2 and 4 provides existene of a mixed strategy equilibrium as the following lemma shows Lemma 5. Suppose assumption 1, 2 and 4 hold. Then, there exists a unique mixed strategy trembling hand perfet Baysian Nash equilibrium in the modified game with the following properties: The seond projet is aepted whenever it is proposed. The first projet is aepted with probability ρ < 1. If the seond stage is reahed, proposal projet 2 in the seond stage gets aepted. Firms propose projet one whenever ψ is suh that 1 < ρ ( 2 ). 4.2 Comparison to the original game All the results of the previous setion have been derived under the assumption that players play the modified game. In this setion, I am going to disuss how these results arry over to the original game. Theorem 1. There exists an equilibrium of the original game that has the following properties: It is outome equivalent to the unique equilibrium of the modified game On the equilibrium path ations are idential to the ones of the modified game The equilibrium does not fail the universal divinity, if any of the two onditions holds i. the aeptane probability ρ is larger than the disount fator δ, or ii. the worst projet is smaller than the outside option w 1 () < 0. The ruial aspet for survival of the refinement is to find beliefs suh that the AA wishes to deny off equilibrium proposals. For the seond proposal this is not that hard sine the firm never wishes to deviate sine that only would inur time osts. Thus, any type is equally likely to deviate. The story is quite different for projet 1. There might be a reason to wait for the firm if it beliefs that in later periods the aeptane probability was large enough. For sure, any off-path aeptane probability that attrats at least some type ˆψ must also attrat all types that annot propose projet 2 beause it is unavailable. However, disriminating between those is not possible, sine they are all only interested in implementing projet 15

16 1. This way, if any type has, for some off-path aeptane probability any inentive to go off-path, also the type ψ = (, ) has an inentive to do so. This is, by definition the worst type in terms of payoff for the AA. Thus, if there is at least one state suh that the AA prefers the outside option, she prefers the outside option to implementing projet 1 under ψ. Sine this type engages in all off-path ativities, it annot be exluded under universal divinity. Thus, if the AA beliefs any deviator is of type ψ she has reason to deny it and the equilibrium sustains. The ruial aspet for the sustainability of the equilibrium is the fat that there exists a positive probability that the better looking projet does not exist. Together with the seond restrition, namely that w 1 () < 0, the non-availability guarantees universal divinity for any length of the game. Thus, the existene of the equilibrium of the modified game arries over to the general game even if we were to restrit our self to a rather narrow set of equilibria. Uniqueness, however, does not neessarily survive. In partiular, the two dimensional type spae of the firm allows for a wide range of equilibria if we onsider the generalized game. Together with the possibility to sent the 0 message as often as possible a wide range of justifiable off-equilibrium beliefs an be onsistent even under strit refinements suh as universal divinity. To understand this reall that the equilibrium of the modified game does not fail universal divinity in the general game simply beause all types for whih projet 2 was not available survive the iterated D2 riterion. That is, if the AA sees a proposal of projet 1 anywhere off the equilibrium path it an, even with strong restritions be of any type (regarding projet 1 s ost funtion). If negotiations go on for suffiiently many rounds, that is if T is large enough, a seond set of equilibria whih I am going to all waiting equilibria would arises. In suh a waiting equilibrium the firm proposes the null-projet for suffiiently many rounds in order to signal that the ex-ante AA-preferred projet is of poor quality or not available. In the traditional bargaining literature these are sometimes the only equilibria that survive even mild refinements and they are typially alled signalling equilibria. 16 In the present ontext signalling also is an issue in the ρ -equilibrium desribed above, whih is why I differ in terminology. The deision whether to aept or rejet an offer made by the firm in those type of equilibria is in fat very similar too the deision rule in the ρ -equilibrium. Whenever the AA believes that the projet offered in the urrent round is at least as good as the disounted value of what she an expet in the future, she would aept the proposed projet, whenever this is not the ase she denies approval. Given this, the firm might delay its proposal 16 Examples would be Admati and Perry (1987) or Cramton (1992) 16

17 strategially under some irumstane while it may prefer to propose something that gets aepted right away instead of suffering the ost of waiting. More preisely, suppose the AA always aepts projet 2 and aepts projet 1 only at time t τ Then for eah 2 it is possible to onstrut a funtion ĉ τ ( 2 ) suh that the firm hooses to propose projet 1 in period τ whenever 1 < ĉ τ ( 2 ) and projet 2 (if possible) in all other ases. More formally this is max{ 1 : δ τ π 1 ( 1 ) π 2 ( 2 )} if it is not ĉ ρ ( 2 ) := if 2 = ( ) else. This funtion almost perfetly orresponds to ρ( 2 ) defined in the beginning of this setion. Consequently, this, too, provides a utoff value for the firm given any τ. As in the other equilibrium the AA optimally hooses τ suh that whenever the firm proposes projet 1 in period τ + 1 it holds that the AA only aepts the proposal if E τ+1 [w 1 H τ+1 ] δe τ+1 [w 2 H τ+1 ]. define the term. Before desribing the possible waiting equilibria let us first properly Definition 2. An equilibrium is alled a waiting equilibrium at τ if it is an equilibrium of the game and the authority uses the following strategy: Projet 2 is aepted in any period Projet 1 is rejeted in any period t τ and aepted in all other periods. Next, it is straightforward to use the utoff funtion ĉ τ derived above to desribe the equilibrium that is losest related to the ρ -equilibrium from above. Lemma 6. Consider a ertain speifiation of the game and fix the equilibrium at ρ. Ignoring the integer onstraint in t there exists a waiting equilibrium at τ where τ = ln(ρ ) ln(1 δ(1 ρ )) ln(δ) that orresponds in the firms deision rule to the ρ -equilibrium, that is ĉ τ ( 2 ) = ρ ( 2 ) for all 2. This equilibrium may be onsidered as the shortest possible waiting equilibrium in a sense that all other waiting equilibria require a waiting time τ τ. Moreover, if and only if ρ > δ, then τ < 1. δ+1 17 To understand why this is τ + 1,reall that a proposal in t=2 is disounted with δ 1 sine partiipants ex-ante payoffs are given in period one values. 17

18 The intuition behind this beomes quite lear if you think about what happens if the AA aepts proposals of projet 1 too early. Then the firm would propose them in a way, suh that the AA better denies it in hope of a better outome on the other projet. From the onstrution of the ρ -equilibrium it is already lear that whenever the firm deides via utoff rules, then there exists a unique deision funtion suh that the AA is indifferent in expetations between the two projets. While this funtion was enfored in the ρ -equilibrium with the aeptane probability, it is now done by appropriately hoosing τ. Sine the two orrespond we just need to look for the parameter in whih the funtions are idential. This is what desribes the smallest τ. In the model, of ourse, time is disrete and thus the above desribed point is generially never an integer. However, as the following proposition shows, for all τ > τ a waiting equilibrium at τ exists. Theorem 2. If T 1 > τ, there exists a set of waiting equilibria at τ for all τ suh that τ τ T 1. All waiting equilibria survive under the refinement of universal divinity if w 1 () < 0 While eah equilibrium on its own is in fat independent of the maximum duration of the game, there is a one to one mapping between numbers of proposal rounds played after τ and the number of waiting equilibria that exist. 4.3 Welfare effets To ompare the different equilibria it is useful to think about how the equilibrium payoffs in the different equilibria behave and to exerise some omparative statis. Following the order in whih the equilibria were haraterized, I start by examining the ρ -equilibrium and thereafter looking at the waiting-equilibria. Besides omparing the equilibria with eah other, a natural benhmark would also be the (ex ante) equilibrium payoffs if the AA had the ability to ommit ex-ante to aepting only projet 2. This benhmark ase is going to be alled a onservative AA throughout the rest of the paper. To simplify notation denote E a [w ρ ] to be the ex-ante payoff expetation of the AA under equilibrium aeptane probability ρ as in the equilibrium of setion 4.1 and let E a [w 0] be the ex-ante expetations if the AA was to deny projet 1 always, i.e. behaving onservatively. Proposition 1. The ρ -equilibrium always assigns a lower payoff to the AA than she gets when ommitting to only aepting the seond projet. 18

19 Proof. The mixed strategy equilibrium inreases the AA s payoff only if E a [w ρ ] > E a [w 0]. The following argues that this is not true. Spelling out the first yields E a [w ρ ] =(1 λ 2 )(1 λ 1 ) w 2 ( 2 ) (1 F 1 ( ρ ( 2 ))) df 2 ( 2 ) (2) + (1 λ 2 )λ 1 E[w 2 ] (3) + (1 λ 2 )(1 λ 1 ) δw 2 ( 2 )F 1 ( ρ ( 2 ))df 2 ( 2 ), (4) where the first is the expeted value in ase both projets are available and projet 2 is proposed given ρ, the seond is the part in whih projet 2 is proposed sine it is the only one available, and the last part overs the expetations whenever 1 is proposed, using the indifferene ondition ( ). Further we may deompose E a [w 0] =(1 λ 2 )E[w 2 ] =(1 λ 2 ) ( λ 1 E[w 2 ] + (1 λ 1 )E [ w 2 ] ) =(1 λ 2 )λ 1 E[w 2 ] + (1 λ 2 )(1 λ 1 ) =(1 λ 2 )λ 1 E[w 2 ] + (1 λ 2 )(1 λ 1 ) =E a [w ρ ] + (1 δ)(1 λ 2 )(1 λ 1 ) w 2 ( 2 )df 2 ( 2 ) w 2 ( 2 ) (F 1 ( ρ ( 2 ) + (1 F 1 ( ρ ( 2 ))) df 2 ( 2 ) w 2 ( 2 )F 1 ( ρ ( 2 ))df 2 ( 2 ). (5) } {{ } 0 The first step just spells out the onditional expetations, the seond divides it into the events in whih projet one is or is not available. The third transforms them into integral form, the fourth multiplies with 1 to bak out the ex-ante expetations of the equilibrium in the last step. Sine expetations in equilibrium are positive when 1 is proposed, the leftover term is positive. Thus, from an ex-ante point of view the AA prefers aepting only projet two if both were available. 19

20 Intuitively, this is rather obvious if one thinks about the different hannels at work in the model. If the AA was able to allow only projet 2 the firm an never exploit the game in her preferred way if opinions about optimal ations differ. Thus, the persuasion hannel is ompletely shut down. Pandering, to the ontrary, is driven to its maximum. 18 Finally, the sequentiality hannel is shut down as well, sine in equilibrium there is no seond period. This sequentiality works, as we saw in the disussion of the equilibrium, as an insurane for the firm to try out her preferred projet. That way, ommitment to pure strategies in fat must work in favour of the AA sine it only shuts down the hannels that benefit the firm. The AA therefore earns something like a onservative ommitment markup. To failitate later omparison it is useful to think of this markup as relative to the expeted payoff if only projet 2 was was proposed. To do so, define and rewrite equation (5) as φ(ρ ) := E[w 0] E[w ρ ] E[w 0] E a [w 0] = (1 φ(ρ ))E a [w 0] + φ(ρ )E a [w 0]. (6) In this ontext φ an be interpreted as the relative onservative ommitment markup (RCCM). With this reformulations it is easier to tell how muh (in relative terms) the AA looses by being unable to ignore any proposals of 1 in equilibrium. Proposition 2. Consider the ρ -equilibrium. An inrease in the time preferene parameter δ leads to a derease in the aeptane probability ρ, a derease in the ex-ante likelihood that projet 1 is being proposed and a derease of the RCCM (and a higher absolute payoff for the AA). While an inrease in δ dereases the ost of waiting one more round for both the firm and the AA, the inreased expeted welfare in the seond period leads the AA to redue its probability of aepting the first projet. That means that the firm redues the states at whih it proposes the first projet and panders more towards the seond. This way it needs 18 In fat ex-ante pandering is optimal from the point of view of the AA. However, on an interim stage there might atually be over-pandering, that is the firm proposes a projet that both firm and AA prefer less than the other one (interim), but that was not preferred ex-ante by the AA. 20

21 0.8 Aeptane Probability 30 Shortest Waiting Equilibrium 4 Commitment Markup δ λ Disount fator δ Aeptane Probability Disount fator δ Shortest Waiting Equilibrium Disount fator δ Commitment Markup Non-availability of projet 2 in % Non-availability of projet 2 in % Non-availability of projet 2 in % Figure 1: Comparative statis with respet to availability of the seond projet λ 2 and disount fator δ. The upper panels deal depit effets on the ρ -equilibrium and the lower panels that on the shortest waiting equilibrium. to give up some of the gains from the inreased δ. Further, the ommitment markup goes down and thus, the AA loses less from not being able to ommit to a pure strategy. 19 Another interesting omparative statis in this model is the effet of non-availability of the seond projet. This sheds light on to the point of how muh the threat of not having an alternative projets works in favour of the firm. Lemma 7. Consider the mixed strategy equilibrium ρ. An inrease in the non availability probability of projet 2, λ 2, leads to an inrease in the aeptane probability ρ in equilibrium an inrease in the ex-ante likelihood that projet 1 is being proposed and an inrease of the RCCM. This result is, of ourse, not very surprising. As it beomes more likely that the seond projet is not available at all, the firm reates, by proposing projet 1, a larger threat to the AA that this is the only projet it has in fat. Thus, given projet 1 has been proposed, there is a higher hane that it is the only one around (in whih ase the AA would want to aept) and the interim expetations of the AA are driven down. She reats by inreasing the aeptane probability. 19 In setion 4.4 I disuss why ommitment to a pure strategy is not optimal either and why this leads to lower redibility of a ommitment assumption on the AA s side. 21

22 In terms of the AAs payoffs under no ommitment, to the ontrary, the story is not so lear sine λ 2 effets both the ex-ante expetations of a onservative strategy as well as those under the equilibrium strategy. The relative mark-up inreases nevertheless, as the interim pressure the firm an put on the AA inreases with a higher probability of projet one being the only one available. Turning to the waiting equilibria, it might at first be interesting to see how the set of possible waiting equilibria hanges with the parameters. While the largest waiting equilibrium (if any suh equilibrium exists) is always defined by T, as Theorem 2 shows, the shortest waiting equilibrium might hange as we hange the parameters of the model. With help of Lemma 6 it is obvious that all parameters exept the disount fator δ effet τ only through ρ. Observe, that τ ρ = 1 ln(δ)ρ 1 δ 1 δ(1 ρ ) < 0. Thus, as ρ inreases τ dereases. The reason an easily be found in the equivalene of the deisions rules of the firm in both equilibria. Therefore, a higher effetive aeptane probability needs to result in a smaller minimum waiting period. 20 The only parameter, that effets τ also in another way than through ρ, is the disount fator δ. Its effet on τ is not all that obvious sine the diret effet ould go either way. Nonetheless, as the following lemma shows, the overall effet of a hange in δ on τ has the opposite sign than a hange of δ has on ρ. Lemma 8. Consider a waiting equilbrium at τ. An inrease in the disount fator δ leads to an inrease in τ. Even if the effet though ρ always supersedes the diret effet of δ, the additional effet that a hange in the disount fator has already points towards the presene of some differene in the analysis of the waiting equilibria ompared to that of the ρ -equilibrium. These differenes are what is going to be onsidered next. To do so, I espeially fous on two effets not present in the ρ -equilibrium, that may, in addition, help to understand the hange in payoff between the different waiting equilibria. First, eah round of waiting dereases the AAs payoff sine the agreement is reahed later in time. The AA is impatient and therefore suffers ost of delay. Seond, and quite 20 In fat, in terms of elastiity, one an show that the movements do not orrespond one-to-one but that τ shrinks less than proportional ompared to the inrease in ρ if δ > ρ sine that implies 0 > τ ρ ρ τ > 1. 22