Lobbying as transactions with evidence

Transcription

1 Lobbying as transactions with evidence Martin Gregor Charles University, Prague March 30, 2015 Abstract We model informational lobbying as bilateral transactions between a lobby and a policy-maker, where the subject of a transaction is disclosure of private verifiable evidence. The policy-maker announces transaction prices, and each lobby agrees or disagrees. In the equilibrium, a negative sign of a transaction surplus implies a prohibitive transaction price, and a non-negative sign implies a non-prohibitive transaction price. We apply this method to a binary setting with two competing lobbies. For most of the information structures, we conveniently construct the signs of the surpluses and thereby identify the equilibrium set of transactions. Through this method, we explain how the existence of countervailing lobbying depends on timing of transactions and the information complementarity of the private evidence. We also identify information structures and timings for which an ex ante disadvantaged lobby may paradoxically end up better than an ex ante advantaged lobby. Finally, we demonstrate that if one lobby gets organized and thereby gains ability to trade with evidence, the equilibrium amount of information may decrease because the policy-maker may replace a more informative transaction by a less informative transaction. Keywords: lobbying, information transaction, access fee, verifiable evidence, disclosure JEL Classification: C72, C78, D72, D83 Institute of Economic Studies, Faculty of Social Sciences, Charles University in Prague. Corresponding address: Opletalova 26, Prague CZ Phone: (00420) martin.gregor@fsv.cuni.cz 1

2 1 Introduction The classic setting of competitive informational lobbying (Austen-Smith and Wright 1992) is as follows: A policy-maker makes a binary decision over an issue. On each side of an issue, special interests are organized into a single lobby with state-independent and opposite (i.e., perfectly negatively correlated) preferences. Each lobby may present a single verifiable piece of evidence. The policy-maker, having collected information from participating lobbies, makes a non-contractible policy choice that accounts for the state-dependent effects external to the lobbies. Two main questions arise: First, will both lobbies participate and present their evidence or will participation (and information) be incomplete? Second, will the ex ante advantaged lobby end up with a larger payoff then the ex ante disadvantaged lobby? Models of competitive informational lobbying differ mainly in the assumptions on the message costs. If the message costs are exogenous, the model is a signaling game (Austen- Smith and Wright 1992, 1994). In the absence of message cost or any extra condition that limits transmission of the evidence, the game is a disclosure game where complete participation is ensured by skeptical beliefs of the policy-maker (Milgrom 1994; Milgrom and Roberts 1996). In this paper, we are interested in the case when the message costs are set by the policy-maker and the costs can be both positive and negative. Models where the policy-maker announces only non-negative message costs are effectively models where money buys access (Cotton 2009 and 2012; Groll and Ellis 2014). In these models, the policy-maker observes evidence only if the lobby agrees to a transfer. The commitment is only unilateral. In contrast, in our setting with both positive and negative message costs, also the lobby is committed; namely, the lobby presents evidence only if she agrees to a transfer (c.f., Gregor 2015). The existence of bilateral commitments characterizes strategic information transmission as a transaction with evidence, which admits both signs of a transfer: The policy-maker now may require a positive transfer (access fee) to extract both information and money from a lobby, but he may also propose a negative transfer (access compensation) to pay for the information. To obtain general results for the model of competitive informational lobbying based on the transactions with evidence, we admit two timings: Either the transactions take place before private signals are observed (ex ante timing), or the transactions take place after private signals are observed (interim timing). Ex ante timing is relevant for long lags between (campaign) contributions and policy-making, while interim timing better captures short lags between contributions and policy-making. We cover all plausible information structures in the binary setting, which generates 20 different classes of interactions. We propose a quasi-bargaining method to identify the equilibrium set of participants and the equilibrium payoffs. The idea is that the uninformed policy-maker proposes transaction prices in the first stage, hence picks up a subgame from a set subgames that differ in the announced prices. The quasi-bargaining method then consists of two steps. The first step is 2

3 to construct, for each subset of lobbies σ, a set of subgames where σ is an equilibrium. Out of this set of subgames, we select the subgame with the transfer-maximizing prices and call this best subgame as σ-subgame. The second step is to compare the policy-maker s values of all σ-subgames. In this step, we know that implementing participation of some lobby S i σ implies not only that the lobby decides to participate, but also that the policy-maker proposes a price such that the lobby s participation condition is met. Hence, both lobby and policy-maker have to agree on participation of the lobby, and both payoffs must comply with outside options of both players. To find out if mutual agreement is possible, we define for each transaction the quasibargaining surplus as the total (transferrable) surplus over the players outside options. This approach provides natural and convenient equilibrium conditions that exploit the signs of the surpluses: Namely, if a lobby is expected to participate, then the bilateral surplus must be non-negative. And vice versa: If a lobby is expected to abstain, then the bilateral bargaining surplus must be negative. Using the quasi-bargaining method, we identify the equilibrium directly for 14 out of 20 classes of models without additional parametrical calculations. For the remaining classes, we demonstrate how to calculate the equilibrium using the parameters. We also demonstrate how to tackle the non-negativity constraint imposed on the levels of the prices. That is, we can handle a unilateral commitment and use the surpluses to derive equilibria of the money buys access models. We have three main objectives. First, we examine whether both lobbies participate or not. Recent descriptive evidence suggests that lobbying participation is very asymmetric; Richter et al. (2009) show that only a small fraction of firms actually lobby, and lobbying expenditures follow a skewed, power-law distribution. Putting aside the entry barriers in the form of large upfront costs (Kerr et al. 2014), common wisdom attributes the absence of informational lobbying to the lack of favorable evidence among the abstaining lobbies. More specifically, a classic persuasion game (Milgrom 1981; Milgrom and Roberts 1986; Bhattacharya and Mukherjee 2013) attributes the absence of communication to the pooling of bad evidence with no evidence. For information transactions, in contrast, the absence of disclosure stems from the policy-maker s unwillingness to initiate communication with certain lobbies. In other words, the revenue-information tradeoff motivates the policy-maker to impose a prohibitively large transaction price. For a single lobby, the revenue-information tradeoff materializes only under very special circumstances. 1 For competitive lobbying, the nature of the policy-maker s tradeoff may be rather different: The policy-maker always meets at least one competing lobby because at least one lobby has a positive willingness to pay and the optimal price is positive. The policy-maker 1 The tradeoff exists only if the price that encourages a lobby to accept an experiment with both favorable and unfavorable outcomes is negative and sufficiently large, i.e., a compensation is prohibitively costly for the policy-maker (Gregor 2015). 3

4 then announces a prohibitive price to the second lobby if an extra meeting with the second lobby excessively reduces the revenues from the meeting with the first lobby. To further elucidate the revenue externality of an extra participating lobby, we examine the information and revenue complementarities of the signals and confirm the intuition that, broadly speaking, extra transaction takes place if benefits of the signals are substitutable, and does not take place if the benefits are complementary. Second, we examine the structure of asymmetric participation. In the access fee model of competitive lobbying by Grossman and Helpman (2001), there is an inverse relationship between the bias of the lobby and participation. In contrast, in Austen-Smith and Wright (1992, 1994), there is a positive relationship, and lobbying of the less biased (ex ante advantaged) lobby is only counteractive. In our setting, the more biased (ex ante disadvantaged) lobby is typically more likely to participate, but the prediction largely depends on the information structure. Additionally, timing is crucial. If transactions are early, the less biased lobby has a very attractive outside option and is likely to not participate. If transactions come late in the game, this ex ante advantage for the less biased lobby diminishes. The attractive outside option of the less biased lobby under early transactions also implies that the less biased lobby always prefers to establish a long-term relationship with the policy-maker in the form of early transactions. Third, we examine the relative payoffs of the two competing lobbies. In a setting with noiseless evidence, which is a corner case of our general noisy structures, the less biased lobby is in the equilibrium more expropriated by the policy-maker and ends up worse off than the more biased lobby. The ex ante advantage of a small bias thus turns into an ex post disadvantage. The curse of the ex ante advantage largely contrasts with strategic communication in the absence of transfers, especially if messages are non-verifiable (Krishna and Morgan 2001). It generates counterintuitive implications, especially in the context of endogenous persuasion, where it predicts that lobbies will strategically deteriorate the quality of their signals or even strategically burn own policy-related assets. This paradox is also behind the counterintuitive predictions that contribution caps may be to the advantage of the richer lobbies and may encourage lobby formation (Cotton 2012). Nevertheless, our analysis for all information structures and alternative timings of access reveals that the curse of the ex ante advantage exists unambiguously in only one class of structures and does not exist in 13 out of 20 classes. Additionally, we distinguish between the curse of an ex ante advantage and the curse of access. While the curse of an ex ante advantage is rare, the curse of access is a robust property of competitive lobbying through transactions where the policy-maker announces prices: Namely, if the equilibrium involves asymmetric participation, then the non-participant is always better off than the participant. The paper proceeds as follows: Section 3 builds a benchmark of a noiseless information structure with interim timing. This basic setting always yields the curse of the ex ante advantage. Section 4 introduces alternative information structures and applies the bilateral 4

5 quasi-bargaining approach to both interim and ex ante timing. Section 5 establishes the specific results for all information structures and for both timings. Section 6 concludes the paper. 2 Related literature Evidence and money are two primary instruments of influence in the toolkit of special interests. The way in which these two instruments are jointly used still remains an open question, with various traditions of lobbying models providing different answers. This paper aims to contribute to the money buys access tradition (Baron 1989; Snyder 1990; Austen-Smith 1995; Ball 1995; Lohmann 1995; Wright 1996; Cotton 2009; Groll and Ellis 2014) and particularly to the family of models that interpret access fees as message fees (Austen-Smith 1998; Cotton 2009, 2012). In such games, the policy-maker establishes prices for meeting and communication, and a lobby either incurs the cost and communicates her private verifiable message or abstains. More precisely, we consider transactions with evidence in the classic context of competitive informational lobbying due to Austen-Smith and Wright (1992). The content of the transaction is verifiable evidence. Verifiability implies that the credibility of the lobby s message is not related to the lobby s bias. The lack of the credibility channel implies that the gains from participation are not necessarily monotonic in the bias as is typical in cheap talk models of competitive lobbying (Grossman and Helpman 2001). The existence of information transactions affects many equilibrium properties of the competitive informational lobbying. For example, Austen-Smith and Wright (1992) model competitive lobbying as a signalling game and observe that the ex ante disadvantaged (extremist) lobby participates more likely than the ex ante advantaged (moderate) strong lobby. This counteracts a popular perception that mainstream lobbies tend to have better access and therefore lobby more. In fact, Austen-Smith and Wright (1992, 1994) coined the term counteractive lobbying to characterize the fact that a moderate lobby participates only to mitigate the effects of lobbying of the extremist lobby. The result holds even if certain equilibrium refinements (especially skeptical beliefs) are lifted (Sloof 1997). In our paper, as we will show, the result indeed holds if transactions are made before private signals are obtained, but not necessarily holds if transactions are made after private signals are obtained. There are two very closely related papers that examine the classic competitive informational lobbying and allow for endogenous prices (message fees). Austen-Smith (1998) considers endogenous noise structures while Cotton (2012) considers exogenous and noiseless information structure. The closest paper is Cotton (2012) as he covers a special case examined our paper, namely interim timing with noiseless evidence. Our paper shows how timing affects the results in Cotton (2012): For interim timing, non-participation may serve as a signal of a having an unfavorable piece of evidence. In contrast, for ex ante timing, non-participation 5

6 does not serve as a signal. Therefore, the outside option of a lobby associated with nonparticipation may be strikingly different, and the results from ex ante timing do not readily translate into interim timing. Robustness to noise in information structures is equally important. Consider two properties of a lobby s strategies: A strategy of a lobby is informative if the policy-maker s action is not constant in the type of the lobby. A message strategy is influential if the policy-maker s action is not constant in the type of the other lobby. While noiseless information structure is both informative and influential, noisy structures may be informative and non-influential, non-informative and influential, or non-informative and non-influential. 2 This paper can be also seen as an extension of the classic disclosure game (Milgrom 1981; Milgrom and Roberts 1986; Bhattacharya and Mukherjee, 2013) by conditional commitments to not listen and not talk. Our setup demonstrates how to calculate the equilibrium participation by means of the quasi-bargaining surpluses, which may serve as a toolkit for modeling disclosure with conditional commitments under more complex settings with more lobbies and a richer state space. Because the model also covers ex-ante timing whereby the participation decision is a decision to run or not run a public experiment, the model can also be combined with lobbying models that include a search for hard evidence with observable outcomes (Bennedsen and Feldman 2006; Dahm and Porteiro 2008a, 2008b). 3 A baseline model The baseline model adopts interim timing and noiseless signals. In the general model, both aspects will be generalized. 3.1 Setup State of nature. The state of the world has two binary dimensions θ i {0, 1}, i = 1, 2. Prior beliefs π i := Pr(θ i = 1) (0, 1) are common knowledge. The priors are distributed independently, π i = Pr(θ i = 1 θ i = 0) = Pr(θ i = 1 θ i = 1). Players, signals. There is a single policy-maker and two lobbies, where the set of lobbies is S = {S 1, S 2 }. Lobby i privately observes a signal t i. Like in Austen-Smith (1998) and Cotton (2012), the signal has a form of verifiable/certifiable evidence that can be hidden but cannot be fabricated; the verifiability property is motivated by the interpretation of a lobbying meeting as a chance to tell the story. We speak of a high-type (H-type) if t i = 1 and a low-type (L-type) if t i = 0. In the noiseless setting, t i is perfectly correlated with θ i. 2 Sloof (1997) adopts a similar point in the analysis of all equilibria in a signaling game in Austen-Smith and Wright (1992). 6

7 Policies, objectives. The policy-maker selects a policy P {P 1, P 2 }. His policy is not contractible, and the set of implementable policies cannot be restricted. Lobbies are in a zero-sum competition: Lobby i s state-independent valuation of P i is one and of P i is zero. (Hence, the valuations are perfectly negatively correlated and symmetric.) In contrast, the policy-maker s valuation is state-dependent. Following Cotton (2012), we assume that the policy-maker considers policy P i incorrect (illegitimate) if and only if θ i = 0 and θ i = 1. Otherwise, the policy is correct (legitimate) and the policy loss is zero. 3 satisfied by a state-dependent policy loss function l(p i, θ i, θ i ) = (1 θ i )θ i. For any pair of posterior beliefs (p 1, p 2 ), the expected loss of policy P i is Asymmetry. L(P i, p 1, p 2 ) := 1 ˆθ 1 =0 ˆθ 2 =0 This property is 1 Pr(θ 1 = ˆθ 1 ) Pr(θ 2 = ˆθ 2 )l(p i, ˆθ 1, ˆθ 2 ) = (1 p i )p i. (1) Without loss of generality, let π 1 > π 2. Consequently, the status-quo policy (the policy-maker s optimal policy under priors) is P 1, L(P 1, π 1, π 2 ) < L(P 2, π 1, π 2 ). The ex ante advantaged Lobby 1 may be called moderate (small bias) and the ex ante disadvantaged Lobby 2 extremist (large bias). 4 Intuitively, the moderate lobby is the player with a mainstream opinion, whereas the extremist lobby holds a non-mainstream opinion. Outcomes. The outcome is a quadruplet of policies, each selected for a particular signal realizations. Let ψ(t 1, t 2 ) be a function that maps the signal realizations into the selected policies. Then, an outcome is the matrix of the signal-specific policies, ( ) ψ(0, 0) ψ(1, 0) O =. ψ(0, 1) ψ(1, 1) Since the policy-maker ex post selects his preferred policy subject to the observed information, the outcome will depend on the amount of information which is equivalent to the number of participating lobbies. Formally, let 2 S be the power set of S, and σ 2 S be the set of participating lobbies. Precisely speaking, for interim timing, only H-types participate, hence σ is the set of lobbies for whom H-types participate. We call O(σ) be the outcome with σ participants. 5 Since the policy-maker announces lobby-specific prices and thereby manipulates participation of the lobbies, we call any participating lobby S i σ an invited lobby, and any non-participating lobby S i σ a non-invited lobby. (Recall that for interim timing, 3 The results can be easily extended to any arbitrary loss function. 4 In Perez-Richet (2014b), Lobby 1 is called strong and Lobby 2 weak. Austen-Smith (1998) calls any lobby with state-independent preferences extremist. 5 Whenever σ appears in the argument of a function, we will avoid the brackets, e.g., we write Λ(S 1) and not Λ({S 1}). 7

8 participation is related only to H-types; a lobby is invited if it is expected that her H-type participates.) To simplify notation, we will introduce the following quintuplet of outcomes: ( ) ( ) ( ) ( ) ( ) P 1 P 1 P 1 P 1 P 1 P 1 P 2 P 1 P 2 P 1 F 0 :=, F 1 :=, F 2 :=, F 3 :=, F 4 := P 1 P 1 P 2 P 1 P 2 P 2 P 2 P 1 P 2 P 2 Indifference-breaking. The noiseless setting features two events when the expected losses are equal (zero) and the policy-maker is ex post indifferent between the policies, namely t 1 = t 2 = 0 and t 1 = t 2 = 1. Given these indifferences, O(S 1, S 2 ) is not unique in noiseless setting. To treat this multiplicity, we suppose that the policy-maker can announce (commit to) any feasible pair of policies for the indifference events. In other words, the policy-maker in a noiseless setting announces O(S 1, S 2 ) from a quadruplet {F 1, F 2, F 3, F 4 }. 6 Prices. The policy-maker also announces discriminatory (lobby-specific) prices c i R, i = 1, 2. A price can be positive (a fee for access) or negative (a compensation for access). A negative price may represent a subsidy or earmark which can be credibly provided to the participating lobby. (Section 5.7 examines situations when the policy-maker cannot compensate the lobbies because the lobbies lack the commitment not to meet, and the price is restrained to be non-negative.) All agents preferences are separable and linear in transfers. Interim timing. In Stage 1, the policy-maker announces prices c i R, i = 1, 2 and O(S 1, S 2 ) {F 1, F 2, F 3, F 4 }. In Stage 2, each Lobby i privately observes t i. In Stage 3, each Lobby i either accepts the price or abstains. In Stage 4, each participating Lobby i discloses her signal t i. In Stage 5, the policy-maker selects his ex post optimal policy. We model Stage 4 such that each participating lobby has to disclose her signal. If not and disclosure is a choice variable of a lobby, we would equivalently impose skeptical policymaker s beliefs that interpret non-participation as L-type (a no news is bad news refinement). With these beliefs, a standard unraveling (full disclosure) result applies (Milgrom, 1981). 3.2 Equilibrium To find the equilibrium outcome O, notice that a policy-maker s announcement in Stage 1 leads to a subgame of Stages 2 5. We will classify the subgames along the set of participating 6 The treatment of indifferences is negligible in the general setting but is of key importance in the noiseless setting. For example, in a noiseless setting in Cotton (2012), O(S 1, S 2) is selected exogenously and O(S 1, S 2) {F 1, F 4}. Here, O(S 1, S 2) is selected by the policy-maker without any exogenous constraints. The advantage of endogeneity is that the comparative statics are not affected by an extra constraint upon O(S 1, S 2). In any case, in a complete parametrical space for the noisy information structures, indifferences will be only knife-edge cases. 8

9 (and signal revealing) lobbies. The benefit of this classification is that the set of signals, the amount of information, and therefore also the outcome are constant within each class. To characterize the amount of information in a subgame, let the expected policy loss in any σ-subgame (called the σ-loss) be Λ(σ). Clearly, Λ(S 1, S 2 ) Λ(S i ) Λ( ) for any i = 1, 2. For noiseless signals, O( ) = F 0, O(S 1 ) = F 3, O(S 2 ) = F 2, so Λ(S 1, S 2 ) = Λ(S 1 ) = Λ(S 2 ) = 0 Λ( ). To identify the equilibrium, notice that the policy-maker effectively selects an optimal subgame out of four σ-classes of subgames. We can immediately eliminate the -class as a candidate for the optimal subgame because the non-informative outcome O( ) with zero revenue is weakly less valuable than any informative outcome with a non-negative revenue. Within each remaining σ-class, we identify the revenue-maximizing prices c i (σ), i = 1, 2. These prices characterize the best σ-subgame within the σ-class. The policy-maker s payoff in this best σ-subgame is W (σ) and Lobby i s payoff is u i (σ). To calculate the policy-maker s payoffs specifically in the baseline setting, we use that the three best σ-subgames are informationally equivalent, Λ(σ) = 0. Then, only revenue matters. Class σ = {S i }. The policy-maker s revenue-maximizing price is c i (S i ) = 1. (The price for a participating Lobby i is set such that the H-type s incentive compatibility condition is just met. The other price c i (S i ) is set prohibitively high so that the H-type of S i is not willing to participate.) The policy-maker s expected revenue is π i c i (S i ), and the expected payoff is Class σ = {S 1, S 2 }. W (S i ) = π i. (2) Revenue-maximizing prices depend on the announced outcome O(S 1, S 2 ); hence, we calculate the revenue-maximizing prices for each O(S 1, S 2 ) and obtain the expected payoff conditional on O(S 1, S 2 ): W O(S 1,S 2 ) π 2 if O(S 1, S 2 ) {F 1, F 2 } (S 1, S 2 ) = π 1 if O(S 1, S 2 ) {F 3, F 4 }. Finally, since π 1 > π 2, W (S 1, S 2 ) = max {W F (S 1, S 2 )} = π 1. (3) O(S 1,S 2 ) In the next step, we derive the σ-subgame and the corresponding outcome O(σ) that the policy-maker prefers in Stage 1. This also defines the equilibrium: σ = arg max W (σ). (4) σ 2S By a straightforward comparison of the expected payoffs in (2) and (3), the policy-maker s optimal outcome is O := O(σ ) {F 3, F 4 }. 9

10 Notice that both outcomes F 3 and F 4 can be achieved by inviting both lobbies and simply announcing the outcome. (Additionally, the outcome F 3 can be alternatively achieved by inviting only Lobby 1.) Thus, the policy-maker does not have to set prohibitive prices to maximize his payoff; a revenue-information tradeoff is absent in the noiseless interim setting. 3.3 Relative payoffs We distinguish between two concepts related to the relative payoffs of lobbies. Lobby i has an ex ante advantage if in the absence of communication (σ = ), her payoff is larger than the payoff of Lobby i, u i ( ) > u i ( ). Lobby i has an ex post advantage if her equilibrium payoff is larger than the equilibrium payoff of Lobby i, u i (σ ) > u i (σ ). Since π 1 > π 2, Lobby 1 has an ex ante advantage. The key result of the noiseless model is that an ex ante advantage (associated with the moderate lobby) is equivalent to an ex post disadvantage. Specifically, u 1 (σ ) = 0 < 1 π 1 = u 2 (σ ). We call this paradoxical property the curse of the ex ante advantage. 7 The importance of the curse of the ex ante advantage stands out once the lobbies characteristics are endogenous; among others, the curse has the potential to generate a race to the bottom in terms of precision of the evidence. Another interesting result is the curse of access: If Lobby i is invited and Lobby i is not invited, σ = {S i }, then the invited lobby has an ex post disadvantage, u i (S i ) < u i (S i ). The curse of access in fact identifies a sufficient condition for the ex post disadvantage. From this perspective, the curse of access is a weaker result than the curse of the ex ante advantage, which identifies both a sufficient and a necessary condition for the ex post disadvantage. In the general model, however, the curse of access appears to be robust whereas the curse of the ex ante advantage is not robust. 4 A general model 4.1 Motivation In this section, we consider all admissible conditionally independent signals. Through the analysis of noisy information structures, we aim to elucidate the nature of the revenueinformation tradeoff of the policy-maker. Namely, we will see how the price and information 7 In a working paper version of the article, we consider also valuation asymmetry and construct a loss function which exhibits the curse for any asymmetry in priors and valuations. If priors are set identical, the curse can be interpreted such that the player with a larger stake (a richer player) always ends up worse off. 10

11 complementarities affect the tradeoff. This tradeoff explains the strategically prohibitive prices and consequently explains information losses in competitive lobbying. The previous Section 3 has shown that the tradeoff is entirely absent in the noiseless information structure with interim access. In this special setting, a single signal is sufficient to generate zero policy loss. A consequence of the zero marginal informational value of the second signal is that the policy-maker s tradeoff associated with a unilateral access restriction is purely in the revenue dimension. Yet, as we have seen in Section 3, the option of a strategic access restriction is then entirely irrelevant for maximization of revenues. A completely different picture arises with noise. The policy-maker compares the effects of incomplete participation on the amount of information and the amount of revenues. For some information structures, incomplete participation decreases both amounts, and the tradeoff does not exist. The interesting information structures are such that the incomplete participation decreases the amount of information but increases the amount of revenues. 4.2 Assumptions Signals. Let each signal t i {0, 1} be characterized by Type-I and Type-II errors. Namely, α = Pr(t 1 = 0 θ 1 = 1), β = Pr(t 1 = 1 θ 1 = 0), γ = Pr(t 2 = 0 θ 2 = 1), and δ = Pr(t 2 = 1 θ 2 = 0). Recall that priors are distributed independently and signals are conditionally independent. To guarantee a natural meaning of the signals, we assume that t i and θ i are positively correlated. Hence, the high state of the world is more likely than the L-type, p i (1) p i (0), which is equivalent to α + β 1 and γ + δ 1. Using the frequencies of H-types f 1 := (1 α)π 1 + β(1 π 1 ) and f 2 := (1 γ)π 2 + δ(1 π 2 ), p 1 (0) = απ 1 1 f 1 π 1 (1 α)π 1 f 1 = p 1 (1), p 2 (0) = γπ 2 1 f 2 π 2 (1 γ)π 2 f 2 = p 2 (1). For brevity, our notation excludes the noise parameters (α, β, γ, δ) from the arguments. Indifference. We disregard information structures with the policy-maker s indifferences; hence, O(S 1, S 2 ) is unique. The reason for this choice is that if the noise parameters are drawn from a joint distribution function with a full support, then the event that the policymaker is indifferent at some signal realization is a measure-zero event. The existence of a unique O(S 1, S 2 ) thus allows us to eliminate the announcement of O(S 1, S 2 ). Timing. The interim timing has been introduced in Section 3. In ex ante timing, Stages 2 and 3 are switched. Specifically, in Stage 2, each Lobby i either pays the price or abstains. Then in Stage 3, each Lobby i privately observes t i. (Therefore, σ is the set of participating lobbies for ex ante timing and the set of lobbies with participating H-types for interim timing.) 11

12 Losses. We know that l(p i, θ i, θ i ) = 0 if θ 1 = θ 2 = 0. As a result, only states of the world in which θ 1 θ 2 are informationally relevant to the policy-maker. Thus, the policy-maker s minimization of the expected loss rewrites into a classic binary-binary (two relevant states and two actions) decision problem (c.f., Börgers et al., 2013). We now parametrically express the expected loss associated with policy P under signal realization (t 1, t 2 ). Using (1) and the conditional probabilities of the signal realizations, ( β f1 ) t1 ( ) 1 t1 ( ) t2 ( ) 1 t2 L(P 1 ; p 1 (t 1 ); p 2 (t 2 )) = (1 p 1 (t 1 ))p 2 (t 2 ) = 1 β 1 γ γ 1 f 1 f 2 1 f 2 (1 π1 )π 2, ( ) t1 ( ) 1 t1 ( ) t2 ( ) 1 t2 L(P 2 ; p 1 (t 1 ); p 2 (t 2 )) = p 1 (t 1 )(1 p 2 (t 2 )) = 1 α α δ 1 δ f 1 1 f 1 f 2 1 f 2 (1 π2 )π 1. The information gain from the policy switch from P i to P i under the signal realization (t 1, t 2 ) is simply measured by the difference in the posteriors, L(P i ; p 1 (t 1 ); p 2 (t 2 )) L(P i ; p 1 (t 1 ); p 2 (t 2 )) = p i (t i ) p i (t i ). Welfare. The application of the utilitarian welfare criterion in a symmetric two-lobby setting is considerably easier than in any asymmetric setting because policy valuations of two pure competitors with symmetric values exactly cancel each other out. 8 Additionally, given zero transaction costs, any monetary transfer is welfare-neutral. Thus, if C i denotes the expected revenues from Lobby i, the utilitarian welfare is measured by the inverse of the policy-maker s expected loss Λ (i.e., the equilibrium amount of information), W (σ)+u 1 (σ)+u 2 (σ) = C 1 (σ)+c 2 (σ) Λ(σ)+Pr(P = P 1 σ) C 1 (σ)+pr(p = P 2 σ) C 2 (σ) = 1 Λ(σ). Given the policy-maker s non-contractible actions, the first-best outcome is thus achieved if access is not restricted, {S 1, S 2 } = arg min σ Λ(σ). Vice versa, only strategic access restriction may create a wedge between the equilibrium outcome O and the first-best outcome O(S 1, S 2 ). 4.3 Information structures Because the power set 2 S has four elements, each information structure generates a quadruplet of outcomes O(σ), σ 2 S. We now classify the information structures into classes, with each class defined by a generic quadruplet of outcomes. Thereby, we will identify 10 different classes of information structure Classification along O(S 1, S 2 ) The first part of the classification of information structures is along O(S 1, S 2 ). By definition, O(S 1, S 2 ) characterizes the outcome if (t 1, t 2 ) is observed. The identification of O(S 1, S 2 ) in 8 For examples of the utilitarian welfare criteria in a single-lobby setting, see Gregor (2014). 12

13 the parametrical space requires a comparison p 1 (t 1 ) p 2 (t 2 ) to be conducted for each out of four pairs of t 1 {0, 1} and t 2 {0, 1}. For the sign of the inequality in (t 1, t 2 ), we introduce a binary indicator: φ t 1,t 2 := 1{p 1 (t 1 ) p 2 (t 2 )}. φ 1,0 = 1 always holds. Using µ := π 2 1 π 2 1 π 1 π 1 R +, where for our asymmetry µ (0, 1), the remaining indicators are identified in the following way: φ 0,0 = 1{α(1 δ) (1 β)γµ} φ 0,1 = 1{αδ (1 β)(1 γ)µ} φ 1,1 = 1{(1 α)δ β(1 γ)µ} The triplet of the binary indicators (φ 0,0, φ 0,1, φ 1,1 ) admits eight combinations; however, not all are feasible. The implication φ 0,0 = 0 φ 0,1 = 0 rules out the triplets (0, 1, 0) and (0, 1, 1). The implication φ 1,1 = 0 φ 0,1 = 0 additionally rules out the triplet (1, 1, 0). The remaining five admissible triplets characterize the support of O(S 1, S 2 ) as {F 0, F 1, F 2, F 3, F 4 } Informativeness indicators The second part of the classification uses the informativeness of the signals. We call a signal t i informative if, ceteris paribus, the policy-maker sets P = P i after observing t i = 0 and P = P i after observing t i = 1. We denote the informativeness indicator of the signal t i as I i. There are three indicators for each signal t i depending on the other signal t i : (i) Ii is for t i being unobserved; (ii) Ii 0 is for a low realization, t i = 0; and (iii) Ii 1 is for a high realization, t i = 1. Informativeness indicators in the absence of the other signal, Ii := 1{p i (0) π i p i (1)}, write parametrically as I 1 I 2 = 1{α + µβ µ 0}, = 1{µγ + δ µ 0}. The informativeness indicators in the presence of the other signal are as follows: Ii k = 1{p i (0) p i (k) p i (1)}. Notice that each quadruplet (I1 0, I1 1, I0 2, I1 2 ) defines a triplet (φ0,0, φ 0,1, φ 1,1 ). Additionally, notice that the triplet (φ 0,0, φ 0,1, φ 1,1 ) is by definition interchangeable with O(S 1, S 2 ). Therefore, the additional classification brought about by the informativeness indicators rests only in the pair (I1, I 2 ). 13

14 Table 1: The admissible information structures Class O(S 1, S 2 )-indicators Informativeness Outcomes φ 0,0 φ 0,1 φ 1,1 I 1 I 2 I 0 1 I 1 1 I 0 2 I 1 2 σ = {S 1, S 2 } σ = {S 1 } σ = {S 2 } C F 0 F 0 F 0 C F 1 F 0 F 0 C F 1 F 0 F 2 C F 1 F 3 F 0 C F 1 F 3 F 2 C F 2 F 0 F 2 C F 2 F 3 F 2 C F 3 F 3 F 0 C F 3 F 3 F 2 C F 4 F 3 F Classes of information structures The classification is obtained by combining O(S 1, S 2 ) and the informativeness indicators (I1, I 2 ). Lemma 1 identifies 10 classes of structures. Lemma 1. There are 10 admissible information structures, defined in Table 1. As a next step, we characterize the outcomes O(σ) based on the set of invited lobbies σ and the information structure. The non-garbled outcome is O(S 1, S 2 ). The garbled outcome in which no signal is observed is O( ) = F 0. (Recall that Lobby 1 has the ex ante advantage.) The non-garbled outcome for σ = {S i } depends upon the indicator Ii : F 3 if I1 O(S 1 ) = = 1, F 2 if I2 O(S 2 ) = = 1, F 0 if I1 = 0. F 0 if I2 = 0. In the equilibrium, O {F 0, O(S 1 ), O(S 2 ), O(S 1, S 2 )}. Table 1 summarizes the garbled and non-garbled outcomes associated with each information structure. Each class of structures characterizes a unique quadruplet of admissible outcomes O(σ), σ 2 S. The quadruplet also characterizes the quadruplet of σ-losses Λ(σ), σ 2 S, and the σ-losses characterize the information complementarity properties of the signals (c.f., Börgers et al., 2013). Definition 1. Let the signals be information complements if σ-losses satisfy Λ(S 1 ) + Λ(S 2 ) > Λ( ) + Λ(S 1, S 2 ). (5) With the opposite strict inequality, the signals are information substitutes. 14

15 4.4 σ-classes of subgames As in the noiseless setting, the equilibrium is characterized by the policy-maker s optimal prices (c 1, c 2 ), which induce the optimal set of participants σ and the optimal outcome O(σ ). The optimum is found in two steps: First, we derive the revenue-maximizing prices for each σ, c 1 (σ) and c 2 (σ). The prices induce a σ-subgame with the policy-maker s expected value W (σ). Second, we compare the policy-maker s expected values W (σ) and select the subgame with the highest value, as in (4). The construction of the revenue-maximizing prices in a σ-class follows the following participation constraints: Participants: For any S i σ, the price c i (σ) is set such that the participant s afterpayment value is equal to the value of the outside option (i.e., non-participation). As in principal-agent models, if the agent s value is below her outside option, the agent (Lobby) deviates. If her value is above her outside option, the principal (policy-maker) can increase the price, transfer part of the value to himself, and still maintain a σ- subgame. Non-participants: For any S i σ, the price c i (σ) is set prohibitively high such the non-participant s value of non-participation is above the value of the outside option (i.e., participation). The exact level is irrelevant because a price imposed upon a nonparticipant is not paid. 4.5 Quasi-bargaining perspective To identify σ requires us to make three pairwise comparisons between the values W (σ). When conducting the pairwise comparisons, we adopt the following quasi-bargaining perspective: Through a vector of prices, the policy-maker selects a σ-class of subgames, where σ describes each Lobby i s participation. Either the price c i is acceptable and Lobby i participates, or the price is unacceptable and Lobby i abstains. (We will alternatively say that the policy-maker either invites Lobby i or does not invite Lobby i.) The invitation of Lobby i requires two participation conditions to be met: The policymaker s participation condition is defined by the policy-maker s outside option of non-invitation. The lobby s participation condition is defined by the lobby s outside option of abstention. The existence of two players participation conditions in which one player (the policy-maker) gives an offer and the second player (the lobby) accepts or rejects the offer constitutes a bilateral bargaining problem. Therefore, we may interpret the equilibrium prices as the policy-maker s bargaining offers in two parallel bilateral bargaining problems. When setting a particular offer for Lobby i, the policy-maker calculates both his partial gains from an invitation and Lobby i s partial gains from an invitation. If the sum of the partial gains (bilateral surplus) is positive, both 15

16 participation conditions can be met and the policy-maker sets a price c i that indeed motivates Lobby i to pay the price and disclose the signal. If the surplus is negative, both participation conditions cannot be met at the same time and the policy-maker sets a prohibitively large price that discourages Lobby i from disclosing the signal. The key in the subsequent analysis is to derive the partial gains. For Lobby i, the partial gain is the change in the frequency of the preferred policy P i relative to her outside option. For the policy-maker, the partial gains consist of an informational gain and the revenue externality upon the other parallel problem, both relative to his outside option of non-invitation. The informational gain is the decrease in the expected policy loss associated with observing an additional signal. The revenue externality is the change in the policy-maker s revenues in the parallel bargaining problem. Simply, if the policy-maker invites Lobby i, he must take into account that he will charge a different price c i because the outside option of the other Lobby i changes. Thus, each bilateral surplus has exactly three components: (i) The change in the frequency of the preferred policy, (ii) the reduction of the policy loss, and (iii) the revenue externality. 9 Each bilateral bargaining problem has a trivial bargaining protocol: The policy-maker makes a single take-it-or-leave-it offer and the lobby agrees or disagrees. As a result, if the surplus is non-negative, the policy-maker extracts the full surplus, and leaves the lobby with the payoff at her outside option. (For convenience, we treat the lobby s indifference such that she always accepts the offer equal to her outside option.) An immediate corollary of the full bargaining power of the policy-maker is that the proposed transaction price c i (σ) is equal to the partial gain from the participation of Lobby i if S i σ, and c i (σ) is above the partial gain from participation of Lobby i if S i σ. In Section 5, we will show in detail that a sign of a bilateral surplus is equivalent to the sign of an inequality between two values W (σ): The surplus for a participant S i is non-negative if and only if W (σ) W (σ \ S i ). Similarly, the surplus is negative for a non-participant S i if and only if W (σ) < W (σ S i ). Therefore, the bargaining perspective generates two easily applicable equilibrium conditions: The bilateral surplus with a participant is non-negative, and the bilateral surplus with a non-participant is negative. Because the quasi-bargaining perspective does not exhaust all of the policy-maker s outside options, the two equilibrium conditions are necessary but not sufficient. In (4), the policymaker compares σ with three alternatives, whereas the bargaining perspective calculates only two surpluses and thereby conducts only two comparisons. 10 Nevertheless, Propositions 1 and 2 reveal that the surplus levels for our zero-sum competition of two lobbies are in fact 9 In a single-lobby setting, the bilateral surplus has only two components (Gregor, 2014). Hence, the difference between a single-lobby and a multiple-lobby setting lies in the existence of the revenue externalities. 10 The single missing equilibrium condition that is not captured by the bargaining perspective is W (σ ) W ( σ ), where σ is a complement to σ. More generally, for n lobbies, the bargaining perspective examines n conditions and omits 2 n n 1 conditions. The bargaining perspective is therefore useful for a low number of lobbies. 16

17 sufficient to characterize σ. In the construction of the surpluses, it is important that the identity of the policy-maker s bargaining partner depends on timing. For ex ante timing, the policy-maker s partner is the lobby as such. For interim timing, the policy-maker s partner is only lobby of H-type. Timing also affects whether the outside options of the two bargaining players constitute a single disagreement event or not. For ex ante timing, any player s non-participation results in the same disagreement event (namely, a different σ-subgame with an abstaining lobby). For interim timing, the policy-maker s non-invitation results in a σ-subgame with an abstaining lobby, while the H-type s non-participation maintains the σ-subgame but the H-type is now interpreted as an L-type. This difference explains why the term quasi-bargaining is used instead of bargaining. 5 Equilibrium prices and payoffs To derive a lobby s willingness to pay (i.e., the partial gain) and consequently the price c i (σ), we need to calculate probability of a favorable event P = P i conditional on the outcome O(σ). To simplify notation, we will be writing Pr(P i σ) := Pr(P = P i O = O(σ)). 5.1 Interim access The policy-maker s expected payoff in the revenue-maximizing σ-subgame is W (σ) = 1{S i σ}f i c i (σ) Λ(σ). (6) i=1,2 The revenue-maximizing prices c i (σ), i = 1, 2, are set as follows: S i σ: The payoff of a participating H-type is u H i (σ) = Pr(P i σ, t i = 1) c i (σ). Her outside option in a separating equilibrium is to pool with a non-participating L-type; hence, it is equal to the payoff of the non-participating L-type, u L i (σ) = Pr(P i σ, t i = 0). We know that the policy-maker exploits the separation and extracts all partial gains of the H-type, where the partial gains under interim access are always non-negative, c i (σ) = Pr(P i σ, t i = 1) Pr(P i σ, t i = 0) 0. (7) S i σ: The payoff of a non-participating H-type is u H i (σ) = Pr(P i σ). Her outside option is a payoff associated with the revelation of t i = 1, namely Pr(P i σ S i, t i = 1) c i (σ). We know that the policy-maker sets a prohibitively large price to discourage participation, c i (σ) > Pr(P i σ S i, t i = 1) Pr(P i σ). 17

18 Now, we construct the bilateral surpluses for interim access. Let Hi be the bilateral surplus between the policy-maker and the H-type of Lobby i if Lobby i does not participate. The surplus consists of an informational gain for the policy-maker at σ = {S i } (his outside option is an informational gain in -subgame) and of a partial gain for the H-type of Lobby (her outside option is σ-subgame with the policy-maker s posterior t i = 0) multiplied by f i. Given full bargaining power for the policy-maker, we know that the partial gain of Lobby i is the price c i (S i ). To obtain the expected revenue in the interim setting, the partial gain of the H-type of Lobby must be multiplied by the frequency of the H-type, f i, because the policy-maker s informational gain is always realized, whereas the Lobby s partial gains are realized only in high realizations. Therefore, Hi = Λ( ) Λ(S i ) + f i c i (S i ) 0. (8) Let Hi e be the bilateral surplus between the policy-maker and the H-type of Lobby i if Lobby i participates. The surplus is derived under σ = {S 1, S 2 } and consists of the two formerly analyzed partial gains plus a revenue externality from participation of S i. More specifically, if S i participates, she affects the parallel bargaining problem with S i. This participation changes the partial gains for the other H-type, which implies a revenue change from f i c i (S i ) into f i c i (S 1, S 2 ). In total, the surplus Hi e is the sum of three components, Hi e = Λ(S i ) Λ(S 1, S 2 ) + f i c i (S 1, S 2 ) + f i c i (S 1, S 2 ) f i c i (S i ). (9) Now, by combining (6), (8) and (9), we find that the surplus is equivalent to the difference between W (σ) and W (σ \ S i ), H i = W (S i ) W ( ), Hi e = W (S 1, S 2 ) W (S i ). Proposition 1 employs the bilateral surpluses to characterize the strategically restricted access. Proposition 1. For interim access, the policy-maker restricts the access of Lobby i if and only if Hi e < 0 and H i H i. Next, we introduce the revenue complementarity of the signals for interim access and combine the revenue and information complementarity properties into a single property. Definition 2. For interim access, let the signals be revenue complements if f 1 c 1 (S 1, S 2 ) + f 2 c 2 (S 1, S 2 ) > f 1 c 1 (S 1 ) + f 2 c 2 (S 2 ). (10) For interim access, let the signals be access complements if f 1 c 1 (S 1, S 2 ) + f 2 c 2 (S 1, S 2 ) Λ( ) Λ(S 1, S 2 ) > f 1 c 1 (S 1 ) + f 2 c 2 (S 2 ) Λ(S 1 ) Λ(S 2 ). (11) With the opposite strict inequality, the signals are revenue substitutes or access substitutes, respectively. 18

19 By inspection of (11), access complementarity determines the sign of Hi e H i, which describes whether the bilateral surplus with one lobby increases or decreases with the invitation of the other lobby. In addition, the sign also characterizes the sign of the overall externality of participation of one lobby upon bargaining with the other lobby. (Exactly the same equivalence also applies in ex ante timing.) More specifically, notice that the invitation of S i generates three externalities on bargaining with S i. The informational externality measures the change in the policy-maker s informational gain in bargaining with S i, [Λ(S i ) Λ(S 1, S 2 )] [Λ( ) Λ(S i )], and the sign is positive if and only if the signals are information complements. The direct revenue externality measures the change in the policy-maker s revenue in bargaining with S i, f i c i (S 1, S 2 ) f i c i (S i ). The indirect revenue externality measures the change in the policy-maker s revenue externality in bargaining with S i, which is f i c i (S 1, S 2 ) f i c i (S i ). The sign of the total revenue externality is positive if and only if the signals are revenue complements. The sum of all three externalities is the overall externality and is equal to Hi e H i. Thus, the overall externality is positive if the signals are access complements, and negative if the signals are access substitutes. Notice that all types of externalities (information externality, total revenue externality and overall externality) are symmetric for the lobbies; an externality of Lobby 1 s participation upon bargaining with Lobby 2 is identical to an externality of Lobby 2 s participation upon bargaining with Lobby 1. For example, symmetry of the overall externality is equivalent to H1 e H1 = He 2 H2. The fact that access complementarity determines whether the bilateral surplus with one lobby increases or decreases with the invitation of the other lobby is used in Corollary 1 to Proposition 1. The corollary identifies a useful special condition for the strategic access restriction under interim timing. Corollary 1. For interim access, a necessary condition for the strategic access restriction is that signals are access substitutes. Equivalently, a sufficient condition for both lobbies being invited is access complementarity. Moreover, in the analysis for particular information structures, it will be useful to see that from the definition of access substitutability, a sufficient condition for access substitutability is information substitutability and revenue substitutability. Similarly, we may use the fact that a sufficient condition for access complementarity is information complementarity and revenue complementarity. 5.2 Ex ante access The policy-maker s expected value in the revenue-maximizing σ-subgame is W (σ) = 1{S i σ}c i (σ) Λ(σ). (12) i=1,2 19