Learning More by Doing Less - CiteSeerX

Learning More by Doing Less Raphael Boleslavsky Christopher Cotton∗

August 8, 2011

Abstract A principal must decide whether to implement each of two independent proposals (e.g., earmark requests, policy reforms, grant funding) of unknown quality. Each proposal is represented by an agent who advocates by producing evidence about quality. Although the principal prefers the most-informative evidence, agents strategically choose less-informative evidence to maximize the probability the principal implements their proposals. In this setting, we show how limited capacity (i.e., the ability of the principal to implement at most one of the two proposals) can motivate agents to produce more-informative evidence in an effort to convince the principal that their proposal is better than the alternative. We derive reasonable conditions under which the principal prefers limited capacity to unlimited capacity.

JEL: D72, D78, D83, L15 Keywords: strategic search, evidence production, persuasion, lobbying ∗

Department of Economics, University of Miami, Coral Gables, Florida 33146. email: [email protected], [email protected]

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Introduction

Imagine that a politician must decide which earmark projects to bring before the appropriations subcommittee. Each project is supported by an interest group, who benefits only if the project is funded. In order to convince the politician that its pet project should be chosen, an interest group supplies evidence of the project’s quality (or popularity) along with its application. Although it may be difficult for the interest group to falsify evidence outright, the evidence that it chooses to produce may not completely reveal all relevant information about the costs and benefits of the project. For example, an interest group decides how many constituents to poll about their support for a project. If it surveys few constituents to make its case, then the survey outcome (whatever it is) is not very informative to the politician about the project’s popularity. If the interest group polls a large portion of constituents, then the resulting evidence is significantly more informative. Given the competitive nature of the earmark process, even producing evidence in support of a project is no guarantee that it will be approved. Alternatively, consider an organization offering grant money to researchers. Many grant applications require some description of preliminary findings, which may be difficult to falsify. However, these findings may vary in how informative they are about the project’s overall promise. Certain highly speculative preliminary findings may show that the research idea has promise, but may not be extremely informative about its ultimate chances of success. In many cases, one may conduct more preliminary research to produce evidence about a proposal’s ultimate chance of success. Depending on the nature of the grant, the organization may have the capability to approve all grant proposals if it believes it is appropriate to do so, or it may be constrained in the number of proposals it is able to fund. Motivated by these (and related) examples, we consider the problem of a principal, who chooses whether to approve or reject two different, independent proposals. The principal prefers to approve proposals if they are good, and reject proposals if they are bad. However, neither the principal nor the agents who advocate on behalf of different proposals know a proposal’s true quality. Prior to making her acceptance decision, the principal observes evidence produced by the agents which reveals either good or bad information about the proposal. This evidence is verifiable in that agents cannot misrepresent or manipulate the evidence they uncover (nor would they choose to keep it hidden). However, agents control how informative their evidence will be: the interest group decides whether to poll large numbers of constituents or only a few, or the researcher decides how much preliminary analysis to conduct prior to applying for a grant. After observing produced evidence, the principal then decides which, if any, proposals to implement. If the principal can accept all proposals if she wants to do so, then she has unlimited capacity to approve proposals. However, as is more typical, the principal may be constrained in her ability to approve proposals. If she is unable to approve all proposals, then she has limited capacity. We first identify a strong conflict of interest between the principal and agents for the case when the principal is unlimited in her ability to implement proposals. The principal wants to maximize the probability of implementing good proposals and not implementing bad proposals. She therefore 2

prefers that agents produce fully-informative evidence, ensuring that she only implements good proposals. Agents, on the other hand, want to maximize the probability that the principal decides to accept their proposal regardless of its true quality. Because of this, the agents produce less-thanfully-informative evidence since doing so maximizes the probability that the politician believes a proposal is beneficial. Indeed, if the principal is initially optimistic about the proposal and would be willing to accept with no additional evidence, there is no reason for the agent to produce any evidence. If the principal is predisposed against a proposal and would reject the proposal in the absence of additional evidence, then the agent must produce informative evidence in his favor for his proposal to be accepted. However, he will choose to supply evidence that is just informative enough for good evidence to sway the principal in favor of the proposal. He will never make the principal fully informed. We show when the principal has unlimited capacity to approve proposals, this conflict of interest between the agents and the principal always leads to agents providing less evidence than the politician prefers. This is true even when there are no direct costs to producing more evidence. The primary goal of our analysis is to show how limited capacity can mitigate the conflict of interest between principal and agents. First, we show that under reasonable condition, limited capacity increases the quality of evidence produced by the agents. Second, we show that these informational benefits can dominate the costs that come with limited capacity, making the principal better off. Under limited capacity, it is not enough for agents to convince the principal that their proposal is likely beneficial. Rather, they also must convince the principal their project is a better choice than the other proposal. In this sense, limited capacity creates a competition in which agents produce more-informative evidence as they vie for priority. In equilibrium, evidence is more informative under limited capacity than under unlimited capacity. However, a more informed principal is not necessarily better off. Limited capacity imposes a capacity constraint that has the potential to make the principal worse off; she cannot accept both proposals even when she expects to benefit from doing so. Our second set of results demonstrate that the informational benefits of limited capacity can dominate the downside of being constrained. Indeed, the principal strictly benefits (expects to make better decisions) from limited capacity when either (i) she is predisposed against implementing proposals that are most-likely harmful, or (ii) when she is predisposed in favor of accepting proposals that are most-likely beneficial and her predispositions are not too strong. Section 2 reviews the relevant literature. Although other articles model adversarial evidence production by competing agents (e.g., Austen-Smith and Wright 1992, Brocas et al. in press, Gul and Pesendorfer in press), the adversarial relationship in those models is exogenous, often arising from agents having opposing preferences on a single policy issue. Within these models, there is no way to compare evidence production under competitive incentives with evidence production in the absence of competitive incentives; the later case is not a possibility. In our framework, on the other hand, the adversarial relationship between the agents arrises due to capacity constraints. This is a subtle difference with significant implications. First, our model illustrates how competitive

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incentives can arise even when support for one option does not necessarily preclude support for the other option as well (e.g., a politician may want to grant an earmark to both a library and a city park; a grant organization might want to fund research on both cancer and unemployment). Second, our framework allow us to compare evidence production in cases with and without competitive pressure (or equivalently, cases of limited and unlimited capacity). In such environments where competitive incentives arise due to limited capacity, we show that the informational advantages often dominate the costs associated with limited capacity. The decision maker often prefers to be able to do less. To best illustrate the affects of limited capacity on evidence production, we allow the agents to choose the quality of evidence he produces, rather than simply whether to produce or not produce evidence (e.g., Austen-Smith and Wright 1992). To keep the analysis tractable and best develop intuition for our results, we focus on the case with two ex ante identical proposals from which to choose. Modeling only two proposal options greatly simplifies the analysis without reducing the insight into the results (most other papers on informational lobbying, for example, assume one or two interest groups). The assumption that the proposals are ex ante identical allows us to consider a situation in which neither proposal has an initial advantage. Allowing for differences in advocate valuations or probability of being good will not change the intuition for the results. In addition to the earmark and grant writing examples, there are a number of other settings the may be described by our framework. Section 8 discusses a number of these applications: a politician choosing type of legislation to introduce, colleges deciding which applicants receive admissions or scholarships, consumers choosing which products to purchase, the FDA choosing which drugs to approve. In each of these situations, limited capacity may be a reality of the decision making process. In legislative decision making, time or staff constraints may prevent a politician from introducing legislation on each policy or project she believes favorable. A grant organization may not be able to fund all projects it believes promising. A college cannot give scholarships to all worthy students. Consumers cannot purchase all goods they believe are beneficial. And our analysis shows that the FDA may approve safer drugs if it is able to approve fewer drugs. Where the costs of limited capacity are largely understood, we demonstrate that these limitations may also have informational benefits. And that these benefits may outweigh the downsides of limited capacity. In the next section, we review the relevant literature. Section 3 presents the model, which is solved for the case of unlimited capacity in Section 4 and for the case of limited capacity in Section 5. Section 6 summarizes the main results of the analysis. Section 7 considers a significant extension of the model in which we allow agents additional freedom in choosing their signal strategies, and show the the results continue to hold. Section 8 considers a number of applications. We conclude in Section 9.

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Related Literature

The seminal work of Austen-Smith and Wright (1992) (henceforth AW) presents a model of adversarial evidence production similar to ours. There are a number of key differences between AW and our model, however. As the introduction discusses, AW focuses on direct competition between two advocates who support mutually exclusive alternatives, where it is impossible to support both alternatives simultaneously (e.g., a politician can not simultaneously believe that it is beneficial to ban smoker and not ban smoking). Believing one option is worthwhile necessarily implies that she believes the other option is not worthwhile. In our model, the alternatives are not by nature mutually exclusive; it is entirely possible for the principal to accept both proposals (e.g., a politician may want to reform both taxes and healthcare). Any competition between the agents arises because budget or time constraints that prevent the principal for implementing all proposals even if she wants to. In addition to describing a variety of other situations which do not necessarily fit into the AW framework (e.g., college admissions, grant accepting, earmarks), our model also allows us to analyze a different set of questions than AW. We focus primarily on the interaction between the principal’s capacity to accept proposals and the accuracy of evidence supplied by interested parties. Our information structure also differs significantly from AW. In AW, interest groups make a binary decision to collect a signal (with a fixed accuracy) or not. In our framework, the agents strategically choose the accuracy of their signals. This allows for the possibility that an agent increases the informativeness of his signal quality in order to provide the principal with a marginally more-informative signal than the other agent. As will become clear in the analysis, some of the most-important results in the paper are related to the strategic choice of signal quality, results that are not possible in a binary game.1 In recent articles, Brocas et al. (in press) and Gul and Pesendorfer (in press) analyze dynamic models of adversarial evidence production, where—like AW—agents have opposing preferences over a single policy and agents do not control the quality of evidence.2 Furthermore, in these models the decision to acquire information are made in a sequentially rational manner. These works are therefore best suited to describing prolonged advertising or persuasion campaigns, rather than to the types of decisions we study. 1

Additional differences between AW and our paper include AW’s assumptions that signal collection is costly, and that agents privately observe their signals before deciding whether to share them with the principal (although the search choice is observed). Instead, for reasons discussed in the analysis, we assume that signal collection is without cost and any signal is publicly observable (although these assumptions are not required for our main results to hold qualitatively). Although the AW assumptions provide a more realistic description of some settings, we do not believe that they would add much to—and would actually distract from—our results. Adding costs of signal accuracy, for example, will make agents less likely to collect informative signals, but even when signals are costless to produce, the agents still prefer to keep the principal less-than-fully informed. 2 Both articles are concerned with the effect of cost differences on equilibrium outcomes. In both analyses, a public draw from a given informative signal is revealed, as long as one party chooses to exert effort. Thus, taking a new draw can be regarded as a decision to reveal new information to the decision maker, affecting her beliefs about which policy is best. However, in these analyses no action directly affects the informativeness of the signal itself. (Note that if players could committ to acquire a certain number of signal realizations initially, the adversaries would be competing in a manner similar to our model.)

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Other articles relate to our underlying evidence production framework but do not consider either direct or indirect competition between agents. Brocas and Carrillo (2007) study a dynamic information acquisition game, in which a leader has the capability to generating information that influences the decision of a follower with different preferences.3 Kamenica and Gentzkow (in press) consider the problem of a sender who tries to influence the action of a receiver by designing a signal whose realization will be observed by the receiver prior to choosing an action. Like Brocas and Carrillo, these authors characterize environments in which the sender benefits from persuasion. These results are closely related to our results for a decision making environment with unlimited capacity.4 However, our focus is quite different: we are primarily concerned with determining the extent to which limited decision making capacity improves the incentives for agents to supply accurate signals. Building on their earlier results, Gentzkow and Kamenica (2011) analyze a persuasion environment with multiple players. They show that (i) moving from collusive to cooperative play, (ii) introducing additional senders, (iii) decreasing alignment of sender preferences, increases the amount of information revealed in equilibrium. We look at an alternative to these remedies. In our environment a restricted action space for the decision maker improves information accuracy. We focus on the agents’ choice of evidence quality, and assume that evidence production is perfectly observable by the principal. Observable search in our model is equivalent to assuming that agents decide whether to show their evidence to the principal when the principal observes search effort, and communication and verification is costless. Henry (2009) considers the impact that mandatory research disclosure rules may have on an agent’s decision to acquire evidence. Che and Kartik (2009) consider how preference alignment between decision makers and agents affects the agents’ incentives to acquire and transmit evidence.5 Other articles assume that agents know their evidence ahead of time, and must choose whether to disclose it (e.g., Milgrom 1981, Milgrom and Roberts 1986, Bull and Watson 2004). Cotton (2009) presents a model in which agents must compete for access to disclose their evidence to a time-constrained decision maker. The preferences of agents in our paper are similar to advocates in Dewatripont and Tirole (1999).

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The Model

A principal and two agents play a two stage game. The risk-neutral principal (she) must decide whether to accept or reject each of two separate proposals. She prefers to accept a proposal only if the proposal is high-quality; that is, if the benefits of accepting the proposal outweigh the costs. Two risk-neutral agents (he), are associated with a separate proposals. An agent wants the principal to accept his proposal regardless of quality. Initially, both the principal and agents are uncertain about the quality of each proposal. Before the principal chooses which proposals to accept, the 3

Brocas and Carrillo’s analysis is best suited to applications with a temporal component, for example, the problem of a committee chairman deciding when to suspend debate and call a vote on a proposal. The authors describe situations in which the leader benefits from controlling the flow of public information. 4 In fact, their setting is more general, because they do not assume a parametric form for the signal structure, as we do. 5 Austen-Smith (1998) considers similar questions in a model of political access.

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agents can produce evidence about the quality of their own proposal. Although we present the model as one in which the principal decides whether to accept different proposals, nothing is changed if we adapt the terminology to refer to the principal awarding funding, admitting applicant, implementing policy, or making other decisions that directly affect agent payoffs. To keep the analysis as straightforward as possible, we assume that each proposal is either “good” (i.e., τi = g) or “bad” (i.e., τi = b). If accepted, all good proposals provide the same net benefit and all bad proposals provide the same net loss to the principal. The true quality of each proposal is unknown to all players, but it is common knowledge that quality is an i.i.d. draw from a Bernoulli distribution: Pr (τi = g) = γ Pr (τi = b) = 1 − γ. If the principal rejects proposal i, she earns proposal-specific payoff wi = 0 independent of τi . If the principal implements proposal i, she earns proposal payoff wi = 1 − q > 0 when the proposal is good and wi = −q < 0 when the proposal is bad. Preference parameter q ∈ (0, 1) represents the “stakes” inherent in the principal’s decision: when q is large the downside of accepting bad proposals is high compared to the upside of accepting good proposals.6 The principal’s total payoff, w, is simply the sum of her payoffs from each proposal: w = w1 + w2 . Agents benefit only if their proposal is accepted, with i receiving ui = 1 if proposal i is accepted and ui = 0 if rejected. In the first stage of the game, each agent simultaneously commissions independent research to produce verifiable information about the quality of its proposal. The research outcome is a publicly observable realization

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Si ∈ {G, B}, of signal si with the following conditional distribution:

Pr (si = G | τi = g) = Pr (si = B | τi = b) = αi Pr (si = G | τi = b) = Pr (si = B | τi = g) = 1 − αi . The signal realization (or outcome) reflects true quality of the proposal with probability αi ∈ [ 12 , 1]. We therefore refer to αi as the “accuracy” of si . Increasing accuracy improve the informativeness of the signal in the sense of Blackwell. A signal with the lowest accuracy, αi = 12 , is completely uninformative; a signal with the highest accuracy, αi = 1, is fully informative about quality. Agent i controls the accuracy of signal si .

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When an agent commissions research, he chooses

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This specification is without loss of generality. If we imagine that the principal’s payoff from accepting a good proposal is v > 0 while the cost of accepting a bad proposal is −c < 0, dividing both payoffs by v + c gives the specification defined in the text. 7 Identical results would hold if the research outcome (signal) were private information for the agent, provided this research outcome was verifiable, i.e. could be witheld but not falsely reported. See Kamenica and Gentzkow (in press) for more information. Identical results would also hold if the signal realization were privately observed by the principal, i.e. if it is a subjective impression, rather than verifiable evidence. 8 In our framework there is no exogenous cost for increasing signal accuracy. Therefore, if agents do not provide fully informative signals, it is because they prefer such signals for strategic reasons, not because more informative

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the ability of the researcher. Higher ability researchers are more likely to produce accurate findings. Alternatively, the agent may influence the informativeness of the signal by choosing the research design directly. Once both agents choose their signal accuracy, these choices are observed by the principal. By implication, the principal is able to observe and correctly interpret the identity, credentials, and reputation of the researcher; alternatively, she is able to observe and effectively analyze the informativeness of the research design.9 Ina later section, we consider an extension in which agents design arbitrary signals, and demonstrate that our results continue to hold. At the beginning of the second stage, the principal observes both signal realizations. On the basis of all available information (prior information, signal accuracies and signal realizations), the principal chooses which proposals, if any, to accept. Once the principal makes her decision, the true quality of any accepted proposal is then revealed, and all payoffs are realized. This is a game of imperfect, but symmetric information, similar to the persuasion game analyzed by Gentzkow and Kamenica (2011) and Brocas et al. (in press). The interaction takes place in one of two decision making environments. The principal may have unlimited capacity to accept proposals: she is able accept neither, either, or both proposals as she sees fit. In this case no externality exists between proposals; the decision to accept each proposal is made independently. Alternatively, the principal may have a limited capacity to accept proposals. If capacity is limited she can accept at most one proposal. This limitation may arise for a variety of reasons: the principal may be constrained by limited budgets, or it may be due to procedural or bureaucratic hurdles that require considerable effort to overcome. When capacity is limited, a decision to accept a proposal precludes the possibility of accepting the other proposal. Therefore, acceptance decisions cannot be made in isolation; the signals and accuracies for both proposals influence the principal’s decision. The capacity of the decision making environment is common knowledge. We solve for Perfect Bayesian Equilibria of this two stage game under the limited and unlimited capacity systems. In the first stage, agents simultaneously choose their respective signal accuracies, α1 and α2 . Once both agents choose their accuracies, these become public. In the second stage, the signals are realized, and the principal decides which proposals to accept (subject to the constraints of the decision making environment).

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Unlimited Capacity

We begin by analyzing a setting in which the principal is unconstrained in her ability to accept proposals. Therefore, the principal accepts any proposal for which the expected payoff from doing so is positive. This expected payoff depends on her posterior belief about the quality of the proposal after observing the realization of the signal. Let γˆ (si , αi ) denote the principal’s belief that proposal signals require too much effort or money to generate. Including an exogenous cost does not alter the qualitative nature of the results. 9 In some fields, research protocols must be registered in a database before subjects are enrolled. Stiff penalties exist for violating the protocols.

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i is good after observing signal realization Si , given accuracy αi : γˆ (G, αi ) =

γαi γαi + (1 − γ) (1 − αi )

γˆ (B, αi ) =

γ (1 − αi ) γ (1 − αi ) + (1 − γ) αi .

The principal accepts proposal i if γˆi (si , αi )(1 − q) − (1 − γˆi (si , αi ))q ≥ 0 ⇐⇒ γˆi (si , αi ) ≥ q. Thus the principal accepts if the updated probability of a good proposal is at least as great as the stakes q. Otherwise, she rejects the proposal. We assume that the principal accepts a proposal if she is indifferent between accepting and rejecting. If γ < q, then the principal is predisposed against accepting: if her decision were based solely on her prior belief, she would reject each proposal. In this case, observing a bad signal realization reinforces the principal’s beliefs that the proposal is bad, and she continues to favor rejection. Observing a good signal realization, however, improves her belief about the quality of the project. When the good realization is generated by a sufficiently accurate signal, it overturns her predisposition and causes her to accept the proposal. This is the case when γˆ (G, αi ) ≥ q ⇔ αi ≥ We define r≡

q (1 − γ) . (1 − q)γ + (1 − γ)q

q (1 − γ) . (1 − q)γ + (1 − γ)q

Therefore, for a good signal realization to persuade the principal to accept a proposal that she is predisposed against, it must be that αi ≥ r. If signal accuracy is sufficiently low, then a good signal outcome is not persuasive enough to overturn the principal’s predisposition to reject. If γ ≥ q, then the principal is predisposed in favor of accepting; her prior is sufficiently optimistic that she would accept each proposal given her prior alone. In this case, observing a good signal realization only strengthens the principal’s beliefs that the proposal is good, and she continues to favor accepting. Observing a bad signal realization, however, weakens her beliefs that the proposal is good. When the signal is sufficiently accurate, the bad realization overturns her predisposition and causes her to reject the proposal. This is the case when γˆ (B, αi ) < q ⇔ αi > 1 −

q (1 − γ) = 1 − r. (1 − q)γ + (1 − γ)q

If signal accuracy is sufficiently low, then a bad realization is not persuasive enough to overturn 9

her predisposition in favor of accepting. The above discussion is summarized in the following lemma. Lemma 4.1 When accepting capacity is unlimited, the principal’s equilibrium strategy is: • If predisposed in favor of accepting, γ ≥ q, then the principal rejects proposal i if and only if si = B and αi ∈ (1 − r, 1]. • If predisposed against accepting, γ < q, then the principal accepts proposal i if and only if si = G and αi ∈ [r, 1]. A persuasive signal has the potential to overturn the principal’s predisposition; no realization of a weak signal affects the principal’s decision. When γ ≥ q, a persuasive signal has accuracy α > 1 − r. When γ < q, a persuasive signal has accuracy α ≥ r. Otherwise, the signal is weak. Agents anticipate the principal’s behavior when they choose signal accuracy. Suppose that the principal is predisposed in favor of accepting, i.e., γ ≥ q. In this case, the principal implements the proposal even if the signal is weak. Choosing persuasive signal introduces the possibility that the principal observes a bad realization and does not implement the reform. In this case, agents strictly prefer weak signals. Alternatively, suppose that the principal is predisposed against accepting, i.e., γ < q. If an agent chooses a weak signal, then the proposal is rejected. If the agent chooses a persuasive signal, then the proposal is accepted if and only if si = G. Let θ(α) denote the probability of observing a good realization given accuracy α: θ(α) ≡ αγ + (1 − α)(1 − γ). When the signal is persuasive, the proposal is accepted with ex ante probability θ(α). An agent’s optimal signal accuracy maximizes θ(α) subject to α ∈ [r, 1]. If γ < 12 , then θ is strictly decreasing in α, and the agent prefers α = r to any higher value. He also prefers α = r to any weak signal, as supplying the marginally persuasive signal leads to a positive probability of the proposal being implemented. If q < 21 , then this is the only possible case. If, however, q > 12 , then we must also consider the possibility that γ is such that

1 2

≤ γ < q. In this case, the principal is predisposed

against accepting proposals that are most likely good (and are more likely than not to generate good signals). In this case, θ(α) is strictly increasing in α and the agent prefers α = 1 to any lower value. These observations lead to the following description of the equilibrium outcome. Lemma 4.2 In equilibrium under unlimited capacity, • If q ≤ γ, then for each i the agent chooses a weak signal, the principal accepts the proposal, and E [wi ] = γ(1 − q) − (1 − γ)q = γ − q 10

and

E [ui ] = 1.

• If γ < q and γ < 21 , then for each i the agent chooses αi = r, the principal accepts the proposal if and only if si = G, and E [wi ] = 0 • If

1 2

and

E [ui ] = θ(r) > γ.

≤ γ < q, then for each i, the agent chooses αi = 1, the principal accepts the proposal if

and only if si = G, and E [wi ] = γ(1 − q)

and

E [ui ] = γ.

Although the principal finds a fully-informative signal optimal, agents choose α = 1 in only one situation: when the principal is predisposed against accepting and proposals are most-likely good (i.e., when

1 2

≤ γ < q). In all other cases, the agent prefers to keep the principal less than fully-

informed, and provides either a weak or marginally-persuasive signal. If the principal is predisposed in favor of accepting, then it is a dominant strategy for the agent to choose a weak signal. Doing so assures that the principal implements the proposal. If the principal is predisposed against accepting and proposals are most-likely bad (γ
12 ), increasing a persuasive signal’s accuracy makes a good realization more likely, which increases the probability that the proposal will be accepted; accuracy is success-enhancing. However, when the signal is most-likely bad (γ < 21 ), θ (α) is decreasing. In this case increasing the accuracy of a persuasive signal decreases the probability of generating a good realization and hurts the probability of proposal implementation; accuracy is therefore success-diminishing. Thus, the prior belief determines which signal accuracy is optimal among the persuasive signals. The second critical factor is the principal’s predisposition toward the proposal, determined by the relationship between q and γ. The principal’s predisposition determines whether or not the agents prefer to produce persuasive or weak signals. If the principal is predisposed towards accepting, agents strictly prefer weak signals to persuasive ones; if the principal is predisposed against accepting, then agents strictly prefer persuasive signals to weak ones. The severity of the conflict of interest between the agents and the principal thus depends critically on the interaction between these two forces. (The problem with) Commitment The principal prefers α = 1 to all lower values, as it allows for a fully-informed decision. The politician’s sequentially rational strategy, however, implements any proposal for which γˆ ≥ q, even if α < 1. The agents recognize this and often prefer to produce a less than fully informative signal.

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If the principal could credibly commit to a (not sequentially rational) strategy at the onset of the game, she could guarantee her ideal outcome by committing to implement proposal i if and only if si = G and αi = 1. That is, she would commit to reject any proposal for which she is less-thanfully informed about its quality. The agents would react by always choosing α = 1, guaranteeing a fully-informed policy decision. In some situations such commitment may be reasonable. In most situations, it is not. In politics, for example, legislators typically cannot (due to legal or political reasons) contract with interest group agents to choose favorable policy conditional upon some action by the agent. Even if they were allowed to, deriving commitment power from writing legally binding contracts, may prove difficult. First, signal accuracy may not be verifiable in court; rendering any contract based on accuracy unenforceable. Second, even if α were verifiable, both the principal and agent would prefer to disregard the contract and allow the proposal to be implemented if the agent produced a favorable, persuasive signal realization with α < 1.10 Without the ability to commit, any promise to reject a proposal that is not associated with a fully revealing signal suffers from a credibility problem. If the agent supplies a persuasive signal and the realization is good, the principal has every reason to implement the proposal, despite the promise to reject it. For the remainder of the analysis, we assume that the principal cannot commit to only accept proposals with perfectly informative signals.

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Limited Capacity

Our analysis has two main results. First we show agents often provide higher-quality evidence (i.e., more accurate signals) when the principal is constrained in the maximum number of proposals she can accept. Limited capacity forces agents to compete for priority in the decision making process. Because proposals are identical ex ante, the only way that an agent can attract attention to his proposal is by supplying a more-accurate signal. Second we show that because of this competitive pressure, the principal often prefers limited capacity to unlimited capacity. That is, the benefits of more-accurate information often outweigh the potential loss caused by being able to implement fewer (potentially beneficial) proposals. In the following sections we characterize the Perfect Bayesian Equilibria of the game when the principal can implement at most one proposal, and compare the outcomes to the corresponding outcomes in the unlimited legislative system. As the principal’s decision is no longer independent for the two proposals, the analysis is a more complicated. We therefore break the analysis into pieces, first solving for equilibrium for two cases when limited capacity increases signal accuracy, then for two cases where limited capacity has no impact on principal information. 10 In politics, for example, legislators typically cannot (due to legal or political reasons) contract with interest group agents to choose favorable policy conditional upon some action by the agent. Even if they were allowed to, deriving commitment power from writing legally binding contracts, it may prove difficult. First, signal accuracy may not be verifiable in court; rendering any contract based on signa accuracy unenforceable. Second, even if α were verifiable, both the principal and agent would prefer to disregard the contract and allow the proposal to be implemented if the agent produced a favorable, persuasive signal with α < 1.

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5.1

When limited capacity improves evidence quality

We begin by considering two cases in which limited capacity strictly improves the accuracy of signals provided by agents. In the first case, the principal is predisposed against the proposals, and the proposals are most likely bad. In the second case, the principal is predisposed in favor of proposals that are most good. For both cases, we show that limited capacity results in agents providing more-informative signals in equilibrium than under unlimited capacity. In the first case, the increase in signal accuracy unambiguously improves principal payoffs; in the second case it improves principal payoffs as long as her priors in favor of acceptance are not too strong. 5.1.1

Predisposed against and proposals most likely bad

The first case that we consider is the one in which the principal is predisposed against accepting (i.e., γ < q), and according to the prior the proposals are most likely bad (i.e., γ < 12 ). We will show that limited capacity strictly improves signal accuracy and principal payoffs. We first characterize principal behavior under limited capacity. Since the principal is predisposed against accepting, she certainly rejects any proposal for which the signal is weak or the realization is bad. If she could, she would accept any proposal that generates a good realization from a persuasive signal. However, as she can accept at most one proposal, if both signals generate good realizations, she accepts the one she believes is more likely to be good. Because a good realization is more informative when the signal is more accurate, when both proposals generate good realizations, the principal implements the proposal with higher signal accuracy. This discussion is summarized in the following lemma. Lemma 5.1 Equilibrium principal behavior under limited capacity when γ < q and γ
αj and accept each proposal with equal probability if αi = αj . This lemma illustrates the impact of limited capacity. By supplying a more accurate signal, an agent gives his issue priority: if both signals are persuasive, and both realizations are good, his proposal will be the one implemented. Because the proposals are most-likely bad, however, increasing accuracy reduces the probability of generating a good signal. In equilibrium, agents trade off the benefit of taking priority against the reduction in the probability of generating a good signal.

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On the basis of Lemma 5.1, we construct each agent’s first period expected payoff as a function of both signal accuracies. Suppose agents choose αi , αj if

αi ∈ [ 21 , r)

then

ui (αi , αj ) = 0

if αi ∈ [r, 1] and αi > αj

then

E [ui (αi , αj )] = θ (αi )

if αi ∈ [r, 1] and αi < αj

then E [ui (αi , αj )] = θ (αi ) (1  − θ (αj )) 

if αi ∈ [r, 1] and αi = αj

then E [ui (αi , αj )] = θ (αi ) 1 −

θ(αj ) 2

Each of these payoffs is straightforward to understand. If an agent produces a weak signal, his proposal is rejected. If an agent produces a persuasive signal with higher accuracy than the other agent, his issue is decided on its merits. It is implemented if and only if it generates a good signal. If agent i produces a persuasive signal that is less accurate than the other agent, his proposal is accepted only if it generates a good signal realization and the other proposal generates a bad realization. Finally, if both agents produce persuasive signals of identical accuracy, then proposal i is accepted if si = G and sj = B. If both proposals generate good signal realizations, then each proposal is accepted with equal probability. A weak signal is strictly dominated by producing a marginally persuasive signal and would never be part of an equilibrium strategy. We first characterize the unique pure strategy equilibrium of this stage game, αi = αj = 1. Suppose that agent j chooses a fully informative signal, αj = 1. In this case, the probability that proposal j generates a good signal realization is θ (1) = γ. Therefore, by choosing αi = 1, agent i expects payoff

 γ γ 1− 2

Any choice of αi ∈ [r, 1) gives expected payoff θ (αi ) (1 − γ). Since γ < 21 , function θ (α) is strictly decreasing and an agent’s optimal choice of α ∈ [r, 1) is α = r, which gives expected payoff θ (r) (1 − γ) =

γ (1 − γ)2 (1 − q)γ + (1 − γ)q

In order for αi = αj = 1 to be a Nash equilibrium, it must be that  γ (1 − γ)2 3γ 2 − 6γ + 2 γ ≥ ⇔q≥ 2 γ 1− 2 (1 − q)γ + (1 − γ)q 2γ − 5γ + 2 No other symmetric pure strategy equilibrium exists, as a deviation to α bi = αj + ε gives higher expected payoff than αi = αj . No asymmetric pure strategy equilibrium exists, as the agent supplying the signal with higher accuracy could always benefit by slightly reducing his signal accuracy. Lemma 5.2 Under the limited capacity legislative system with γ < q and γ < 21 , • If q ≥

3γ 2 −6γ+2 2γ 2 −5γ+2

then the unique Nash equilibrium of the first stage game is for each agent to

choose a fully informative signal, α1 = α2 = 1. 14

• No other pure strategy Nash equilibrium of the first stage game exists for any value of q. If both agents choose α = 1, the most profitable deviation for an agent is to produce a marginally persuasive signal, ceding full priority to the other agent but maximizing the probability of generating a good signal realization. Such a deviation is profitable if the probability of generating a good realization when choosing the marginally persuasive signal is high; this in turn is the case when the marginally persuasive signal; accuracy r is close to

1 2.

When the stakes q increase, the cost

of mistakenly implementing a bad proposal is high; thus the principal requires a more informative signal in order to overturn her predisposition towards rejection. In other words, increases in q increase the value of r. Thus a sufficiently high q ensures that both agents produce fully-informative signals in equilibrium. Next, we provide a partial characterization of all mixed strategy equilibria of the first stage game. This partial characterization is based on several straightforward observations about the nature of the equilibrium. For our main result, the partial characterization is sufficient; we provide a complete description of the mixed strategy Nash equilibrium in the Appendix. Consider a mixed strategy Nash equilibrium in which each agent’s signal accuracy is the realization of random variable A. We assume that this mixed strategy Nash equilibrium involves randomization, i.e. it is not the pure strategy equilibrium α1 = α2 = 1 already characterized. Lemma 5.3 Under the limited capacity legislative system with γ < q and γ < 21 , in any symmetric mixed strategy Nash equilibrium of the stage game, the following four properties must hold: 1. Weak signals are outside of the support of A. 2. The only possible mass point is α = 1. 3. The smallest signal accuracy inside the support of A is r. 4. If disjoint intervals [x1 , x2 ] and [x3 , x4 ] are in the support of A, then the entire interval [x1 , x4 ] is inside the support of A. This Lemma is not difficult to understand. As weak signals are dominated they are outside of the support of the mixed strategy equilibrium. No interior mass point can exist in a symmetric mixed strategy equilibrium; if both strategies have the same mass point, then agents choose the same accuracy with positive probability. If both proposals generate good realizations and the accuracies are the same (which happens with nonzero probability), a tie occurs, and each proposal is accepted with probability

1 2.

By just slightly increasing the accuracy associated with the mass point, an

agent can assure that all ties break in his favor, which increases his payoff by a discrete amount, and is therefore profitable. The only possible mass point is therefore at α = 1. If the smallest signal accuracy in the support of the mixed strategy α were strictly greater than r, then whether agent i chose αi = r or chose αi =α, his accuracy is always less than the accuracy of the other agent, which always gives the other agent decision making priority. Thus whether αi = r or αi =α, proposal i is 15

implemented if sj = B and si = G. However, because increased accuracy reduces the probability of generating a good signal (and does not affect the probability that sj = B), the signal with αi = r has a higher probability that si = G. Therefore if α> r then choosing αi = r dominates the mixed strategy. The intuition for the last part of the Lemma is similar: under the assumptions, pure strategies x2 and x3 give an agent the same priority in the decision making process, but accuracy x2 is more likely to generate a good signal realization. Only two types of mixed strategies are consistent with Lemma 5.3. The first type of mixed strategy calls for agents to choose their signal strengths from a continuous CDF with support on [r, α ¯ ], where α ¯ ≤ 1. The second type of mixed strategy equilibrium calls for agents choosing signal strengths from a CDF with continuous support on [r, α ¯ ] and mass point on 1.11 Under both feasible mixed strategies, both agents choose signal qualities greater than r with probability one, since the mixed strategy CDFs do not allow for a mass point on r and r is the smallest accuracy in the support. Compared to the case of unlimited capacity, here limited capacity almost always results in a more informed principal. From here, we can conclude that the principal always prefers an environment of limited capacity to unlimited capacity. Recall, that with unlimited capacity, agents supply marginally persuasive signals, and the principal is, at best, indifferent between implementing and rejecting proposals. Her equilibrium expected payoff under the unlimited capacity system is zero. If capacity is limited, agents almost always supply more accurate (persuasive) signals than they do when capacity is unlimited. This results in a better informed acceptance decision, which benefits the principal. Furthermore, there is no cost to the principal of limited capacity: because she is at best indifferent between accepting and rejecting in the unlimited system, the option to accept a proposal in this system is worthless. Thus the limited capacity system is always preferred. Proposition 5.4 When γ < q and γ < 12 , • With probability 1, agents provide more accurate signals under limited capacity than under unlimited capacity. • The principal prefers limited capacity to unlimited capacity. Unlike the case analyzed in this section, if the principal is predisposed in favor of the proposal, the option to accept a proposal has ex ante value. As limited capacity precludes the option to implement both proposals, it imposes a cost on the principal. Limited capacity would only be preferred if competition for priority stimulates a sufficient increase in the equilibrium signal accuracy. We analyze this case in the next section. 5.1.2

Predisposed in favor and proposals most likely good

We turn to the case in which the principal is predisposed in favor accepting (i.e., q < γ), and proposals are most likely good (i.e., γ > 12 ). Here supplying a persuasive signal is not a prerequisite 11

Readers interested in more details about these mixed strategies should consult the appendix.

16

for a proposal to be implemented. This works against the supply of accurate signals. Indeed, in this case under unlimited capacity, agents always produce weak signals rather than persuasive ones, and the principal always accepts both proposals. However, signal accuracy is success-enhancing; therefore, some potential exists for agents to send accurate signals in equilibrium. We begin by characterizing the principal’s equilibrium behavior in the second stage. The principal prefers to accept proposals that she believes have sufficiently high probability of being good (i.e., those for which γˆ ≥ q). Because the principal is predisposed toward accepting, she is definitely willing to implement any proposal with a good signal realization. Thus, if only one proposal generates a good signal realization, that proposal is accepted, regardless of its signal accuracy. If both proposals generate good realizations, the principal prefers to accept the proposal with the more-accurate signal. Furthermore, even if both proposals generate bad realizations, the principal may still be sufficiently optimistic to implement one of the proposals as long as the bad realization was not strong enough to overturn her predisposition. Of two proposals with bad signal realizations, she is more optimistic about the proposal drawn from the less accurate signal. She will be willing to implement this proposal, provided the signal the generated the bad realization is weak or only marginally persuasive. This discussion is summarized in the following lemma: Lemma 5.5 Equilibrium principal behavior under limited capacity when q < γ and γ >

1 2

is as

follows: • If si = G and sj = B accept proposal i. • If si = G and sj = G accept proposal i accept proposal i if αi > αj . If αi = αj accept each proposal with probability 12 . • If si = B and sj = B accept proposal i if and only if αi < αj and αi ≤ 1−r. If αi = αj ≤ 1−r accept each proposal with probability 12 . Following this lemma, it is straightforward to construct each agent’s expected payoff as a function of both signal accuracies. Suppose agents choose αi , αj , if

αi > αj

if αi ∈ (1 − r, 1] and αi < αj [ 21 , 1

then

E [ui (αi , αj )] = θ (αi )

then E [ui (αi , αj )] = θ (αi ) (1 − θ (αj ))

if

αi ∈

if

αi ∈ (1 − r, 1] and αi = αj

then E [ui (αi , αj )] = θ (αi ) 1 −

if

αi ∈ [ 12 , 1 − r] and αi = αj

then

− r] and αi < αj

then

E [ui (αi , αj )] = 1 −  θ (αj )

θ(αj ) 2



E [ui (αi , αj )] = 12 .

If agent i produces a more accurate signal than agent j, proposal i is implemented if and only if proposal i generates a good signal. If agent i produces a less informative signal than agent j and this signal is persuasive, then proposal i is implemented if and only if issue j generates a bad signal and i generates a good signal. On the other hand if i’s signal is less informative than j and is also weak, then proposal i is implemented if and only if j generates a bad signal. If both agents 17

produce a signal with the same persuasive accuracy, then proposal i is implemented whenever si = G and sj = B; furthermore, if both proposals generate good realizations, each is accepted with equal probability. Finally, if both agents produce the same weak signal, then i is implemented if it generates a good realization and j does not; otherwise each proposal is implemented with probability 21 . In this case, the probability of i being implemented is just θ (α) (1 − θ (α)) +

 1 1 θ (α)2 + (1 − θ (α))2 = 2 2

As in the previous case, an agent who supplies a more informative signal receives priority in the decision making process. Furthermore, because accuracy is success enhancing, it is easy to see that supplying a fully informative signal αi = 1 is a best reply to any less than fully informative signal αj < 1.12 Thus, in any pure strategy Nash equilibrium, at least one agent supplies a fully informative signal. If the best reply to a fully informative signal is also a fully informative signal, then α1 = α2 = 1 is the unique Nash equilibrium of the first stage game. This is not always the case, however. If agent i replies to a fully informative signal by also issuing a fully informative signal, his payoff is

 γ γ 1− 2

When both agents produce fully-informative signals, both proposals have equal priority, and a good signal realization is required for a proposal to be accepted. Rather than supply a fully informative signal as a response to a fully informative signal, agent i may prefer to produce a weak signal. In doing so, agent i cedes priority to agent j; however, if proposal j generates a bad signal realization, proposal i will be implemented, regardless of i’s signal realization. If a good signal realization is not sufficiently likely when α = 1 (i.e. γ is not large enough), an agent may prefer to sacrifice priority in order to remove the requirement that his proposal generate a good realization in order to be implemented. A fully revealing signal from each agent α1 = α2 = 1 is the unique Nash equilibrium of the first stage game if and only if  √ γ γ 1− ≥ 1 − γ ⇐⇒ γ ≥ 2 − 2. 2 √ If, on the other hand, γ < 2 − 2, then the best response to a fully-informative signal, is a weak   signal α ∈ 12 , 1 − r . Because agent j prefers to produce a fully-informative signal whenever agent i produces a weak signal, a multitude of asymmetric equilibria exist. In each of these equilibria one of the agents supplies a fully revealing signal and the other agent supplies a weak signal. Lemma 5.6 Under limited capacity with q < γ and γ > 12

1 2

Suppose αj < 1. Because θ (α) is increasing, αi = 1 is clearly better than any other αi > αj . Among accuracies less than αj
, weak or marginally persuasive accuracies are preferred to strictly persuasive values. Furthermore, because θ 21 = 12 and θ (α) is increasing, θ (α) > 21 . Hence 1 − θ (α), the payoff to any weak signal, is less than 12 , which is itself less than the payoff to a fully revealing signal.

18

• If γ ≥ 2 −

√

2 then the unique Nash equilibrium of the first stage game is for each agent to

produce a fully informative signal, α1 = α2 = 1. • If γ < 2 −

√

2 then many asymmetric Nash equilibria of the first stage game exist. In each

Nash equilibrium, one agent produces a fully revealing signal αi = 1, and the other agent produces a weak signal, αj ∈ [ 21 , 1 − r]. • If γ < 2 −

√

2 a symmetric mixed strategy Nash equilibrium exists. Each agent chooses

αi = 1 − r with probability p =

γ 2 −4γ+2 , γ 2 −2γ+1

and αi = 1 with probability 1 − p,

The first two parts of the lemma follow directly from the preceding discussion. The third part is established in the appendix. To determine whether the principal prefers limited or unlimited capacity, recall that under unlimited capacity agents supply weak signals, and the principal approves both proposals, E [w] = 2 (γ − q). Unlike the case in which the principal was predisposed against the proposals, if the principal is predisposed in favor of the proposals, the option to accept a proposal has (ex ante) value in equilibrium. Limited capacity therefore imposes an expected cost on the principal, as she does not have the option to implement one of the proposals that she would choose to implement if capacity were unlimited. For the limited capacity system to be preferred, it must result in a sufficiently large increase in signal accuracy; accepting at most one proposal with better information about its quality must dominate accepting both proposals with no additional information about √ their quality. In the case where γ ≥ 2 − 2, limited capacity induces an equilibrium in which both signals are fully revealing. If at least one of the proposals is good, then a good proposal is √ implemented. Thus when γ ≥ 2 − 2 the principal prefers the limited capacity system (ex ante) if and only if
(1 − q) 2γ − γ 2 ≥ 2 (γ − q) ⇐⇒ q ≥ Next consider the case when γ < 2 −

√

γ2 γ 2 − 2γ + 2

2. If limited capacity induces a fully-informative signal

by only one of the agents (as always happens in the asymmetric pure strategy equilibrium and sometimes happens in the symmetric mixed strategy equilibrium) the principal implements the proposal associated with the fully informative signal if and only if it is a good proposal. If the realization of the fully informative signal is bad, she implements the proposal about which she is uninformed. If she anticipates that agents play the asymmetric pure strategy equilibrium in the √ case γ < 2 − 2, then for such γ, the principal prefers limited capacity if and only if γ (1 − q) + (1 − γ) (γ − q) ≥ 2 (γ − q) ⇔ q ≥ γ 2 If the principal anticipates that agents play the symmetric mixed strategy equilibrium in the case

19

γ 21 , • In any pure strategy Nash equilibrium of the first stage game, no agent provides a less accurate signal, and at least one agent provides a more-accurate signal, when capacity is limited than when capacity is unlimited. • In the mixed strategy Nash equilibrium of the first stage game, agents never choose weak signals, and choose fully revealing signals with positive probability. • For each γ, a value q˜ < γ exists, such that the principal prefers limited capacity to unlimited capacity if and only if q ∈ [˜ q , γ). The benefit of limited capacity is an increase in evidence quality (i.e. signal accuracy). However, because she is predisposed in favor of the proposals, limited capacity has a cost: the principal is constrained to implement fewer policies than she would like to implement if capacity were unlimited. If proposals have relatively little downside risk (small q) and relatively high probability of being good (high γ), then the principal has a strong prior predisposition in favor of accepting proposals. In this case, the option to implement a reform is very valuable. In this case, the informational benefits associated with limited capacity will not be large enough to overcome the costs of not being able to support both proposals. When q is sufficiently close to γ, however, the informational benefits dominate the downside associated with the capacity constraint. As q approaches γ, the principal’s predisposition in favor of proposals becomes weaker, and the ex ante expected benefit from being able to implement a given proposal decreases. Decreasing the ex ante benefit from implementation in turn decreases the relative downside of limited capacity, which eventually allows the informational benefits of more accurate signals to dominate the costs from being constrained on the number of proposals that can be accepted. The threshold value q˜ is derived in the analysis √ √ γ2 leading up to Proposition 5.7. If γ ≥ 2 − 2 then q˜ = γ 2 −2γ+2 . If γ < 2 − 2 and the principal anticipates that agents play one of the asymmetric pure strategy Nash equilibria of the first stage, √ then q˜ = γ 2 . If γ < 2 − 2 and the principal anticipates that agents play the mixed strategy Nash equilibrium in the first stage, then q˜ =

5γ 4 −17γ 3 +15γ 2 −4γ . 5γ 3 −11γ 2 +8γ−2

Whenever q ∈ [˜ q , γ) it is better

to implement a single proposal with access to better information about proposal quality, than to decide to implement both proposals based solely on prior information.

20

5.2

When limited capacity does not improve evidence quality

We turn our attention to the two cases in which limited capacity does not improve the accuracy of signals provided by agents. Because signal accuracy is not improved, in these cases limited capacity does not improve principal payoffs. The first case occurs when the principal is predisposed in favor of proposals that are most likely bad (q < γ and γ < 12 ). In this case both critical factors are stacked against the supply of accurate signals, and only completely uninformative signals are supplied in equilibrium. In the second case, the principal is predisposed against proposals that are most likely good (q > γ, γ < 21 ). In this case, each agent supplies a fully informative signal, even when capacity is unlimited. Hence, there is no benefit from limited capacity. 5.2.1

Predisposed in favor and proposals most likely bad

Consider now the case in which the principal is predisposed in favor of proposals (i.e., q < γ) which are most likely bad (i.e., γ
12 , q < γ, agents play the symmetric mixed strategy rather than the asymmetric pure strategy equilibrium. While we find the symmetric equilibrium more appealing, this choice makes little difference for the qualitative nature of the results.

22

Figure 1: Range of parameters for which the principal prefers limited capacity

7

General Signals

The previous analysis demonstrates that limited capacity improves the informativeness of evidence supplied by agents, and that this benefit can outweigh the cost associated with limited capacity. In demonstrating this result, we confined agents to choose their signals from a particular parametric class, in which a binary realization is equal to the true quality of the proposal with probability α. In this section we demonstrate that our results continue to hold if we allow agents to design signals in any matter they see fit. Formally, a signal consists of a pair of random variables (Σg , Σb ). A signal realization (also called evidence) is a public realization of one of these random variables. If the true type of the proposal is g, then a realization of Σg is observed; otherwise a realization of Σb is observed. Because the true quality of each proposal is unknown, any signal realization s ∈ Support (Σi ) induces a posterior belief that the proposal is good, consistent with Bayes’ Rule. This posterior belief depends on the prior, the signal realization, and on the signal structure itself. Before the signal is realized, the principal’s posterior belief is a random variable Γ, with distribution function G. According to the law of total expectation the expected value of posterior belief must be equal to the prior belief, i.e. E [Γ] = γ. Moreover, according to Kamenica and Gentzkow (in press), this is the only restriction on the random variable Γ.14 For any random variable Γ with support in the unit interval and expected value γ, there exists a signal (Σg , Σb ) for which G is the distribution of the principal’s posterior belief. This insight considerably simplifies the analysis of this game. Instead of designing a signal 14

Although in the body of their paper they argue this only for signals with a finite number of realizations, they argue in the online appendix that their main result extends to the case in which realizations are drawn from a compact metric space.

23

directly, each agent i simultaneously chooses a random variable Γi (or a distribution function Gi ), with support in the unit interval and expectation γ. This choice represents the ex ante distribution of the principal’s posterior belief about project quality, and is equivalent to a choice of one of many signals that generate the same distribution of posterior beliefs. Once both agents have chosen their Γi , these random variables are realized (i.e. the principal observes the evidence and rationally updates her beliefs). On the basis of the realized posterior beliefs, the principal decides which proposals to accept, respecting the capacity constraints of the decision making environment.15 The principal’s equilibrium behavior is simple. Given a realized posterior belief γ bi that proposal i is good, her expected payoff from accepting proposal i is γ bi − q. Therefore, if her capacity is unlimited, then the principal will accept any proposal for which the realized value of the posterior belief is no less than q. If capacity is limited, she will accept the proposal that generates the highest posterior belief, provided this belief is no less than q. If γ < q and Support (Γi ) ⊂ [0, q), then no realization will generate a posterior belief that overturns the principal’s predisposition against the proposal. Similarly, if γ ≥ q and Support (Γi ) ⊂ [q, 1] then no signal realization can overturn the principal’s predisposition in favor of the proposal. In both of these cases, the signals are weak, otherwise, they are persuasive. If a signal is fully-revealing the principal can determine the quality of the proposal from the signal: the principal’s posterior belief is therefore 1 with probability γ, and 0 with probability 1 − γ.

7.1

Unlimited Capacity

To analyze the game with unlimited capacity, observe first that it is sequentially rational for the principal to accept all projects for which the realized posterior belief is at least as large as q. Thus, in equilibrium each agent chooses random variable Γi to maximize Pr (Γi ≥ q) subject to E [Γi ] = γ and Pr (0 ≤ Γi ≤ 1) = 1. If the principal is predisposed in favor of implementing proposals (γ ≥ q), then each agent chooses a signal for which the entire support of Γi is above q. The simplest way to do that is to concentrate all mass on γ so that Pr (Γi = γ) = 1, but any weak signal achieves the same result: each proposal is always accepted. Consistent with the results of the previous section, when the principal is predisposed in favor of accepting and capacity is unlimited agents supply weak signals. If the principal is predisposed against proposals, (γ < q) then agents must choose a persuasive signal for a proposal to have a chance to be accepted. Notice however, that signals with nonzero probability mass on posterior beliefs strictly above q are dominated.16 Similarly, any posterior distributions which put non-zero probability mass on realizations between 0 and q are dominated.17 15

Ganuza and Penalva (2010) use a similar representation of signals in their analysis of information disclosure in auctions. 16 By concentrating all mass above q in a mass point on q, the agent generates a new random variable with the same probability of being greater than or equal to q but with a smaller mean; the agent can then move additional probability mass from realizations below q to the mass point on q, in order to satisfy the constraint. Doing so increases the agent’s payoff. 17 By moving the probability mass between 0 and q into a mass point on 0, the agent can increase the probability mass that is weakly above q without violating the constraint.

24

Thus, the optimal signal concentrates mass on only two posterior beliefs, 0 and q. In order to satisfy the constraint on the expectation, the probability of generating the posterior belief q must be equal to φ =

γ q.

Thus, the optimal signal requires only two realizations. One signal realization reveals

that the proposal is bad for certain, while the good realization leaves the principal just indifferent between accepting and rejecting the proposal. This is qualitatively similar to the results of Section 4.18 Lemma 7.1 In equilibrium under unlimited capacity and general signals • If q ≤ γ, then for each i the agent chooses a weak signal, the principal accepts the proposal, and E [wi ] = γ − q

and

E [ui ] = 1.

• If γ < q then for each i the agent chooses a signal such that the posterior belief Γi is equal to q with probability

γ q

and zero with probability 1 − γq , the principal accepts if and only if the

posterior is equal to q, and E [wi ] = 0

and

E [ui ] =

γ . q

This result demonstrates that the same conflict of interest exists with general signal structures: although agents have the capacity to fully reveal the quality of their proposals, in equilibrium they never choose to do so. Moreover, the evidence that they choose to provide is effectively worthless to the principal. She would obtain the same expected payoff if she did not observe the signal realization and simply acted according to her prior belief.

7.2

Limited Capacity

We establish that our main results continue to hold when agents are allowed to design arbitrary signals. Limited capacity forces agents to compete for priority, which improves the informativeness of the signals they choose to supply. We also show that for a wide range of parameters the principal prefers the limited capacity system to the unlimited capacity system. In fact, for every value of γ, there exists a threshold value such that, if q exceeds this threshold, the principal prefers the limited capacity system. With limited capacity, the game between agents can be described as follows. Each agent simultaneously chooses a random variable Γi with support contained in the unit interval and mean γ. Each random variable is then realized. The agent with the higher realization receives payoff one, provided this realization exceeds q. If the realizations are equal and exceed q, then agents each receive payoff 12 . Otherwise, an agent’s payoff is zero. A games with similar features is analyzed 18

The optimal signal in the parametric class of 4, however, is not optimal, because the principal’s belief following a bad realization is not zero, as it is under the optimal signal.

25

by Wagman and Conitzer (2011).19 We first present a characterization of Nash Equilibria of the game. Details of the arguments are presented in the appendix. Lemma 7.2 In a symmetric equilibrium under unlimited capacity and general signals • If • If

2−2γ 2−γ

√

≤ q then both agents send fully informative signals.

1 − 2γ ≤ q ≤

2−2γ 2−γ

then each agent selects random variable Γ, a mixture between 0, 1,

and a uniform random variable on [q, q 1 ] :

Γ=

   

0

with probability

U [q, q 1 ]

with probabiliy

  

with probability   p where q 1 = 2 − γ + q 2 + γ 2 . • If q ≤

√

1

q 2−q 1 q 1 −q 2−q 1 2−2q 1 2−q 1

1 − 2γ then each agent selects random variable Γ, a mixture between 0, and a uniform

random variable on [q, q 2 ] : ( Γ=

0

2γ q+q 2 2γ q+q 2

with probability 1 −

U [q, q 2 ]

with probabiliy p where q 2 = γ + q 2 + γ 2 .

The basic intuition for this result is similar to the intuition of the previous section: competition forces agents to compete for priority in the decision making process. Because of this competitive pressure agents send revealing or partially revealing signals, which benefits the principal. In fact, the distribution of posterior beliefs that is part of the equilibrium under limited capacity is a mean preserving spread of the distribution under unlimited capacity.20 As argued by Ganuza and Penalva (2010), the signals generated in equilibrium under limited capacity can not be worse for a decision maker, regardless of his preferences. Therefore, there are informational benefits to limiting capacity. Furthermore, whenever γ < q the principal always strictly prefers a limited capacity system; this is because under unlimited capacity, her payoff is the same as if she acted according to her prior, rejecting all projects. Under a limited capacity system, the competition for priority causes agents to send more informative signals, and the principal does accepts a proposal with non-zero probability, giving her a positive expected payoff. When γ ≥ q , however, the principal has a positive expected payoff under the unlimited capacity system, 2 (γ − q). Therefore the loss of capacity imposes a 19

The main difference with Wagman and Conitzer (2011) is the existence of an upper bound on the support of random variables in our analysis. Otherwise the game is identical to the game analyzed in the appendix of Wagman and Conitzer. This upper bound has significant implications for the results. 20 In making this claim, we assume that in the case q ≥ γ, agents choose an uninformative signal, concentrating all mass on γ.

26

Figure 2: Range of parameters for which the principal prefers limited capacity cost on the decision maker. However, as q approaches γ, this cost vanishes, but the informational benefits are always bounded away from zero. Thus we are led to the following Proposition 7.3 For every γ there exists some threshold qe (γ) < γ such that for q ≥ qe (γ), the principal prefers the limited capacity system. Although we do not explicitly calculate this function, we construct this function numerically in Figure 2. The shaded region depicts the parameter values for which the principal prefers limited capacity. Thus our main results, that limited capacity improves the informativeness of equilibrium signals, and that the principal often prefers limited capacity continue to hold when agents can design arbitrary signal structures.

8 8.1

Applications Legislative Decision Making

Limited capacity is a reality of the legislative process. An appropriation committee may not be willing to allot individual legislators funds to support all potential earmarks they believe beneficial. For example, Frisch and Kelly (2010, 2011) present evidence that during the 2006 budget cycle the Chairman of the US House Appropriations Subcommittee for Labor-Health and Human Services allowed each rank-and-file member of the U.S. House to request up to $400,000 in earmark funding from his subcommittee. The allotted amount increased systematically for subcommittee members, principals in at-risk districts and those in leadership positions. If a legislator requested a larger 27

amount of funding from the subcommittee, the funding was rejected or cut down to the allotted amount. This process means that legislators must carefully decide which earmark projects for which to request funding. Legislators often respond to their own limited capacity to request earmarks by setting up an application process through which potential beneficiaries may apply, and lobby for funding. Similarly, time and staff constraints may prevent a politician from introducing legislation on each policy proposal she or her constituents support. In an attempt to convince the legislator that its policy proposal is worth the legislator’s time and effort, an interest group can collect evidence confirming the merits of its proposal. Our story is consistent with ?’s story of “lobbying as legislative subsidy,” where special interest and lobby groups promise assistance (e.g., provide help conducting research or writing legislation) to time constrained politicians in an effort to convince the politicians to take up their cause. In our model, some of the assistance—conduction research and helping the politician better understand the implications of a policy—may come before the legislator decides which policies to pursue. We have shown that limited legislative capacity can entice special interest groups to produce more informative evidence about the merits of their projects or policies. These constraints have the potential to improve politician (and constituent) wellbeing, even though they may sometimes prevent good policies from being implemented.

8.2

Grant Writing

The second motivating example from the introduction involved grant writing. A funding organization must choose which research or community development proposals to accept. If the funding organization could back all projects that it believes worthy of funding, then applicants with ex ante promising projects have no incentive to produce additional evidence about the merits of its project, and applicants with projects the funding organization is predisposed not to accept will collect just enough evidence about the quality of their project to change the funding decision in their favor. Funding organizations, however, rarely have the ability to back all projects. The organization often must decide which of the promising projects is most-promising. The Robert Wood Johnson Foundation makes this clear on their website: “Due to the volume of proposals we receive, many excellent projects that meet our criteria still do not receive funding.” One way that applicants make their proposals stand out is by producing evidence about the likely success or contributions of their project. For researchers, this means providing a moredetailed description of their qualifications, research methods and policy implications, or producing preliminary results. Before applying for funding for a large scale field experiment, for example, researchers often run smaller trials. The number of treatments and the number of subjects in their experiments affect the informativeness of the trials about the eventual success of the project. Even

28

in theoretical work, the researchers decide whether to develop a formal model and preliminary propositions prior to submitting an application.

8.3

College Admissions

The process of admission to elite undergraduate colleges has become increasingly competitive. With the increased competition has come a greater emphasis on extracurricular activities. In our model, a qualified applicant meets the minimum academic requirements for admission, i.e., generates a good signal realization of his or her academic potential. However, simply generating a good signal realization is not enough for admission when there are too many applicants. Consistent with the predictions of our model, limited capacity to admit students leads applicants to engage in extracurricular activities in order to improve the informativeness of their excellent academic transcripts. Not only does an applicant wish to demonstrate excellent academic achievement, but to the extent possible takes actions to make these achievements seem more difficult, and thus more informative about his quality. Extracurricular achievements send the message that not only is the applicant so capable that he or she can perform well academically, but he or she can do so even under time pressures and distractions. Thus, extracurricular achievements improve the informativeness of the academic achievements of a candidate.21

8.4

Firm Expansion

A firm executive may be looking to expand operations, and not know which divisions offer the most-promise for expansion. The capital available for expansion may be enough to fund expansion or only one division, even if the executive believes that multiple divisions are worth expanding. Prior to choosing which division to expand, the division managers may propose strategic plans and compile verifiable evidence about the profitability of the expansion of their divisions. Our results suggest that strategically limiting the amount of funds available for investment may lead to better investment decisions regarding these funds. That is, limited resources causes competition between the division managers, incentivizing them to produce more informative information about the future plans and profitability of their divisions.

8.5

Product Information and Sampling

Firms produce different, independent products. Each firm allows a consumer to have a certain level of interaction with the good prior to purchase (e.g., hands on interaction with iPads at the Apple Store, test drives at car dealerships, free samples at a chocolate shop). The more interaction the firm allows prior to purchase, the more informative the consumer’s impression is about the quality of the good. 21 One of the authors is reminded of his high school valedictorian, who got a perfect score on the SAT, twice. He took the exam a second time to demonstrate that the perfect score he earned on the first attempt was no accident, increasing the informativeness of his signal.

29

If consumers are without budget constraints, the producers would need to only provide consumers with enough interaction with the products to convince them that it is most-likely in their best interest to make a purchase. When consumers are budget constrained, however, convincing them that the project is most-likely worthwhile is not enough. Producers must convince consumers that their product is likely a better purchase than the other products. Our model predicts that limited consumer budgets lead to increased pre purchase interactions with products. These results do not require that the products be substitutes. In fact, our model best explains the competition between producers of products that are independent, non-mutually exclusive purchase decisions. For example, a consumer may be able to afford either a new bicycle or an iPad. The more hands-on time she has with the iPad and the longer of test ride on the bicycle, the more informed she will be about her potential benefits of the two items when she chooses which to buy.

8.6

FDA Approval

The US Food and Drug Administration (FDA) regulates pharmaceuticals. Included in this is the control of whether drugs are approved for over-the-counter or prescription use. To gain FDA approval, drug companies conduct clinical trials to provide sufficient evidence that their produce is safe and effective for human use. There is currently no official limit to the number of new drugs that the FDA can approve every year. Our analysis shows that limiting the number of annual new drug approvals could theoretically increase consumer wellbeing. This would be because competition for a limited number of approvals could incentivize drug companies to conduct more-extensive reviews of their product before applying for FDA approval. Of course, we are ignoring the financial costs and potential health effects of increased time to market associated with additional trials, and the political and public relations problems that may arise from not approving drugs that the evidence suggests will most-likely lives.

9

Conclusion

We develop a model of persuasion in which a principal must decide whether to implement each of two independent proposals (e.g., earmark requests, policy reforms, grant funding). Agents advocating on behalf of the proposals can produce evidence about the quality of their respective proposal, enabling the principal to make a more informed decision. The principal prefers the agents to produce the most-informative evidence possible. Agents, however, will strategically choose evidence quality to maximize the probability that the principal implements their proposal (maximizing their own payoffs rather than the principal’s payoff). When the principal can implement all proposals she believes are worthwhile (i.e., the case of unlimited capacity), the agents typically will not produce fully informative evidence, leaving the principal less than fully informed about proposal quality. Within this framework, we show that the principal will often be both better informed and better off when she is constrained in the number of proposals she can implement. When the principal is unable to implement all proposals, the agents are concerned about their proposal being 30

given priority over the alternatives in the event that the principal would like to implement more proposals than she has capacity. If the principal observes evidence in favor of two proposals, but can only implement one of them, then she will give priority to the one with the more-informative evidence—that is, the one she is more certain is high quality. The agents react to this by increasing the quality of the evidence that they produce. Under limited capacity, the agents produce more informative evidence compared to the case in which the legislator had unlimited capacity to implement proposals. We then derive reasonable conditions under which this informational benefit dominates the expected cost, and the principal prefers to be limited in the number of proposals she can implement. The model highlights an informational benefit of limited capacity and competitive advocacy that has not been focused on before in the literature. The framework may be applied to understand incentives in a variety of settings, lending insight into informational lobbying, consumer sampling, and preliminary research in grant applications, among others.

10

Appendix

Here we provide additional details about the analysis. We omit formal proofs for results that follow directly from the analysis in the body of the paper.

10.1

Proof of Lemma 5.3

Lemma 5.3 Under the limited capacity legislative system with γ < q and γ < 12 , in any symmetric mixed strategy Nash equilibrium of the stage game, the following four properties must hold: 1. Weak signals are outside of the support of A. 2. The only possible mass point is α = 1. 3. The smallest signal accuracy inside the support of A is r. 4. If disjoint intervals [x1 , x2 ] and [x3 , x4 ] are in the support of A, then the entire interval [x1 , x4 ] is inside the support of A. Proof. (1) Uninformative signals are strictly dominated. (2) If the mixed strategy equilibrium strategy has a mass point at a, then it can be described as follows: with probability φ each agent selects α = a. With probability 1 − φ each agent draws a signal accuracy from CDF F (x) (which is possibly discontinuous itself, i.e. has mass points). Consider the following deviation for agent i. With probability φ agent selects α = a + ε. With probability 1 − φ agent i draws signal 2 accuracy from CDF F (x) With probability (1 − φ) both choose to draw from F (x), in this case the payoff is unchanged. With probability (1 − φ) φ the other agent plays his mass point, while the deviating agent plays from F (x). In this case the deviator’s payoff is unchanged. With probability (1 − φ) φ the other agent plays from F (x) while the deviator plays the mass point. Because the mass point under the deviation is arbitrarily close to the original mass point the payoff is also arbitrarily close to the original payoff. With probability φ2 both players  play the mass point. In this case, the deviation gives expected payoff θ (a + ε)  . Thus for small values of ε this deviation causes a discrete increase in the agent’s instead of θ (a) 1 − θ(a) 2

31

2

expected payoff. When ε is very small this increase is approximately θ(a) > 0. This deviation is profitable. 2 The only value of a for which no such deviation exists is a = 1. (3) Imagine that α, the lowest element of the support of A, were strictly greater than r. Compare α to r. As there is no mass point at α, with probability 1 the other agent’s signal accuracy is strictly higher than both α and r. Therefore, the expected payoff from choosing α for certain is θ (α) (1 − E [θ (αj )]), while the payoff to choosing r is θ (r) (1 − E [θ (αj )]). Because γ < 12 , θ (x) is decreasing; hence, θ (r) (1 − E [θ (αj )]) > θ (α) (1 − E [θ (αj )]) Therefore, if the smallest element of the support is larger than r, then playing the mixed strategy is dominated by choosing r. (4) Imagine that disjoint intervals [x1 , x2 ] and [x3 , x4 ] are in the support of A, but interval (x2 , x3 ) is outside the support of A. Compare playing x2 to x3 . Both of these signal accuracies have the same probability of being larger than the other player’s accuracy, and both accuracies have the same probability of being less than the other player’s signal accuracy. However, in both of these cases, the payoff associated with pure strategy x2 is higher than the payoff associated with x3 . This contradicts the indifference condition.

10.2

Characterization of mixed strategy equilibrium when γ < q, γ
θ 21 = 12 . Thus 1 − θ (1 − r) < 21 . Any pure strategy α ei < 1 − r has a smaller expected payoff than αi = 1 − r against this mixed strategy. Next we compare the

33

payoff of playing pure strategy 1 − r < α ei < 1 to playing pure strategy αi = 1 against this mixed strategy. If the other agent chooses αj = 1 − r then playing α ei < 1 is dominated by playing αi = 1. If the other agent plays αj = 1, then the payoff to playing αi = 1 is γ − 21 γ 2 . The payoff to playing α ei < 1 is θ (e αi ) (1 − γ) . As θ (e αi ) < θ (1) = γ it follows that 1 γ − γ 2 > γ (1 − γ) > θ (α) (1 − γ) 2 Therefore playing any pure strategy 1 − r < α ei < 1 is worse than playing αi = 1 against this mixed strategy. Hence, αi = 1 and αi = 1 − r, give the same payoff when played against this mixed strategy, and all other possible pure strategies give a worse payoff when played against this mixed strategy.

10.4

Characterization of pure strategy equilibrium when q < γ, γ
12 is a weak signal, that
is strictly less than αj . Observe that θ 12 = 12 > θ (α) for any α > 12 . Suppose first that the other agent chooses αj > 12 . By choosing αi ∈ [ 12 , r] and αi < αj , agent i assures himself payoff 1 − θ (αj ) .Notice that 1 − θ (αj ) >

θ (αi ) >

1 > θ (αi ) 2 and θ (αi ) (1 − θ (αj ))

and   θ (αj ) θ (αi ) > θ (αi ) 1 − . 2 Thus the payoff of choosing any αi ∈ [ 21 , r] and αi < αj against αj > 12 is greater than the payoff of choosing any other value of αi . If αj = 12 , then agent i’s payoff from choosing αi = 21 is just 12 which is greater than his payoff of choosing any αi > 12 . Thus α1 = α2 = 21 is a Nash equilibrium, and no other Nash equilibrium exists.

10.5

General Signal Structures

We begin with a simple, but important observation: in this game any random variable Γi with non-zero probability mass in the open interval (0, q) is dominated. Indeed, shifting all mass in this interval to zero allows the agent to put more mass on realizations greater than or equal to q, which strictly increases his expected payoff. (1) We now describe conditions under which it is a Nash Equilibrium for each agent to supply a fully revealing signal. Suppose that agent j supplies a fully revealing signal: Pr (Γj = 1)

= γ

Pr (Γj = 0)

=

1−γ


If i also supplies a fully revealing signal, his expected payoff is γ 1 − γ2 . Imagine that agent i deviates from a fully revealing signal. Because any Γi that places non-zero probability mass on (0, q) is dominated, any possible best response can put positive probability mass on 0, in the interval [q, 1) and on 1. Suppose Γi = 0

34

with some probability φL , Γi ∈ [q, 1) with probability φM , and Γi = 1 with probability φH . The agent’s best response to a fully revealing signal is thus defined by the following maximization problem  γ max φM (1 − γ) + φH 1 − φM ,φH ,g 2 subject to φM E [Γi |q ≤ Γi < 1] + φH = γ φ M + φH ≤ 1 φi ≥ 0 Let g = E [Γi |Γi ∈ [q, 1)] . We first prove that in the problem characterizing the best response to a fully revealing signal, it must be the case that g = q

 γ max φM (1 − γ) + φH 1 − φM ,φH ,g 2 subject to φM g + φH = γ φM + φH ≤ 1 φi ≥ 0 g ∈ [q, 1) Suppose to the contrary that g > q , and imagine that φM , φH , g is the solution to this problem. According to the constraints it must be that φM g + φH = γ φM + φH = x where x ≤ 1. It therefore follows that "

φH = γ−gx 1−g φM = x−γ 1−g

#

In order for φM to be positive it must be that x ≥ γ and for φH to be positive, g ≤ γx . Consider an alternate mechanism   φeH = γ−qx 1−q   e  φM = x−γ  1−q

g=q This alternative mechanism is feasible, as φeH q + φeM = γ φeM + φeH = x x ≥ γ → φeM ≥ 0 q < g ≤ γx → φeH ≥ 0

35

The alternative mechanism leads to a higher payoff, however. To see this calculate the difference       γ γ − qx γ γ − gx γ − gx  γ − qx  1− + x− 1− + x− (1 − γ) − (1 − γ) 1−q 2 1−q 1−g 2 1−g 1 x−γ = γ (g − q) ≥0 2 (1 − g) (1 − q) Thus, the problem can be reduced vy substituting g = q as follows:  γ max φM (1 − γ) + φH 1 − φM ,φH 2 subject to φM q + φH = γ φ M + φH ≤ 1 φi ≥ 0 Consider any feasible solution (φM , φH ). It must be the case that φM q + φH = γ φM + φH = x where x ≤ 1. These conditions imply that φH = γ−qx 1−q φM = x−γ 1−q Non-negativity then implies that x ≤ x ≥

γ q γ

The payoff from such (φM , φH ) is therefore  γ φM (1 − γ) + φH 1 − = 2 =

x−γ γ − qx  γ (1 − γ) + 1− 1−q 1−q 2 1 γ2 (2q + 2γ − qγ − 2) x − 2q − 2 2q − 2

If the coefficient on x is negative the optimum is to set x to its lower bound γ, which implies φH = γ, φnM = o0; which is just the fully revealing signal. Otherwise the solution is to set x to its upper bound, min 1, γq . In either case, the fully revealing signal is not a best response. Thus, the best response to a fully revealing signal is a fully revealing signal if and only if 2 − 2γ 1 (2q + 2γ − qγ − 2) ≤ 0 ⇔ q ≤ 2q − 2 2−γ QED (2) Observe first that in a symmetric Nash Equilibrium, there can be no interior mass point: the only possible mass points are on realizations 0 and 1. No mass point exists in (0, q) as a strategy that puts

36

positive probability in this interval is dominated. If agent j uses a strategy with mass point in [q, 1), then by choosing a strategy with a mass point on a marginally higher realization and culling a small amount of mass from elsewhere in the distribution to preserve the constraint, agent i could ensure that when both agents play mass points, agent i always wins. Because both agents play their mass points with positive probability, ensuring that all ties break in his favor creates a discontinuous increase in his payoff, at the cost of a marginal reduction in the probability of winning otherwise. Such a deviation is therefore profitable, and no interior mass points can exist in equilibrium. 22 This observation allows us to represent any candidate equilibrium strategy in the following way:    0 Γ= X   1

with probability with probability with probability

1 − φM − φH φM φH

    

where X is random variable with support contained in interval [q, 1] and continuous distribution function Φ (x). 23 Next, imagine that agent j plays the following strategy   

0 Γj = X˜U [q, q] , Φ (x) =   1

with probability with probability with probability

x−q q−q

q=γ+

p

 1−φ   2γ φ = q+q   0

q2 + γ 2

Observe first that q≤q and q≤1⇔q≤

p

1 − 2γ

and 0≤φ≤1 so that Γj is indeed a random variable provided 0 ≤ q ≤

√

1 − 2γ. Furthermore,



E [Γj ]

= = =

 q+q 2   2γ q+q q+q 2 γ φ

so that Γj is a valid strategy. Any best response to this strategy does not put probability mass in the 22 No mass point can exist in the interval (0, q) as startegies with non-zero probability mass in this interval are dominated. Imagine that a mass point exists in interval [q, 1), i.e. with probability φ, Γj = γ b ∈ [q, 1) and with probability 1 − φ, Γj is the realization of some random variable X. Suppose that agent i chooses an alternative strategy which plays γ b + ε1 and reduces φ by ε2 to preserve the mean constraint. If both agents play X, or one agent plays his mass point and the other plays X, the payoff of the deviation is within some ε3 of the payoff to using the strategy; however, when both agents play their mass points, the deviator ensures that all ties break in his favor which leads to a discrete increase in his payoff. Such a deviation is therefore profitable. 23 Because no interior mass points exist in equilibrium, and the posibility that Γ has mass point at 1 is accounted for, X has no mass points.

37

interval (q, 1). Indeed, collecting this probability mass into an atom concentrated on q does not reduce the probability of winning but does reduce the mean. This allows more probability mass to be allocated to higher realizations which increases the probability of winning. No best response could put mass in interval (0, q) or in interval (q, 1). Thus the best response to this strategy must be some random variable Γi where  0     Y distributed according to F (y) Γi =  with support contained in [q, q]    1

with probability with probability with probability

 1−θ     θ     0

and Γi satisfies the constraint on the expected value E [Γi ]

= γ ⇔ θE [y] = γ

E [Y ]

=

γ θ

Consider the expected payoff from choosing any such Γi . Z Pr (Γi ≥ Γj and Γi ≥ q)

!

q

= θ (1 − φ) + φ

Φ (y) dF (y) q

Z

q

= θ 1−φ+φ q

 = θ 1−φ+

! x−q dF (y) q−q

φ q−q

 Z

!!

q

xdF (y) − q q

Because Support (Y ) ⊂ Support (X) , q

Z

xdF (y) = E [Y ] q

Therefore,  Pr (Γi ≥ Γj and Γi ≥ q)

=



θ 1−φ+

φ q−q



 (E [Y ] − q)



q (1 − φ) − q

γφ + θ (q (1 − φ) − q) q−q

=

 φ γ θ 1−φ+ −q q−q θ

=

γφ + θ (q (1 − φ) − q) q−q

=



 p γ + q2 + γ 2 1−

=

0

=

γφ q−q

38

γ+q+

2γ p

! q2 + γ 2

−q

This payoff is independent of θ and F (y). Thus, all random variables Γi that could be best responses to Γj (because they allocate no mass in (q, 1) and (0, q)), give the same expected payoff when played against Γj . All such random variables are therefore best responses. As Γj is also of this type, Γj is a best response to itself, and is therefore a Nash Equilibrium. To derive the CDF of the equilibrium strategy, observe that this CDF is simply   

1−φ G1 (x) = (1 − φ) + φ x−q q−q   1

for for for

x ∈ [0, q] x ∈ [q, q] x ∈ [q, ∞]

    

Substituting and simplifying gives the distribution function stated in the proposition.

G1 (x) =

        

√q 2

q +γ 2 √x2 2 γ+ q +γ γ+

1

for for for

x ∈ [0, q] i h p x ∈ q, γ + q 2 + γ 2 h i p x ∈ γ + q2 + γ 2 , ∞

        

QED (3) As in (2) any candidate equilibrium strategy can be represented in the following way:    0 Γ= X   1

with probability with probability with probability

 1 − φM − φH   φM   φH

where X is random variable with support contained in interval [q, 1] and continuous distribution function Φ (x). Next, imagine that agent j plays the following strategy   

with probability 1 − φM − φH = x−q q−q with probability φM = 2−q q−q 2−2q with probability φH = 2−q   p q = 2 − γ + q2 + γ 2

0 Γj = X˜U [q, q] , Φ (x) =   1

Observe first that p

1 − 2γ

≤ q≤

2 − 2γ ⇔q≤q≤1 2−γ

and q

≤ q ≤ 1 ⇒ φM ≥ 0, φH ≥ 0, φM + φH ≤ 1

39

q 2−q

    

so that Γj is a random variable, and 

 q+q E [Γj ] = φM + φH 2    q−q q+q 2 − 2q = + 2−q 2 2−q =

=


1 q 2 − 4q − q 2 + 4 2 (2 − q)   2     p p 1   − 4 2 − γ + q2 + γ 2 − q2 + 4 2 − γ + q2 + γ 2 p 2 γ + q2 + γ 2

= −

 p
3 1  2p 2 2 − q 2 + γ 2 2 − 2q 2 γ + γ 2 q 2 + γ 2 q q + γ 2q 2

= γ Thus, Γj is a valid strategy. As in part (2) no random variable that puts positive probability mass in intervals (q, 1) or (0, q) could be a best response. Therefore, any best response to this strategy must be in the following class   0 with probability 1 − θM − θH        Y distributed according to F (y)  with probability θM Γi =   with support contained in [q, q]       1 with probability θH and E [Γi ]

= γ ⇔ θH + θM E [Y ] = γ

E [Y ]

=

γ − θH θM

Consider now the expected payoff of playing strategy Γi against Γj .

Z (1 − φM − φH ) + φM

θM

!

q

Φ (y) dF (y) q

θM

(1 − φM

φM − φH ) + q−q

Z

!!

q

ydF (y) − q q

As Support (Y ) ⊂ [q, q] Z

q

ydF (y) = E [Y ] q

Continuing the calculation

40

  φH + θH 1 − = 2

 + θH

φH 1− 2



    φM φH θM (1 − φM − φH ) + (E [Y ] − q) + θH 1 − = q−q 2  θM

φM q−q

(1 − φM − φH ) +



γ − θH −q θM

 + θH

  φH 1− = 2

1 (2γφM − θM (2 (q − q) φH + 2qφM − 2 (q − q)) − θH ((q − q) φH + 2φM − 2 (q − q))) 2 (q − q) Next, observe that 2 (q − q) φH + 2qφM − 2 (q − q)  2 (q − q)

2 − 2q 2−q



 + 2q

q−q 2−q

=

 − 2 (q − q)

=

0

and (q − q) φH + 2φM − 2 (q − q)  (q − q)



2 − 2q 2−q

 +2

q−q 2−q

=

 − 2 (q − q)

=

0

Thus the expected payoff from playing any Γi that can be a best response is γφM (q − q) and is independent of the choices of θM , θH , or F (y) and is therefore an equilibrium. The distribution function of the equilibrium strategy is

G2 (x) =

          

Thus

G2 (x) =

              

q 2−q

q 2−q

for 

x−q q−q q−q 2−q



for

 x ∈ [0, q]     x ∈ [q, q] 

+ for x ∈ [q, 1) 1 for x ∈ [1, ∞)    p q = 2 − γ + q2 + γ 2

√q 2

γ+

+

q 2−q q−q 2−q

q

√x2

γ+ q +γ 2 √2 − γ+ q 2 +γ 2

1

for

x ∈ [0, q] h  i p x ∈ q, 2 − γ + q 2 + γ 2   p x ∈ [2 − γ + q 2 + γ 2 , 1)

for

x ∈ [1, ∞)

for

+γ 2

for 1

    

              

QED (4) We check that equilibrium strategy with limited capacity is a mean preserving spread of the equilib-

41

rium strategy with unlimited capacity. We assume that with unlimited capacity, agents send uninformative signals, concentrating all probability mass on γ. Any random variable with mean γ is a mean preserving spread of γ. Therefore, to establish the result it is enough to show that the integral of Gi from minuns infinity to any x are greater than the integral of ( F (x) = over the same domain. Z

1− 1

( 

x

F (z) dz =

γ q

1−

for for

x ∈ [0, q) x ∈ [q, ∞]



for

γ q

x

)

x ∈ [0, q)

)

x−γ for x ∈ [q, ∞]  √q x for γ+ q 2 +γ 2      Rx 2 q√ +x2 √x2 2 dx = 12 + q for 2 2 2

−∞

Z

x

G1 (z) dz = −∞

             

Z

x

G2 (x) = −∞



2

√q 2

γ+

q +γ

γ+

q +γ

γ+

x−γ

         

 1 2

√q 2

γ+ 

q +γ

for

      h i  p x ∈ q, γ + q 2 + γ 2   h i  p   2 2 x ∈ γ + q + γ ,∞  x ∈ [0, q]



q +γ 2

2 q√ +x2 γ+ q 2 +γ 2

x 

for

x ∈ [0, q] h  i p x ∈ q, 2 − γ + q 2 + γ 2   p x ∈ [2 − γ + q 2 + γ 2 , 1)

for

x ∈ [1, ∞)

for for

     2  √ − 1 (x − 1) + (1 − γ)   γ+ q 2 +γ 2    x−γ

                  

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