A bs ract

11
Introduction
Preparing for a variety of natural, social, and economic shocks is an important task of every government. Many governments appropriate a large amount of money on research into the forecasting of such natural shocks as hurricanes, snow storms and other extreme weather conditions, earthquakes, epidemic outbreaks, and so o nn. 1 Along with forecasting, a government's strategies to prepare for those shocks typically involve two forms of interventions. The first is a direct intervention that is implemented at t he government's own cost. The second is an indirect intervention that consists of raising public awareness of the risk of the shocks and advising the public to take preparatory actions themselves. In the case of an epidemic outbreak, for example, the direct interventions include stricter quarantine control, building depressurized rooms at hospitals, increasing the stock of anti-virus medicines, and so on. On the other hand, an indirect intervention consists of advice to the public to receive vaccinations, avoid traveling and exercise hygiene practices.  Likewise, against earthquakes, direct interventions include enforcing stricter building codes and reinforcing public buildings such as schools and highways, while indirect inter- ventions include advice to the public to reinforce their own houses, prepare food stocks, and purchase earthquake insurance.  Unlike direct interventions, it is the public themselves who bear the cost of the advised action. 2  The e ssential feature of many of these preparatory actions is that they are specific investment in the sense that they have value only when the shock strikes. It is argued by some that the policy of spending much money on forecasting shocks and at t the same time advising the public to take preparatory measures is inconsistent. 3   One interpretation of this claim is as follows:  If the accurate forecasting of a shock is possible, then the public is led to think that timely direct interventions will save them costly efforts.  On the other hand, from the point of view of the government, indirect interventions are much less costly and the public's own action is often more effective in mitigating the damage. The purpose of this paper is to provide a formal examination of the above logic in a stylized model where a principal (government) acquires information and then discloses it t to an agent (the public). We present the possibility of strategic obscurity by showing that the principal may find it optimal to acquire imperfect information when the agent can free-ride on his effort. A more detailed description of the model is as follows: Facing the risk of a shock, the principal chooses whether to acquire information about the shock and if he does, what forecasting technology to use to generate his information.  The forecasting technology determines the quality of his private information about the risk of the shock.  The set of technologies available to the principal contains the perfect technology that yields a perfectly informative signal about the occurrence of the shock as well as a continuum of imperfect technologies that yield a noisy signal about it. Choice of any technology is costless. Upon acquiring information, the principal determines whether to take a preparatory action, and at the same time advises the agent on whether he should take a costly preparatory action. The preparatory actions yield mutual benefits only when the shock strikes so that taking no action is dominant for both parties in the event of no shock. In the event of a sure shock, on the other hand, we suppose that the net benefits of the actions are such that the principal has a dominant strategy of taking action, whereas the agent finds it optimal to take action if and only if the principal does not. This creates the fundamental commitment problem for the principal who is better off when the agent makes unilateral effort than when he takes unilateral action himself. We first show that when the prior probability of the shock is moderately high, acquiring no information is better for the principal than acquiring perfect informa- tion.  When the prior probability is low, however, perfect information is superior to no information. Our question hence is: Does the optimal policy entail imperfect information even when the prior probability is low?  To answer this question, we consider a simple model with binary signals. As mentioned above, the principal first chooses and commits to the forecasting technology and disclosure rule. Regardless of the choice of the information technology, taking no action is a dominant strategy for him when he observes the low-risk signal. Hence, the technology determines his incentive only at t he high-risk signal. When the technology is close to perfect, the
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high-risk signal is a strong indicator of the occurrence of the shock, and forces the principal to take action. This eliminates the incentive of the agent to take action, and there exists no disclosure rule that induces action from him.  On the other hand, when the technology is sufficiently imperfect, the high-risk signal is a mild indicator of the occurrence of the shock, and allows the principal to take no action himself while making it incentive compatible for the agent to take action. In other words, the imperfect information technology generates a signal that can be used as a commitment device for the principal to implement the unilateral action by the agent.  When the marginal benefit of the agent's unilateral action to the principal is significantly larger than that of the principal's own unilateral action to himself, we show that the optimal policy indeed entails imperfect information, and also that the optimal technology is the most accurate one that does not interfere with the prin- cipal's incentive to take no action at the high-risk signal. Furthermore, the optimal disclosure rule pools advice in the sense that it advises the agent to take action at the high-risk signal, and also at t he low-risk signal with positive probability. We further show that even if the principal must fully disclose his private signal, the same imperfect information technology is optimal under the same payoff condition. In this sense, we show that strategic obscurity results from information acquisition and not from information disclosure. The paper is organized as follows: After the discussion of the related literature in the next section, we formulate a model of information acquisition and disclosure in Section 3.  Section 4 presents some preliminary analysis.  Section 5 analyzes the principal's payoffs under the perfect information and no information policies. Optimality of imperfect information is established in Section 6.  We conclude in Section 7 with a discussion.
2
Related  Literat ure
Decision making in the face of a natural shock is a classical subject in both the theoretical and empirical literature.  Nelson and Winter (1964) study the weather forecasting system that maximizes the welfare of its user who must decide whether to take a protective action against rain.  Howe and Cochrane (1974) study the decision problem faced by authorities under a snow storm forecast. Their empirical observation on the "reluctance on the part of snow removal authorities to be sensitive to any but very severe forecasts in making operation decisions" is consistent with the
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optimal policy in the current paper. Brookshire et al. (1985) show that the expected utility hypothesis is a reasonable description of decision-making behavior facing a low-probability, high-loss event of an earthquake.  Lewis and Nickerson (1989) study the interaction of self-insurance and public interventions against natural disasters. Information acquisition and disclosure is an increasingly popular topic in the theoretical literature. Combination of the following elements is a distinguishing feature of the present model and has not been studied together to the best of our knowledge. ● The principal commits ex ante to information acquisition and information disclosure policies. 4 ● Information acquisition is costless. ● The principal has a continuous choice of information quality. 5 Principal-agent models of information acquisition in the literature are divided into two groups depending on who acquires information. 6  Cremer et al. (1998a, b), Kessler (1998), Lewis and Sappington (1997), Szalay (2005, 2009), and Dai et al. (2006) study the design of an optimal contract when an agent can privately in- vest resources to acquire information either before or after the contract is signed. 7 In these models, positive cost of information acquisition is a critical element that determines the form of an optimal contract as well as the agent's decision to become informed. The second class of models assume information acquisition by the princi- pal and examine whether ignorance helps the principal commit to some decision in a subsequent interaction with the agent.  Among others, Dewatripont and Maskin (1995) show that simple contracts based on the limited observation of variables may be superior to more complete contracts when renegotiation is possible, and Cremer (1995) shows that the principal may choose to acquire no information about the agent's productivity in a dynamic model with adverse selection.8  Like these models, we assume that the principal acquires information and then plays a game against the agent.9  We show that even when complete ignorance cannot serve as a commit- ment device, a variable degree of incomplete ignorance (i.e., acquisition of imperfect information) may still be a useful commitment device. The oice of signal quality in information disclosure problems is studied by Lewis and Sappington (1994) and Bergemann and Pesendorfer (2007), who both analyze a seller's problem when he chooses the quality of buyers' private signals. In these models, hence, the player who controls the signal quality does not observe the resulting information.  Kamenica  and  Gentzkow (2011) study information ac- quisition by a sender when the signal is publicly observable as in the case of full disclosure in our model. When the sender has no action to take, they ask whether or not acquisition of some information dominates no information. 10  In contrast, our focus is on the comparison between the acquisition of imperfect information and that of perfect information when the sender of information also has an action to take. Finally, it is also possible to relate our finding to the literature on government transparency, which asks whether disclosure of a government's private information induces inefficient coordination by the public and creates uninsurable risks.  Our conclusion points to the possibility that even full disclosure takes place, the con- tent of information may be less than what is potentially available if information acquisition is endogenous. 11
3
Model
There are a principal (player 1) and an agent (player 2) facing the risk of a shock. The shock corresponds to one of the two states of the world ω ∈ Ω = {0,1}: The shock occurs in state ω = 1 and does not in state ω = 0. The prior probability of the shock equals p = P(ω = 1) ∈ (0, 1). Before the state is realized, each player i either "takes action" (a; = 1) or not (a; = 0) against the shock. Denote by A; = {0,1} the set of actions of player i. We suppose that actions are taken simultaneously after the disclosure.  The players' payoffs depend on the action profile and the state. Specifically, player i's payoff under the action profile a = (a i,a ) in state ω is given by
Hence, the players benefit from the actions only when there is a shock (ω = 1), but incur the cost c; > 0 of taking action even when there is no shock. Let
d 1 is the marginal benefit of his own action a 1 = 1 to the principal when it is unilaterally taken, and d  is the marginal benefit of a 1 = 1 when the agent also chooses a2 = 1. m2 and mi are the marginal benefits of the agent's action to the principal when the principal himself chooses α 1 = 0 and α 1 = 1, respectively. d 2, d 2 , m 2 , and m 2 are the corresponding quantities for the agent. We also introduce notation for the net benefits of one's own and the other's actions normalized by the own cost of taking action as follows:
Using this notation, we can express the simultaneous-move game played by the two parties as:
in the no shock state (ω = 0), and
77 in the shock state (ω = 1). We assume that
(3) and (4) show that the two players' actions are strategic substitutes: The marginal benefit of the own action is higher when it is unilateral. Furthermore, (3) says that a 1 = 1 is a dominant action for the principal (player 1) in the event of a sure shock, and (4) says that the agent (player 2)'s best response is to take action when the principal does not, and vice versa, again in the shock state.  (5) says that for the principal, the marginal benefit of the agent's unilateral action is higher than the net marginal benefit of his own unilateral action. We also assume that
In other words, when normalized by its cost, the agent's unilateral action raises his own utility more than the principal's unilateral action raises his. This is a natural assumption to make in view of the fact that the public's own preparatory actions are often small-scale but effective, whereas the government's intervention is often designed to protect the entire population of a region or a country and hence is costly. When the principal chooses to acquire information, he observes signal θ in a finite set Θ. His forecasting technology r determines the level of accuracy of θ in a sense made precise below. Let fω,r(θ) denote the probability of signal θ in state ω when the forecasting technology is r. The timing of events is as follows.  First, the principal chooses his forecasting technology r and disclosure rule g, which determines his non-binding advice to the agent as a function of the signal θ. The agent observes both r and g. The principal observes θ, and then discloses it t o the agent in the form of advice to the agent on whether or not he should take action. After the disclosure, both parties choose actions simultaneously. Finally, the state is realized and the players receive payoffs. The principal's choice of an advice given the observation of θ is expressed by a disclosure rule g :  Θ  →  [0,1]:  g(θ) is the probability that action a2 = 1 is advised to the agent when θ is observed.  The principal's policy is a pair (r,g) of his forecasting technology and disclosure rule.  By the generalized revelation principle of Myerson (1982), the present formulation that the disclosure takes the
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form of advice to the agent is without loss of generality. 12  We assume that the policy ( r,g) is chosen in advance and is publicly announced.  Public observability of the forecasting technology r is a reasonable assumption given that it usually entails publicly observable activities such as launching a satellite, building a supercomputer or a network of sensors, and so o n. We also asoume that the principal commits to his disclosure rule g in the sense that for any signal θ, his advice is generated according to the distribution (1 - g(θ), g(θ)) over A2 = {0, 1}. 13 To summarize, the principal with private signal θ believes that game Gω in (1) and (2) is played with probability P ( ω | θ ), where the probability distribution P depends on his forecasting technology r. On the other hand, the agent who received the advice α 2 believes that game G ω is played with probability P(ω | α 2 ), where the probability distribution P now depends on the principal's policy ( r g ). Unless θ is publicly observable, hence, the two parties attach different probabilities to the two games. Given a policy ( r, g ), each player's strategy is defined as follows. The principal's strategy σ 1 : Θ × A 2 → {0,1} chooses an action as a function of the observed signal 0 as well as the realization of his random advice to the agent. On the other hand, the agent's strategy σ2 : {0,1} → {0,1} chooses an action as a function of the principal's advice.  Denote by α 2 the random variable representing the principal's advice to the agent.  Then σ 1 (θ,α 2 ) denotes the random variable representing the principal's action.  Let σž be the obedient strategy such that σž ( α ) = α 2 for any α 2 ∈ {0,1}. We denote by π i (σ | r,g) player i's ex ante expected payoff under the strategy profile σ = (σ 1 ,σ 2 ) and the policy ( r,g). Explicitly, they are given by

9
where the expectation is taken over the state ω, the signal θ as well as the random advice α 2 . The strategy profile σ is a (Bayes-Nash) equilibrium under (r,g) if π i σ | r, g) ≥ π i (σ ′ , σ j | r,g) for any σ ′ and i̸= j. 14  A policy ( r,g) is incentive compatible if there exists a strategy σ 1 of the principal such that (σ 1 , σ 2 ) is an equilibrium under ( r g ). For an incentive compatible policy ( r g ), if σ 1 is understood, we simply write π ; ( r , g ) for the equilibrium payoff π ; (σ ι , σ ž | r , g ). An incentive compatible policy (r,g) is optimal if there exists no other incentive compatible policy that yields a strictly higher equilibrium payoff. In other words, ( r,g ) is optimal if there exists σ 1 such that σ = ( σ 1,σ ž ) is an equilibrium under ( r g ), and for any policy ( r ′, g ′ ) under which σ ′ = ( σ ′ , σ ♂ ) is an equilibrium for some σ 1  we have

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Preliminary  Analysis:  Publicly  Observable  θ
As a benchmark, we first consider the equilibrium action profile when the signal θ is publicly observable. We suppose in this section that the strategies σ 1 and σ 2 of both the principal and the agent are functions of θ alone and are pure.  Let the technology r be given. When the signal is θ, the principal chooses a 1 = 1 if
Upon simplification, we see that
Since δ | ≤ δ  by assumption, it follows that a 1 = 1 is a dominant action for the principal if the likelihood ratio Conversely, a 1 = 0 is a dominant action for the principal if the likelihood ratio > 声- , . As for the agent, he chooses α2 = 1 if
Equivalently, we have

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Since δ 1 < 0 by assumption, the agent never chooses a 2 = 1 when a 1 = 1. To sum- marize, we can describe the equilibrium action profile as follows when θ is publicly observa ble:
This exercise shows that the likelihood ratio of the two states given the signal θ is what determines the equilibrium behavior at θ.
5
Optimal Policy with Imperfect Information
We now return to our or ginal setup where the principal's signal θ is his private information. In what follows, we suppose for concreteness that θ is drawn from the binary set Θ = { ℓ, h } according to the following distribution conditional on ω: 15
The forecasting technology r satisfies r ∈ [0, f ] . We can interpret h and ℓ as high and low risk signals, respectively, because r < 4 implies that the likelihood ratios are ordered as:
Note that r = 0 corresponds to perfect information since ω = 0 results in θ = ℓ, and ω = 1 results in θ = h. Let us first suppose that the principal (player 1) acquires no information. In this case, the equilibrium is determined by the prior probability alone. Since we can identify = 1 under no information, (8) implies that the equilibrium action is given by
Since u 1 (1,0) – c 1 < u 1 (0,1) by (5), the principal is better off with (a 1,a 2 ) = (0,1) than with (a 1,a 2) = (1,0). It follows that the optimal policy in the second case above should have ( a 1 , a 2 ) = (0 , 1). Hence, the action profile under the optimal no information policy is given by 16
and the principal's payoff is given by
Suppose next that the principal acquires perfect information r = 0. In this case, fr o (0) = 0 for θ = h and = ∞ for θ = ℓ. Note from (8) that in the likelihood ratio If , II neither case, the agent (player 2) chooses a2 = 1 in equilibrium. It follows that an incentive compatible policy must advise no action for any θ. Therefore, the action profile under the perfect information policy is given by
and the principal's ex ante expected equilibrium payoff equals π 1 (σ | r,g) = p{u 1 (1,0) − c 1 } ≡ π 1 .

1 2
Intuitively, if the agent (player 2) knows that the principal (player 1) knows the state, he will not choose a2 = 1 because he knows that the principal chooses a1 = 1 in state 1. Since the principal advises no action at any θ , the optimal policy under perfect information entails no disclosure.  On the other hand, the above profile is also equivalent to what happens when the principal acquires perfect information and then fully discloses it t o the agent. Comparison of the principal's payoff under no information in (9) and that un- der perfect information in (10) is summarized in the following proposition and is illustrated in Figure 1. Proposition 1 a) If p < # or p > # , then perfect information yields the greater

    expected payoff to the principal than no information. 47, 4 ), then no information yields the greater expected payoff than perfect b) Ifp E information. Intuitively, no information dominates perfect information if the prior p is in the intermediate range so that under no information, the principal can commit to no action and induce the agent to take action unilaterally. When p is above this range, the shock is too likely for the principal to commit to no action, and when it is below
1 3
this range, the shock is too unlikely for the agent to take action even unilaterally. In these cases, perfect information is better than no information. We can interpret the observation in Proposition 1(b) as one expression of the value of strategic ignorance mentioned in Section 2. Our focus in subsequent sections is hence on the case where p is small so that complete ignorance is inferior to perfect information. We are now ready to analyze imperfect information policies. In this case, we show that the principal chooses his forecasting technology so as to control the likelihood ratio of the two states given some signal realization. In particular, we show that he uses a high-risk signal as a commitment device to implement (α ι ,α 2 ) = (0 ,1 ). Given the conclusion above, we assume in the remainder of this section that the prior probability p of the shock state ω = 1 is low and satisfies
This in particular implies that under no information, the agent does not take action even unilaterally. For any technology r < 4, taking no action a 1 = 0 is dominant for him at θ = ℓ since where the first inequality follows from r < 1 , the second from (11), and the third from (6). Note that we can represent the disclosure rule g by a pair (y,z) where y,z E [0, 1] denote the probabilities that action a2 = 1 is suggested for signals h and ℓ, respectively:
According to the classification in (8), we consider the following four possibilities regarding the likelihood ratio at the high-risk signal θ = h : 17
Case 1.
In this case, taking action a 1 = 1 is dominant for the principal at θ = h by (8). On the other hand, he cannot induce action a2 = 1 from the agent: If the agent learns that θ = ℓ , he will choose α 2 = 0 since
and if he learns that θ = h, he will again choose a2 = 0 since
It follows that for no signal realization, the agent will choose a 2 = 0. This in turn implies that no disclosure rule (12) induces a 2 = 1 if r satisfies (13). The maximum payoff in this case hence is achieved under perfect information r = 0. Case 2.
In this case, the agent has no incentive to take action even unilaterally if he learns that θ = h. By the same logic as in the first case, he has no incentive to take action if he learns that θ = ℓ. Hence, no disclosure rule (12) induces α 2 = 1 if r satisfies (14). Case 3.
Case 4.
We will deal with these two cases simultaneously. In Case 4, the principal (player 1) observing θ = h chooses α 1 = 0 and advises the agent (player 2) to choose α 2 = 1. Hence, when θ = h, the action profile ( α 1 , α 2 ) = (0 , 1) is played with probability
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one y = 1. In Case 3, on the other hand, the principal observing θ = h chooses a 1 = 1 when the agent chooses a 2 = 0, and chooses a 1 = 0 when the agent chooses a2 = 1. Hence, when θ = h, the probability g(h) = y with which action profile (α 1 ,α 2 ) = (0 , 1) is played can be less than one, and (α 1 ,α 2 ) = (1 , 0) is played with probability 1 – y. Hence,
In both cases, the principal's ex ante expected payoff under ( r, g ) is given by
On the other hand, the agent's incentive compatibility condition when advised to take action is given by
The agent's incentive compatibility condition when advised to take no action is implied by (18) under (11). After some algebra, we see that (18) is equivalent to
We can hence write the principal's optimization problem as:
In what follows, we solve this maximization problem in two steps. First, we fix r and solve for the optimal disclosure rule g x = ( y x , z x ) conditional on r . We then solve for the optimal value of r. Lemma 2 For any r satisfying (15) or (16), if g*, = (y*, z*) is the optimal disclo- sure rule conditional on r, then a) y, = 1 so that a2 = 1 is suggested with probability one at θ = h.

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Proof. See the Appendix.  ■ The intuition behind Lemma 2 is as follows: For the first observation, when θ = h and r satisfies (15) or (16), the agent has an incentive to take action when the principal does not.  When θ = h, hence, the principal can induce a 2 = 1 for free without violating the agent's incentive conditions (18). As for the second observation, the principal wants to induce a 2 = 1 as much as possible even if θ = ℓ . He does this by randomly advising a 2 = 1 when θ = ℓ to the extent that there is no slackness in the agent's incentive condition (18). Let π 1 (r) = π 1 (r,g, s) be the principal's payoff given r when the disclosure rule is optimally chosen as in Lemma 2. The following proposition verifies that π 1 (r) is strictly decreasing in r and hence maximized when r is at the lower end of its admissible range (15) or (16).
Proof. We can readily verify that is increasing in r . Hence,
The above proposition shows that the principal chooses r so that the likelihood fi, r (h) at signal θ = h is just at the critical level where he is indifferent between taking action and not when a 2 = 1. In other words, he would prefer a more accurate signal so long as it does not interfere with his own incentive to take no action. Since the likelihood ratio at θ = h is proportional to r for r small, and the critical value  p8 . is proportional to p for p small, it follows that the optimal imperfect information technology r* is proportional to p when p is small. In particular, r* → 0 as p → 0. Having identified the optimal imperfect information policy in Proposition 3, we now identify the sufficient conditions under which it dominates the perfect informa- tion policy and hence is globally optimal. For this, we consider an alternative policy
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(r*, ˆ g) that uses the same r* but advises a 2 = 1 only when θ = h and advises a 2 = 0 otherwise: g = ( j, z ) = (1,0). Note that ( r *, g ) is itself incentive compatible, and hence that the principal's payoff under the optimal policy (r*,g*) is strictly larger than that under (r*, g):
The following theorem presents a sufficient condition for ( r*, g*) to dominate perfect information by comparing the principal's payoff under ( r *, ˆ g ) with that under perfect information π 1 = p { u 1 (1,0) = c 1 }. Theorem 4  Suppose that
Then the imperfect information policy (r*,g*) in Proposition 3 dominates the perfect information policy and is optimal. Proof. Note that π 1 (r *, g) > p {u 1 (1,0) - c 1 } if and only if
This holds since r* < H and (21) is equivalent to H < 1 - As noted above, any r satisfying (15) or (16) coupled with the disclosure rule ˆ g = (ˆ g, ˆ z ) = (1,0) is incentive compatible. This disclosure rule, on the other hand, is equivalent to full disclosure of θ since under full disclosure, the action profile (α 1,α 2 ) = (0,1) is an equilibrium if and only if θ = h by (8). Note also by (8) that if r satisfies (16) with strict inequalities, then (a1,a2) = (0,1) is the unique equilibrium when θ = h. Since any such r also satisfies r < f, the same condition (21) also ensures that (r, ˆ g) dominates perfect information. The corollary below summarizes this observation. Corollary 5 Suppose that (21) holds. Then there exists an equilibrium under the full disclosure of θ with the imperfect forecasting technology r* that dominates the perfect information policy. Furthermore, for any r satisfying
the full disclosure of θ with the imperfect forecasting technology r induces the unique equilibrium that dominates the perfect information policy.
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The sufficient condition (21) is a strengthening of (5) and holds when the agent's unilateral action has a significantly larger impact on the principal's payoff than his own unilateral action. For example, suppose that the principal's payoff is written in the form: u1(a1,a2) = k1a1 + k2a2 — a1a2, where the cross product term represents strategic substitution. Then the benefit of the agent's unilateral action for the principal equals m 1 = u 1 (0,1) - u 1 (0,0) = k 2 , and that of his own unilateral action equals d 1 = u 1 (1,0) - u 1 (0,0) = k 1 . Hence, (21) holds when k 2 > 2(k 1 − c 1 ). The above conclusion extends to alternative setups where the principal and the agent move sequentially. Suppose first that the principal moves first. By choosing a1 = 0, he reveals no additional information and hence can force a2 = 1 on the agent when r = r*. It follows that the imperfect information policy in Theorem 4 remains optimal under (21). Suppose next that the agent moves first after the disclosure. In this case, if
then the agent has no incentive to take action under r = r* since then he can force the principal to take action by taking no action himself. 18 However, if the principal adopts a less accurate forecasting technology r that satisfies (22), then (α 1 ,α 2 ) = (0 , 1) can be implemented as a unique equilibrium when θ = h even if the agent is the first mover. Hence, the imperfect information policy (r,g,*) for any such r again dominates perfect information again under (21). 19
6
Conclusion
In a model of information acquisition and disclosure, we show that endogenous information about the risk of a shock may be imperfect when the agent may free ride on the principal's preparation efforts. For a shock with moderately high prior probability, the principal prefers no information to perfect information.  On the
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other hand, for a shock with small prior probability, the principal prefers perfect information to no information, but the optimal policy may entail a strictly positive degree of imperfection. Specifically, we show that the full disclosure of imperfect information may outperform perfect information. The model adopts an extreme assumption that a perfectly informative signal is costlessly available to the principal. Of course, if acquisition of more accurate information is more costly, then it only reinforces the main conclusion of the paper. When acquisition of perfect information is technologically infeasible, the relevant question is whether the optimal information is less precise than what is techno- logically feasible. The answer naturally depends on the parameters, but the basic intuition of the present analysis continues to be valid. The scientific assessment of a risk is often very difficult to communicate to non- Furthermore, it is often observed that individuals overreact to a small experts. probability risk in some cases, and undermine a moderately high probability risk in other cases. In this sense, the biggest challenge for the sender of information may be to induce the right action from the receivers taking into account the imperfection and bias in their information processing.20  Theoretical investigation into such a process would be an interesting topic of future research.
Appendix
Proof of Lemma 2.   a) If r satisfies (16), then y r = 1 by (17). Suppose then that r satisfies (15).  Let (r, g) be any incentive compatible policy with g(h) < 1. We show that (r, g) with
is also incentive compatible and yields a strictly higher payoff to the principal. By our choice of r , we have
Add side-by-side (23) to the incentive compatibility condition (19) for (r,g). The resulting inequality is the incentive compatibility condition for (r,g). That (r,g)
20
yields a strictly higher payoff to the principal is implied by (5): u 1 (0,1) > u 1 (1,0) – cl b) The conclusion follows immediately once we substitute y = 1 into π 1 ( r, g ), and rewrite the optimization problem (20) with respect to z as:
[1] Masaki Aoyagi (2013), Strategic Obscurity in the Forecasting of Disasters, working paper. [2] Dirk Bergemann and Martin Pesendorfer (2007), Information Structures in Op- timal Auctions, J. Econ. Theory, 137, 580-609. [3] Helmut Bester and Roland Strausz (2001), Contracting with Imperfect Com- mitment and the Revelation Principle: The single agent case, Econometrica, 69, 1077-1098. [4] David S. Brookshire, Mark A. Thayer, John Tschirhart, and William D. Schulze (1985), A Test of the Expected Utility Model: Evidence from Earthquake Risks, J. Polit. Econ., 93(2), 369-389. [5] Juan D. Carrillo and Thomas Mariotti (2000), Strategic Ignorance as a Self- Disciplining Device, Rev. Econ. Stud., 67(3), 529-544. [6] Jacques Cremer (1995), Arm's Length Relationship, Quart. J. Econ., 110(2), 275-295. [7] Jacques Cremer, Fahad Khalil, and Jean-Charles Rochet (1998), Strategic In- formation Gathering Before a Contract is Offered, J. Econ. Theory, 81, 163-200. [8] Jacques Cremer, Fahad Khalil, and Jean-Charles Rochet (1998), Contracts and Productive Information Gathering, Games Econ. Behav., 25, 174-193. [9] Chifeng Dai, Tracy Lewis, and Giuseppe Lopomo (2006), Delegating Manage- ment to Experts, RAND J. Econ., 37(3), 503-520. [10] Mattias Dewatripont and Eric Maskin (1994), Contractual Contingencies and Renegotiation, RAND J. Econ., 26(4), 704-719. [11] Sara L. Eggers, and Baruch Fischhoff (2004), Setting Policies for Consumer Communications: A Behavioral Decision Theory Approach, J. Public Policy Mark, 23(1), 14-27. [12] Baruch Fischhoff (1994), What Forecasts (Seem to) Mean, Int. J. Forecasting, 10, 387-403. [13] Baruch Fischhoff (2011), Communicating about the Risk of Terrorism (or Any- thing Else), Amer. Psychol., 66(6), 520-531. [14] Emir Kamenica and Matthew Gentzkow (2011), Bayesian Persuasion, Amer. Econ. Rev., 101, 2590-2615. [15] Jack Hirschleifer (1971), The Private and Social Value of Information and the Reward from Incentive Activity, Amer. Econ. Rev., 61, 561-574. [16] Charles W. Howe and Harold C. Cochrane (1976), A Decision Model for Ad- justing to Natural Hazard Events with Application to Urban Snow Storms, Rev. Econ. Stat., 58, 50-58. [17] Anke S. Kessler (1998), The Value of Ignorance, RAND J. Econ., 29(2), 339- 354. [18] Tracy Lewis and David Nickerson (1989), Self-insurance against Natural Dis- asters, J. Environ, Econ, Manag, 16, 209-223. [19] Tracy Lewis and David E. M. Sappington (1993), Ignorance in Agency Prob- lems, J. Econ. Theory, 61, 169-183. [20] Tracy Lewis and David E. M. Sappington (1997), Information Management in Incentive Problems, J. Polit. Econ., 105(4), 796-821. [21] Tracy Lewis and David E. M. Sappington (1994), Supplying Information to Facilitate Price Discrimination, Int. Econ. Rev., 35(2), 309-327. [22] Steven Matthews and Andrew Postlewaite (1985), Quality Testing and Disclo- sure, RAND J. Econ., 16(3), 328-340. [23] Roger Myerson (1982), Optimal Coordination Mechanisms in Generalized Principal-Agent Problems, J. Math. Econ., 10, 67-81. [24] Stephen Morris and Hyun-Song Shin (2002), Social Value of Public Information, Amer. Econ. Rev., 92, 1521-1534. [25] Richard R. Nelson and Sidney G. Winter (1964), A Case Study in Economics of Information and Coordination: The Weather Forecasting System, Quart. J. Econ., 78, 420-441. [26] Roy Radner and Joseph E. Stiglitz (1984), A Nonconcavity in the Value of Information, in Bayesian Models in Economic Theory, M. Boyer and R. E. Kihlstrom, Eds., Elsevier Science Publishers B.V. [27] Makoto Saito (2008), Social Science of Earthquake Forecasting, Shosai-no-Mado 580, 12-24 (in Japanese). [28] Emmanuel Skoufias (2003), Economic Crises and Natural Disasters: Coping Strategies and Policy Implications, World Dev., 31(3), 1087-1101. [29] Vasiliki Skreta (2006), Sequentially Optimal Mechanisms, Rev. Econ. Stud., 73, 1085-1111. [30] Lars, E. O. Svensson (2006), Social Value of Public Information: Morris and Shin (2002) is Actually Pro Transparency, Not Con, Amer. Econ. Rev., 96, 448-451. [31] Dezsö Szalay (2005), Economics of Clear Advice and Extreme Options, Rev. Econ. Stud., 72, 1173-1198. [32] Dezsö Szalay (2009), Contracts with Endogenous Information, Games Econ. Behav., 65, 586-625. [33] Carl E. Walsh (2007), Announcements and the Role of Policy Guidance, work- ing paper.