## Abstract / Description of output

We consider a Bayesian persuasion problem where a sender's utility depends only on the expected state. We show that upper censorship that pools the states above a cutoff and reveals the states below the cutoff is optimal for all prior distributions of the state if and only if the sender's marginal utility is quasi-concave. Moreover, we show that it is optimal to reveal less information if the sender becomes more risk averse or the sender's utility shifts to the left. Finally, we apply our results to the problem of media censorship by a government.

Original language | English |
---|---|

Pages (from-to) | 561-585 |

Number of pages | 25 |

Journal | Theoretical Economics |

Volume | 17 |

Issue number | 2 |

Early online date | 25 May 2022 |

DOIs | |

Publication status | Published - May 2022 |

## Keywords / Materials (for Non-textual outputs)

- Bayesian persuasion
- censorship
- D82
- D83
- information design
- L82
- media

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*Theoretical Economics*,

*17*(2), 561-585. https://doi.org/10.3982/TE4071

**Censorship as optimal persuasion**. In: Theoretical Economics. 2022 ; Vol. 17, No. 2. pp. 561-585.

}

*Theoretical Economics*, vol. 17, no. 2, pp. 561-585. https://doi.org/10.3982/TE4071

**Censorship as optimal persuasion.**/ Kolotilin, Anton; Mylovanov, Timofiy; Zapechelnyuk, Andriy.

In: Theoretical Economics, Vol. 17, No. 2, 05.2022, p. 561-585.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - Censorship as optimal persuasion

AU - Kolotilin, Anton

AU - Mylovanov, Timofiy

AU - Zapechelnyuk, Andriy

N1 - Funding Information: Our application to media censorship fits into the literature on media capture that addresses the problem of media control by governments, political parties, or lobbying groups. Besley and Prat (2006) pioneered this literature by studying how a government's incentives to censor free media depend on a plurality of media outlets and on transaction costs of bribing the media. In related work, Gehlbach and Sonin (2014) consider a setting with a government-influenced monopoly media outlet that trades off mobilizing the population for some collective goal and collecting the revenue from subscribers who demand informative news.9 The models of Besley and Prat (2006) and Gehlbach and Sonin (2014) have two states of the world and either a single media outlet or a few identical media outlets. Hence, the government uses the same censorship policy for each outlet. We are the first to consider a richer model of media censorship with a continuum of states and heterogeneous media outlets. As a result, the government optimally discriminates media outlets by permitting sufficiently supportive ones and banning the remaining ones. As in Suen (2004), Chan and Suen (2008), and Chiang and Knight (2011), we assume that media outlets use binary approval policies that communicate only whether the state is above some standard. This assumption reflects a cursory reader's preference for simple messages such as positive or negative opinions and yes or no recommendations. Our results, however, do not rely on this assumption. As we have shown, when the government's utility is S-shaped in the expected state, as in the majoritarian and proportional cases, upper-censorship policies are optimal among all possible signals about the state of the world. Thus, the government cannot benefit from introducing new media outlets or using more complex forms of media control such as aggregating information from multiple media outlets and possibly adding noise. In our model, each citizen observes messages from all permitted media outlets. However, among all these media outlets, only one is pivotal for a citizen's choice between two actions. Therefore, our results are not affected if each citizen could follow only his most preferred media outlet, as in Chan and Suen (2008). More generally, the government cannot benefit from information discrimination of citizens, as follows from Kolotilin et al. (2017). Our results can be easily extended to the case of a finite number of media outlets. When the government's utility is S-shaped in the expected state, it is again optimal to censor all sufficiently opposing media outlets and permit the rest. In our model, each citizen's optimal action does not depend on what other citizens do. Otherwise the citizens' optimal actions would constitute an equilibrium of the game induced by the available information. Imposing some equilibrium selection criterion, we can still define the government's utility as a function of the expected state and apply our results when the citizens' and government's utilities are linear in the state. There are other aspects that can be relevant in media economics. Citizens can incur some cost of following media outlets. While we have already mentioned that citizens gain no benefit from following more than one outlet, it is entirely possible for them to stop watching news altogether if it is sufficiently uninformative. Moreover, it can be costly for the government to censor media outlets. So another important question is how the government should prioritize censoring. These extensions are nontrivial and left for future research. We now show that the media censorship problem can be formulated as a persuasion problem, in which the government is a sender and a representative citizen is a receiver. Observe that a government's upper-censorship policy with cutoff ω* is equivalent to an upper-censorship signal that reveals the states below ω* and pools the states above ω*. Consequently, when upper censorship is optimal among all signals, it is also optimal among signals induced by censoring media outlets. Let m be the expected state induced by messages from the permitted media outlets. A citizen of type r chooses ar=1 if and only if r≤m. Consequently, the aggregate action ā is the mass of all citizens whose types are at most m, so ā=G(m). Next, using the citizens' optimal behavior, we derive the government's expected utility conditional on the expected state m: 9V(m)=E[B(ā)+ρ∫01u(ar,ω,r)g(r)dr|m]=B(G(m))+ρ∫0m(m−r)g(r)dr=∫0G(m)b(r)dr+ρ∫0mG(r)dr. We thus obtain the persuasion problem in Section 2, where the sender's Bernoulli utility is given by (9). By Theorem 1, upper censorship is optimal among all censorship policies if V is S-shaped. Consider two special cases of interest: majoritarian and proportional. In the majoritarian case, the government cares only about the aggregate action, so ρ=0 and V(m)=B(G(m)). For example, this can reflect the government's utility when it is supported by the fraction G(m) of voters in an upcoming election. 3 Theorem Let V be given by (9) with ρ=0. If b and g are (strictly) log-concave on [0,1], then upper censorship is (uniquely) optimal. Theorem 3 shows that an upper-censorship policy is optimal if both the population density g and the marginal benefit b from convincing citizens to choose a=1 are log-concave and thus unimodal. This condition on b holds if the government's top priority is to reach a certain approval threshold such as a simple majority, so that B is a smooth approximation of a step function. Let V be given by (9) with ρ=0. If b and g are (strictly) log-concave on [0,1], then upper censorship is (uniquely) optimal. The second special case of interest is where the government's benefit is proportional to the share of supporters, B(ā)=ā. Then V is the same as (6) in Section 4 and, thus, upper censorship is optimal when g is log-concave, by Theorem 2. Propositions 2 and 3 imply that in both the majoritarian and proportional cases, the government optimally censors more media outlets if the citizens are more supportive of the government, so the cutoff approval standard ω* decreases if the distribution of the citizens' types is shifted to the left. Moreover, in the proportional case, the government censors more media outlets if it cares more about the approval and less about the social welfare, so ω* decreases if ρ becomes smaller. There is a continuum of heterogeneous citizens indexed by r∈[0,1] distributed with G that has a strictly positive and twice continuously differentiable density g. The state of the world ω∈[0,1] has a distribution F that has a strictly positive density f. Each citizen chooses between a=0 and a=1. The utility of a citizen of type r is given by u(ar,ω,r)=(ω−r)ar, where ar∈{0,1} is the citizen's action. There can be various interpretations of the citizen's action a=1, such as voting for the government, supporting a government's policy, or taking an individual decision that benefits the government. The state of the world ω is a common benefit of action a=1, whereas r is a type-specific cost of this action. Thus, the state ω can be interpreted as the government's quality or valence, and the type r can be interpreted as a citizen's ideological position or preference parameter. There is a government that cares about a weighted average of the social utility and the government's intrinsic benefit from the aggregate action. For a given state ω, the government's utility is given by B(ā)+ρ∫01u(ar,ω,r)g(r)dr, where ā=∫01arg(r)dr is the aggregate action in the society and B is a twice continuously differentiable function with a strictly positive derivative b. Parameter ρ captures the alignment of the government's preferences with those of the citizens, and the term B(ā) captures the government's intrinsic benefit from the aggregate action as well as possible externalities of the citizens' actions. Since the marginal benefit b from increasing the aggregate action is strictly positive and, in addition, more citizens prefer action a=1 when the state ω is higher, a high state is good news for the government. Citizens obtain information about the unobservable state ω through media outlets. There is a continuum of media outlets indexed by c∈[0,1]. Each media outlet c approves the pro-government action a=1 if ω≥c and criticizes it if ω<c. We thus can interpret c as the media outlet's standard of approval. A media outlet with a higher approval standard c is more opposing to the government because it criticizes the pro-government action on a larger set of states. The government can censor media outlets to stop them from broadcasting. Censored media outlets are uninformative to the public. The government's censorship policy is a set of the media outlets X⊂[0,1] that are censored. The other media outlets in [0,1]\X continue to broadcast. The government's censorship of media outlets can take various forms, including banning access to internet sites, withdrawing licenses, and even arresting editors and journalists using broadly formulated legislation on combating extremism. Importantly, the government implements a censorship policy before the state is realized. This assumption reflects that daily news coverage is the privilege of the media, and the government cannot routinely interfere in this process.8 A government censorship policy is upper censorship with cutoff ω*∈[0,1] if it censors all sufficiently opposing media outlets. Specifically, media outlets whose approval standards are below ω* are permitted and the rest are censored, so X=[ω*,1]. Note that full censorship is an extreme form of upper censorship where all media outlets are censored (ω*=0), while free media is a degenerate form of upper censorship where (almost) no media outlets are censored (ω*=1). The timing is as follows. First, the government chooses a set X⊂[0,1] of censored media outlets. Second, the state ω is realized, and each permitted media outlet approves or criticizes action a=1 according to its approval standard. Finally, each citizen observes messages from all permitted media outlets, updates his beliefs about ω, and chooses an action. In this section, we present a model of media censorship by the government and show that it can be represented as a persuasion problem with a privately informed receiver in Section 4.1. We apply our results to provide conditions for the optimality of upper-censorship policies that censor all media outlets except the most pro-government ones. There is a continuum of heterogeneous citizens indexed by r∈[0,1] distributed with G that has a strictly positive and twice continuously differentiable density g. The state of the world ω∈[0,1] has a distribution F that has a strictly positive density f. Each citizen chooses between a=0 and a=1. The utility of a citizen of type r is given by u(ar,ω,r)=(ω−r)ar, where ar∈{0,1} is the citizen's action. There can be various interpretations of the citizen's action a=1, such as voting for the government, supporting a government's policy, or taking an individual decision that benefits the government. The state of the world ω is a common benefit of action a=1, whereas r is a type-specific cost of this action. Thus, the state ω can be interpreted as the government's quality or valence, and the type r can be interpreted as a citizen's ideological position or preference parameter. There is a government that cares about a weighted average of the social utility and the government's intrinsic benefit from the aggregate action. For a given state ω, the government's utility is given by B(ā)+ρ∫01u(ar,ω,r)g(r)dr, where ā=∫01arg(r)dr is the aggregate action in the society and B is a twice continuously differentiable function with a strictly positive derivative b. Parameter ρ captures the alignment of the government's preferences with those of the citizens, and the term B(ā) captures the government's intrinsic benefit from the aggregate action as well as possible externalities of the citizens' actions. Since the marginal benefit b from increasing the aggregate action is strictly positive and, in addition, more citizens prefer action a=1 when the state ω is higher, a high state is good news for the government. Citizens obtain information about the unobservable state ω through media outlets. There is a continuum of media outlets indexed by c∈[0,1]. Each media outlet c approves the pro-government action a=1 if ω≥c and criticizes it if ω<c. We thus can interpret c as the media outlet's standard of approval. A media outlet with a higher approval standard c is more opposing to the government because it criticizes the pro-government action on a larger set of states. The government can censor media outlets to stop them from broadcasting. Censored media outlets are uninformative to the public. The government's censorship policy is a set of the media outlets X⊂[0,1] that are censored. The other media outlets in [0,1]\X continue to broadcast. The government's censorship of media outlets can take various forms, including banning access to internet sites, withdrawing licenses, and even arresting editors and journalists using broadly formulated legislation on combating extremism. Importantly, the government implements a censorship policy before the state is realized. This assumption reflects that daily news coverage is the privilege of the media, and the government cannot routinely interfere in this process.8 A government censorship policy is upper censorship with cutoff ω*∈[0,1] if it censors all sufficiently opposing media outlets. Specifically, media outlets whose approval standards are below ω* are permitted and the rest are censored, so X=[ω*,1]. Note that full censorship is an extreme form of upper censorship where all media outlets are censored (ω*=0), while free media is a degenerate form of upper censorship where (almost) no media outlets are censored (ω*=1). The timing is as follows. First, the government chooses a set X⊂[0,1] of censored media outlets. Second, the state ω is realized, and each permitted media outlet approves or criticizes action a=1 according to its approval standard. Finally, each citizen observes messages from all permitted media outlets, updates his beliefs about ω, and chooses an action. We now show that the media censorship problem can be formulated as a persuasion problem, in which the government is a sender and a representative citizen is a receiver. Observe that a government's upper-censorship policy with cutoff ω* is equivalent to an upper-censorship signal that reveals the states below ω* and pools the states above ω*. Consequently, when upper censorship is optimal among all signals, it is also optimal among signals induced by censoring media outlets. Let m be the expected state induced by messages from the permitted media outlets. A citizen of type r chooses ar=1 if and only if r≤m. Consequently, the aggregate action ā is the mass of all citizens whose types are at most m, so ā=G(m). Next, using the citizens' optimal behavior, we derive the government's expected utility conditional on the expected state m: 9V(m)=E[B(ā)+ρ∫01u(ar,ω,r)g(r)dr|m]=B(G(m))+ρ∫0m(m−r)g(r)dr=∫0G(m)b(r)dr+ρ∫0mG(r)dr. We thus obtain the persuasion problem in Section 2, where the sender's Bernoulli utility is given by (9). By Theorem 1, upper censorship is optimal among all censorship policies if V is S-shaped. Consider two special cases of interest: majoritarian and proportional. In the majoritarian case, the government cares only about the aggregate action, so ρ=0 and V(m)=B(G(m)). For example, this can reflect the government's utility when it is supported by the fraction G(m) of voters in an upcoming election. 3 Theorem Let V be given by (9) with ρ=0. If b and g are (strictly) log-concave on [0,1], then upper censorship is (uniquely) optimal. Theorem 3 shows that an upper-censorship policy is optimal if both the population density g and the marginal benefit b from convincing citizens to choose a=1 are log-concave and thus unimodal. This condition on b holds if the government's top priority is to reach a certain approval threshold such as a simple majority, so that B is a smooth approximation of a step function. The second special case of interest is where the government's benefit is proportional to the share of supporters, B(ā)=ā. Then V is the same as (6) in Section 4 and, thus, upper censorship is optimal when g is log-concave, by Theorem 2. Propositions 2 and 3 imply that in both the majoritarian and proportional cases, the government optimally censors more media outlets if the citizens are more supportive of the government, so the cutoff approval standard ω* decreases if the distribution of the citizens' types is shifted to the left. Moreover, in the proportional case, the government censors more media outlets if it cares more about the approval and less about the social welfare, so ω* decreases if ρ becomes smaller. Our application to media censorship fits into the literature on media capture that addresses the problem of media control by governments, political parties, or lobbying groups. Besley and Prat (2006) pioneered this literature by studying how a government's incentives to censor free media depend on a plurality of media outlets and on transaction costs of bribing the media. In related work, Gehlbach and Sonin (2014) consider a setting with a government-influenced monopoly media outlet that trades off mobilizing the population for some collective goal and collecting the revenue from subscribers who demand informative news.9 The models of Besley and Prat (2006) and Gehlbach and Sonin (2014) have two states of the world and either a single media outlet or a few identical media outlets. Hence, the government uses the same censorship policy for each outlet. We are the first to consider a richer model of media censorship with a continuum of states and heterogeneous media outlets. As a result, the government optimally discriminates media outlets by permitting sufficiently supportive ones and banning the remaining ones. As in Suen (2004), Chan and Suen (2008), and Chiang and Knight (2011), we assume that media outlets use binary approval policies that communicate only whether the state is above some standard. This assumption reflects a cursory reader's preference for simple messages such as positive or negative opinions and yes or no recommendations. Our results, however, do not rely on this assumption. As we have shown, when the government's utility is S-shaped in the expected state, as in the majoritarian and proportional cases, upper-censorship policies are optimal among all possible signals about the state of the world. Thus, the government cannot benefit from introducing new media outlets or using more complex forms of media control such as aggregating information from multiple media outlets and possibly adding noise. In our model, each citizen observes messages from all permitted media outlets. However, among all these media outlets, only one is pivotal for a citizen's choice between two actions. Therefore, our results are not affected if each citizen could follow only his most preferred media outlet, as in Chan and Suen (2008). More generally, the government cannot benefit from information discrimination of citizens, as follows from Kolotilin et al. (2017). Our results can be easily extended to the case of a finite number of media outlets. When the government's utility is S-shaped in the expected state, it is again optimal to censor all sufficiently opposing media outlets and permit the rest. In our model, each citizen's optimal action does not depend on what other citizens do. Otherwise the citizens' optimal actions would constitute an equilibrium of the game induced by the available information. Imposing some equilibrium selection criterion, we can still define the government's utility as a function of the expected state and apply our results when the citizens' and government's utilities are linear in the state. There are other aspects that can be relevant in media economics. Citizens can incur some cost of following media outlets. While we have already mentioned that citizens gain no benefit from following more than one outlet, it is entirely possible for them to stop watching news altogether if it is sufficiently uninformative. Moreover, it can be costly for the government to censor media outlets. So another important question is how the government should prioritize censoring. These extensions are nontrivial and left for future research. A special case that appears in much of the persuasion literature is where the sender wishes to maximize the probability that the receiver accepts the proposal. This is the case of ρ=0, so v(ω,r)=1 and V(m)=G(m). Thus, V is S-shaped if and only if the density g of the outside option is unimodal, where the inflection point of V is the mode of g. By Theorem 1, this unimodality is necessary and sufficient for the optimality of upper censorship. In this case, the comparative statics results of Propositions 1 and 2 are also easy to interpret. Let g1 and g2 be unimodal densities of distributions G1 and G2 of the outside option. First, observe that G2 is more risk averse than G1 if and only if g2/g1 is decreasing, meaning that the distribution G2 of the outside option is smaller in the likelihood ratio order than the distribution G1. Second, observe that G2 is a (location–scale) shift to the left of G1 if and only if each r is replaced with a smaller value α+βr, meaning that each realization of the outside option is smaller proportionally and by a constant. Thus, the sender optimally reveals less information if the receiver becomes more willing to accept the proposal, in that the distribution of the outside option decreases either in the likelihood-ratio order (Proposition 1) or in the location–scale shift order (Proposition 2). To build intuition, consider the limit case where the density g is symmetric and highly concentrated around its mode x>E[ω], meaning that the outside option is close to x with high probability. In this case the optimal censorship cutoff ω*<x is such that the expected state of the pool m*=E[ω|ω≥ω*] is slightly above x, so that the receiver is very likely to accept the proposal. On the one hand, a further increase in m* reduces the probability that the state belongs to the pooling interval [ω*,1], but only negligibly increases the acceptance probability conditional on ω∈[ω*,1]. On the other hand, setting m*≤x drops the acceptance probability to below a half. This illustrates Lemma 2 asserting that x∈(ω*,m*). Now suppose that the distribution of the outside option either decreases in the likelihood-ratio order or in the location–scale shift order, so that the outside option is now close to some x˜<x with high probability. By the intuition provided above, the optimal censorship cutoff should then decrease so that the expected state of the pool falls from slightly above x to slightly above x˜. This illustrates Propositions 1 and 2. This section focuses on the additive representation of V given by 6V(m)=G(m)+ρ∫0mG(r)dr, where ρ∈R is a parameter, and G is a probability distribution function that has a strictly positive and continuously differentiable density g on [0,1]. To motivate this representation, Section 4.1 presents a micro-founded model with a privately informed receiver who chooses between two actions. Section 4.2 sets the stage by reinterpreting the results of Section 3 in the simple case where ρ=0. Section 4.3 specifies explicit conditions on ρ and G for upper censorship to be optimal and analyzes how changes in ρ and G affect Blackwell informativeness of the optimal signal.6 There are two players: a sender and a receiver. The receiver chooses whether to accept a proposal (a=1) or to reject it (a=0). The proposal has an uncertain value (state) ω∈[0,1] for the receiver. By accepting the proposal, the receiver forgoes an outside option worth r∈[0,1], which is the receiver's private information. The state and outside option are independent random variables whose probability distributions F and G have strictly positive probability densities f and g, with g being continuously differentiable. The receiver's and sender's utilities from a=0 are normalized to zero, and their utilities from a=1 are given by 7u(ω,r)=ω−randv(ω,r)=1+ρ(ω−r), where ρ∈R is an alignment parameter. That is, the sender's utility is a weighted sum of the action a=1 and the receiver's utility. The alignment parameter ρ captures the relative weight that the sender assigns to the receiver's utility. A sender with such preferences may correspond to a partially benevolent government as in Section 5 or to a recommender system that cares about both producer and consumer welfare. The timing is as follows. First, the sender publicly chooses a signal s, arbitrarily correlated with ω but independent of r. Then realizations of ω, r, and s are drawn. Finally, the receiver observes the realizations of his outside option r and the signal s, and then chooses between a=0 and a=1. Conditional on the expected state m, the receiver's expected utility from a=1 is m−r. So the receiver optimally chooses a=1 if and only if r≤m. Consequently, conditional on m, the sender's posterior expected utility is V(m)=∫0m(1+ρ(m−r))g(r)dr=G(m)+ρ∫0mG(r)dr. That is, the sender's Bernoulli utility function is indeed given by (6). A special case that appears in much of the persuasion literature is where the sender wishes to maximize the probability that the receiver accepts the proposal. This is the case of ρ=0, so v(ω,r)=1 and V(m)=G(m). Thus, V is S-shaped if and only if the density g of the outside option is unimodal, where the inflection point of V is the mode of g. By Theorem 1, this unimodality is necessary and sufficient for the optimality of upper censorship. In this case, the comparative statics results of Propositions 1 and 2 are also easy to interpret. Let g1 and g2 be unimodal densities of distributions G1 and G2 of the outside option. First, observe that G2 is more risk averse than G1 if and only if g2/g1 is decreasing, meaning that the distribution G2 of the outside option is smaller in the likelihood ratio order than the distribution G1. Second, observe that G2 is a (location–scale) shift to the left of G1 if and only if each r is replaced with a smaller value α+βr, meaning that each realization of the outside option is smaller proportionally and by a constant. Thus, the sender optimally reveals less information if the receiver becomes more willing to accept the proposal, in that the distribution of the outside option decreases either in the likelihood-ratio order (Proposition 1) or in the location–scale shift order (Proposition 2). To build intuition, consider the limit case where the density g is symmetric and highly concentrated around its mode x>E[ω], meaning that the outside option is close to x with high probability. In this case the optimal censorship cutoff ω*<x is such that the expected state of the pool m*=E[ω|ω≥ω*] is slightly above x, so that the receiver is very likely to accept the proposal. On the one hand, a further increase in m* reduces the probability that the state belongs to the pooling interval [ω*,1], but only negligibly increases the acceptance probability conditional on ω∈[ω*,1]. On the other hand, setting m*≤x drops the acceptance probability to below a half. This illustrates Lemma 2 asserting that x∈(ω*,m*). Now suppose that the distribution of the outside option either decreases in the likelihood-ratio order or in the location–scale shift order, so that the outside option is now close to some x˜<x with high probability. By the intuition provided above, the optimal censorship cutoff should then decrease so that the expected state of the pool falls from slightly above x to slightly above x˜. This illustrates Propositions 1 and 2. We now provide necessary and sufficient conditions for the optimality of upper censorship for the additive representation of V given by (6), without restrictions on ρ. A probability density g is (strictly) log-concave on a given interval if lng is (strictly) concave on that interval. Log-concavity is a common assumption in a variety of economic applications, such as voting, signalling, and monopoly pricing (see Section 7 in Bagnoli and Bergstrom (2005)). Log-concave densities exhibit nice properties, such as unimodality and hazard rate monotonicity. Many familiar probability density functions are log-concave (see Table 1 in Bagnoli and Bergstrom (2005)). The next theorem connects the optimality of upper censorship and the log-concavity of the density g. 2 Theorem Let V be given by (6). If g is (strictly) log-concave on [0,1], then upper censorship is (uniquely) optimal for each f and each ρ. Conversely, if g is not (strictly) log-concave on [0,1], then there exist f and ρ such that upper censorship is not (uniquely) optimal. Let V be given by (6). If g is (strictly) log-concave on [0,1], then upper censorship is (uniquely) optimal for each f and each ρ. Conversely, if g is not (strictly) log-concave on [0,1], then there exist f and ρ such that upper censorship is not (uniquely) optimal. The proof of Theorem 2 is immediate by Theorem 1 and the following lemma that links the S-shapedness of V and the log-concavity of g.7 3 Lemma Let V be given by (6). If g is (strictly) log-concave on [0,1], then V is (strictly) S-shaped on [0,1]. Conversely, if g is not (strictly) log-concave on [0,1], then there exists ρ such that V is not (strictly) S-shaped on [0,1]. Let V be given by (6). If g is (strictly) log-concave on [0,1], then V is (strictly) S-shaped on [0,1]. Conversely, if g is not (strictly) log-concave on [0,1], then there exists ρ such that V is not (strictly) S-shaped on [0,1]. Proof By (6), using the assumption that g is strictly positive on [0,1], we have, for m∈[0,1], 8V″(m)=g′(m)+ρg(m)=g(m)(g′(m)g(m)+ρ). If g is (strictly) log-concave on [0,1], then g′/g is (strictly) decreasing on [0,1]. So V″ (strictly) crosses the horizontal axis from above at most once on [0,1] and, thus, V is (strictly) S-shaped on [0,1]. Conversely, if g is not (strictly) log-concave on [0,1], then there exist m1<m2 such that g′/g is strictly (weakly) increasing on [m1,m2]. Choosing ρ=−g′(m)/g(m) for some m∈(m1,m2), we obtain that V is not (strictly) S-shaped on [0,1]. □ By (6), using the assumption that g is strictly positive on [0,1], we have, for m∈[0,1], 8V″(m)=g′(m)+ρg(m)=g(m)(g′(m)g(m)+ρ). If g is (strictly) log-concave on [0,1], then g′/g is (strictly) decreasing on [0,1]. So V″ (strictly) crosses the horizontal axis from above at most once on [0,1] and, thus, V is (strictly) S-shaped on [0,1]. Conversely, if g is not (strictly) log-concave on [0,1], then there exist m1<m2 such that g′/g is strictly (weakly) increasing on [m1,m2]. Choosing ρ=−g′(m)/g(m) for some m∈(m1,m2), we obtain that V is not (strictly) S-shaped on [0,1]. □ The conditions for the optimality of lower censorship are symmetric to those presented in Theorem 2. Specifically, if V is given by (6) and g is log-convex, then lower censorship is optimal. Thus, if g is both log-concave and log-convex (so it is exponential), then an optimal signal is both upper censorship and lower censorship. The only two signals with this property are full disclosure and no disclosure. This shows that the optimal signal is polarized between full disclosure and no disclosure (as, for example, in Lewis and Sappington (1994) and Johnson and Myatt (2006)) only in the knife edge case where the receiver's outside option has an exponential distribution. 2 Corollary Let V be given by (6). If g is exponential, in that g′(r)/g(r)=−λ for some λ∈R, then, for each density f, full disclosure is uniquely optimal for ρ>λ, no disclosure is uniquely optimal for ρ<λ, and all signals are optimal for ρ=λ. Conversely, if g is not exponential, then there exist f and ρ such that neither full disclosure nor no disclosure is optimal. Let V be given by (6). If g is exponential, in that g′(r)/g(r)=−λ for some λ∈R, then, for each density f, full disclosure is uniquely optimal for ρ>λ, no disclosure is uniquely optimal for ρ<λ, and all signals are optimal for ρ=λ. Conversely, if g is not exponential, then there exist f and ρ such that neither full disclosure nor no disclosure is optimal. We now adapt Propositions 1 and 2 to obtain comparative statics results for the additive representation. 3 Proposition Let V be given by (6) and let g be log-concave. The sender optimally reveals less information (i)if the alignment parameter ρ decreases (ii)if the distribution G of the outside option shifts to the left. Let V be given by (6) and let g be log-concave. The sender optimally reveals less information (i)if the alignment parameter ρ decreases (ii)if the distribution G of the outside option shifts to the left. The proof of part (i) shows that if ρ decreases, then V becomes more risk averse. Intuitively, if the sender puts a lower weight ρ on the receiver's preferences, she optimally endows the receiver with a lower utility by disclosing less information. The proof of part (ii) shows that if G shifts to the left, then V also shifts to the left and, in addition, becomes more risk averse. The intuition for part (ii) is the same as in the state-independent case of Section 4.2. Following Rayo and Segal (2010) and Kamenica and Gentzkow (2011), there has been a rapid growth of the literature on Bayesian persuasion (see Bergemann and Morris (2019) and Kamenica (2019) for excellent reviews). In a standard persuasion problem, a sender designs a signal about an uncertain state of the world to persuade a receiver. Much of this literature focuses on a special linear case where the utilities depend only on the expected state. A wide range of applications includes clinical trials, bank stress tests, school grading policies, quality certification, advertising strategies, transparency in organizations, persuasion of voters, and media control. In many of these applications, it is optimal to censor the states on one side of a cutoff and reveal the states on the other side of the cutoff. In this paper, we characterize necessary and sufficient conditions under which such censorship signals are optimal. A linear persuasion problem is described by a prior distribution of a one-dimensional state and the sender's expected utility (under the receiver's optimal action) as a function of the expected state. We show that if the sender's utility is S-shaped (that is, the sender's marginal utility is quasi-concave) in the expected state, then, and only then, an upper-censorship signal that pools the states above a cutoff and reveals the states below the cutoff is optimal for all prior distributions of the state.1 In addition, we perform comparative statics on the informativeness of the optimal censorship signal. When the sender's utility is S-shaped, the sender optimally reveals less information if she becomes more risk averse and if her utility shifts to the left. To interpret an S-shaped utility, we consider a setting where a privately informed receiver chooses one of two actions: to accept or to reject a proposal. The sender reveals information about the proposal's value to the receiver. By accepting the proposal, the receiver forgoes a privately known outside option. If the sender wishes to maximize the probability that the receiver accepts the proposal, then the sender's utility is equal to the distribution function of the outside option. It is S-shaped (and thus upper censorship is optimal) if and only if the probability density of the outside option is unimodal. Furthermore, the sender is more risk averse (and thus optimally reveals less information) if and only if the density of the outside option decreases in the likelihood ratio order. More generally, when the sender cares about a weighted sum of the receiver's utility and the probability that the proposal is accepted, the sender's utility is S-shaped for all possible weights if and only if the density of the outside option is log-concave. When this is the case, the sender optimally reveals less information if the weight she places on the receiver's utility falls and if each realization of the outside option decreases proportionally or by a constant, so that the receiver is more willing to accept the proposal. We apply our results to the problem of media censorship. We consider a stylized setting with heterogeneous citizens and media outlets. Media outlets differ in their approval standards. A partially benevolent government influences citizens' actions by choosing which media outlets to permit and which ones to censor. Each permitted media outlet approves the government when the state is above its approval standard and criticizes it otherwise. In this context, upper censorship means that the government censors only media outlets with sufficiently high approval standards. We show that the government optimally censors more media outlets if the society experiences an ideology shock in favor of the government and if influencing the society's decisions becomes relatively more important than maximizing the citizens' individual welfare. Related Literature. This paper is the first to provide necessary and sufficient conditions for the optimality of censorship that are prior-independent (as well as bias-independent in the setting with a privately informed receiver). This characterization unifies and generalizes sufficient conditions scattered across the literature. Under our conditions, the sender's problem is simply to find an optimal censorship cutoff. The independence of our conditions of the fine details of the environment enables a novel comparative statics analysis on the optimal censorship cutoff. In related work, Kolotilin (2018) characterizes prior-dependent conditions for the optimality of interval disclosure, which includes censorship as a special case. Moreover, Alonso and Câmara (2016b) provide sufficient conditions for the optimality of censorship in a more specific setting with discrete states. They also perform comparative statics with respect to a parallel shift, which is a special case of a location–scale shift considered in our paper. Our comparative statics results with respect to the sender's risk aversion and bias are novel. More broadly, this paper contributes to the literature on the linear persuasion problem highlighted by Kamenica and Gentzkow (2011).2 Using Blackwell's theorem, Gentzkow and Kamenica (2016) and Kolotilin et al. (2017) characterize outcomes implementable by public signals and private persuasion mechanisms, respectively. By formulating the linear persuasion problem as a linear program, Kolotilin (2018), Dworczak and Martini (2019), and Dizdar and Kováč (2020) establish strong duality, thereby providing the verification tool for the optimality of a candidate signal. Finally, Arieli, Babichenko, Smorodinsky, and Yamashita (2021) and Kleiner, Moldovanu, and Strack (2021) characterize extreme solutions to a linear persuasion problem, thereby narrowing down the set of candidate optimal signals. There is a diverse literature where censorship policies emerge as optimal signals in specific instances of the linear persuasion problem, starting from the prosecutor–judge example, as well as lobbying and product advertising examples, in Kamenica and Gentzkow (2011). Other contexts where censorship is optimal include grading policies (Ostrovsky and Schwarz (2010)), media control (Gehlbach and Sonin (2014), Ginzburg (2019), Gitmez and Molavi (2020)), clinical trials (Kolotilin (2015)), voter persuasion (Alonso and Câmara (2016a,b)), transparency benchmarks (Duffie, Dworczak, and Zhu (2017)), stress tests (Goldstein and Leitner (2018), Orlov, Zryumov, and Skrzypach (2021)), online markets (Romanyuk and Smolin (2019)), attention management (Lipnowski, Mathevet, and Wei (2020), Bloedel and Segal (2021)), quality certification (Zapechelnyuk (2020)), and relational communication (Kolotilin and Li (2021)). Finally, our application to media censorship contributes to the media economics literature, which we discuss in Section 5.3.

PY - 2022/5

Y1 - 2022/5

N2 - We consider a Bayesian persuasion problem where a sender's utility depends only on the expected state. We show that upper censorship that pools the states above a cutoff and reveals the states below the cutoff is optimal for all prior distributions of the state if and only if the sender's marginal utility is quasi-concave. Moreover, we show that it is optimal to reveal less information if the sender becomes more risk averse or the sender's utility shifts to the left. Finally, we apply our results to the problem of media censorship by a government.

AB - We consider a Bayesian persuasion problem where a sender's utility depends only on the expected state. We show that upper censorship that pools the states above a cutoff and reveals the states below the cutoff is optimal for all prior distributions of the state if and only if the sender's marginal utility is quasi-concave. Moreover, we show that it is optimal to reveal less information if the sender becomes more risk averse or the sender's utility shifts to the left. Finally, we apply our results to the problem of media censorship by a government.

KW - Bayesian persuasion

KW - censorship

KW - D82

KW - D83

KW - information design

KW - L82

KW - media

UR - http://www.scopus.com/inward/record.url?scp=85130592802&partnerID=8YFLogxK

U2 - 10.3982/TE4071

DO - 10.3982/TE4071

M3 - Article

AN - SCOPUS:85130592802

SN - 1933-6837

VL - 17

SP - 561

EP - 585

JO - Theoretical Economics

JF - Theoretical Economics

IS - 2

ER -