- Email: [email protected]
A model of Bayesian persuasion with transfers
Accepted Manuscript A model of Bayesian persuasion with transfers Cheng Li
PII: DOI: Reference:
S0165-1765(17)30412-3 https://doi.org/10.1016/j.econlet.2017.09.036 ECOLET 7792
To appear in:
Economics Letters
Received date : 6 July 2017 Revised date : 15 September 2017 Accepted date : 29 September 2017 Please cite this article as: Li C., A model of Bayesian persuasion with transfers. Economics Letters (2017), https://doi.org/10.1016/j.econlet.2017.09.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A sender wants to persuade a receiver to take an action.
The sender first produces information about the benefits of the action and then offers monetary transfers to the receiver.
We characterize a sender-optimal information structure.
We show that limiting the sender’s monetary payments may incentivize the sender to produce more information.
*Title Page
A Model of Bayesian Persuasion with Transfers Cheng Li*
Abstract We analyze a persuasion game in which a sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of taking the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information. Keywords: Bayesian persuasion; monetary transfers; signal informativeness JEL Codes: D83
* Mississippi
State University, Department of Finance and Economics, Mississippi State, MS 39762.
Email: [email protected].
1
*Manuscript Click here to view linked References
A Model of Bayesian Persuasion with Transfers Cheng Li*
Abstract We analyze a persuasion game in which a sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of taking the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information. Keywords: Bayesian persuasion; monetary transfers; signal informativeness JEL Codes: D83
* Mississippi
State University, Department of Finance and Economics, Mississippi State, MS 39762.
Email: [email protected].
1
1
Introduction
The seminal work by Kamenica and Gentzkow (2011) studies a Bayesian persuasion model in which a sender chooses which information to communicate to a receiver, who then takes an action that affects the payoffs of both players. In their model, the sender is not allowed to make monetary transfers to the receiver. In many situations, however, an agent can influence a decision maker’s action through both information production and monetary transfers. For example, a lobbying group can influence a politician’s policy decision by providing both policy relevant information and campaign contributions. A salesman can persuade a consumer to buy a product by offering both product information and price discounts. Motivated by these examples, we develop a simple model of Bayesian persuasion with transfers. A sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information.
2
Model
Consider a persuasion game with two players: Sender (he) and Receiver (she). Sender wants to persuade Receiver to take action A. In state 1, action A is “beneficial” to Receiver: it gives her benefits B > 0. In state 2, action A is “harmful” to Receiver: it gives her a negative payoff −L < 0. If Receiver does not take action A, her payoff is 0.
We use a ∈ {1, 0} to denote Receiver’s action. a = 1 when Receiver takes action A, and
a = 0 otherwise. We denote the state of the world by ω ∈ {1, 2}. Although ω is ex ante unknown to all players, it is common knowledge that Pr(ω = 1) = µ0 ∈ (0, 1). We use q0 to denote Receiver’s ex ante expected benefits from taking action A as follows: q0 = µ0 B − (1 − µ0 )L.
(1)
Sender prefers Receiver to take action A regardless of ω. He receives benefits of v > 0 when Receiver takes action A and receives benefits of 0 otherwise. In the first stage of the game, Sender designs signal S, which produces a public realization s. 2
Signal S consists of a pair of state-dependent random variables: S1 and S2 . When ω = 1, s is drawn from S1 , and when ω = 2, s is drawn from S2 . The key difference between Kamenica and Gentzkow (2011) and our model is that we allow monetary transfers from Sender to Receiver. After observing signal realization s, Sender offers Receiver monetary payment c ≥ 0, which he commits to pay if Receiver takes action A.1 Sender’s payoff uS is v − βc when a = 1 uS (a, c) = 0 when a = 0 ,
(2)
where β > 0 represents Sender’s cost of monetary payments. Receiver’s payoff uR is B + αc when a = 1 and ω = 1 uR (a, ω, c) = −L + αc when a = 1 and ω = 2 0 when a = 0 ,
(3)
where α > 0 represents Receiver’s value of money. The game takes place in three stages. First, Sender designs signal S. Second, signal realization s is publicly revealed, and Sender offers monetary payment c to Receiver. Third, Receiver observes signal S, signal realization s, and monetary offer c and then decides whether to take action A. After observing a signal realization, Receiver updates her beliefs about ω with the Bayes’ rule. We use µ to denote Receiver’s posterior belief that ω = 1 and q to denote Receiver’s ex post expected benefits of action A: q = µB − (1 − µ)L.
(4)
We use random variable Q to represent the distribution of q generated by signal S. Because Q’s realization is q ∈ [−L, B], its support must be a subset of interval
[−L, B]. Additionally, according to the law of total expectation, the expected value of Q must be equal to the prior, that is, E[Q] = q0 . We say that random variable Q is Bayes-plausible if its support is a subset of interval [−L, B] and E[Q] = q0 . 1 The
contingent payment assumption is standard in the influence games studied in Grossman and
Helpman (1994) and Bennedsen and Feldmann (2006).
3
Lemma 1 For any Bayes-plausible random variable Q, there exists a signal S for which Q is the distribution of q. For any signal S, there exists a unique Bayes-plausible random variable Q that represents the distribution of q generated by S. Lemma 1 is adapted from Cotton and Li (2016). It greatly simplifies our analysis. To determine Sender’s optimal signal, it is sufficient to find Sender’s optimal Bayesplausible random variable.
3
Equilibrium
We use backward induction to solve the subgame perfect equilibrium of the game.
3.1
Monetary payments
Given the expected benefits of action A and Sender’s monetary offer, Receiver takes action A when q + αc ≥ 0.
(5)
The equilibrium of the second-stage game involves c = 0, a = 1 when q ≥ 0, q
c = − α , a = 1 when − αv β ≤ q < 0, c = βv , a = 0 when q < − αv β .
3.2
Optimal signal
In the first stage of the game, Sender designs signal S to maximize his expected payoff
v when q ≥ 0 β αv EuS (q) = v + α q when − β ≤ q < 0 0 when q < − αv β
(6)
Since q ∈ [−L, B], we can divide our analysis into two cases. Figure 1 shows EuS (q)
when α < βL/v. In this case, the concave closure of EuS is v v + L q0 if q0 ∈ [−L, 0) C(q0 ) = v if q0 ∈ [0, B] 4
(7)
The concave closure C(q0 ) is the smallest concave function that is no less than EuS everywhere. It gives the highest payoff that Sender can achieve at prior q0 . EuS (q), C(q0 ) v
0 −L
− αv β
0
B
q, q0
Figure 1: EuS (q) (blue line) and its concave closure C(q0 ) (red line) when α < βL/v. When q0 ≥ 0, Sender has multiple optimal signals. All signals that always generate
non-negative realizations give Sender the highest payoff v. The most straightforward signal of this type is a completely uninformative one (i.e., Q = q0 with probability one). If q0 < 0, Sender has a uniquely optimal signal, denoted by S ∗ . This signal induces random variable Q that generates q = −L with probability −q0 /L and q = 0 with probability 1 + q0 /L. Probability −q0 /L ∈ [0, 1] because L > 0 and q0 ∈ [−L, 0]. In the
Appendix, we prove that signal S ∗ is uniquely optimal when α < βL/v and q0 < 0.
Figure 2 shows EuS and its concave closure when α ≥ βL/v. It is straightforward
to see that Sender has multiple optimal signals when q0 ≥ 0, because all signals that always generate non-negative realizations give Sender the highest payoff v.
When q0 < 0, Sender also has multiple optimal signals. Any Bayes-plausible random variable Q that always generates non-positive realizations gives Sender the expected payoff v + q0 , which is the highest the Sender can achieve given prior q0 (i.e., C(q0 )). Among these signals, the most informative signal is S ∗ , which induces q = −L
with probability −q0 /L and q = 0 with probability 1 + q0 /L. S ∗ is more informative
than other optimal signals because it leads to a more disperse distribution of q. Our measurement of signal informativeness follows Ganuza and Penalva (2010), who measure signal informativeness based on the property that more informative signals 5
EuS (q), C(q0 ) v
β
v − αL 0 −L
0
B
q, q0
Figure 2: EuS (q) (blue line) and its concave closure C(q0 ) (also in blue) when α ≥ βL/v. lead to greater variability of conditional expectations. Proposition 1 Sender’s optimal signal is as follows: 1. When q0 ≥ 0, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-negative realizations.
2. When q0 < 0 and α ≥ βL/v, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-positive realizations.
3. When q0 < 0 and α < βL/v, Sender has a uniquely optimal signal S ∗ that induces −L with probability − q0 /L Q= 0 with probability 1 + q0 /L
4
Limits on monetary transfers
In this section, we consider a limit on monetary transfers, such as a campaign contribution limit. We denote the limit by c¯ and assume that c¯ is binding: c¯ < v/β. Such a limit changes the above analysis in that we need to replace v/β, which ¯ which represents Sender’s represents Sender’s maximum willingness to pay, with c, maximum allowed payment.
6
Proposition 2 With monetary transfer limit c¯ < v/β, Sender’s optimal signal is as follows: 1. When q0 ≥ 0, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-negative realizations.
2. When q0 < 0 and c¯ ≥ L/α, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-positive realizations.
3. When q0 < 0 and c¯ < L/α, Sender has a uniquely optimal signal S ∗ that induces −L with probability − q0 /L Q= 0 with probability 1 + q0 /L Next, we show that a limit on monetary transfers may incentivize Sender to produce more information. If q0 < 0 and α ≥ βL/v, Sender has multiple optimal
signals when there is no limit on monetary transfers. Among these signals, the most informative one is S ∗ . When q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α makes Sender always choose signal S ∗ . Therefore, when q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α
either incentivizes Sender to choose a more informative signal or has no impact on Sender’s signal design. Specifically, the limit c¯ < L/α incentivizes Sender to choose a more informative signal if Sender chooses signals other than S ∗ in the game without monetary transfer limit, and there is no impact on Sender’s signal design if Sender chooses signal S ∗ in the game without monetary transfer limit. Proposition 3 When q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α incentivizes Sender to
choose a more informative signal if Sender chooses signals other than S ∗ in the game without monetary transfer limit, and there is no impact on Sender’s signal design if Sender chooses signal S ∗ in the game without monetary transfer limit.
5
Conclusions
The contribution of the paper is twofold. First, we characterize Sender’s optimal signal in a model of Bayesian persuasion with transfers. Second, we show that limiting monetary transfers may incentivize Sender to produce more information.
7
Acknowledgments I appreciate comments from an anonymous referee and research assistance from Brandon Bolen. All errors are my own.
References Bennedsen, Morten, and Sven E. Feldmann. 2006. “Informational Lobbying and Political Contributions.” Journal of Public Economics, 90: 631–656. Cotton, Christopher, and Cheng Li. 2016. “Clueless Politicians.” Queen’s University Working Paper. Ganuza, Juan-Jose, and Jose S. Penalva. 2010. “Signal Orderings Based on Dispersion and Private Information Disclosure in Auctions.” Econometrica, 78(3): 1007– 1030. Grossman, Gene M., and Elhanan Helpman. 1994. “Protection for Sale.” American Economic Review, 84(4): 833–850. Kamenica, Emir, and Matthew Gentzkow. 2011. “Bayesian Persuasion.” American Economic Review, 101(6): 2590–2615.
Appendix In the Appendix, we prove that signal S ∗ is uniquely optimal when α < βL/v and q0 < 0. Suppose S ∗ generates posterior belief distribution Q∗ . First, we show that Q∗ does not generate realizations of q ∈ (−αv/β, 0). Suppose Q∗ generates a realization
˜ q˜ ∈ (−αv/β, 0). q˜ gives Sender payoff v + β q/α. Now suppose Sender shifts probability
weight from realization q˜ to realizations −L and 0. To keep Q∗ Bayes-plausible, Sender ˜ on realization −L and probability 1 + q/L ˜ on realization must put probability −q/L ˜ which is higher than v + β q/α ˜ 0. This gives Sender the expected payoff of v + v q/L, when q˜ < 0 and α < βL/v. This contradicts that Q∗ generates q˜ ∈ (−αv/β, 0).
Now suppose that Q∗ generates realizations of q ≥ 0 with probability r and
realizations of q ≤ −αv/β with probability 1 − r. Such a Q∗ gives Sender expected 8
payoff rv. An optimal signal must give Sender expected payoff C(q0 ) = v + vq0 /L. By setting rv = v + vq0 /L, we have r = 1 + q0 /L. Therefore, E[Q∗ ] = (1 +
q0 q αv )E[q|q ≥ 0] − 0 E[q|q ≤ − ]. L L β
(8)
In equation (8), E[q|q ≥ 0] ≥ 0 and E[q|q ≤ −αv/β] ≥ −L. This implies that E[Q∗ ] ≥
(−q0 /L)(−L) = q0 . Bayes-plausibility requires E[Q∗ ] = q0 . This is the case if and only if E[q|q ≥ 0] = 0 and E[q|q ≤ −αv/β] = −L. Therefore, Q∗ is unique: it generates q = 0
with probability 1 + q0 /L and q = −L with probability −q0 /L.
9
PII: DOI: Reference:
S0165-1765(17)30412-3 https://doi.org/10.1016/j.econlet.2017.09.036 ECOLET 7792
To appear in:
Economics Letters
Received date : 6 July 2017 Revised date : 15 September 2017 Accepted date : 29 September 2017 Please cite this article as: Li C., A model of Bayesian persuasion with transfers. Economics Letters (2017), https://doi.org/10.1016/j.econlet.2017.09.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A sender wants to persuade a receiver to take an action.
The sender first produces information about the benefits of the action and then offers monetary transfers to the receiver.
We characterize a sender-optimal information structure.
We show that limiting the sender’s monetary payments may incentivize the sender to produce more information.
*Title Page
A Model of Bayesian Persuasion with Transfers Cheng Li*
Abstract We analyze a persuasion game in which a sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of taking the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information. Keywords: Bayesian persuasion; monetary transfers; signal informativeness JEL Codes: D83
* Mississippi
State University, Department of Finance and Economics, Mississippi State, MS 39762.
Email: [email protected].
1
*Manuscript Click here to view linked References
A Model of Bayesian Persuasion with Transfers Cheng Li*
Abstract We analyze a persuasion game in which a sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of taking the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information. Keywords: Bayesian persuasion; monetary transfers; signal informativeness JEL Codes: D83
* Mississippi
State University, Department of Finance and Economics, Mississippi State, MS 39762.
Email: [email protected].
1
1
Introduction
The seminal work by Kamenica and Gentzkow (2011) studies a Bayesian persuasion model in which a sender chooses which information to communicate to a receiver, who then takes an action that affects the payoffs of both players. In their model, the sender is not allowed to make monetary transfers to the receiver. In many situations, however, an agent can influence a decision maker’s action through both information production and monetary transfers. For example, a lobbying group can influence a politician’s policy decision by providing both policy relevant information and campaign contributions. A salesman can persuade a consumer to buy a product by offering both product information and price discounts. Motivated by these examples, we develop a simple model of Bayesian persuasion with transfers. A sender wants to persuade a receiver to take an action. The sender first produces information about the benefits of the action and then offers monetary transfers to the receiver. We characterize a sender-optimal information structure and show that limiting monetary payments may incentivize the sender to produce more information.
2
Model
Consider a persuasion game with two players: Sender (he) and Receiver (she). Sender wants to persuade Receiver to take action A. In state 1, action A is “beneficial” to Receiver: it gives her benefits B > 0. In state 2, action A is “harmful” to Receiver: it gives her a negative payoff −L < 0. If Receiver does not take action A, her payoff is 0.
We use a ∈ {1, 0} to denote Receiver’s action. a = 1 when Receiver takes action A, and
a = 0 otherwise. We denote the state of the world by ω ∈ {1, 2}. Although ω is ex ante unknown to all players, it is common knowledge that Pr(ω = 1) = µ0 ∈ (0, 1). We use q0 to denote Receiver’s ex ante expected benefits from taking action A as follows: q0 = µ0 B − (1 − µ0 )L.
(1)
Sender prefers Receiver to take action A regardless of ω. He receives benefits of v > 0 when Receiver takes action A and receives benefits of 0 otherwise. In the first stage of the game, Sender designs signal S, which produces a public realization s. 2
Signal S consists of a pair of state-dependent random variables: S1 and S2 . When ω = 1, s is drawn from S1 , and when ω = 2, s is drawn from S2 . The key difference between Kamenica and Gentzkow (2011) and our model is that we allow monetary transfers from Sender to Receiver. After observing signal realization s, Sender offers Receiver monetary payment c ≥ 0, which he commits to pay if Receiver takes action A.1 Sender’s payoff uS is v − βc when a = 1 uS (a, c) = 0 when a = 0 ,
(2)
where β > 0 represents Sender’s cost of monetary payments. Receiver’s payoff uR is B + αc when a = 1 and ω = 1 uR (a, ω, c) = −L + αc when a = 1 and ω = 2 0 when a = 0 ,
(3)
where α > 0 represents Receiver’s value of money. The game takes place in three stages. First, Sender designs signal S. Second, signal realization s is publicly revealed, and Sender offers monetary payment c to Receiver. Third, Receiver observes signal S, signal realization s, and monetary offer c and then decides whether to take action A. After observing a signal realization, Receiver updates her beliefs about ω with the Bayes’ rule. We use µ to denote Receiver’s posterior belief that ω = 1 and q to denote Receiver’s ex post expected benefits of action A: q = µB − (1 − µ)L.
(4)
We use random variable Q to represent the distribution of q generated by signal S. Because Q’s realization is q ∈ [−L, B], its support must be a subset of interval
[−L, B]. Additionally, according to the law of total expectation, the expected value of Q must be equal to the prior, that is, E[Q] = q0 . We say that random variable Q is Bayes-plausible if its support is a subset of interval [−L, B] and E[Q] = q0 . 1 The
contingent payment assumption is standard in the influence games studied in Grossman and
Helpman (1994) and Bennedsen and Feldmann (2006).
3
Lemma 1 For any Bayes-plausible random variable Q, there exists a signal S for which Q is the distribution of q. For any signal S, there exists a unique Bayes-plausible random variable Q that represents the distribution of q generated by S. Lemma 1 is adapted from Cotton and Li (2016). It greatly simplifies our analysis. To determine Sender’s optimal signal, it is sufficient to find Sender’s optimal Bayesplausible random variable.
3
Equilibrium
We use backward induction to solve the subgame perfect equilibrium of the game.
3.1
Monetary payments
Given the expected benefits of action A and Sender’s monetary offer, Receiver takes action A when q + αc ≥ 0.
(5)
The equilibrium of the second-stage game involves c = 0, a = 1 when q ≥ 0, q
c = − α , a = 1 when − αv β ≤ q < 0, c = βv , a = 0 when q < − αv β .
3.2
Optimal signal
In the first stage of the game, Sender designs signal S to maximize his expected payoff
v when q ≥ 0 β αv EuS (q) = v + α q when − β ≤ q < 0 0 when q < − αv β
(6)
Since q ∈ [−L, B], we can divide our analysis into two cases. Figure 1 shows EuS (q)
when α < βL/v. In this case, the concave closure of EuS is v v + L q0 if q0 ∈ [−L, 0) C(q0 ) = v if q0 ∈ [0, B] 4
(7)
The concave closure C(q0 ) is the smallest concave function that is no less than EuS everywhere. It gives the highest payoff that Sender can achieve at prior q0 . EuS (q), C(q0 ) v
0 −L
− αv β
0
B
q, q0
Figure 1: EuS (q) (blue line) and its concave closure C(q0 ) (red line) when α < βL/v. When q0 ≥ 0, Sender has multiple optimal signals. All signals that always generate
non-negative realizations give Sender the highest payoff v. The most straightforward signal of this type is a completely uninformative one (i.e., Q = q0 with probability one). If q0 < 0, Sender has a uniquely optimal signal, denoted by S ∗ . This signal induces random variable Q that generates q = −L with probability −q0 /L and q = 0 with probability 1 + q0 /L. Probability −q0 /L ∈ [0, 1] because L > 0 and q0 ∈ [−L, 0]. In the
Appendix, we prove that signal S ∗ is uniquely optimal when α < βL/v and q0 < 0.
Figure 2 shows EuS and its concave closure when α ≥ βL/v. It is straightforward
to see that Sender has multiple optimal signals when q0 ≥ 0, because all signals that always generate non-negative realizations give Sender the highest payoff v.
When q0 < 0, Sender also has multiple optimal signals. Any Bayes-plausible random variable Q that always generates non-positive realizations gives Sender the expected payoff v + q0 , which is the highest the Sender can achieve given prior q0 (i.e., C(q0 )). Among these signals, the most informative signal is S ∗ , which induces q = −L
with probability −q0 /L and q = 0 with probability 1 + q0 /L. S ∗ is more informative
than other optimal signals because it leads to a more disperse distribution of q. Our measurement of signal informativeness follows Ganuza and Penalva (2010), who measure signal informativeness based on the property that more informative signals 5
EuS (q), C(q0 ) v
β
v − αL 0 −L
0
B
q, q0
Figure 2: EuS (q) (blue line) and its concave closure C(q0 ) (also in blue) when α ≥ βL/v. lead to greater variability of conditional expectations. Proposition 1 Sender’s optimal signal is as follows: 1. When q0 ≥ 0, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-negative realizations.
2. When q0 < 0 and α ≥ βL/v, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-positive realizations.
3. When q0 < 0 and α < βL/v, Sender has a uniquely optimal signal S ∗ that induces −L with probability − q0 /L Q= 0 with probability 1 + q0 /L
4
Limits on monetary transfers
In this section, we consider a limit on monetary transfers, such as a campaign contribution limit. We denote the limit by c¯ and assume that c¯ is binding: c¯ < v/β. Such a limit changes the above analysis in that we need to replace v/β, which ¯ which represents Sender’s represents Sender’s maximum willingness to pay, with c, maximum allowed payment.
6
Proposition 2 With monetary transfer limit c¯ < v/β, Sender’s optimal signal is as follows: 1. When q0 ≥ 0, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-negative realizations.
2. When q0 < 0 and c¯ ≥ L/α, Sender has multiple optimal signals. The optimal signals induce Bayes-plausible random variable Q with non-positive realizations.
3. When q0 < 0 and c¯ < L/α, Sender has a uniquely optimal signal S ∗ that induces −L with probability − q0 /L Q= 0 with probability 1 + q0 /L Next, we show that a limit on monetary transfers may incentivize Sender to produce more information. If q0 < 0 and α ≥ βL/v, Sender has multiple optimal
signals when there is no limit on monetary transfers. Among these signals, the most informative one is S ∗ . When q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α makes Sender always choose signal S ∗ . Therefore, when q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α
either incentivizes Sender to choose a more informative signal or has no impact on Sender’s signal design. Specifically, the limit c¯ < L/α incentivizes Sender to choose a more informative signal if Sender chooses signals other than S ∗ in the game without monetary transfer limit, and there is no impact on Sender’s signal design if Sender chooses signal S ∗ in the game without monetary transfer limit. Proposition 3 When q0 < 0 and α ≥ βL/v, a limit of c¯ < L/α incentivizes Sender to
choose a more informative signal if Sender chooses signals other than S ∗ in the game without monetary transfer limit, and there is no impact on Sender’s signal design if Sender chooses signal S ∗ in the game without monetary transfer limit.
5
Conclusions
The contribution of the paper is twofold. First, we characterize Sender’s optimal signal in a model of Bayesian persuasion with transfers. Second, we show that limiting monetary transfers may incentivize Sender to produce more information.
7
Acknowledgments I appreciate comments from an anonymous referee and research assistance from Brandon Bolen. All errors are my own.
References Bennedsen, Morten, and Sven E. Feldmann. 2006. “Informational Lobbying and Political Contributions.” Journal of Public Economics, 90: 631–656. Cotton, Christopher, and Cheng Li. 2016. “Clueless Politicians.” Queen’s University Working Paper. Ganuza, Juan-Jose, and Jose S. Penalva. 2010. “Signal Orderings Based on Dispersion and Private Information Disclosure in Auctions.” Econometrica, 78(3): 1007– 1030. Grossman, Gene M., and Elhanan Helpman. 1994. “Protection for Sale.” American Economic Review, 84(4): 833–850. Kamenica, Emir, and Matthew Gentzkow. 2011. “Bayesian Persuasion.” American Economic Review, 101(6): 2590–2615.
Appendix In the Appendix, we prove that signal S ∗ is uniquely optimal when α < βL/v and q0 < 0. Suppose S ∗ generates posterior belief distribution Q∗ . First, we show that Q∗ does not generate realizations of q ∈ (−αv/β, 0). Suppose Q∗ generates a realization
˜ q˜ ∈ (−αv/β, 0). q˜ gives Sender payoff v + β q/α. Now suppose Sender shifts probability
weight from realization q˜ to realizations −L and 0. To keep Q∗ Bayes-plausible, Sender ˜ on realization −L and probability 1 + q/L ˜ on realization must put probability −q/L ˜ which is higher than v + β q/α ˜ 0. This gives Sender the expected payoff of v + v q/L, when q˜ < 0 and α < βL/v. This contradicts that Q∗ generates q˜ ∈ (−αv/β, 0).
Now suppose that Q∗ generates realizations of q ≥ 0 with probability r and
realizations of q ≤ −αv/β with probability 1 − r. Such a Q∗ gives Sender expected 8
payoff rv. An optimal signal must give Sender expected payoff C(q0 ) = v + vq0 /L. By setting rv = v + vq0 /L, we have r = 1 + q0 /L. Therefore, E[Q∗ ] = (1 +
q0 q αv )E[q|q ≥ 0] − 0 E[q|q ≤ − ]. L L β
(8)
In equation (8), E[q|q ≥ 0] ≥ 0 and E[q|q ≤ −αv/β] ≥ −L. This implies that E[Q∗ ] ≥
(−q0 /L)(−L) = q0 . Bayes-plausibility requires E[Q∗ ] = q0 . This is the case if and only if E[q|q ≥ 0] = 0 and E[q|q ≤ −αv/β] = −L. Therefore, Q∗ is unique: it generates q = 0
with probability 1 + q0 /L and q = −L with probability −q0 /L.
9