- Email: [email protected]
Sequential cheap talks
Games and Economic Behavior 90 (2015) 128–133
Contents lists available at ScienceDirect
Games and Economic Behavior www.elsevier.com/locate/geb
Note
Sequential cheap talks Keiichi Kawai 1 University of New South Wales, UNSW Business School, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 18 October 2011 Available online 14 February 2015 JEL classification: C72 D82
a b s t r a c t In this note, we analyze a multidimensional cheap talk game where two senders sequentially submit messages. We provide a necessary and sufficient condition for the existence of a fully-revealing equilibrium. © 2015 Elsevier Inc. All rights reserved.
Keywords: Multidimensional cheap talk Sequential messages
1. Introduction In this note, we analyze a multidimensional cheap talk game where two senders sequentially submit messages. We show that a fully-revealing equilibrium exists if and only if the senders’ biases are opposing. That is, the product of the vectors that represent senders’ biases are negative. As shown in Krishna and Morgan (2001), when the state space is one-dimensional Euclidean space, a fully-revealing equilibrium exists if and only if two senders’ biases are opposing. We show that the analysis of Krishna and Morgan (2001) for the one-dimensional Euclidean space can easily be extended to a general n-dimensional Euclidean state space. It is well known that multidimensional cheap talk games have positive results on information transmission when the senders simultaneously send messages.2 When messages are submitted sequentially, one might think it is impossible to achieve the fully-revealing equilibrium, even when the biases are opposing, because there exists an action that both senders strictly prefer over the receiver’s ideal action.3 We nonetheless show that a fully-revealing equilibrium exists if and only if the senders’ biases are opposing. The result follows from an observation that the conflict of interests between the decision maker and the second sender is essentially one-dimensional. More precisely, we can always transform the coordinate system so that the decision maker’s actions that are ideal for the decision maker and the second sender differ in only one coordinate. This observation implies that the existence of a fully-revealing equilibrium boils down to whether or not the decision maker can successfully solicit information regarding the coordinate on which the decision maker and the second sender have conflict of interests. This in turn enables us to fully utilize the concept of self-serving messages characterized in Krishna and Morgan (2001), i.e., the set of second sender’s messages that the decision maker disregards. As a result, the set
E-mail address: [email protected]. The author acknowledges the comments from Shintaro Miura, Carlos Oyarzun and Satoru Takahashi. The author greatly appreciates an anonymous referee for the valuable comments, which led to a substantial improvement on the previous version of the article. 2 See Battaglini (2002). 3 Note that such an action does not exist when the state space is one-dimensional. 1
http://dx.doi.org/10.1016/j.geb.2015.02.007 0899-8256/© 2015 Elsevier Inc. All rights reserved.
K. Kawai / Games and Economic Behavior 90 (2015) 128–133
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Fig. 1. Normalization of x B .
of second sender’s messages that the decision maker disregards in this note has a direct analogy to the one-dimensional Euclidean space model. This is a clear contrast to the constructive proof by Miura (2014) that has independently shown the same result as in this note. In Miura (2014), the construction of the set of second sender’s messages that the decision maker disregards heavily relies on the nature of quadratic-loss payoff functions. Consequently, the analysis in Miura (2014) has two limitations. First, it does not provide a direct analogy to the one-dimensional Euclidean space model. Moreover, it is only applicable to quadratic-loss payoff functions. In contrast, even though we only present the result for the quadratic-loss payoff functions, the analysis presented in this note can easily be generalized to more general payoff functions. 2. Model We consider the following n-dimensional cheap talk game with sequential communication. There are three players: two senders, A and B, and a decision maker (DM, henceforth). By u θi (a), we denote player i’s payoff when the true state is θ and action taken by the DM is a. We assume that both the state space and action space of the DM are Rn , and payoff n functions are quadratic-loss, that is, u θi (a) = − j =1 (a − (θ + xi ))2 , i = D M , A , B, and x D M = 0.4 The game proceeds as follows: 1. 2. 3. 4.
Senders learn the true state θ ; Sender A sends a message m A : θ → m A ∈ Rn ; Sender B sends a message m B : θ × m A → m B ∈ Rn ; DM takes an action a : m A × m B → a ∈ Rn .
The equilibrium concept we use is a perfect Bayesian equilibrium. 3. Fully-revealing equilibrium For the rest of the paper, we normalize x B = (x B1 , 0, 0, · · · , 0), x B1 ≥ 0. That is, we use an orthogonal basis so that the conflict of interests between the DM and Sender B is about the first-dimension alone. This normalization is without loss of generality, but simplifies notations.5 With such a normalization, the existence of a fully-revealing equilibrium boils down to the question of whether the DM can elicit the truthful report with respect to the first-dimension. As it turns out, a fully-revealing equilibrium exists if and only if two senders’ biases are opposing, i.e., x A · x B ≤ 0. Theorem 1. There exists a fully-revealing equilibrium if and only if x A · x B ≤ 0. To understand the intuition behind the “if” part of the result, it is useful to review the case where n = 1. As shown in Krishna and Morgan (2001), a fully-revealing equilibrium exists if and only if x A < 0 ≤ x B , i.e., x A · x B ≤ 0. A strategy profile that supports the fully-revealing equilibrium is as follows:
4 The result in this note can be easily generalized for more general payoff functions. But for expositional simplicity, we limit our attention to the quadratic-loss payoff functions. 5 This can be done by simply “rotating” the coordinates. See Fig. 1 for n = 2 case.
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K. Kawai / Games and Economic Behavior 90 (2015) 128–133
Fig. 2. m A ∈ [θ − x B , θ).
m A (θ) = θ
⎧ θ + xB ⎪ ⎪ ⎨ m A + 2x B m B (θ, m A ) = mA ⎪ ⎪ ⎩ θ + xB m A if m B a (m A , m B ) = mB
if m A if m A if m A if m A
< θ − xB ∈ [θ − x B , θ) ∈ [θ, θ + x B ) ≥ θ + xB
∈ (m A , m A + 2x B ) otherwise
Suppose the DM tentatively believes that m A = θ . Then under the hypothesis that Sender A is telling the truth, adoption m m of m B strictly benefits Sender B if and only if m B ∈ (m A , m A + 2x B ). That is, u B A (m B ) > u B A (m A ) if and only if m B ∈ (m A , m A + 2x B ). Krishna and Morgan (2001) term such a message “self-serving.” In the strategy profile described above, the DM ignores any self-serving message, but follows any non-self-serving message. Note that an untruthful message by Sender A will be countered by Sender B’s non-self-serving message. To see why it is optimal for Sender B to do so, note that following m A , Sender B can induce any action m B ∈ / (m A , m A + 2x B ). Therefore, Sender B chooses a message m B ∈ / (m A , m A + 2x B ) that is closest to Sender B’s ideal point θ + x B . For example, if m A ∈ [θ − x B , θ), then it is optimal for Sender B to send m B = m A + 2x B .6 When m A = θ , however, there exists no non-self-serving message m B such that u θB (m B ) > u θB (m A ). That is, Sender B cannot do better than truthfully reporting θ . Now we see why Sender A does not send m A < θ . If m A ∈ [θ − x B , θ), then such a message will be countered by a non-self-serving message m A + 2x B . Since the DM adopts Sender B’s message, the resulting payoff is u θA (m A + 2x B ), which is lower than u θA (θ). If Sender A sends m A = θ instead, Sender B does not have any incentive to send a non-self-serving message, and hence sends m B = θ . Therefore, Sender A yields u θA (θ). Similarly, Sender A does not send m A < θ − x B . That Sender A has no incentive to send m A > θ is straightforward. This is how a fully-revealing equilibrium is sustained when n = 1. The intuition behind the case with n ≥ 2 is quite similar. Since the generalization of the result from n = 2 to n ≥ 3 is straightforward, we restrict our attention to the case with n = 2 for expositional simplicity. To see why there is a fully-revealing equilibrium when x A · x B ≤ 0, consider the following equilibrium strategy profiles for n = 2.
m A (θ) = θ;
⎧ if m A1 < θ1 − x B1 θ + x B1 ⎪ ⎪ ⎨ 1 m A1 + 2x B1 if m A1 ∈ [θ1 − x B1 , θ1 ) m B1 (θ, m A ) = m A1 if m A1 ∈ [θ1 , θ1 + x B1 ) ⎪ ⎪ ⎩ if m A1 ≥ θ1 + x B1 θ1 + x B1 m B2 (θ, m A ) = θ2 ; m A m B ∈ S (m A ) a (m A , m B ) = mB
otherwise
where S (m A ) ≡ {m B : m B1 ∈ (m A1 , m A1 + 2x B1 )}.7 To see why this constitutes an equilibrium, first observe that for a given m A , Sender B can induce any action a ∈ / S (m A ) by sending m B = a. Also recall that Sender B does not have any bias in the second dimension, i.e., x B2 = 0. Moreover, u θB (m B1 , m B2 ) < u θB (m B1 , θ2 ) for m B2 = θ2 . Therefore, the DM can always elicit θ2 from Sender B. Thus, we only need to see if the DM can successfully elicit θ1 or not. The reason that Sender A does not send m A1 < θ1 is the same as n = 1 case. If m A1 < θ1 , then Sender B sends a message such that m B1 = max {θ1 + x B1 , m A1 + 2x B1 }. As shown in Fig. 4(a), since the DM follows Sender B’s message as m B ∈ / S (m A ), Sender A’s payoff is lower than sending m A = θ . One may wonder why Sender A would not send m A1 ∈ (θ1 , θ1 + x B1 ) such that u θA (m A ) > u θA (θ) and u θB (m A ) > u θB (θ).8 The point m A in Fig. 4(b) represents such a message. Recall that any message by Sender B satisfies m B2 = θ2 . That is, 6 See Fig. 2. Solid line segments in the bottom line represent the set of actions Sender B can induce following m A . Among those, Sender B chooses a message that is closest to θ + x B . When, for example, m A ∈ [θ − x B , θ), then B chooses m B = m A + 2x B . 7 See Fig. 3. The shaded area represents S (m A ). 8 Note that for n ≥ 2 when the true state is θ , there may be an action which both senders prefer over a = θ . That is,
P (θ) ≡ a : u θA (a) > u θA (θ) and u θB (a) > u θB (θ) = ∅ in general. When n = 1, however, P (θ) = ∅ if and only if x A · x B > 0.
K. Kawai / Games and Economic Behavior 90 (2015) 128–133
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Fig. 3. S (m A ).
Fig. 4. m A and m B .
Sender B responds to such an m A by m B = (m A1 , θ2 ), and a (m A , m B ) = m B because m B ∈ / S (m A ). Then, by assumption that x A · x B ≤ 0, we have u θA (m A1 , θ2 ) < u θA (θ). Therefore, even though both Sender A and B are better off when the DM adopts such an m A than θ , Sender A does not send such a message knowing it will induce Sender B to send m B = (m A1 , θ2 ); and the DM to take a = m B . m m Now note the set of self-serving message is included in S (m A ).9 That is, {m B : u B A (m B ) > u B A (m A )} S (m A ). Therefore, if Sender B’s message is self-serving, then the DM follows Sender A’s message. Moreover, m B (θ, m A ) ∈ / S (m A ) for any θ and m A in the equilibrium described above. The implication of this is two-fold: (i) Sender B’s self-serving message will be ignored, which deters Sender B from sending an untruthful message; and (ii) any untruthful message by Sender A will be countered by Sender B’s non-self-serving message; which deters Sender A from sending an untruthful message. Remark 1. In the fully-revealing equilibrium above, the DM follows (i) Sender A’s message if m B is self-serving; and (ii) Sender B’s message only if m B is non-self-serving.10 One may be wondering if we can construct a fully-revealing equilibrium in which the DM follows (i’) Sender A’s message if m B is self-serving; and (ii’) Sender B’s message if m B is non-self-serving. The answer is negative in general. To see this, suppose that x A · x B < 0. Moreover, suppose that the DM uses the strategy
9 10
See Fig. 3. The elements inside of circle represent the set of self-serving messages. That is, the DM may still follow Sender A’s message even if Sender B’s message is non-self-serving. This happens when Sender B sends m B such that
m
m
m B ∈ S (m A ) \ m B : u B A (m B ) > u B A (m A ) .
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K. Kawai / Games and Economic Behavior 90 (2015) 128–133
Fig. 5. Remark 1.
Fig. 6. x A · x B > 0.
characterized by (i’) and (ii’). Then as depicted in Fig. 5, Sender A can induce action a by sending message m A , which Sender B counters by a non-self-serving message m B . Now we see why there is no fully revealing equilibrium when x A · x B > 0. To see this, consider two states 0 and θ˜ = (θ˜1 , 0) for a very small θ˜1 as depicted in Fig. 6. Note that when the true state is 0, both senders prefer a = θ˜ over a = 0, that is, ˜ > u 0 (0) and u 0 (θ˜ ) > u 0 (0). u 0A (θ) B B A We argue that if a = θ˜ when θ = θ˜ in an equilibrium, then a = 0 when θ = 0 in the equilibrium. That is, there is no-fully ˜
revealing equilibrium. To see this, let mθA be Sender A’s equilibrium message when θ = θ˜ that results in a = θ˜ . Since a = θ˜
when θ = θ˜ in the equilibrium, Sender B cannot induce any action in the interior of the shaded circle in Fig. 6. That is, ˜
˜
˜
˜
˜
˜ . Furthermore, observe that u θ (a) ≤ u θ (θ) ˜ implies following mθA , any action a Sender B can induce satisfies u θB (a) ≤ u θB (θ) B B ˜
˜ .11 That is, following mθ , Sender B cannot induce any action in the interior of the smaller circle that lies in u 0B (a) ≤ u 0B (θ) A ˜
the shaded circle. The implication of this is that when θ = 0 and Sender A’s message is mθA , Sender B sends a message that ˜ = 0, Sender A prefers to send mθA rather than a message that leads to
in turn induces a = θ˜ . Since u 0A (θ˜ ) > u 0A (0), when θ a = 0. This proves that there is no fully-revealing equilibrium when x A · x B > 0. 4. Conclusion
In this note, we analyzed a sequential cheap talk game with n-dimensional Euclidean state space, and showed that the result by Krishna and Morgan (2001) can be naturally extended to n-dimensional case. That is, a fully-revealing equilibrium
11
˜ . The interior of the smaller circle that lies inside of the shaded circle in Fig. 6 represents {a : u 0B (a) > u 0B (θ)}
K. Kawai / Games and Economic Behavior 90 (2015) 128–133
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exists if and only the senders’ biases are opposing. We conclude by noting that Theorem 1 can be easily generalized to more general payoff functions as long as the set of actions Sender B cannot induce following Sender A’s message, S (m A ), is well-defined.12 References Battaglini, M., 2002. Multiple referrals and multidimensional cheap talk. Econometrica 70, 1379–1401. Krishna, V., Morgan, J., 2001. A model of expertise. Quart. J. Econ. 116, 747–775. Miura, S., 2014. Multidimensional cheap talk with sequential messages. Games Econ. Behav. 87, 419–441.
12
For example, if the payoff functions satisfy the following four conditions, then the “if” part of Theorem 1 follows.
Condition 1. The payoff functions u θD M , u θA , and u θB are quasi-concave, single-peaked functions with peaks at θ, θ + x A , and θ + x B for any θ , respectively; θ+y
Condition 2. For any θ and y, u θi (a) = u i
(a + y ), i = D M , A , B;
Condition 3. There exists a β > 1 such that u 0B (0) = u 0B (β x B );
Condition 4. Let x˜ i be a vector such that ˜xi = 1, and x˜ i · xi = 0, i = A , B. Then u 0i (1 + γ )xi + γ˜ x˜ i is strictly decreasing in γ and γ˜ ; Condition 1 is commonly used in the literature. Condition 2 states that the shape of an indifference curve does not depend on the true state θ . To understand Condition 3, suppose that the true state is θ . If the DM’s action a is far away from θ in the direction of Sender B’s bias x B , i.e., a = θ + β x B for a sufficiently large β , then Sender B prefers a = θ over a . To see the implication of Condition 4, first note that for any a, there exist γ , γ˜ > 0 and a vector x˜ i orthogonal to xi with ˜xi = 1 such that a = θ + γ xi + γ˜ x˜ i . Therefore, if u i is “smooth,” and the DM chooses a = θ + γ xi , then xi , the bias vector of player i, is orthogonal to the hyperplane tangent to the indifference curve of player i on a = θ + γ xi .
Contents lists available at ScienceDirect
Games and Economic Behavior www.elsevier.com/locate/geb
Note
Sequential cheap talks Keiichi Kawai 1 University of New South Wales, UNSW Business School, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 18 October 2011 Available online 14 February 2015 JEL classification: C72 D82
a b s t r a c t In this note, we analyze a multidimensional cheap talk game where two senders sequentially submit messages. We provide a necessary and sufficient condition for the existence of a fully-revealing equilibrium. © 2015 Elsevier Inc. All rights reserved.
Keywords: Multidimensional cheap talk Sequential messages
1. Introduction In this note, we analyze a multidimensional cheap talk game where two senders sequentially submit messages. We show that a fully-revealing equilibrium exists if and only if the senders’ biases are opposing. That is, the product of the vectors that represent senders’ biases are negative. As shown in Krishna and Morgan (2001), when the state space is one-dimensional Euclidean space, a fully-revealing equilibrium exists if and only if two senders’ biases are opposing. We show that the analysis of Krishna and Morgan (2001) for the one-dimensional Euclidean space can easily be extended to a general n-dimensional Euclidean state space. It is well known that multidimensional cheap talk games have positive results on information transmission when the senders simultaneously send messages.2 When messages are submitted sequentially, one might think it is impossible to achieve the fully-revealing equilibrium, even when the biases are opposing, because there exists an action that both senders strictly prefer over the receiver’s ideal action.3 We nonetheless show that a fully-revealing equilibrium exists if and only if the senders’ biases are opposing. The result follows from an observation that the conflict of interests between the decision maker and the second sender is essentially one-dimensional. More precisely, we can always transform the coordinate system so that the decision maker’s actions that are ideal for the decision maker and the second sender differ in only one coordinate. This observation implies that the existence of a fully-revealing equilibrium boils down to whether or not the decision maker can successfully solicit information regarding the coordinate on which the decision maker and the second sender have conflict of interests. This in turn enables us to fully utilize the concept of self-serving messages characterized in Krishna and Morgan (2001), i.e., the set of second sender’s messages that the decision maker disregards. As a result, the set
E-mail address: [email protected]. The author acknowledges the comments from Shintaro Miura, Carlos Oyarzun and Satoru Takahashi. The author greatly appreciates an anonymous referee for the valuable comments, which led to a substantial improvement on the previous version of the article. 2 See Battaglini (2002). 3 Note that such an action does not exist when the state space is one-dimensional. 1
http://dx.doi.org/10.1016/j.geb.2015.02.007 0899-8256/© 2015 Elsevier Inc. All rights reserved.
K. Kawai / Games and Economic Behavior 90 (2015) 128–133
129
Fig. 1. Normalization of x B .
of second sender’s messages that the decision maker disregards in this note has a direct analogy to the one-dimensional Euclidean space model. This is a clear contrast to the constructive proof by Miura (2014) that has independently shown the same result as in this note. In Miura (2014), the construction of the set of second sender’s messages that the decision maker disregards heavily relies on the nature of quadratic-loss payoff functions. Consequently, the analysis in Miura (2014) has two limitations. First, it does not provide a direct analogy to the one-dimensional Euclidean space model. Moreover, it is only applicable to quadratic-loss payoff functions. In contrast, even though we only present the result for the quadratic-loss payoff functions, the analysis presented in this note can easily be generalized to more general payoff functions. 2. Model We consider the following n-dimensional cheap talk game with sequential communication. There are three players: two senders, A and B, and a decision maker (DM, henceforth). By u θi (a), we denote player i’s payoff when the true state is θ and action taken by the DM is a. We assume that both the state space and action space of the DM are Rn , and payoff n functions are quadratic-loss, that is, u θi (a) = − j =1 (a − (θ + xi ))2 , i = D M , A , B, and x D M = 0.4 The game proceeds as follows: 1. 2. 3. 4.
Senders learn the true state θ ; Sender A sends a message m A : θ → m A ∈ Rn ; Sender B sends a message m B : θ × m A → m B ∈ Rn ; DM takes an action a : m A × m B → a ∈ Rn .
The equilibrium concept we use is a perfect Bayesian equilibrium. 3. Fully-revealing equilibrium For the rest of the paper, we normalize x B = (x B1 , 0, 0, · · · , 0), x B1 ≥ 0. That is, we use an orthogonal basis so that the conflict of interests between the DM and Sender B is about the first-dimension alone. This normalization is without loss of generality, but simplifies notations.5 With such a normalization, the existence of a fully-revealing equilibrium boils down to the question of whether the DM can elicit the truthful report with respect to the first-dimension. As it turns out, a fully-revealing equilibrium exists if and only if two senders’ biases are opposing, i.e., x A · x B ≤ 0. Theorem 1. There exists a fully-revealing equilibrium if and only if x A · x B ≤ 0. To understand the intuition behind the “if” part of the result, it is useful to review the case where n = 1. As shown in Krishna and Morgan (2001), a fully-revealing equilibrium exists if and only if x A < 0 ≤ x B , i.e., x A · x B ≤ 0. A strategy profile that supports the fully-revealing equilibrium is as follows:
4 The result in this note can be easily generalized for more general payoff functions. But for expositional simplicity, we limit our attention to the quadratic-loss payoff functions. 5 This can be done by simply “rotating” the coordinates. See Fig. 1 for n = 2 case.
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K. Kawai / Games and Economic Behavior 90 (2015) 128–133
Fig. 2. m A ∈ [θ − x B , θ).
m A (θ) = θ
⎧ θ + xB ⎪ ⎪ ⎨ m A + 2x B m B (θ, m A ) = mA ⎪ ⎪ ⎩ θ + xB m A if m B a (m A , m B ) = mB
if m A if m A if m A if m A
< θ − xB ∈ [θ − x B , θ) ∈ [θ, θ + x B ) ≥ θ + xB
∈ (m A , m A + 2x B ) otherwise
Suppose the DM tentatively believes that m A = θ . Then under the hypothesis that Sender A is telling the truth, adoption m m of m B strictly benefits Sender B if and only if m B ∈ (m A , m A + 2x B ). That is, u B A (m B ) > u B A (m A ) if and only if m B ∈ (m A , m A + 2x B ). Krishna and Morgan (2001) term such a message “self-serving.” In the strategy profile described above, the DM ignores any self-serving message, but follows any non-self-serving message. Note that an untruthful message by Sender A will be countered by Sender B’s non-self-serving message. To see why it is optimal for Sender B to do so, note that following m A , Sender B can induce any action m B ∈ / (m A , m A + 2x B ). Therefore, Sender B chooses a message m B ∈ / (m A , m A + 2x B ) that is closest to Sender B’s ideal point θ + x B . For example, if m A ∈ [θ − x B , θ), then it is optimal for Sender B to send m B = m A + 2x B .6 When m A = θ , however, there exists no non-self-serving message m B such that u θB (m B ) > u θB (m A ). That is, Sender B cannot do better than truthfully reporting θ . Now we see why Sender A does not send m A < θ . If m A ∈ [θ − x B , θ), then such a message will be countered by a non-self-serving message m A + 2x B . Since the DM adopts Sender B’s message, the resulting payoff is u θA (m A + 2x B ), which is lower than u θA (θ). If Sender A sends m A = θ instead, Sender B does not have any incentive to send a non-self-serving message, and hence sends m B = θ . Therefore, Sender A yields u θA (θ). Similarly, Sender A does not send m A < θ − x B . That Sender A has no incentive to send m A > θ is straightforward. This is how a fully-revealing equilibrium is sustained when n = 1. The intuition behind the case with n ≥ 2 is quite similar. Since the generalization of the result from n = 2 to n ≥ 3 is straightforward, we restrict our attention to the case with n = 2 for expositional simplicity. To see why there is a fully-revealing equilibrium when x A · x B ≤ 0, consider the following equilibrium strategy profiles for n = 2.
m A (θ) = θ;
⎧ if m A1 < θ1 − x B1 θ + x B1 ⎪ ⎪ ⎨ 1 m A1 + 2x B1 if m A1 ∈ [θ1 − x B1 , θ1 ) m B1 (θ, m A ) = m A1 if m A1 ∈ [θ1 , θ1 + x B1 ) ⎪ ⎪ ⎩ if m A1 ≥ θ1 + x B1 θ1 + x B1 m B2 (θ, m A ) = θ2 ; m A m B ∈ S (m A ) a (m A , m B ) = mB
otherwise
where S (m A ) ≡ {m B : m B1 ∈ (m A1 , m A1 + 2x B1 )}.7 To see why this constitutes an equilibrium, first observe that for a given m A , Sender B can induce any action a ∈ / S (m A ) by sending m B = a. Also recall that Sender B does not have any bias in the second dimension, i.e., x B2 = 0. Moreover, u θB (m B1 , m B2 ) < u θB (m B1 , θ2 ) for m B2 = θ2 . Therefore, the DM can always elicit θ2 from Sender B. Thus, we only need to see if the DM can successfully elicit θ1 or not. The reason that Sender A does not send m A1 < θ1 is the same as n = 1 case. If m A1 < θ1 , then Sender B sends a message such that m B1 = max {θ1 + x B1 , m A1 + 2x B1 }. As shown in Fig. 4(a), since the DM follows Sender B’s message as m B ∈ / S (m A ), Sender A’s payoff is lower than sending m A = θ . One may wonder why Sender A would not send m A1 ∈ (θ1 , θ1 + x B1 ) such that u θA (m A ) > u θA (θ) and u θB (m A ) > u θB (θ).8 The point m A in Fig. 4(b) represents such a message. Recall that any message by Sender B satisfies m B2 = θ2 . That is, 6 See Fig. 2. Solid line segments in the bottom line represent the set of actions Sender B can induce following m A . Among those, Sender B chooses a message that is closest to θ + x B . When, for example, m A ∈ [θ − x B , θ), then B chooses m B = m A + 2x B . 7 See Fig. 3. The shaded area represents S (m A ). 8 Note that for n ≥ 2 when the true state is θ , there may be an action which both senders prefer over a = θ . That is,
P (θ) ≡ a : u θA (a) > u θA (θ) and u θB (a) > u θB (θ) = ∅ in general. When n = 1, however, P (θ) = ∅ if and only if x A · x B > 0.
K. Kawai / Games and Economic Behavior 90 (2015) 128–133
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Fig. 3. S (m A ).
Fig. 4. m A and m B .
Sender B responds to such an m A by m B = (m A1 , θ2 ), and a (m A , m B ) = m B because m B ∈ / S (m A ). Then, by assumption that x A · x B ≤ 0, we have u θA (m A1 , θ2 ) < u θA (θ). Therefore, even though both Sender A and B are better off when the DM adopts such an m A than θ , Sender A does not send such a message knowing it will induce Sender B to send m B = (m A1 , θ2 ); and the DM to take a = m B . m m Now note the set of self-serving message is included in S (m A ).9 That is, {m B : u B A (m B ) > u B A (m A )} S (m A ). Therefore, if Sender B’s message is self-serving, then the DM follows Sender A’s message. Moreover, m B (θ, m A ) ∈ / S (m A ) for any θ and m A in the equilibrium described above. The implication of this is two-fold: (i) Sender B’s self-serving message will be ignored, which deters Sender B from sending an untruthful message; and (ii) any untruthful message by Sender A will be countered by Sender B’s non-self-serving message; which deters Sender A from sending an untruthful message. Remark 1. In the fully-revealing equilibrium above, the DM follows (i) Sender A’s message if m B is self-serving; and (ii) Sender B’s message only if m B is non-self-serving.10 One may be wondering if we can construct a fully-revealing equilibrium in which the DM follows (i’) Sender A’s message if m B is self-serving; and (ii’) Sender B’s message if m B is non-self-serving. The answer is negative in general. To see this, suppose that x A · x B < 0. Moreover, suppose that the DM uses the strategy
9 10
See Fig. 3. The elements inside of circle represent the set of self-serving messages. That is, the DM may still follow Sender A’s message even if Sender B’s message is non-self-serving. This happens when Sender B sends m B such that
m
m
m B ∈ S (m A ) \ m B : u B A (m B ) > u B A (m A ) .
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Fig. 5. Remark 1.
Fig. 6. x A · x B > 0.
characterized by (i’) and (ii’). Then as depicted in Fig. 5, Sender A can induce action a by sending message m A , which Sender B counters by a non-self-serving message m B . Now we see why there is no fully revealing equilibrium when x A · x B > 0. To see this, consider two states 0 and θ˜ = (θ˜1 , 0) for a very small θ˜1 as depicted in Fig. 6. Note that when the true state is 0, both senders prefer a = θ˜ over a = 0, that is, ˜ > u 0 (0) and u 0 (θ˜ ) > u 0 (0). u 0A (θ) B B A We argue that if a = θ˜ when θ = θ˜ in an equilibrium, then a = 0 when θ = 0 in the equilibrium. That is, there is no-fully ˜
revealing equilibrium. To see this, let mθA be Sender A’s equilibrium message when θ = θ˜ that results in a = θ˜ . Since a = θ˜
when θ = θ˜ in the equilibrium, Sender B cannot induce any action in the interior of the shaded circle in Fig. 6. That is, ˜
˜
˜
˜
˜
˜ . Furthermore, observe that u θ (a) ≤ u θ (θ) ˜ implies following mθA , any action a Sender B can induce satisfies u θB (a) ≤ u θB (θ) B B ˜
˜ .11 That is, following mθ , Sender B cannot induce any action in the interior of the smaller circle that lies in u 0B (a) ≤ u 0B (θ) A ˜
the shaded circle. The implication of this is that when θ = 0 and Sender A’s message is mθA , Sender B sends a message that ˜ = 0, Sender A prefers to send mθA rather than a message that leads to
in turn induces a = θ˜ . Since u 0A (θ˜ ) > u 0A (0), when θ a = 0. This proves that there is no fully-revealing equilibrium when x A · x B > 0. 4. Conclusion
In this note, we analyzed a sequential cheap talk game with n-dimensional Euclidean state space, and showed that the result by Krishna and Morgan (2001) can be naturally extended to n-dimensional case. That is, a fully-revealing equilibrium
11
˜ . The interior of the smaller circle that lies inside of the shaded circle in Fig. 6 represents {a : u 0B (a) > u 0B (θ)}
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exists if and only the senders’ biases are opposing. We conclude by noting that Theorem 1 can be easily generalized to more general payoff functions as long as the set of actions Sender B cannot induce following Sender A’s message, S (m A ), is well-defined.12 References Battaglini, M., 2002. Multiple referrals and multidimensional cheap talk. Econometrica 70, 1379–1401. Krishna, V., Morgan, J., 2001. A model of expertise. Quart. J. Econ. 116, 747–775. Miura, S., 2014. Multidimensional cheap talk with sequential messages. Games Econ. Behav. 87, 419–441.
12
For example, if the payoff functions satisfy the following four conditions, then the “if” part of Theorem 1 follows.
Condition 1. The payoff functions u θD M , u θA , and u θB are quasi-concave, single-peaked functions with peaks at θ, θ + x A , and θ + x B for any θ , respectively; θ+y
Condition 2. For any θ and y, u θi (a) = u i
(a + y ), i = D M , A , B;
Condition 3. There exists a β > 1 such that u 0B (0) = u 0B (β x B );
Condition 4. Let x˜ i be a vector such that ˜xi = 1, and x˜ i · xi = 0, i = A , B. Then u 0i (1 + γ )xi + γ˜ x˜ i is strictly decreasing in γ and γ˜ ; Condition 1 is commonly used in the literature. Condition 2 states that the shape of an indifference curve does not depend on the true state θ . To understand Condition 3, suppose that the true state is θ . If the DM’s action a is far away from θ in the direction of Sender B’s bias x B , i.e., a = θ + β x B for a sufficiently large β , then Sender B prefers a = θ over a . To see the implication of Condition 4, first note that for any a, there exist γ , γ˜ > 0 and a vector x˜ i orthogonal to xi with ˜xi = 1 such that a = θ + γ xi + γ˜ x˜ i . Therefore, if u i is “smooth,” and the DM chooses a = θ + γ xi , then xi , the bias vector of player i, is orthogonal to the hyperplane tangent to the indifference curve of player i on a = θ + γ xi .