# Coordination and Private Information Revelation

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Model

**Stage 1: Revelation:**Agent A decides whether or not to reveal her type $\theta $. Agent A can either reveal her exact type by providing hard information, or decline to offer any information. The message space of a type $\theta $ agent A is, therefore, $M\left(\theta \right)=\left\{\theta \right\}\cup ${Not Reveal her Type}3. Thus, the set of random messages of agent A of type $\theta $ is given by $\Delta \left(M\right(\theta \left)\right)$, where $\Delta \left(M\right(\theta \left)\right)=\left\{m\right|m\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\phantom{\rule{0.277778em}{0ex}}\mathrm{probability}\phantom{\rule{0.277778em}{0ex}}\mathrm{distribution}\phantom{\rule{0.277778em}{0ex}}\mathrm{over}\phantom{\rule{0.277778em}{0ex}}M\left(\theta \right)\}$. We define $M={\cup}_{\theta}M\left(\theta \right)=[{\theta}_{l},{\theta}_{h}]\cup ${Not Reveal her Type}.**Stage 2: Coordination:**The agents play a coordination game, where agent i chooses an action from {Adopt i, Switch to j, $i,j\in \{A,B\},j\ne i\}$. If both the agents choose to adopt their own idea, there is no coordination: agent A obtains $\theta $ and agent B obtains b. On the other hand, if both the agents switch to the other’s idea, then they both have a payoff of 0. If they coordinate on A, then the payoff vector is $(\theta +c,c)$, whereas it is $(c,b+c)$ if they coordinate on B.4

**Assumption**

**1.**

- The strategy of a type $\theta $ agent A in the revelation stage, i.e., Stage 1, denoted by ${\alpha}^{I}\left(\theta \right)$, is a probability distribution over $M\left(\theta \right)$.
- The strategies of the agents A and B in Stage 2, i.e., the coordination stage, are:
- -
- The strategy of a type $\theta $ agent A in the coordination stage is a mapping ${\alpha}^{C}\left(\theta \right)$ from her decision in Stage 1 to a probability distribution over the action space $\{adopt\phantom{\rule{0.277778em}{0ex}}A,switch\phantom{\rule{0.277778em}{0ex}}to\phantom{\rule{0.277778em}{0ex}}B\}$, i.e., ${\alpha}^{C}\left(\theta \right):M\left(\theta \right)\to [0,1].$
- -
- Following a history where, in Stage 1, agent A revealed her type to be $\theta $, let ${q}_{R}\left(\theta \right)$ denote a mixed strategy of agent B where he plays “Adopt B” with probability ${q}_{R}\left(\theta \right)$.
- -
- Similarly, following a history where agent A played “Not Reveal $\theta $” in Stage 1, ${q}_{NR}$ denotes a mixed strategy of agent B, where he plays “Adopt B” with probability ${q}_{NR}$.

- The belief of agent B at the coordination stage, i.e., Stage 2, is a mapping $\overline{B}$ from the message sent by agent A in the revelation stage to a probability distribution over agent A’s type space $[{\theta}_{l},{\theta}_{h}]$, i.e., $\overline{B}:M\to \Delta [{\theta}_{l},{\theta}_{h}]$5.
- Finally, agent A’s strategy in Stage 2 is said to be a cut-off strategy iff there is some $\widehat{\theta}\in [{\theta}_{l},{\theta}_{h}]$ such that she adopts A iff $\theta \ge \widehat{\theta}$.

## 4. The Analysis

#### 4.1. Full Disclosure Equilibria in the Absence of Coordination Failure

**Proposition**

**1.**

- (a)
- In this equilibrium, in Stage 1, all A agents reveal their own types. In Stage 2:
- (i)
- In case of non-revelation by the A agent in Stage 1, agent B believes that agent A is of type ${\theta}_{h}$.
- (ii)
- Fix ${\theta}^{\prime}$, where ${\theta}^{\prime}\in [{\theta}_{l},{\theta}_{h}]$. The agents coordinate on A if $\theta \ge {\theta}^{\prime}$, and on B if $\theta <{\theta}^{\prime}$.

- (b)
- For ${\theta}^{\prime}=b$, the equilibrium so described is efficient.

**Proof.**

- (a)
- Clearly, the Stage 2 strategies constitute an equilibrium, so that it is sufficient to examine whether agent A’s revelation strategy is optimal or not.Please note that following disclosure of her type, agent A’s payoff is at least c since disclosure ensures coordination in Stage 2. Whereas in case agent A opts not to disclose, then given the belief, agent B plays a completely mixed strategy, adopting B with probability $\frac{{\theta}_{h}+c}{2c}$. It is straightforward to check that for all $\theta $, it is optimal for agent A to switch to B, so that her payoff is $\frac{{\theta}_{h}+c}{2}$, which is less than her payoff from revealing her type (which is at least c).
- (b)
- Please note that there is coordination on A iff $\theta \ge b$, which is efficient.

#### 4.2. Non-Revelation Equilibria in the Presence of Coordination Failure

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

- (i)
- Agent A adopts her own idea, i.e., A, if and only if $\theta \ge \widehat{\theta}$, where $\widehat{\theta}={F}^{-1}\left(\frac{c-b}{2c}\right)$ and ${\theta}_{l}<\widehat{\theta}<{\theta}_{h}$, and switches to B otherwise.
- (ii)
- Agent B adopts her own idea, i.e., B, with probability $\frac{1}{2}+\frac{{F}^{-1}\left(\frac{c-b}{2c}\right)}{2c}$.
- (iii)
- In case agent A does not reveal her type in Stage 1, agent B believes that the type of agent A is distributed over $[{\theta}_{l},{\theta}_{h}]$ with distribution $F\left(\theta \right)$. In case agent A does reveal her type to be θ, say, the belief of agent B puts probability 1 on agent A being of type θ.
- (iv)
- In this PBEM, a type θ agent A has an expected payoff of:$${\pi}_{A}\left(\theta \right)=\left(\right)open="\{"\; close>\begin{array}{c}\theta +\frac{c-\widehat{\theta}}{2},\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{0.277778em}{0ex}}\theta \widehat{\theta},\hfill \\ \frac{c+\widehat{\theta}}{2},\phantom{\rule{4pt}{0ex}}\mathit{otherwise},\hfill \end{array}$$

**Proof.**

**Proposition**

**4.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 5. Purification of Agent B’s Mixed Strategies

- The strategy of a type $\theta $ agent A in Stage 1, denoted ${\beta}^{I}\left(\theta \right)$, is a probability distribution over $M\left(\theta \right)$.
- Strategies of the agents in Stage 2, i.e., the coordination stage, are:
- -
- In Stage 2, the strategy of a type $\theta $ agent A, i.e., ${\beta}_{A}^{C}\left(\theta \right)$, maps from her message is Stage 1, to the set of probability distributions over her action space. Hence, ${\beta}_{A}^{C}\left(\theta \right):M\left(\theta \right)\to [0,1]$.
- -
- In Stage 2, agent B’s strategy is a mapping from his own type space $\mathsf{\Xi}$, and the disclosure made by A in Stage 1, to the set probability distributions over his own actions. Hence, ${\beta}_{B}^{C}:\mathsf{\Xi}\times M\to [0,1]$.
- -
- Agent A’s strategy in Stage 2 is said to be a cut-off strategy if there exists $\widehat{\theta}\in \phantom{\rule{3.33333pt}{0ex}}\Theta $ such that A adopts A iff $\theta \ge \widehat{\theta}$.
- -
- Agent B’s strategy in Stage 2, ${\beta}_{B}^{C}$, is said to be a cut-off strategy if there exists $\widehat{b}\in \phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$ such that B adopts B iff $b\ge \widehat{b}$.

- The belief of a type b agent B at the coordination stage, i.e., Stage 2, is a mapping ${\overline{B}}_{b}$ from the message sent by agent A in the revelation stage to a probability distribution over agent A’s type space $[{\theta}_{l},{\theta}_{h}]$, i.e., ${\overline{B}}_{b}:M\to \Delta [{\theta}_{l},{\theta}_{h}]$10.

#### 5.1. Analysis: Non-Revelation by the A Agent in Stage 1

**Definition**

**2.**

**Lemma**

**3.**

- (a)
- In any PBEC of this continuation game, the cut-off for agent B ($\widehat{b}\left(\theta \right)$) and the probability that agent A adopts A, i.e., $\alpha \left(\theta \right)$, solves:$$\begin{array}{ccc}\hfill \widehat{b}\left(\theta \right)& =& (\alpha -\frac{c+\theta}{2c})c,\hfill \\ \hfill \alpha \left(\theta \right)& =& \frac{(c+\theta )}{2c}+\frac{{G}^{-1}((c-\theta )/2c)}{c}.\hfill \end{array}$$
- (b)
- A θ-type A-agent’s payoff from revealing her type is $\frac{c+\theta}{2}$.
- (c)
- The equilibrium cut-off for the B-types is unique and interior, i.e., $0<\widehat{b}<1$. Moreover, it generates the same probability distribution over the two choices, i.e., A and B, as that under the PBEM equilibrium following type revelation by agent A in the baseline model.

**Proof.**

- (a)
- Consider the continuation game following the A agent revealing her type to be $\theta $. Let the B-agents adopt a cut-off strategy involving a cut-off of $\widehat{b}\left(\theta \right)$. Let $l\left(\theta \right)$ be the fraction of B-agents that adopt B with $l\left(\theta \right)\equiv 1-G\left(\widehat{b}\left(\theta \right)\right)$. Let $\alpha \left(\theta \right)$ be the probability that the A-agent adopts A.

- (b)
- Using (13), the A-agent’s expected payoff is given by $cl\left(\theta \right)=\frac{c+\theta}{2}$.
- (c)
- Please note that $G\left(\widehat{b}\right)=1-l\left(\theta \right)=\frac{c-\theta}{2c}$. Given that $G\left(b\right)$ is strictly increasing, $\widehat{b}$ is unique. Given that $c>{\theta}_{h}\ge \theta $, it is straightforward to check that $0<G\left(\widehat{b}\right)<1$, so that ${b}_{l}<\widehat{b}<{b}_{h}$.

**Proposition**

**5.**

- (a)
- Consider the Stage 2 continuation game where agent A does not reveal her type in Stage 1. In any PBEC, agent A plays a cut-off strategy in Stage 2.
- (b)
- In Stage 1, all types of agent A, with the possible exception of one type, strictly prefer non-revelation to revelation.
- (c)
- If $F\left(\theta \right)+\frac{\theta}{2c}+\frac{{G}^{-1}((c-\theta )/2c)}{c}$ is monotonic in θ, then this game has a unique PBEC.

**Proof.**

- (a)
- The proof mimics that of Lemma 1 earlier.
- (b)
- Given Proposition 5(a), we restrict attention to PBEC where, following non-revelation the A agent plays a cut-off strategy. Let the cut-offs following non-revelation be ${\widehat{\theta}}_{NR}$ for the A, and ${\widehat{b}}_{NR}$ for the B agents. Equating the payoffs from switching to A and adopting B for the indifferent B agent, i.e., of type ${\widehat{b}}_{NR}$, we have that$${\widehat{b}}_{NR}=(G\left({\widehat{b}}_{NR}\right)-F\left({\widehat{\theta}}_{NR}\right))c.$$

- At $\theta ={\widehat{\theta}}_{NR}$, ${\pi}_{A}\left(NR\right)={\widehat{\theta}}_{NR}+c.G\left({\widehat{b}}_{NR}\right)={\widehat{\theta}}_{NR}+c.\left(\frac{c-{\widehat{\theta}}_{NR}}{2c}\right)=\frac{{\widehat{\theta}}_{NR}+c}{2}={\pi}_{A}\left(R\right)$.
- For all types of $\theta >{\widehat{\theta}}_{NR}$, ${\pi}_{A}\left(NR\right)-{\pi}_{A}\left(R\right)=\frac{\theta}{2}-c[\frac{1}{2}-G\left({\widehat{b}}_{NR}\right)]$. Please note that $\frac{\partial [{\pi}_{A}\left(NR\right)-{\pi}_{A}\left(R\right)]}{\partial {\theta}_{NR}}=1/2>0$, and at $\theta ={\theta}_{h}$, this difference ${\pi}_{A}\left(NR\right)-{\pi}_{A}\left(R\right)=\frac{{\theta}_{h}}{2}-c[\frac{1}{2}-G\left({\widehat{b}}_{NR}\right)]$. This expression is positive, iff ${\widehat{b}}_{NR}>{G}^{-1}\left(\frac{c-{\theta}_{h}}{2c}\right)$. Finally, from Equation (16), ${\widehat{b}}_{NR}={G}^{-1}\left(\frac{c-{\widehat{\theta}}_{NR}}{2c}\right)>{G}^{-1}\left(\frac{c-{\theta}_{h}}{2}\right)$, since ${\widehat{\theta}}_{NR}<{\theta}_{h}$ and $G\left(B\right)$ is strictly increasing. As discussed earlier, ${\pi}_{A}\left(R\right)={\pi}_{A}\left(NR\right)$ at $\theta ={\widehat{\theta}}_{NR}$.Therefore, in the range $({\widehat{\theta}}_{NR},{\theta}_{h}]$, ${\pi}_{A}\left(R\right)<{\pi}_{A}\left(NR\right)$.
- Next, consider $\theta $ in the range $[{\theta}_{l},\widehat{\theta})$. Over this range ${\pi}_{A}\left(NR\right)$ is independent of $\theta $, whereas ${\pi}_{A}\left(R\right)$ is strictly decreasing in $\theta $. Given that ${\pi}_{A}\left(NR\right)$ equals ${\pi}_{A}\left(R\right)$ at ${\widehat{\theta}}_{NR}$, it follows that ${\pi}_{A}\left(NR\right)>{\pi}_{A}\left(R\right),\forall \theta \in [{\theta}_{l},{\widehat{\theta}}_{NR})$.

- (c)
- Given that $F\left(\theta \right)+\frac{\theta}{2c}+\frac{{g}^{-1}((c-\theta )/2c)}{c}$ is monotonic in $\theta $, from (17) it follows that ${\widehat{\theta}}_{NR}$ is unique. This in turn ensures that ${\widehat{b}}_{NR}$ is unique.

## 6. Extensions

#### 6.1. Two-Sided Private Information

**Proposition**

**6.**

**Proof.**

**Remark**

**3.**

#### 6.2. Imprecise Information Revelation

**Proposition**

**7.**

**Proof.**

#### 6.3. Mandatory Disclosure

**Proposition**

**8.**

**Example**

**1.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Correlated Equilibrium

## Appendix B. Uniqueness of the Non-Revletion PBEM Over a Larger Class of Disclosure Strategies

**Lemma**

**A1.**

- 1.
- $0<{q}_{nr}<1$ is constant. ${q}_{R}$ increases linearly with θ.
- 2.
- ${q}_{R}$ intersects ${q}_{nr}$ from below at the coordination cutoff point $\widehat{\theta}$.
- 3.
- ${q}_{nr}$ increases linearly in $\widehat{\theta}$.

**Proof.**

- The proof follows from equations the fact that ${q}_{nr}=\frac{\widehat{\theta}+c}{2c}$ whereas ${q}_{R}=\frac{\theta +c}{2c}$, as discussed in the PBEM of the one-shot game.
- Please note that ${q}_{nr}=\frac{1}{2}+\frac{\widehat{\theta}}{2c}$ and ${q}_{R}=\frac{1}{2}+\frac{\theta}{2c}$. Therefore, ${q}_{nr}={q}_{R}$ at $\widehat{\theta}$. As ${q}_{nr}$ is constant and ${q}_{R}$ increases with $\theta $, for all $\theta >\widehat{\theta}$, ${q}_{nr}<{q}_{R}$. Therefore, ${q}_{R}$ has to intersect ${q}_{nr}$ from below at $\widehat{\theta}$.
- As ${q}_{nr}=\frac{\widehat{\theta}+c}{2c}$, we get that $\frac{\partial {q}_{nr}}{\partial \widehat{\theta}}=\frac{1}{2c}>0$. Thus, ${q}_{nr}$ increases linearly in $\widehat{\theta}$.

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Proposition**

**A1.**

**Proof.**

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1 | For the sake of clarity, we assume feminine gender for agent/player A and masculine gender for B. |

2 | There is, by now, a large literature on cheap talk games. For reasons of brevity we refrain from discussing this literature. |

3 | We would like to thank an anonymous reviewer in this journal for pointing out that the message space (revealing only hard evidence) becomes type-dependent: $M=M\left(\theta \right)$. |

4 | The coordination continuation game follows [11]. |

5 | Given that A can only mix between the true type and no announcement, note that a rational belief for B should put zero mass on all types other than what was announced. Pooling among different types can only occur if “no announcement” realizes. We thank an anonymous reviewer of this journal for pointing this out. |

6 | Of course there exist equilibria where the agents coordinate on, say, A in Stage 2, but there is no information revelation in the first stage. |

7 | In Section 5, we provide a purification argument that provides a foundation for the mixed strategic equilibrium that we examine here. |

8 | We are indebted to an anonymous referee for suggesting an argument that shortened the proof. |

9 | In fact, we can prove that the non-revelation outcome is unique in the class of revelation strategies, where A reveals its type over finite unions of disjoint sets of the type space. The proof is in Appendix B. |

10 | Given that A can only mix between the true type and no announcement, note that a rational belief for B should put zero mass on all types other than what was announced. Pooling among different types can only occur if “no announcement” realizes. As mentioned earlier, we thank an anonymous reviewer of this journal for pointing this out. |

11 | Can one provide sufficient conditions such that $\alpha <1?$ This is equivalent to showing that $\frac{{G}^{-1}((c-\theta )/2c)}{(c-\theta )/2c}<c$. Clearly, one sufficient condition is that $G\left(b\right)$ satisfies both (a) $\frac{{G}^{-1}\left(x\right)}{x}$ be increasing in $x,$ and (b) $\frac{{G}^{-1}\left(x\right)}{x}\le 1,\phantom{\rule{0.277778em}{0ex}}\forall x$. Please note that this is satisfied whenever $G\left(b\right)$ is uniform. |

12 | Can one provide sufficient conditions such that ${\widehat{\theta}}_{NR}>{\theta}_{l}?$ This is equivalent to showing that $\frac{{G}^{-1}((c-{\widehat{\theta}}_{NR})/2c)}{(c-{\widehat{\theta}}_{NR})/2c}<c$. Clearly, one sufficient condition is that $G\left(b\right)$ satisfies both that (a) $\frac{{G}^{-1}\left(x\right)}{x}$ is increasing in $x,$ and (b) $\frac{{G}^{-1}\left(x\right)}{x}\le 1,\phantom{\rule{0.277778em}{0ex}}\forall x$. Please note that this is satisfied whenever $G\left(b\right)$ is uniform. |

13 | It is straightforward to show that there exists an equilibrium where neither agent reveals in Stage 1, and, in Stage 2, switches if and only if ${\theta}_{i}\le \widehat{\theta}$. |

14 | Such imprecise revelation may be attractive in scenarios where the technologies may possibly be copied if revealed. Note, however, that while we model the possibility of imprecise information revelation, it is not assumed to yield any gain in utility. |

Switch to A | Adopt B | |
---|---|---|

Adopt A | $\theta +c$, c | $\theta $, b |

Switch to B | 0, 0 | c, $b+c$ |

Switch | Adopt B | |
---|---|---|

Adopt A | $\theta +c(1-l)$, $c(1-l)$ | $\theta +c(1-l)$, b |

Switch | $cl$, 0 | $cl$, $b+cl$ |

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**MDPI and ACS Style**

Saha, D.; Roy Chowdhury, P.
Coordination and Private Information Revelation. *Games* **2018**, *9*, 64.
https://doi.org/10.3390/g9030064

**AMA Style**

Saha D, Roy Chowdhury P.
Coordination and Private Information Revelation. *Games*. 2018; 9(3):64.
https://doi.org/10.3390/g9030064

**Chicago/Turabian Style**

Saha, Debdatta, and Prabal Roy Chowdhury.
2018. "Coordination and Private Information Revelation" *Games* 9, no. 3: 64.
https://doi.org/10.3390/g9030064