# Coordination and Private Information Revelation
## 6 Extensions
### Imprecise Information Revelation  
Please note that the informed agent can either reveal her own type truthfully, or can refuse to reveal. This message space does not allow for imprecise disclosures, unlike e.g., ref [10] who not only allows for imprecise revelation, but also cheap talk. Given that strategic uncertainty in coordination is at the heart of this paper, in this sub-section we allow for imprecise information revelation though in the presence of strategic uncertainty, showing that the non-revelation equilibrium does survive. Consider a scenario where agent A is allowed to report that her type lies within a set, rather than the exact value. Information is still assumed to be hard however, so that agent A can only make truthful claims.14 For technical reasons, we restrict attention to disclosures within closed sets only. In Stage 1, agent A chooses an element from \(\overline{\Theta}\), where \(\overline{\Theta}\) is the set of all closed subsets of \([\theta_{l},\theta_{h}]\). However information is hard in the sense that if an agent of type \(\theta\) chooses \(\Theta(\theta)\in\ \overline{\Theta}\) in Stage 1, then it must be the case that \(\theta\in\Theta(\theta)\). Please note that since \([\theta_{l},\theta_{h}]\in\ \overline{\Theta}\), revealing no information is also an option. Footnote 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. Consider the PBEM of the baseline model described in Propositions 1 and 2. Define \(q_{NR}^{*}\) (respectively \(q_{R}^{*}(\theta)\)) to be the probability that agent B adopts \(B\) in the second stage, given that agent A chooses not to reveal any information (respectively reveals her type \(\theta\)). Recall that  
\[q_{NR}^{*}=\frac{1}{2}+\frac{F^{-1}(\frac{\varsigma-\hat{b}}{2c})}{2c},\ \text{and}\ q_{R}^{*}(\theta)=\frac{1}{2}+\frac{\theta}{2c}.\]  
Furthermore, (a) \(q_{NR}^{*}>q_{R}^{*}(\theta)\) if and only if \(\theta<\hat{\theta}\), and (b) \(q_{R}^{*}(\theta)\) intersects \(q_{NR}^{*}\) from below at the coordination cut-off point \(\hat{\theta}\). We then prove that there exists an equilibrium where there is no information revelation by agent A. **Proposition 7**: _There exists an equilibrium where there is no information revelation by agent A in Stage 1 (except possibly by a single type)._  
Proof. : The argument is by construction. Consider the following strategies:  
In Stage 1, agent A reveals no information. In case agent A selects any other \(\Theta^{\prime}\in\ \bar{\Theta}\) instead, then agent B believes that [PERSON]'s type is \(\sup\Theta^{\prime}\). In Stage 2, the agents play completely mixed strategies with agent B's beliefs described above. It remains to check if any type A agent has an incentive to deviate and reveal her type in Stage 1. Suppose an agent of type \(\theta^{\prime}\) deviates and chooses \(\Theta^{\prime}\in\overline{\theta}\), where \(\theta^{\prime}\in\ \Theta^{\prime}\). Let \(\theta^{\prime\prime}=\sup\ \Theta^{\prime}\). Recall that agent A's payoff from selecting \(\Theta^{\prime}\), is \(\theta^{\prime}+c-q_{R}^{*}(\theta^{\prime\prime})c\), where \(q_{R}^{*}(\theta^{\prime\prime})=\frac{1}{2}+\frac{\theta^{\prime\prime}}{2c}\). Given agent B's belief, and the fact that agent A's payoff is decreasing in \(q_{R}^{*}(\theta)\), agent A cannot do any better than to announce \(\theta^{\prime}\) itself (more generally announce a set with \(\theta^{\prime}\) as its supremum). However, if \(\theta^{\prime}>\hat{\theta}\), then agent A would prefer not to reveal and adopt, rather than reveal, since \((\theta^{\prime}+c)-q_{NR}^{*}c>(\theta^{\prime}+c)-q_{R}^{*}(\theta^{\prime})c\). This is because \(q_{NR}^{*}<q_{R}^{*}(\theta)\) for all \(\theta>\hat{\theta}\).