# Coordination and Private Information Revelation
## Appendix B Uniqueness of the Non-Revelation PBEM Over a Larger Class of Disclosure Strategies  
We now establish that non-revelation of information is unique in the class of disclosure strategies, where A reveals its type over finite unions of disjoint sets of A's type space. This is a very large class of disclosure strategies, over which we show non-revelation holds uniquely in equilibrium. Consider the possible disclosure set of A to be R, where R is either a continuous interval or a finite union of disjoint intervals. Therefore,  
\[R=\cup_{s=1}^{n}[\theta_{s}^{\prime},\theta_{s}^{\prime\prime}].\]Let \(R^{c}\) be the set (either an interval or a finite union of disjoint intervals in the type space \(\Theta\)) over which no type of agent A reveal their type. Please note that \(\hat{\theta}\) is the cut-off type of A in \(R^{c}\) who is indifferent between adopting A and switching to B. The completely mixed strategy of B, \(q_{R}\) (if A reveals) or \(q_{nr}\) (if A does not reveal) obeys a single crossing property in the type space of A, so that \(q_{nr}>q_{R}\)\(\forall\)\(\theta<\hat{\theta}\) and \(q_{nr}<q_{R}\)\(\forall\)\(\theta>\hat{\theta}\). 1. \(0<q_{nr}<1\) is constant. \(q_{R}\) increases linearly with \(\theta\). 2. \(q_{R}\) intersects \(q_{nr}\) from below at the coordination cutoff point \(\hat{\theta}\). 3. \(q_{nr}\) increases linearly in \(\hat{\theta}\). Proof. :
1. The proof follows from equations the fact that \(q_{nr}=\frac{\hat{\theta}+c}{2c}\) whereas \(q_{R}=\frac{\theta+c}{2c}\), as discussed in the PBEM of the one-shot game. 2. Please note that \(q_{nr}=\frac{1}{2}+\frac{\hat{\theta}}{2c}\) and \(q_{R}=\frac{1}{2}+\frac{\theta}{2c}\). Therefore, \(q_{nr}=q_{R}\) at \(\hat{\theta}\). As \(q_{nr}\) is constant and \(q_{R}\) increases with \(\theta\), for all \(\theta>\hat{\theta}\), \(q_{nr}<q_{R}\). Therefore, \(q_{R}\) has to intersect \(q_{nr}\) from below at \(\hat{\theta}\). 3. As \(q_{nr}=\frac{\hat{\theta}+c}{2c}\), we get that \(\frac{\hat{\alpha}q_{nr}}{\hat{\alpha}\theta}=\frac{1}{2c}>0\). Thus, \(q_{nr}\) increases linearly in \(\hat{\theta}\). Proof. : Agent A of type \(\theta_{h}\) will not deviate and reveal, as agent B's strategy would become \(q_{R}=\frac{\theta_{h}+c}{2c}\) upon revelation as discussed earlier in the PBEM. By not revealing and adopting its idea, [PERSON]'s payoff would be \((\theta_{h}+c)-q_{nr}c\), which is strictly greater than the revelation payoff of \((\theta_{h}+c)-q_{R}c=\frac{\theta_{h}+c}{2}\). Thus, type \(\theta_{h}\) would not reveal. We have, thus, shown that the non-revelation set \(R^{c}\) contains \(\theta_{h}\). Consider the subset \(R^{h}\) of \(R^{c}\) which contains \(\theta_{h}\). So \(R^{h}=[\theta^{\prime},\theta_{h}]\subset R^{c}\). No type in this subset reveals in equilibrium. Now, suppose \(R^{h}\) is contiguous with a revelation range. Therefore, \(\theta^{\prime}\), the infimum of the set \(R^{h}\), has to be indifferent between revelation and non-revelation. If a revelation range is contiguous with \(R^{c}\), then the infimum of the set containing \(\theta_{h}\) must coincide with \(\hat{\theta}\), i.e., \(\theta^{\prime}=\hat{\theta}\). Proof. : If \(\theta^{\prime}>\hat{\theta}\), then agent A would prefer not to reveal and adopt rather than reveal as \((\theta^{\prime}+c)-q_{nr}c>(\theta^{\prime}+c)-q_{R}c\). This is because \(q_{nr}<q_{R}\) for all \(\theta>\hat{\theta}\). If \(\theta^{\prime}<\hat{\theta}\), then type \(\theta^{\prime}\) would prefer not to reveal and switch getting a payoff of \(q_{nr}c\) as opposed to \(q_{R}c\) if it revealed, where \(q_{nr}>q_{R}\) for all \(\theta<\hat{\theta}\). It is only for \(\hat{\theta}=\theta^{\prime}\) that \(q_{R}=q_{nr}\) and the payoffs from revelation and non-revelation are the same, making \(\theta^{\prime}\) indifferent between these strategies. There cannot be a contiguous revelation range with \(R^{h}\). \(R^{c}\) is an continuous interval with \(\theta_{h}\) in it. Proof.