# Coordination and Private Information Revelation
## 5 Purification of Agent B's Mixed Strategies
### Analysis: Non-Revelation by the A Agent in Stage 1
Clearly, one sufficient condition is that \(G(b)\) satisfies both (a) \(\frac{G^{-1}(x)}{x}\) be increasing in \(x\), and (b) \(\frac{G^{-1}(x)}{x}\leq 1\), \(\forall x\). Please note that this is satisfied whenever \(G(b)\) is uniform. (b) Using (13), the A-agent's expected payoff is given by \(cl(\theta)=\frac{c+\theta}{2}\). (c) Please note that \(G(\hat{b})=1-l(\theta)=\frac{c-\theta}{2c}\). Given that \(G(b)\) is strictly increasing, \(\hat{b}\) is unique. Given that \(c>\theta_{h}\geq\theta\), it is straightforward to check that \(0<G(\hat{b})<1\), so that \(b_{l}<\hat{b}<b_{h}\). Next observe that the fraction of B-agents adopting \(B\), i.e., \(l(\theta)=1-G(\hat{b})=\frac{\theta_{l}+c}{2c}\) (from (11)). From Lemma 1, recall that \(q_{R}(\theta)=\frac{\theta_{l}+c}{2c}\), where \(q_{R}(\theta)\) is the probability with which B adopts \(B\) in the our baseline model for the continuation game where agent A reveals her type to be \(\theta\). Thus, \(l(\theta)=q_{R}(\theta)\). Proposition 5 below is the central result in this section. We find that in the revelation stage of the modified game, agent A does not reveal her type, showing that the non-revelation result is robust to this modification. Proposition 5: Consider any PBEC of the modified game. (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 \(F(\theta)+\frac{\theta}{2c}+\frac{G^{-1}((c-\theta)/2c)}{c}\) is monotonic in \(\theta\), then this game has a unique PBEC. (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 \(\hat{\theta}_{NR}\) for the A, and \(\hat{b}_{NR}\) for the B agents. Equating the payoffs from switching to \(A\) and adopting \(B\) for the indifferent B agent, i.e., of type \(\hat{b}_{NR}\), we have that  
\[\hat{b}(\theta) = (\alpha-\frac{c+\theta}{2c})c,\]0  
Similarly, equating the payoffs from \(A\) and \(B\) for the indifferent A agent, i.e., of type \(\hat{\theta}_{NR}\), we have that  
\[\hat{b}(\theta) = (\alpha-\frac{c+\theta}{2c})c,\]1  
Given that \(c>\theta_{h}\geq\hat{\theta}\), it follows that \(0<G(\hat{b}_{NR})<1\), so that \(b_{l}<\hat{b}_{NR}<b_{h}\). Next, from Equations (15) and (16), we have that  
\[\hat{b}(\theta) = (\alpha-\frac{c+\theta}{2c})c,\]2  
It is straightforward to check that \(F(\hat{\theta}_{NR})<1\), so that \(\hat{\theta}_{NR}<\theta_{h}\).12 Given \(\hat{\theta}_{NR}\), \(\hat{b}_{NR}\) can then be solved using Equation (15). Furthermore, if \(\hat{\theta}_{NR}=\theta_{l}\), then from Equation (15), \(\hat{b}_{NR}=c.G(\hat{b}_{NR})\)In all cases, for all \(\theta\geq\hat{\theta}_{NR}\), A adopts \(A\) and for all \(\theta<\hat{\theta}_{NR}\), A switches to \(B\) upon non-revelation of \(\theta\) in Stage 1. Please note that for \(\hat{\theta}_{NR}>\theta_{l}\), the expected payoff for A, denoted \(\pi_{A}(NR)\), is:  
\[\pi_{A}(NR)=\left\{ \begin{array}{ll}\theta+c.G(\hat{b}_{NR}), \text{ if }\theta\geq\hat{\theta},\\ c[1-G(\hat{b}_{NR})],\text{ if }\theta<\hat{\theta}.\end{array} \right. \tag{18}\]  
Whereas, if \(\hat{\theta}_{NR}=\theta_{l}\), then from Equations (16) and (18), the expected payoff for A is:  
\[\pi_{A}(NR)=\theta+\frac{c-\theta_{l}}{2},\ \forall\ \theta. \tag{19}\]  
We first consider \(\hat{\theta}_{NR}>\theta_{l}\):  
1. At \(\theta=\hat{\theta}_{NR}\), \(\pi_{A}(NR)=\hat{\theta}_{NR}+c.G(\hat{b}_{NR})=\hat{\theta}_{NR}+c. (\frac{c- \theta_{NR}}{2c})=\frac{\theta_{NR}+c}{2c}=\pi_{A}(R)\). 2. For all types of \(\theta>\hat{\theta}_{NR}\), \(\pi_{A}(NR)-\pi_{A}(R)=\frac{\theta}{2}-c[\frac{1}{2}-G(\hat{b}_{NR})]\).