# Coordination and Private Information Revelation
## 5 Purification of Agent B's Mixed Strategies
### Analysis: Non-Revelation by the A Agent in Stage 1  
Given the complementarities inherent in this game, cut-off strategies arise naturally for all types of A, as well as the B-agents. In the continuation game following type revelation by the A agent, we, therefore, focus on equilibria where the B-agents play a cut-off strategy with a cut-off of \(\hat{b}\). Next, consider the continuation game following non-revelation by agent A. Let \(\hat{\theta}_{NR}\) denote the cut-off for agent A and \(\hat{b}_{NR}\) denote the cut-off for the B-agents in the subsequent coordination stage. **Definition 2**. : _A PBEC for the modified game is a perfect Bayesian equilibrium where, in every continuation game, the B-agents play cut-off strategies with strictly interior cut-offs in the coordination stage of the game._  
_We first argue that in the unique PBEC for the continuation game following type revelation by agent \(A\), the probability that the B-agents adopt \(B\) is the same as that under the PBEM equilibrium where agent \(B\) plays a completely mixed strategy (see Lemma 1)._  
**Lemma 3**. : _Consider the Stage 2 continuation game where agent A reveals her type \(\theta\) in Stage 1._ \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Switch** & **Adopt \(B\)** \\ \hline Adopt \(A\) & \(\theta+c(1-l)\), \(c(1-l)\) & \(\theta+c(1-l)\), \(b\) \\ Switch & \(cl\), \(0\) & \(cl\), \(b+cl\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Modified payoff matrix for the one-shot game. _In any PBEC of this continuation game, the cut-off for agent \(B\) (\(\hat{b}(\theta)\)) and the probability that agent \(A\) adopts \(A\), i.e., \(\alpha(\theta)\), solves:_  
\[\hat{b}(\theta) = (\alpha-\frac{c+\theta}{2c})c,\] \[\alpha(\theta) = \frac{(c+\theta)}{2c}+\frac{G^{-1}((c-\theta)/2c)}{c}.\]  
_A \(\theta\)-type \(A\)-agent's payoff from revealing her type is \(\frac{c+\theta}{2}\)._  
_The equilibrium cut-off for the B-types is unique and interior, i.e., \(0<\hat{b}<1\). 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. :
1. 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 \(\hat{b}(\theta)\). Let \(l(\theta)\) be the fraction of B-agents that adopt \(B\) with \(l(\theta)\equiv 1-G(\hat{b}(\theta))\). Let \(\alpha(\theta)\) be the probability that the A-agent adopts \(A\). First, consider the decision problem facing a B-agent with private valuation \(b\). His expected payoff when he switches to \(B\), is:  
\[\pi_{B}(B)=(1-\alpha).(b+cl)+\alpha.b, \tag{8}\]  
whereas his expected payoff when he adopts \(A\), is:  
\[\pi_{B}(A)=\alpha.c(1-l). \tag{9}\]  
For the indifferent type \(\hat{b}\), Equating (8) and (9), we get:  
\[\hat{b}=(\alpha-l)c. \tag{10}\]  
Next consider the decision problem facing the A-agent with private valuation \(\theta\). Her expected payoff from adopting \(A\) is:  
\[\pi_{A}(A)=\theta+c(1-l), \tag{11}\]  
whereas her expected payoff from switching to \(B\):  
\[\pi_{A}(B)=c.l. \tag{12}\]  
For the A-agent of type \(\theta\) to be indifferent between \(A\) and \(B\), Equations (11) and (12) have to yield  
\[l(\theta)=\frac{c+\theta}{2c}. \tag{13}\]  
Given that \(c>\theta_{h}\geq\theta\), it is straightforward to check that \(0<l(\theta)<1\). Therefore, \(\hat{b}(\theta)=(\alpha-\frac{c+\theta}{2c})c\). Solving (10) and (13) simultaneously, we find that  
\[\alpha=l(\theta)+\frac{\hat{b}}{c}=\frac{(c+\theta)}{2c}+\frac{G^{-1}((c- \theta)/2c)}{c}>0, \tag{14}\]using the fact that \(l(\theta)=\frac{c+\theta}{2}\) (from (13)), and \(G(\hat{b}(\theta))=1-l(\theta)=\frac{c-\theta}{2c}\).11  
Footnote 11: Can one provide sufficient conditions such that \(\alpha<17\) This is equivalent to showing that \(\frac{G^{-1}((c-\theta)/2c)}{(c-\theta)/2c}<c\).