# Coordination and Private Information Revelation
## 4 The Analysis
### Non-Revelation Equilibria in the Presence of Coordination Failure
The value of this cut-off \(\hat{\theta}\), given that agent B is playing a completely mixed strategy, must make agent B indifferent between adopting \(B\) and switching to \(A\), so that  
\[(1-F(\hat{\theta}))c=(1-F(\hat{\theta}))b+F(\hat{\theta})(b+c)=b+F(\hat{\theta })c. \tag{6}\]  
This yields \(\hat{\theta}=F^{-1}(\frac{c-b}{2c})\), where \(\hat{\theta}\) is well defined since \(0<\frac{c-b}{2c}<1\) (given that \(c>b\)). Furthermore, given that agent B adopts \(B\) with probability \(q_{NR}\), the cut-off strategy of agent A in the coordination stage will be optimal if, at \(\hat{\theta}\), we get the following expression for \(q_{nr}\):  
\[\frac{\hat{\theta}+c}{2c}=q_{nr}. \tag{7}\]  
Given that \(\hat{\theta}=F^{-1}(\frac{c-b}{2c})\), the result follows. Figure 1 provides a graphical representation of agent A's payoffs under the three strategic options, not revealing her information, revealing her information and then playing \"Adopt A\", and revealing her information and then playing \"Switch to B\". Figure 1: Expected payoff for A with information revelation dominated by non-revelation payoffs. We next examine efficiency properties of this PBEM. Proposition 4: _The PBEM discussed in Propositions 2 and 3 is inefficient._  
Proof: Since agent B plays a completely mixed strategy, there is a positive probability that agent B will choose \(B\) even when \(\theta\geq b\), as well as choose \(A\) even when \(\theta<b\). At this point a few remarks would be useful. Remark 1: _We find that the full disclosure results in Section 4.1, together with the non-revelation results in Section 4.2, jointly suggest that it is the presence, or absence of strategic uncertainty that determines whether there is information non-revelation or not. Thus, both sets of results are of interest. Even so, in future work we plan to examine if one can use some selection mechanism to isolate the non-revelation equilibrium._  
_One strand of the literature on equilibrium selection considers global games and the role of higher order beliefs. The global games framework, as proposed by [30], has been extensively used to study, among others, equilibrium selection in coordination games arising in the pricing of debt ([31]), and to problems of stochastic common learning ([32]). In this context, ref [33] examines the role of higher order beliefs and the precision of signals about private vs. public information in equilibrium selection. However, note that in the coordination continuation game, not only are the pure strategy equilibria strict, they are also not dominance solvable. Hence, we cannot use the method of global games for selecting any one of the pure strategy equilibria over the other._  
Remark 2: _We next examine scenarios where the parameter values do not satisfy Assumption 1. Please note that if \(b>c\), then adopting \(B\) is a dominant strategy for agent \(B\). Similarly, if \(c<\theta_{l}\), then adopting \(A\) is a dominant strategy for agent \(A\). In either of these cases, the coordination problem disappears, and agent \(A\)'s payoff is the same irrespective of whether she reveals her own type in Stage 1 or not. Thus, the interesting case is if \(b<c\) and \(\theta_{l}<c<\theta_{l}\). In this case there is an equilibrium with partial information revelation. Let \(\hat{F}(\theta)\) denote the probability distribution derived from \(F(\theta)\) conditional on \(\theta\) being less than \(c\). It is now straightforward to construct an equilibrium where all \(A\) agents with \(\theta\geq c\) reveal their type and choose \(A\) in the coordination stage, whereas the other \(A\) agents do not reveal their type. In particular, we can mimic the argument in Propositions 2 and 3 (replacing \(F(\theta)\) with \(\hat{F}(\theta)\)), to construct equilibrium strategies for all \(A\) agents with \(\theta<c\). Interestingly, as discussed earlier in the introduction, ref [8] also demonstrates the existence of equilibrium with partial disclosure in the presence of hard information._