# Coordination and Private Information Revelation
## 4 The Analysis
### Non-Revelation Equilibria in the Presence of Coordination Failure
Recall from Lemma 2 that if an agent A of type \(\theta\) reveals her type in Stage 1, then her payoff in any PBEM is \(\frac{\theta+c}{2}\). Therefore for such revelation to be optimal for agent A, it must be that  
\[\frac{\theta+c}{2}\geq\text{Max}[q_{NR}c,\theta+c-q_{NR}c]. \tag{3}\]  
However, we note that  
\[\frac{\theta+c}{2}>q_{NR}c\Rightarrow\frac{\theta+c}{2}<\theta+c-q_{NR}c. \tag{4}\]  
Therefore, \(\frac{\theta+c}{2}\) cannot be bigger than both \(q_{NR}c\) and \(\theta+c-q_{NR}c\). Thus, for information revelation to be optimal it must be that  
\[\frac{\theta+c}{2}=q_{NR}c. \tag{5}\]  
Given that \(q_{NR}\) is independent of \(\theta\), (5) can hold for at most one \(\theta\), so that the probability of revelation is zero.8  
Footnote 8: We are indebted to an anonymous referee for suggesting an argument that shortened the proof. The intuition for non-revelation has to do with coordination possibilities in the second stage game. As argued earlier, for an agent A with a high realization of \(\theta\), non-revelation followed by choosing to adopt \(A\) is optimal. This follows as revelation would lead the other agent to follow extremely aggressive strategies. Whereas, if \(\theta\) is low, then agent A's private benefit from adopting \(A\) itself is low compared to the possible coordination benefits from \(c\). Consequently, agent A is more interested in coordination itself rather than the identity of the idea on which coordination takes place. Furthermore, given that \(b\) is large relative to \(\theta\), agent B will put a relatively 'large' probability on adopting \(B\) in case of information revelation, even if the revealed \(\theta\) turns out to be small. Therefore, agent A has little to gain by revealing information so as to encourage coordination on \(A\). She would rather ensure that coordination takes place on \(B\), since while she loses because agent B's idea is selected, she more than makes up for it because the probability associated with coordination on \(B\) is larger. We now characterize the equilibrium, showing that there is a 'unique' PBEM.9  
Footnote 9: In fact, we can prove that the non-revelation outcome is unique in the class of revelation strategies, where A reveals its type over finite unions of disjoint sets of the type space. The proof is in Appendix B. A unique PBEM exists. In this equilibrium, there is no information revelation in Stage 1. In Stage 2:1. _Agent_ \(A\) _adopts her own idea, i.e.,_ \(A\)_, if and only if_ \(\theta\geq\hat{\theta}\)_, where_ \(\hat{\theta}=F^{-1}(\frac{c-b}{2c})\) _and_ \(\theta_{l}<\hat{\theta}<\theta_{hr}\)_, and switches to_ \(B\) _otherwise._
2. _Agent_ \(B\) _adopts her own idea, i.e.,_ \(B\)_, with probability_ \(\frac{1}{2}+\frac{F^{-1}(\frac{c-b}{2c})}{2c}\)_._
3. _In case agent_ \(A\) _does not reveal her type in Stage_ 1_, agent_ \(B\) _believes that the type of agent_ \(A\) _is distributed over_ \([\theta_{l},\theta_{h}]\) _with distribution_ \(F(\theta)\)_. In case agent_ \(A\) _does reveal her type to be_ \(\theta\)_, say, the belief of agent_ \(B\) _puts probability_ \(1\) _on agent_ \(A\) _being of type_ \(\theta\)_._
4. _In this PBEM, a type_ \(\theta\) _agent_ \(A\) _has an expected payoff of:_ \[\pi_{A}(\theta)=\left\{ \begin{array}{ll}\theta+\frac{c-\hat{\theta}}{2},& \text{if }\theta>\hat{\theta},\\ \frac{c+\hat{\theta}}{2},&\text{otherwise},\end{array} \right.\] _and agent_ \(B\) _has an expected payoff of_ \(\frac{b+c}{2}\)_._  
Proof. : From Proposition 2, we know that with probability \(1\) there will be no revelation in the first stage. Next consider Stage 2. Given Lemma 1, we know that agent A will be playing a simple cut-off strategy where she adopts \(A\) if and only if her type is larger than some cut-off value, call it \(\hat{\theta}\).