# Coordination and Private Information Revelation
## 5 Purification of Agent B's Mixed Strategies  
The non-revelation result in Propositions 2 and 3 are open to the critique that we examine equilibria where agent B plays a completely mixed strategy whenever A reveals her private information about her own type. In an effort to address this issue, we next argue that there exists a natural extension of our framework such that sustaining the non-revelation result does not require the B agent to play mixed strategies in the coordination stage.  
To this end, we extend the baseline framework to allow for a unit mass of B-agents with different realizations over their private benefit \(b\), where the exact realization of \(b\) for any given B agent is private information. We then argue that a version of [PERSON]'s purification theorem goes through: following type revelation by the A agent, the continuation pure strategy played by the B-type agents generates the same probability distribution as the mixed-strategy equilibrium in the baseline model (see Lemma 3 later). Subsequently, we use this result to demonstrate (in Proposition 5 later on), that the non-revelation results goes through under this re-formulation. Formally, the B agents' private benefit from adopting \(B\) is distributed over the compact, continuous type space \(\Xi\), where \(\Xi=[b_{l},b_{h}]\subset\mathbb{R}_{+}\), with distribution \(G(b)\), where \(b_{l}<b_{h}\), and \(G(b)\) is non-degenerate and strictly increasing. Furthermore, for this section we assume that an analogue of Assumption 1 goes through, i.e., \(c>\max\{\theta_{h},b_{h}\}\).  
We next turn to modeling the coordination benefits in this setup. The coordination benefit on technology \(i\), \(i\in\{A,B\}\), arises if and only if the A agent, as well as a positive measure of B-agentsadopt this technology. Thus, conditional on the A agent adopting technology \(i\), the total coordination benefit from \(i\) is given by \(c.x\), where \(x\) denotes the fraction of B-agents opting for this technology. The modified payoff matrix in this new game is shown in Table 2.  
We next introduce some definitions that we need for the analysis:  
The strategy of a type \(\theta\) agent A in Stage 1, denoted \(\beta^{I}(\theta)\), is a probability distribution over \(M(\theta)\). Strategies of the agents in Stage 2, i.e., the coordination stage, are:  
* In Stage 2, the strategy of a type \(\theta\) agent A, i.e., \(\beta_{A}^{C}(\theta)\), maps from her message is Stage 1, to the set of probability distributions over her action space. Hence, \(\beta_{A}^{C}(\theta):M(\theta)\rightarrow[0,1]\).
* In Stage 2, agent B's strategy is a mapping from his own type space \(\Xi\), and the disclosure made by [PERSON] in Stage 1, to the set probability distributions over his own actions. Hence, \(\beta_{B}^{C}:\Xi\times M\rightarrow[0,1]\).
* Agent A's strategy in Stage 2 is said to be a cut-off strategy if there exists \(\hat{\theta}\in\ \Theta\) such that [PERSON] adopts \(A\) iff \(\theta\geq\hat{\theta}\).
* Agent B's strategy in Stage 2, \(\beta_{B}^{C}\), is said to be a cut-off strategy if there exists \(\hat{b}\in\ \Xi\) such that [PERSON] adopts \(B\) iff \(b\geq\hat{b}\).  
The belief of a type \(b\) agent B at the coordination stage, i.e., Stage 2, is a mapping \(\bar{B}_{b}\) from the message sent by agent A in the revelation stage to a probability distribution over agent A's type space \([\theta_{I},\theta_{h}]\), i.e., \(\bar{B}_{b}:M\rightarrow\Delta[\theta_{I},\theta_{h}]\)10.  
Footnote 10: Given that A can only mix between the true type and no announcement, note that a rational belief for B should put zero mass on all types other than what was announced. Pooling among different types can only occur if “no announcement” realizes. As mentioned earlier, we thank an anonymous reviewer of this journal for pointing this out.  
One can define a perfect Bayesian equilibrium of this game in the usual manner.