# Coordination and Private Information Revelation
## 4 The Analysis
### Non-Revelation Equilibria in the Presence of Coordination Failure  
We now turn to equilibria where there is coordination failure in Stage 2, focusing on equilibria where agent B plays a completely mixed strategy in the coordination stage. Our investigation is in line with many papers investigating coordination problems, e.g., ref [27] on corporate take-overs, ref [28] on repeated coordination games, as well as [29] on international environment agreements for addressing greenhouse gas emissions (modeled as a participation game). Ref [29], for example, defend their investigation of mixed strategy equilibria on the grounds that it captures the uncertainty of countries in coordinating an effective climate change treaty.7  
Footnote 7: In Section 5, we provide a purification argument that provides a foundation for the mixed strategic equilibrium that we examine here. We begin by examining equilibria in the coordination stage with the following definition. Definition 1: A _PBEM_ denotes a perfect Bayesian equilibrium where agent B plays a completely mixed strategy in the coordination stage, i.e., Stage 2. We begin by examining equilibria in the coordination stage. We first examine [PERSON]'s strategy following non-revelation of her own type. Lemma 1: _Consider the Stage 2 continuation game where agent A does not reveal her type in Stage 1. In any PBEM, agent A plays a cut-off strategy in Stage 2._  
Proof: Consider the Stage 2 continuation game following non-revelation of her type by agent A in Stage 1. Consider a PBEM where, in Stage 2, agent B plays adopt \(B\) with probability \(q_{NR}\), \(0<q_{NR}<1\). Next, note that the payoff to agent A in this continuation game from adopting \(A\) (call it \(\pi_{A}\) (Not Reveal and Adopt A)) is increasing in her type \(\theta\). Thus,  
\[\pi_{A}\left(\text{Not Reveal and Adopt A}\right)=(1-q_{NR})c+\theta. \tag{1}\]  
Similarly,  
\[\pi_{A}(\text{Not Reveal and Switch to B})=\text{q}_{\text{NR}}c. \tag{2}\]  
Let \(\hat{\theta}\) be the minimum of all \(\theta\in[\theta_{l},\theta_{h}]\) such that \(\pi_{A}(\text{ Not Reveal and Adopt A})\leq\pi_{A}\left(\text{ Not Reveal and Switch to B}\right)\). Hence, for all types \(\theta\geq(<)\hat{\theta}\), it is optimal to adopt \(A\) (switch to \(B\)) following non-revelation, given that \(0<q_{NR}<1\). In Lemma 2 below we consider the equilibrium outcome in Stage 2 following revelation of her type by agent A in Stage 1. Given that we have a standard Battle-of-the-Sexes game, we omit the proof (which is routine). Lemma 2: _Consider any candidate PBEM such that agent A reveals her type \(\theta\) in Stage 1. In Stage 2, agent B plays \"Adopt B\" with probability \(q_{R}(\theta)=\frac{\theta+c}{2c}\), and has a payoff of \(\frac{b+c}{2}\), whereas agent A of type \(\theta\) plays \"Adopt A\" with probability \(\frac{b+c}{2c}\) and has a payoff of \(\frac{\theta+c}{2}\)._From Lemma 2 note that \(q_{R}(\theta)\) is increasing in \(\theta\), so that agent B becomes more aggressive in adopting his own idea as \(\theta\) increases. This follows from the intuition of mixed strategies itself, which requires the choice of \(q_{R}(\theta)\) to be such that [PERSON] is indifferent between her two pure strategies. If agent A with high \(\theta\) reveals her type, agent B will respond much more aggressively compared to the case when agent A does not (since in this case agent B's response will be based on the expected value of \(\theta\)). This intuition has important implications for the coordination possibilities in the second stage game, and, as we shall find, plays an important role in Proposition 2 (to follow). Proposition 2 is the central result of this section, showing that in the presence of coordination issues there is no information revelation by agent A (except possibly by a single type), providing a new insight as to why 'unraveling' may not occur. Consider any PBEM. In Stage 1, all types of agent A, with the possible exception of one type, strictly prefer non-revelation to revelation.