# Coordination and Private Information Revelation
## 3 The Model  
Two agents A and B each have an idea/technology of their own, denoted \(A\) and \(B\) respectively. There is one-sided asymmetric information: agent A has some private information regarding her own payoff from adopting \(A\). We analyze a two stage game with an initial information revelation stage, where agent A may or may not reveal her private information. This is followed by a version of the battle-of-the sexes game, where both the agents choose, simultaneously, whether to adopt their own idea, or to switch to the idea of the other agent. The outcome in Stage 2 of course depends on the information revealed earlier, if any, in Stage 1. In case an agent adopts her own idea, she obtains a private benefit. She also obtains an additional coordination benefit in case the other agent coordinates on the same idea as well. In case she switches to the idea of the other agent, she obtains no private benefit, but will obtain the coordination benefit in case both choose the same idea. Formally, agent A's private benefit from adopting \(A\), denoted \(\theta\), is distributed over the compact, continuous type space \(\Theta\), where \(\Theta=[\theta_{l},\theta_{h}]\subset\mathbb{R}_{+}\), with distribution \(F(\theta)\), with \(F(\theta)\) being non-degenerate and strictly increasing. The exact realization of \(\theta\) is however private information of agent A. Agent B's benefit from operating \(B\), denoted \(b\), is however deterministic. Furthermore, both agents obtain a coordination benefit \(c\) in case they both choose the same idea. The timing of the game is as follows:  
1. **Stage 1: Revelation:** Agent A decides whether or not to reveal her type \(\theta\). Agent A can either reveal her exact type by providing hard information, or decline to offer any information. The message space of a type \(\theta\) agent A is, therefore, \(M(\theta)=\{\theta\}\cup[\text{Not Reveal her Type}]\)3. Thus, the set of random messages of agent A of type \(\theta\) is given by \(\Delta(M(\theta))\), where \(\Delta(M(\theta))=\{m|m\) is a probability distribution over \(M(\theta)\}\). We define \(M=\cup_{\theta}M(\theta)=[\theta_{l},\theta_{h}]\cup[\text{Not Reveal her Type}]\). Footnote 3: We would like to thank an anonymous reviewer in this journal for pointing out that the message space (revealing only hard evidence) becomes type-dependent: \(M=M(\theta)\). 2. **Stage 2: Coordination:** The agents play a coordination game, where agent \(i\) chooses an action from \(\{\text{Adopt }i\), Switch to \(j,i,j\in\{A,B\},j\
eq i\}\). If both the agents choose to adopt their own idea, there is no coordination: agent A obtains \(\theta\) and agent B obtains \(b\). On the other hand, if both the agents switch to the other's idea, then they both have a payoff of 0. If they coordinate on \(A\), then the payoff vector is \((\theta+c,c)\), whereas it is \((c,b+c)\) if they coordinate on \(B\).4 Footnote 4: The coordination continuation game follows [11]. The payoff matrix for the Stage 2 game is given in Table 1 below:  
Assumption 1 below allows us to focus on the interesting case where coordination benefits are large enough such that the possibility of coordination failure is a significant strategic consideration. In Remark 2 later, we briefly examine the outcome for other parameter values. **Assumption 1**. : \(c>max\{\theta_{h},b\}\)_._  
We need some notations:  
The strategy of a type \(\theta\) agent A in the revelation stage, i.e., Stage 1, denoted by \(\alpha^{I}(\theta)\), is a probability distribution over \(M(\theta)\). The strategies of the agents A and B in Stage 2, i.e., the coordination stage, are:  
* The strategy of a type \(\theta\) agent A in the coordination stage is a mapping \(\alpha^{C}(\theta)\) from her decision in Stage 1 to a probability distribution over the action space \(\{\text{\emph{adopt }}A,\text{\emph{switch to }}B\}\), i.e., \(\alpha^{C}(\theta):M(\theta)\rightarrow[0,1]\).