# Coordination and Private Information Revelation
Letting \(\theta\) denote the value of her own idea to agent A, if agent A reveals a relatively high \(\theta\), agent B becomes more aggressive in adopting his own idea and choosing \(B\) with a relatively high probability in the ensuing coordination game. Thus, if an agent A with high valuation 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 average value of agent A's type). For such an agent A, non-revelation is optimal. Whereas if agent A has a relatively 'bad' idea,then agent A is more interested in coordination itself, rather than the identity of the idea on which coordination takes place. Therefore, agent A has little to gain by revealing information and ensuring coordination on her own idea. She would rather ensure that coordination takes place on agent B's idea (\(B\)) and still gain through a higher payoff due to the higher probability associated with coordination on \(B\). So, agent A with 'bad' idea (low values of \(\theta\)) would also prefer not to reveal. Please note that the non-revelation outcome involves agent B playing a completely mixed strategy in the coordination stage, with [PERSON]'s adoption of a mixed strategy reflecting his lack of knowledge regarding both [PERSON]'s type as well as action. It has been argued, most notably by [11], that in the presence of coordination uncertainties, mixed strategies can help capture such uncertainties. In a similar vein, ref [12] argues that randomization in mixed strategies reflects the uncertainty in the mind of a player about the opponent's strategy, rather than a deliberate mixing of pure strategies. It is natural to ask, however, if the non-revelation result is critically dependent on the fact that agent B plays a completely mixed strategy in equilibrium. To that end we adopt a purification argument akin to [PERSON]'s defense of mixed-strategy equilibria [13], and examine a modified version of our baseline framework where agent B's type is also private information (though agent B cannot reveal his type as hard information). We find that the coordination game has an equilibrium where each B-type plays a pure strategy, with each B-type opting for technology \(B\) iff his valuation exceeds a critical cut-off. Interestingly, this strategy generates the same probability distribution over agent B's actions, as that under the mixed-strategy equilibrium in our baseline framework. We use this equilibrium to demonstrate that the non-revelation result holds in the purification setup as well, even though the coordination game does not involve any type of agent B playing mixed strategies. Finally, we go on to examine several extensions of the baseline model. We allow for both sided asymmetric information (where both agents can reveal hard information) as well as the possibility of imprecise information revelation. The non-revelation result is robust to these extensions. Furthermore, in case of mandatory disclosure of information, we find that the overall probability of coordination (on either A or B) may or may not exceed that in the absence of such mandatory disclosure, with the result depending on the nature of the distribution of \(\theta\). In line with the literature on persuasion games, this paper restricts itself to revelation of hard evidence. With soft information, we would be in the realm of signaling games (cheap talk games, to be precise). It is well-known that in this class of games a babbling equilibrium necessarily exists, but equilibria with partial information revelation also exist as demonstrated by [14] in their path-breaking contribution.2 Our result is different in that we show, modulo there being strategic uncertainty in the coordination continuation game, non-revelation or babbling follows as the _unique_ outcome. It would, however, be of interest to examine if the results are robust to allowing for soft information.