Abstract This paper studies two-player games in continuous time with imperfect public monitoring, in which information may arrive both gradually and continuously, governed by a Brownian motion, and abruptly and discontinuously, according to Poisson processes. For this general class of two-player games, we characterize the equilibrium payoff set via a convergent sequence of differential equations. The differential equations characterize the optimal trade-off between value burnt through incentives related to Poisson information and the noisiness of incentives related to Brownian information. In the presence of abrupt information, the boundary of the equilibrium payoff set may not be smooth outside the set of static Nash payoffs. Equilibrium strategies that attain extremal payoff pairs as well as their intertemporal incentives are elicitable from the limiting solution. The characterization is new even when information arrives through Poisson events only.

This paper endogenizes intervention in financial crises as the strategic negotiation between a regulator and creditors of distressed banks. Incentives for banks to contribute to a voluntary bail-in arise from their exposure to financial contagion. In equilibrium, a bail-in is possible only if the regulator’s threat to not bail out insolvent banks is credible. Contrary to models without intervention or with government bailouts only, sparse networks enhance welfare for two main reasons: they improve the credibility of the regulator’s no-bailout threat for large shocks, and they reduce free-riding incentives among bail-in contributors when the threat is credible.