The Dilemma of the Cypress and the Oak Tree
I study repeated games with mediated communication and frequent actions. I derive Folk Theorems with imperfect public and private monitoring under minimal detectability assumptions.
Even in the limit, when noise is driven by Brownian motion and actions are arbitrarily frequent, as long as players are sufficiently patient they can attain virtually efficient equilibrium outcomes, in two ways: secret monitoring and infrequent coordination. Players follow private strategies over discrete blocks of time. A mediator constructs latent Brownian motions to score players on the basis of others' secret monitoring, and gives incentives with these variables at the end of each block to economize on the cost of providing incentives.
This brings together the work on repeated games in discrete and continuous time in that, despite actions being continuous, strategic coordination is endogenously discrete. As an application, I show how individual full rank is necessary and sufficient for the Folk Theorem in the Prisoners' Dilemma regardless of whether monitoring is public or private.