This paper develops new recursive, set based methods for studying repeated games with private monitoring. For any finite-state strategy profile, we find necessary and sufficient conditions for whether there exists a distribution over initial states such that the strategy, together with this distribution, form a correlated sequential equilibrium (CSE). Also, for any given correlation device for determining initial states (including degenerate cases where players’ initial states are common knowledge), we provide necessary and sufficient conditions for the correlation device and strategy to be a CSE, or in the case of a degenerate correlation device, for the strategy to be a sequential equilibrium. We also consider several applications. In these, we show that the methods are computationally feasible, and how to construct and verify equilibria in a secret price-setting game.