This paper considers dynamic games in which multiple principals contract sequentially and non-cooperatively with the same agent and provides characterization results useful for applications. Our benchmark model is one of private contracting in which downstream principals do not observe upstream mechanisms, nor the decisions taken in these mechanisms. We show that any equilibrium outcome that can be sustained with any arbitrary strategy space for the principals can also be sustained by restricting the principals to offer excended direct mechanisms. In these mechanisms, the agent first reports his extended type (i.e. his exogenous private information along with the endogenous payoff-relevant decisions contracted upstream), the principal then responds by offering the agent a (possibly degenerate) menu of contracts that are payoff-equivalent for that extended type, and finally the agent selects a contract from this menu and the contract is executed. We also show that characterizing equilibria through extended direct mechanisms is facilitated by the fact that (i) each principal can be restricted to offer a single mechanism; (ii) when the agent's strategy is Markov (i.e. it depends only on payoff-relevant information), each mechanism can be restricted to offer a single contract to each extended type; and (iii) restricting the agent's strategy to be Markov is without loss in the case of deterministic decisions, e.g. when the contracts are deterministic and the agent does not mix over effort. We finally show how the aforementioned results must be adjusted to occommodate alternative assumptions on the observability of upstream histories and/or the sequence of contracting examined in the literature.
