In this paper, we formulate in a game-theoretic framework three coordination schemes for analyzing DSO-TSO interactions. This framework relies on a reformulation of the power flow equations by introducing linear mappings between the state and the decision variables. The first coordination scheme, used as a benchmark, is a co-optimization problem where an integrated market operator activates jointly resources connected at transmission and distribution levels. We formulate it as a standard constrained optimization problem. The second one, called shared balancing responsibility, assumes bounded rationality of TSO and DSOs which act simultaneously and is formulated as a non-cooperative game. The last one involves rational expectation from the DSOs which anticipate the clearing of the transmission market by the TSO, and is formulated as a Stackelberg game. For each coordination scheme, we determine conditions for existence and uniqueness of solutions. On a network instance from the NICTA NESTA test cases, we span the set of Generalized Nash Equilibria solutions of the decentralized coordination schemes. We determine that the decentralized coordination schemes are more profitable for the TSO and that rational expectations from the DSOs gives rise to a last-mover advantage for the TSO. Highest efficiency level is reached by the centralized co-optimization, followed very closely by the shared balancing responsibility. The mean social welfare is higher for the Stackelberg game than under shared balancing responsibility. Finally, under imperfect information, we check that the Price of Information, measured as the worst-case ratio of the optimal achievable social welfare with full information to the social welfare at an equilibrium with imperfect information, is a stepwise increasing function of the coefficient of variation of the TSO and reaches an upper bound.