  This paper studies a multi-period demand response management problem in the
smart grid where multiple utility companies compete among themselves. The
user-utility interactions are modeled by a noncooperative game of a Stackelberg
type where the interactions among the utility companies are captured through a
Nash equilibrium. It is shown that this game has a unique Stackelberg
equilibrium at which the utility companies set prices to maximize their
revenues (within a Nash game) while the users respond accordingly to maximize
their utilities subject to their budget constraints. Closed-form expressions
are provided for the corresponding strategies of the users and the utility
companies. It is shown that the multi- period scheme, compared with the
single-period case, provides more incentives for the users to participate in
the game. A necessary and sufficient condition on the minimum budget needed for
a user to participate is provided.
