  We study properties of quantum strategies, which are complete specifications
of a given party's actions in any multiple-round interaction involving the
exchange of quantum information with one or more other parties. In particular,
we focus on a representation of quantum strategies that generalizes the
Choi-Jamio{\l}kowski representation of quantum operations. This new
representation associates with each strategy a positive semidefinite operator
acting only on the tensor product of its input and output spaces. Various facts
about such representations are established, and two applications are discussed:
the first is a new and conceptually simple proof of Kitaev's lower bound for
strong coin-flipping, and the second is a proof of the exact characterization
QRG = EXP of the class of problems having quantum refereed games.
