  A preparation game is a task whereby a player sequentially sends a number of
quantum states to a referee, who probes each of them and announces the
measurement result. The measurement setting in each round, as well as the final
score of the game, are decided by the referee based on the past history of
settings and measurement outcomes. Many experimental tasks in quantum
information, such as entanglement quantification or magic state detection, can
be cast as preparation games. In this paper, we introduce general methods to
design $n$-round preparation games, with tight bounds on the average game
scores achievable by players subject to constraints on their preparation
devices. We illustrate our results by devising new adaptive measurement
protocols for entanglement detection and quantification. Surprisingly, we find
that the standard procedure in entanglement detection, namely, estimating $n$
times the average value of a given entanglement witness, is in general
sub-optimal for detecting the entanglement of a specific quantum state. On the
contrary, there exist $n$-round experimental scenarios where detecting the
entanglement of a known state optimally requires adaptive measurement schemes.
