Currently, card plays are visible to all players. Given that the player to last play a card gets narrative control (within the confines of the strong/weak ending), this creates a discentive towards playing a card first.

I believe a solution could be to provide a function for players to play cards without revealing to the group (apart from the narrator) that cards have been played. Although you can see the other players moves, and know their general Strengths and Weaknesses, you won't know if the challenge has been met until the Narrator reveals such.

This approach would obviously not always be appropriate, and so I suggest it be a Narrator controlled option.