The question, quite obviously, is of a philosophical nature and, as such, doesn't have a correct answer. Rather, the way you answer it reflects on your personality traits and, possibly, leads to self-analysis, soul searching and other meaningless crap.
Meanwhile, I am NOT a woman and so I am not qualified to answer this question, and neither is Benzo, I hope.
Had I been a man in the arena, I'd look long and hard into having an AK-47 ready. Put yourself in his shoes: if he follows her advice, any outcome is quite bad! If he sees the tiger behind that door, it means his beloved is a selfish fuck. If he sees a lady, it means the princess loves him so much that the life with another woman must be unbearable for him (I am assuming affairs are punishable by the same arena choice in that kingdom).
Like I said, the correct answer for a male is as follows:
1. Get your hands on AK-47 and a lot of extra clips in the bag.
2. Open any door you like. In fact, don't even look at the princess (you don't want to know the truth about how she feels anyway, what with women being about as steady as, say, a spring weather). Open the doors ignoring the ladies and looking for a tiger. Shoot the crap out of the tiger, thus demonstrating your resolve and tactical capabilities. Than turn to the king's lounge and announce, in a grave voice: "I will be leaving now, OK? But I WILL BE BACK". Curtains.
*takes a bow*