Extensive Form Games ( The game of “ chicken”) Two cars are driving at each other at great speeds.
If nobody changes directions, in 3 seconds they will collide and die a gruesome death,
yielding payoffs of −100 for both players. At every second they have to (simultaneously)
decide wether to continue or not (i.e. there are 3 periods to this game). Payoffs are as
follows: If both players choose continue for each of the three periods, then they get −100.
If at any period one player choses “not continue” and the other chooses “continue” the
player that chose not to continue gets 0 and the other gets 100. If at some point both
simultaneously choose not to continue they both get 0.

True or False: Any equilibrium has the property that one of the players chooses
not to continue, but this only happens in the last period (as in the movies!).

I just wanted to double check my answer with someone. 

My answer: I think that the answer is false. Because the strategy profile where player 1 plays continue at every stage and player 2 plays not continue at every stage is a subgame perfect Nash equilibrium I believe? Because based on the one shot deviation principle, there is not point in the history where player 2 can deviate from this strategy at one node profitably. And (continue, not continue) is a Nash equilibrium for the individual game. Thus, this strategy would result in a subgame perfect Nash equilibrium without getting to the third stage of this game.
I believe you are correct -- though I will say appealing to the one-shot deviation principle here seems a little overpowered. There are only three stages to this game so checking for all equilibria (and not just your conjectured one) in each proper subgame is doable. You do seem to be applying it correctly, however!