Regarding the recent question about which player it is better to be in a game of Russian Roulette (http://math.stackexchange.com/questions/96331/in-russian-roulette-is-it-best-to-go-first), where it was answered that neither of the players had the advantage as the overall odds where the same.
A game that can stop on the first move (and has the same overall odds for each player by the end) surely must provide an "initial advantage" for one of the players...
The game begins with player #1 pulling the trigger (1/6 chance dying) and player #2 waiting (0/6 chance dying).
Clearly, player #2 has the initial advantage (yet same overall odds) in this game.
Isn't it best to be player #2, since player #2 is guaranteed to live on the first move (which is player #1's move) and this game can stop right here on this move.
Rules of the game: 2 players, more than 2 chambers (win/lose or continue), one bullet, no shuffle between moves, player can't shoot other player, player can only lose when his turn produces the chamber with the bullet on trigger pull.
Initial Advantage: beginning advantage in game (somewhat independant of overall / last-move odds).