How to Win at “Shotgun” (or not)Posted: August 28, 2011
I’ve known about this game for quite a while. I learned it in camp as a kid, and loved it at the time. In college, some of my friends rediscovered this game, and would play it at the dining hall table. So, I thought I would write about it in a post.
This game is a little bit more obscure than some of the others I’ve covered, so let me go through some of the details (which can be found here, anyway). Basically, you move simultaneously, by slapping your lap twice, then doing one of three moves: reload, shoot, or shield. Reloading gives your gun an additional bullet. Shooting means you’re trying to kill the other guy. Shielding blocks the (potential) bullet that the other guy is shooting at you. You win by shooting the other guy while he’s reloading; if you shoot each other simultaneously, the game ends in a draw.
Let’s assign the winner 1 point, the loser 0 points, and each player 1/2 a point in case of a draw. That seems pretty reasonable, right?
So, first thing, we can see that there’s a very simple subgame perfect Nash equilibrium (SPNE): each player always shields, no matter how many bullets he/she or the other player have. That this is a SPNE is pretty clear: nobody can do better by ever doing something else, since they’ll never get the opportunity to successfully shoot the other guy anyway, since his shield will always be up. Thus the game ends in a draw (or goes on forever, you pick). It’s pretty boring, but it works.
The question is, can we come up with something more interesting? Can anyone actually ever win in an SPNE? As we’ll see, the answer is NO.
Notice that in the SPNE, the expected number of points at any time cannot be less than 1/2 for each player: if it were, then they could do better by simply shielding forever, guaranteeing them at least 1/2 a point. Since the total number of points is 1, this means that each player can always expect exactly 1/2 a point.
This immediately implies that neither player will ever reload when the other guy has a bullet in the chamber of his gun (at least, not with a non-zero probability). If there ever were the slightest chance that he would, the other guy could shoot him, guaranteeing himself an expected number of points greater than 1/2. This is because, in the slight probability that he was reloading, the other guy would win (and get one point); if he ended up shielding while the other guy shot, the other guy could just shield forever. But this goes against the point in the previous paragraph: in any SPNE, the expected number of points for each player is exactly 1/2. Thus one will never reload, when there’s a possibility one could be shot, in any SPNE.
But this means that no one ever wins. Sounds pretty boring. I don’t think I’ll be playing this anymore.