My sketch of the snowball fight reminded Eddie Dekel of a popular children’s game.  After he described it to me, I recognized it as a game I have seen my own kids play.  It works like this.  Two kids face off.  At each turn they simultaneously choose one of three actions: load, shoot, defend. (They do this by signaling with their arms: cock your wrist to load, make a gun with your fist to shoot, cross your arms across your chest to defend.  They first clap twice to synchronize their choices, just like in rock-paper-scissors.)

If you shoot when the other is loading you win.  You cannot shoot unless you have previously loaded.  If you shoot unsuccessfully (because the opponent either defended or also shot) your gun is empty and you must reload again.  (Your gun holds only one bullet.  But Eddie mentioned a variant in which guns have some larger, but still finite, capacity.)

The game goes on until someone wins.  In practice it usually ends pretty quick.  But what about in theory?

First a little background theory.  This is a symmetric, zero-sum, complete information multi-stage game.  If we assign a value of 1 to winning and 0 to losing, the symmetric zero-sum nature means that each player can guarantee an expected payoff of 1/2.  In that respect the game is similar to rock-scissors-paper.  Indeed the game appears to be a sort-of dynamic extension of RSP.

But, despite appearances, it is actually much less interesting than RSP.  In RSP, the ex ante symmetry (each player expects a payoff of 1/2) is typically broken ex post (often one player wins and the other loses, although sometimes it is a draw.)  By contrast, with best play LSD (load, shoot, defend silly I actually don’t know if it has an official name) is never decisive and in fact it never ends.

Here’s why.  The game has four “states” corresponding to how many bullets (zero or one) the two players currently have in their guns.  Obviously the game cannot end when the state is (0,0) and since playing load is either forbidden (depending on the local rules) or dominated when the state is (1,1), the game cannot end there either.

So it remains to figure out what best play prescribes when the game is imbalanced, either state (1,0) or (0,1).  The key observation is that just as at the beginning of the game, where symmetry implied that each player had an expected payoff of 1/2, it is still true at this state of the game that even the weaker player can guarantee an expected payoff of 1/2. Simply defend.  Forever if need be.  There is no reason to think that this is an optimal strategy but still its one strategy at your disposal so you certainly can’t do worse than that.

The surprising thing is that best play requires it.  To see why, suppose that the weaker player chooses load with positive probability.  Then the opponent can play shoot with probality 1 and the outcome is either (shoot, load) [settle down Beavis] in which case the opponent wins, or (shoot, defend) in which case the game transits to state (0,0).  Since the value of the first possibility is 1 and the value of the second is 1/2 (just as at the start of the game), this gives an expected payoff to the opponent larger than 1/2.  But since payoffs add up to 1, that gives the weaker player an expected payoff less than 1/2 which he would never allow.

So the weaker player must defend with probability 1 and this means that the game will never end.  Pretty boring for the players, but rather amusing for the spectators.

We can try to liven it up a bit for all involved.  The problem is that its not really like RSP or its 2-action cousin Matching Pennies which work the way they do because of the cyclical relation of their strategies.  (Rock beats Scissors which beats Paper which beats Rock…)  We can add in an element of that by removing the catch-all defend action and replace it with two actions defend-left and defend-right (say the child leans to either side.)  And then instead of plain-old shoot, we have shoot-left and shoot-right. Your shot misses only when you shoot to the opposite side that he leans.  There are a number of ways to rule on what happens when I shoot-left and he loads, but I would guess that anything sensible would produce a game that is more interesting than LSD.