If this is all obvious, forgive me I came late to this (I grew up in Orange County, CA where it last snowed in December of Yeah Right.)

The first thing to do, obviously is to make a snowball.  Your enemy combatant will do the same.  You each now have one snowball in your stockpile.  What next?

If you throw your snowball you will be unarmed and certain to pay the consequences.  So you don’t.  Neither does she.  You are at a standoff, but very soon you figure out what to do while you wait for the standoff to resolve.  Make another snowball.  Of course she does the same.

Now you each have an arsenal of two snowballs.  Two is very different from one however because if you throw your snowball you still have one to defend yourself with.  But you will have one fewer than she.  This still puts her at an advantage because once you use your last snowball you are again unarmed.  So you will only throw your first snowball if you have a reasonable chance of landing it.

The alternative is to make another snowball.  Which of these is the better option depends on what she is expecting.  If she knows you will throw, she is prepared to dodge it and then press her advantage.  If she knows you will make another one she will wait for you to reach down into the snow when you are most vulnerable and she will draw first blood.

So you have to randomize.  So does she.  There are two possible outcomes of these independent randomizations.  First, one or two snowballs may fly resulting in a sequence of volleys which eventually deplete your stocks down to one or two snowballs left.  The second possibility is that both of you increase your stockpile by one snowball.

Thus, equilibrium of a well-played snowball fight gives rise to the following stochastic process.  At each stage, with a certain positive probability, the stockpiles both increase by one snowball.  This continues without bound until, with the complementary probability in each stage, a fight breaks out depleting both stockpiles and beginning the process again from zero.

Special mention should be made of a third strategy which is to be considered only in special circumstances.  Rather than standing and throwing, you can charge at her and take a shot from close range.  This has the obvious advantages but clearly leaves you defenseless ex post.  Running away should be ruled out because you will be giving up your entire store of snowballs and eventually you will have to come back.  No, the only option at this point is to tackle her, landing you both deep in the snow.  With the right adversary, this mutually assured destruction could be the best possible outcome.

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