Dreidel maximization

I'm not a genius at math, but I was sitting with a couple of friends who are naturally gifted. They are not Jewish, and I was explaining to them how one plays dreidel. (Or "the dreidel game," as most Googlable references seem to call it.) They started talking about how one could do best at it.

That is, how should one set about to win the most money through dreidel? (That not being the point of the game, but never mind about that for now.) Game theory generally involves maximizing expected earnings through decision making; the problem with dreidel is that precious little decision is involved. Let's try and describe the game and then see what variables one might be able to play with to tweak the expectations.

The four possible dreidel outcomes, together with what happens to one's earnings, go more or less like this:

N (nun): {null}
G (gimel): +x
H (hey): +x/2
Sh (shin): -2R

You get x (the size of the pot) if the dreidel lands on gimel, and 1/2x (half the pot) if it lands on hey; if the dreidel lands on shin, you put in some ante (usually an even number, in my experience), hence 2R.

Some simple things even people like me can see straight off. The first is that your earnings from gimel and hey depend on how much there is in the pot. That is, x changes with the progression of the game, so I really should have written it as a function of time, f(t,x). But that would be getting ridiculous.

One way of thinking would be that to maximize your earnings you should sit as far away from the luckiest (or craftiest) dreidelist as possible, so that the pot has a chance to fill up again after he or she spins his out-of-proportion gimels.

Another simple observation is that the relative "influences" on the game of gimel, hey, and shin depend on how much the ante is. If there's only a finite supply of money (or peanuts, raisins, or whatever currency you've chosen) available to the player, and 2R (the size of the ante required when Shin is spun) is a considerable fraction of that supply, then one shin can lay you low no matter how many nuns or gimels you've gotten.

One way of modifying the game might be to make the shin-ante (R) vary with the number of players, the size of the pot as it stands, or the time elapsed.

Or not. 35 days till Chanukah.

PS: Of course, a real mathematician has made some progress on the problem. He has conjectured, based on some simple simulations, that the length of a two-person dreidel game is on the order of the square of the number of nuts (or whatever tokens are used), and the length of a k-person game -- on the order of that number of tokens to the kth power.

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