December 27, 2005
The dreidel is always rigged
This is not widely known, but back in 1976, Robert Feinerman proved the following theorem about the dreidel:
Let X_n be the payoff on the nth spin of the dreidel and let p be the number of players. Then, the expected value of X_n, E(X_n), is:
(p / 4) + ((5 / 8) ^ (n - 1)) * ((p - 2) / 8)
That is, if there are more than two people playing a game of dreidel, there is a noticeable first player advantage in gelt. (I suspect at least one regular commenter knows this through empirical study.)
However, in 1996, Felicia Moss Trachtenberg came up with a simple way to tweak dreidel to give fair payoffs: adjust the penalty to ante ratio so that it is equal to the number of players divided by 2. Thus, if there are three players, the penalty should be three chocolate coins and the ante two chocolate coins (or six and four, or thirty to twenty, et cetera). If there are four players, the penalty could be two chocolate coins to an ante of one, since 2 / 1 = 4 / 2 .
More recently, Doron Zeilberger of Rutgers University conjectured that the length of a game of dreidel was of the order of the number of nuts (chocolate coins, whatever) squared. This was proved by his Rutgers colleagues Thomas Robinson and Sujith Vijay last year.
Y'all know what to do.
Non-linked references:
Feinerman, R., The American Mathematical Monthly, vol. 83, no. 8, (October 1976), pages 623-625.
Trachtenberg, F. M., The College Mathematics Journal, vol. 27, no. 4, (September 1996), pages 278-281.
Both papers are available on JSTOR, if you're lucky enough to have access.
Posted by coyu at December 27, 2005 02:43 AM