Holiday Math: Dreidel Simulation

Happy Hanukkah, Center of Math fans! Since today's the first day (in Cambridge) of the holiday, I decided to take a mathematical/computational look at the dreidel game, one of the many traditions practiced on Hanukkah. Keep reading to learn how the game is played, and to see what I found out with my simulation of the game—the results may surprise you!

A dreidel, the object of import in the dreidel game, is a four-sided top, where each of the sides are marked, in circulation about the dreidel from right to left, with the Hebrew letters נ (Nun), ג (Gimel), ה (He), and ש (Shin).

In the dreidel game itself, each player starts with their own pool of coins, often the foil-wrapped chocolate coins called Hanukkah gelt, and contributes a coin to the central pot. Someone goes first, and on each player's turn, they spin the dreidel, and act based on what letter is showing when it falls. If the dreidel lands on נ, nothing happens; if it lands on ג , the player takes the whole pot and everyone contributes a coin to form a new pot; on ה, the player takes half the pot; and upon ש, the player must contribute a coin to the pot. Each of the letters respectively stands for the Yiddish words "nisht," "ganz," "halb," and "shtel ayn," meaning "nothing," "all," "half," and "put in"—these are of course the actions assigned to each face of the dreidel. If someone's out of money and has to contribute a coin, they're out. The dreidel is passed around the circle of usually around 4 players, and each player continues spinning it in succession until either everyone gets bored or everyone except one player is eliminated.

While the dreidel game's complete lack of player decision-making perhaps might decrease its appeal as a serious game to play, it in turn makes the game much easier to simulate and study using computational methods. So, I wrote a Python program which simulates games of dreidel, with various adjustable parameters regarding how the game is played. Here are some of the results I found. 

Firstly, as expected, increasing each players' wealth makes the game go longer. However, if the players keep playing until everyone but a single person has been eliminated, the games tend to go very long.

Each set of data here comprises ten thousand game simulations: all game simulations assumed four players, and that, being particularly dedicated, the players would quit only after a thousand turns had gone by. The number of coins for each data set indicates the number of coins each player began the game with, minus the first coin they contributed to the pot. Even starting with only three coins, over a third of the time the game would last over a hundred turns. Increasing the number of coins increases the modal game length, and when each player started with ten coins, nearly half the time the game would reach 1000 turns with at least two players still in the game.

Additionally, for these data, I checked for which player won, that is, which player was left standing, or who had the most coins at the end of the game, where the players' names are their order of play as the game begins:

No matter how many coins are used in the game, the first player seems to have a notable probabilistic advantage over the fourth player, though that does decrease with increasing numbers of coins used. How might it be possible to balance these odds out? Would increasing the penalty incurred by rolling a ש possibly lessen the advantage the first player has in the possibility of rolling a ג on their first turn?  A common variant is a penalty of three coins upon a roll of ש, one for each of the Shin's stems. The following graphs show the distributions of the winning player for games with 5 and 10 starting coins, where a roll of ש incurs a penalty of 1, 2, 3, or 4 coins.

For both game-wealths, this change appears to decrease, but doesn't totally eliminate the earlier players' advantage over the later players: the stabilization seems higher when the number of coins each player possesses is greater. Oddly enough, the second player also seems to gain a particular advantage for higher penalties, and the third player wins less frequently than the fourth. Of course, this also has the effect of making the games go quicker: here are the frequencies of game-lengths for 10-coin games with this modification. With a three or four coin penalty for a Shin, the longest games, which occur fairly frequently for lower penalties, are all but eliminated.

Though this change can speed up and balance out your games, my advice is still to play first. And once again, the Center would like to wish you a happy holiday season.