Fair Play and Dreidel

For centuries, the game of dreidel has been part of the festivities associated with Hanukkah. Surprisingly, it turns out that this ancient pastime is also an unfair game.

The dreidel is a four-cornered top or spinner having sides labeled with the Hebrew letters Nun, Gimel, He, and Shin.

Any number of players can participate, with each player contributing a penny or some other unit to the pot to start the game. The players then take turns spinning the dreidel until each player has had a turn. Then they spin again in the same order. The game continues until the players reach some mutually agreed stopping point.

Each spin has four equally likely outcomes. If the letter Nun (N) comes up, there's no payoff, and play passes to the next player. If Gimel (G) comes up, the player collects the entire pot, and everyone contributes a penny to form a new pot. If He (H) comes up, the player collects half the pot. If Shin (S) comes up, the player adds a penny to the pot.

In 1976, mathematician Robert Feinerman proved, as reported in the article "An Ancient Unfair Game," that the first player has a greater expected payoff than the second player, who in turn has a greater expected payoff than the third player. Thus, the first player has an unfair advantage over the second player, and so on.

"Furthermore, this unfairness is accentuated if a stopping rule is used which does not guarantee an equal number of turns to each player," Feinerman noted.

Feinerman derived the following formula for the expected value of the payoff on the nth spin with N players:

Some two decades later, Felicia Moss Trachtenberg extended Feinerman's results and worked out a way to make the game fair. The key is to change the ratio of the amount a each player puts in the pot to begin the game (or collects after spinning G) and the amount p of the penalty paid for spinning S.

"The modified game of dreidel will be fair just when p/a = N/2, where N is the number of players," Trachtenberg said. Thus, for four players, if the ante is 1 penny, the penalty should be 2 pennies.

To see why this ratio works, notice that the amount in the pot when the first player spins is Na. The player's possible payoffs are 0, Na, Na/2, and −p, depending on which side of the dreidel comes up. For the game to be fair for the second player, the expected payoff must remain constant from the first to the second spin, and that can happen only if the ratio p/a equals N/2.

In the standard version, the ratio is less than N/2, which biases the game toward the first player. It's also possible to bias the game toward the last player by making the ratio greater than N/2.

Trachtenberg evidently learned the lesson of her research, at least with regard to the standard game. The brief biography that appears with Trachtenberg's article in the September 1996 College Mathematics Journal noted: "In the future, Felicia intends to go first when playing dreidel, especially against her husband, Ari Trachtenberg, a renowned dreidel enthusiast."

Originally posted December 2, 1996