Ivars Peterson’s MathLand
December 2, 1996
Fair Play and Dreidel
For centuries, Jews have played the game of dreidel as part of the
festivities associated with Hanukkah. Surprisingly, it turns out that this
ancient game is also an unfair game. The dreidel is a four-cornered top or
spinner having sides labeled with the Hebrew letters Nun, Gimel, Hay, 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, at which time the players spin again in the same order. The game
continues until 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 Hay (H) comes up, the player collects half the pot. If
Shin (S) comes up, the player adds a penny to the pot.
In 1976, Robert Feinerman of Herbert H. Lehman College (CUNY) in Bronx,
N.Y., proved 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 derived the following formula for the expected value of the payoff
on the nth spin with N players: N/4 + [(5/8)^( N-1)][(N-2)/8].
A few years ago, as an undergraduate math major at MIT, 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 after G is spun and the amount pof 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 says. 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 holds.
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 is now a graduate student in statistics at the University of
Illinois at Urbana-Champaign. The brief bio that appears with her article in
the September College Mathematics Journal notes: “In the future, Felicia
intends to go first when playing dreidel, especially against her husband,
Ari Trachtenberg, a renowned dreidel enthusiast.”
Copyright C 1996 by Ivars Peterson.
Feinerman, Robert. 1976. An ancient unfair game. American Mathematical
Trachtenberg, Felicia Moss. 1996. The game of dreidel made fair. College
Mathematics Journal 27(September):278-281.
You can play a computerized dreidel game via an “automated random
dreidel-server” at the web site http://www.jcn18.com/spinner.htm.
Comments are welcome. Please send messages to Ivars Peterson at
firstname.lastname@example.org. Ivars Peterson is the mathematics and physics writer and
online editor at Science News (http://www.sciencenews.org/). He is the
author of The Mathematical Tourist, Islands of Truth, Newton’s Clock, and
Fatal Defect. His current work in progress is The Jungles of Randomness (to
be published in 1997 by Wiley