EV Example: Dreidel

Dreidel is a game played during Hanukkah where each player starts by putting 1 piece into the pot and then players rotate around spinning the 4-sided dreidel. If the pot goes to 0 or 1 then all players again put 1 unit into the pot.

Dreidel Spins and EV

The outcomes are:

Dreidel Spin	Probability	Game Outcome
נ (nun)	$\frac{1}{4}$	Do nothing
ג (gimel)	$\frac{1}{4}$	Take whole pot
ש (shin)	$\frac{1}{4}$	Take half pot
ה (hei)	$\frac{1}{4}$	Put 1 unit into pot

The general expected value of a spin with pot size $p$ is:

$$\begin{equation} \begin{split} \mathbb{E}[\text{Dreidel Spin}] &= [\text{nun result}]*Pr(\text{nun}) + [\text{gimel result}]*Pr(\text{gimel}) \\ &\quad + [\text{shin result}]*Pr(\text{shin}) + [\text{hei result}]*Pr(\text{hei}) \\ \mathbb{E}[\text{Dreidel Spin}] &= 0*\frac{1}{4} + p*\frac{1}{4} + \frac{p}{2}*\frac{1}{4} + (-1)*\frac{1}{4} \\ &= 0 + \frac{p}{4} + \frac{p}{8} + (-0.25) \\ &= \frac{3p}{8} - 0.25 \end{split} \end{equation}$$

Example Scenario

Suppose it’s your turn to spin and the pot has 6 pieces in it.

The possible outcomes are:

Dreidel Spin	Probability	Game Outcome
נ (nun)	$\frac{1}{4}$	0
ג (gimel)	$\frac{1}{4}$	+6
ש (shin)	$\frac{1}{4}$	+3
ה (hei)	$\frac{1}{4}$	-1

We can now compute the expected value of the spin:

$$\begin{equation} \begin{split} \mathbb{E}[\text{Dreidel Spin}] &= [\text{nun result}]*Pr(\text{nun}) + [\text{gimel result}]*Pr(\text{gimel}) \\ &\quad + [\text{shin result}]*Pr(\text{shin}) + [\text{hei result}]*Pr(\text{hei}) \\ \mathbb{E}[\text{Dreidel Spin}] &= 0*\frac{1}{4} + 6*\frac{1}{4} + 3*\frac{1}{4} + (-1)*\frac{1}{4} \\ &= 0 + 1.5 + 0.75 + (-0.25) \\ &= 2 \end{split} \end{equation}$$

(It turns out that the game is “painfully slow” such that a 4 person game where each player starts with 10 units and each spin takes 10 seconds would take an average of 2 hours and 23 minutes (860 spins). Ben Blatt has suggested improvements.)