From time to time, a journalist may face a soul-shattering dilemma. A scoop so shocking it cannot be withheld, yet so terrible it cannot be told.
And what goes for journalists, goes double for stick-figure cartooning math teachers. Thus, as one who loves truth even at its ugliest, I choose to divulge a fact sure to rattle your faith in humanity itself:
The game of dreidel is built on a lie.
Dreidel, of course, is a beloved Chanukah game. (Happy Chanukah, everybody!) First, each player places a chocolate coin in the center. Then, you take turns spinning a four-sided top (the dreidel), obeying the commands that appear on its ides:
The top functions like a die, with an equal chance of landing on each side—at least, in theory.
The reality is far more sinister.
Fearless and groundbreaking research by Robert and Eva Nemiroff reveals that on the typical dreidel, not all sides are equally likely.
I quote here from their startling abstract:
all three dreidels tested—a cheap plastic dreidel, an old wooden dreidel, and a dreidel that came embossed with a picture of Santa Claus—were not fair… it is conjectured that hundreds of pounds of chocolate have been distributed during Chanukah under false pretenses.
It I worth asking: Why?
No, not “why does a Jewish toy come embossed with a picture of Santa Claus,” although this too is a vexing matter. I mean: Why is the dreidel unfair?
Is it shoddy craftsmanship?
A manufacturer’s deviousness?
Anti-Chanukah sabotage?
The likeliest answer: none of these. It seems that, across the board, spinning is a poor randomization process. A classic study by three Stanford researchers called Dynamical Bias in the Coin Toss found that spinning coins on a table was less effective for randomization than flipping them through the air.
One can imagine why. The long duration of a spin, from rapid beginning to wobbly end, allows time enough to amplify a tiny difference in weight distribution. The heavier side falls down. The lighter side lands up. Invisible deviations in density become visible disparities in chocolate allocation.
What’s the solution?
One drastic measure: change randomizers. Use a tetrahedral die, or two coins (with HH, HT, TH, and TT as the four outcomes). But this would remove the dreidel from dreidel. Unacceptable. When a patient comes with chest pains, you don’t yank out her heart.
Instead, I have a different solution: each turn, you spin the dreidel three times, and interpret the outcome according to this table:
Each row follows the same pattern. It consists of four permutations: one without nun, one without shin, one without he, and one without gimmel. Because order does not affect the probability of a permutation, each row is therefore equally likely.
Via this system, the underlying probabilities of the dreidel itself are rendered irrelevant. Even a grossly asymmetric dreidel can be used to play a fair and balanced game.
Now, is this hyper-complicated? Yes.
Liable to confuse and alienate children? Almost certainly.
Totally unnecessary, given that nobody cares whether the four sides of the dreidel come up with equal likelihood? Perhaps.
But mathematics has never been about “understandable” or “desired.” It has always been about insinuating itself, over all manner of protests, into nostalgic memories and cherished holiday sentiments. And I refuse to let that tradition die.
As shown in Feinerman [1976], even if the physical dreidel itself acted as a fair randomizer, the game remains unfair. Player one has an advantage over the rest of the players, player two has an advantage over all players except 1, et cetera.
Perhaps this ancient game is actually a thinly disguised lesson in how life isn’t fair, and “them that has, gets.”
Fascinating – I can’t tell if that feature is very un-Jewish or very Jewish (but either way I suspect I myself am not Jewish enough to judge).
Actually, Feinerman assumed the amount of money per player was continuous, so serious problems with his assumptions. However, I have (unpublished) work that was presented at the MOVES 2015 conference, building Markov chains to analyse who wins. For example, with 2 players, player 1 wins more often, but only 52.8% of the times with 2 coins each to start, down to 50.8% with 7 each. 3 players with 5 coins each player 1 36.2%, player 2 32.7%, player 3 31.1%. with 12 coins, 34.5%, 33.2%, 32.4%. As a function of coins, player 1 comes down, 2 and 3 go up, we 3 always worst. Same sort of decreasing chance to win with 4 players.
Love this! We were playing dreidel last night, and pondering how it could be that gimmel came up so rarely. I even proposed recording our data and seeing whether it was, in fact, conforming to the expected probability, but the darn kids nixed that plan.
As a mom, I’m glad to have an excuse to keep them from eating all the gelt with which they gamble. I thought maybe rigged dreidels were invented by mean moms like myself who restrict sugar…
🙂
I think it would take a *lot* of spins to establish the probability differences, but hey, you’ve got eight nights, and every study needs replication!
Note that the dreydl’s four letters are traditionally understood as שנגה, an acronym for נס גדול היה שם ‘a great miracle happened there’. Perhaps the miracle is that the game can be made to work correctly even when made of inherently unreliable parts, which is one of the messages of Judaism and its multi-millennial survival. (Dreydls sold in Israel have פ instead of ש, making it ‘a great miracle happened here.)
That makes me really want an Israeli dreidel (even though shin is a cooler-looking letter than pe).
The obvious solution is to flip the dreydel in the air like a coin, no?
Yeah, Kate Nowak proposed this on Twitter; others raised concerns that a clay dreidel might break when thrown. (Kate was not impressed.)
This is the perfect mixture of math and Jewish traditions… Fairness and gambl– I mean candy. 😊👍
This is reminding me of something I saw related to 20 sided dice used to play Dungeons and Dragons.
D+D dice are not made to the same standards as casino dice.
The dice are nowhere near perfect icosahedra. There is some sort of bias that should be be expected. But, to test the dice by random trials to determine what that bias might be, (with any moderately high degree of confidence) requires so many rolls, that the dice will be further deformed by all of these rolls. Whatever bias that might have been measured no longer apply.
Which gets to a question of philosophy. If you know that the driedel (or dice) is biased, but you know one knows what that bias is, is it still a fair game? (as has been discussed above the game is not fair due to order of play, but the larger point still holds.)
The traditional solution to making an unfair game into a fair one is to require the player who has an advantage to pay the other players a fee for participating…
And here I thought the solution was to call the player with an unfair advantage “the house.”
upvote
Proposed alternative title: Dreidelytics.
So the procedure shown here is a multi-symbol generalization of the Von Neumann debiaser algorithm for 2-values symbols (like coins, or bits). This has shortcomings. For example, I just spun a coin on my desk 8 times and the result was HTHTHTHT. This is completely unbiased, but has a serial correlation coefficient of -1.0. A reasonable hypothesis is that it’s the way I’m holding it – I pick it up and spin it with the winning side facing me and it lands on the opposite side. So the spins are not independent. The Von Neuman debiaser and derivatives such as the Yuval Perez debiaser require that the input samples be independent. When they are not, the result is not unbiased – for example the Von Neuman debiaser, when fed with the above sequence would yield TTTT. Hardly unbiased. Classic results in cryptography tells us that there is no deterministic algorithm than can turn a steam of dependent bits into an unbiased stream. However we can get close with MAC (message authentication code) algorithms using an independent seed. Close enough for crypto and close enough for apportioning chocolate.
Ack – I didn’t even consider the independence of the spins, which in hindsight (and as your tabletop experiment shows) is an obvious concern.
Nice to know the name of the algorithm – I’d seen it for coins, of course, but not with Von Neuman’s name attached.
I’m glad that the cryptographers have an epsilon-good-enough solution, though. More for their sake than for mine. Although who knows, maybe Jewish children can be sold on the merits of MAC.
Brilliant!
Happy Chanukah!
Math is my achilles heel. And the drawings, not bad at all. Quaint. ☺
Informative blog 😊
Really nice👏👍
The equalizer for fairness should be fairly obvious: Make the dreidels out of chocolate…and eat them after each spin.
Or you could play dreidel a different way…
http://www.majorleaguedreidel.com/
Very interesting analysis. Pretty cool.
Gre
Great info. I never understood how this game worked.
https://forward.com/culture/326379/the-true-history-of-the-dreidel/
Very interesting origin story. The idea that this was originally played in bars helps to justify how such an uninteresting game could survive (players were drunk and didn’t notice). I’m still curious how it came to be associated with Chanukah.
http://bit.do/netbij-com
Since this post is about Chanuccah, written on Hanukkah, it is worth mentioning that Chanuka is probably the holiday with the most number of possible English spellings. This is despite the fact that it has mainly one pronunciation (Sephardi, the Ashkenazi one is rare these days).
Happy Khanuka, everyone
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Eric Schwitzgebel, a philosophy professor at UC Riverside, has a nice (& funny) piece on Dreidel, which also explains why the wild unfairness of physical dreidels isn’t a problem, given the “real” point of the game:
“Dreidel: A seemingly foolish game that contains the moral world in miniature“
http://schwitzsplinters.blogspot.com/2017/12/dreidel-seemingly-foolish-game-that.html?m=1
Such a wonderful and funny piece! It’s smart, but it hit the jackpot for me when it encorporated humor too. Thank you for this.
Good
Tip lad tip tops for you 5p
I am a Jew who never played the game. I do own a few dreidels but they have collected dust and therefore throw the math off. Anyway we eat the chocolates with abandonment. I never understood what I was. A Jew with little background, barely given a bat mitzvah and knowing a few Yiddish words. Yet I am a Jew and married a Jew and insisted my kids be Jewish. And they in turn married non Jews and had children who are not Jewish. Throws me for a loop.
Every culture and religion just might have a built-in way to teach children that life is not always fair. The Santa Claus on the dreidel, though, that’s harsh.