# Non-transitive Dice

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I recall reading that when Warren and Charlie first met they spoke about non-transitive dice at length.  Attached is a nice presentation on a few of the patterns for non-transitive dice.

Cheers

JEast

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I thought that was Buffett and Gates.

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In the same category: http://en.wikipedia.org/wiki/Flexagon A hexahexaflexagon is cool and easy to make (plenty of tutorials online), but more complicated stuff also exists. Fascinating stuff

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Actually this event with Buffett & Gates took place somewhere in the early 1990's...

... personally I enjoy this phenomenon the same way as the Monty-Hall problem, which was presented in the movie "21 the movie" about the MIT Blackjack team.

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In 1996, Bill Gates wrote a review of Roger Lowenstein's book, "Buffett: The Making of an American Capitalist".

Gates' review, which was titled, "What I Learnt from Warren Buffett" was published by Harvard Business Review in early 1996 and later by the Fortune magazine. In that review, Gates wrote a small passage on his and Buffett's love of mathematics, which I am reproducing below:

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"One area in which we do joust now and then is mathematics. Once Warren presented me with three unusual dice, each with a unique combination of numbers (from 1 to 12) on its six sides. He proposed that we each choose one of the dice, discard the third, and wager on who will roll the highest number most often. He gratiously offered to let me choose the die first.

"Okay," Warren said, "because you get to pick first, what kind of odds will you give me?"

I knew something was up. "Let me look at those dice," I said.

After studying the numbers on their faces for a moment, I said, "This is a losing proposition. You choose first."

Once he chose a die, it took me a couple of minutes to figure out which remaining die to choose in response. Because of the careful selection of the numbers on each die, they were nontransitive. Each of the three dice could be beaten by one of the others: die A would tend to beat die B, die B would tend to beat die C, and die C would tend to beat die A. This means that there was no winning first choice of a die, only a winning second choice. It was counter-intuitive, like a lot of things in the business world."

http://mathworld.wolfram.com/images/eps-gif/EfronsDice_700.gif

SUNDAY, NOVEMBER 06, 2005

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Nontransitive dice @ Wikipedia

A set of dice is nontransitive if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it's not true that A rolls higher than C more than half the time. In other words, a set of dice is nontransitive if its "rolls a higher number than more than half the time" relation is not transitive.

It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. Using such a set of dice, one can invent games which are biased in ways that people used to transitive dice might not expect (see Example).

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Non-transitive Dice

by James Grime

http://s14.postimage.org/70pfequ9t/image.jpg

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There are actually three sources on the net that sell these non-transitive dice.

All online shops are in the U.K.

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Grand Illusions in the UK (Online Shop)

Non-Transitive Dice - Set 1 (£3.99)

http://www.grand-illusions.com/acatalog/non_transitive_dice_1.jpg

Non-Transitive Dice - Set 2 (£3.99)

http://www.grand-illusions.com/acatalog/non_transitive_dice_2.jpg

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MathGear.co.uk in the UK(Online Shop)

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Highland-Games.co.uk in the UK (Online Shop)

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Once he chose a die, it took me a couple of minutes to figure out which remaining die to choose in response. Because of the careful selection of the numbers on each die, they were nontransitive. Each of the three dice could be beaten by one of the others: die A would tend to beat die B, die B would tend to beat die C, and die C would tend to beat die A. This means that there was no winning first choice of a die, only a winning second choice. It was counter-intuitive, like a lot of things in the business world."

This is a great topic and I remember when I read this story originally.

Perhaps it is obvious to others, but do other posters see why / how this is relevant to business and investing?

I have my opinion.  I haven't previously seen the transitive dice discussion subsequently connected to why this might be interesting to the history's greatest investor.  I think it is fascinating and learned something about how to think about investing when I first read the story.

It is particularly interesting w/r/t investing in the case of Buffett but, in my opinion, it is also interesting in the case of Gates and Microsoft.

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Once he chose a die, it took me a couple of minutes to figure out which remaining die to choose in response. Because of the careful selection of the numbers on each die, they were nontransitive. Each of the three dice could be beaten by one of the others: die A would tend to beat die B, die B would tend to beat die C, and die C would tend to beat die A. This means that there was no winning first choice of a die, only a winning second choice. It was counter-intuitive, like a lot of things in the business world."

This is a great topic and I remember when I read this story originally.

Perhaps it is obvious to others, but do other posters see why / how this is relevant to business and investing?

I have my opinion.  I haven't previously seen the transitive dice discussion subsequently connected to why this might be interesting to the history's greatest investor.  I think it is fascinating and learned something about how to think about investing when I first read the story.

It is particularly interesting w/r/t investing in the case of Buffett but, in my opinion, it is also interesting in the case of Gates and Microsoft.

It has to do with information asymmetry and the possibility of  scienter.  Simply, avoiding playing when the deck might be stacked against you or at least waiting until the informational advantage no longer resides with others.  In other words, waiting at bat for the three two count with the bases loaded in a game when you know the pitcher will then be forced to throw the perfect pitch.  If the pitch isn't perfect you can still win because the rules are skewed in your favor: no called strikes.

Interestingly, the probabilities in the Monte Hall game or the earlier The Lady or the Tiger game are very different if the person in charge of opening the door has optionality and evil intent.

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Interestingly, the probabilities in the Monte Hall game or the earlier The Lady or the Tiger game are very different if the person in charge of opening the door has optionality and evil intent.

Thanks twa...I hadn't thought of this particular similarity.  Interesting.

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I remember that Buffet and Ed Thorpe (the guy who actually came up with "black scholes" as well as how to count cards in blackjack) discussed non-transitive dice when they first met.  Here is a short discussion of the dice, his conversations with Buffet on the matter, and some of the implications:

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I remember that Buffet and Ed Thorpe (the guy who actually came up with "black scholes" as well as how to count cards in blackjack) discussed non-transitive dice when they first met.  Here is a short discussion of the dice, his conversations with Buffet on the matter, and some of the implications:

Thanks for posting this document. Haven't looked in a long time at Thorp's site. Only knew some references in Fortunes Formular or Ziemba's Kelly book with certain remarks about Thorp's first meeting with Buffett in the 1960's something??? Thorp said that he already knew that WEB would someday be the richest person on earth. Cheers!

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I think one of the investing take aways here is that the highest EV isn't necessarily always the best decision.  You have to look at a range of outcomes and then compare that to another range of options.

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I remember that Buffet and Ed Thorpe (the guy who actually came up with "black scholes" as well as how to count cards in blackjack) discussed non-transitive dice when they first met.  Here is a short discussion of the dice, his conversations with Buffet on the matter, and some of the implications:

T-bone1,

Great article / write-up.  Thanks for posting.  The board previously discussed "Fortune's Formula" -- that's the book where I first heard of Thorpe -- hadn't seen this.

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