Sunday, November 06, 2005

Billionaires' Wager With Loaded Dice

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:

"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."

Then, in January 1997, Time magazine did a cover story titled "In Search of the Real Bill Gates" for which Buffett gave the magazine an interview. He spoke about the game-of-dice incident. However, his account was slighly different from that of Gates. Here is the relevant extract:

"He loves games that involve problem solving," Buffett says. "I showed him a set of four dice with numbers arranged in a complex way so that any one of them would on average beat one of the others. He was one of three people I ever showed them to who figured this out and saw the way to win was to make me choose first which one I'd roll." (For math buffs: the dice were nontransitive. One of the others who figured it out was the logician Saul Kripke.)

Three dice or four, it does not matter. What matters is: (1) how the numbers were arranged on those dice; and (2) what general lesson can be drawn from the wager?

There are several ways in which numbers on nontransitive dice can be arranged. Here are two ways involving four dice:

Take a look at the upper deck of four dice. A will tend to beat B. Why? Because four out of six times die A will land the number 4 and two out of six times it will land on the number 0. However, die B will land 3 on every roll because all its six sides carry the number 3. So, two-thirds of the time A will beat B and one thirds of the time B will beat A.

Similar analysis shows that B will beat C two-thirds of the time and C will beat D two-thirds of the time. The nontransitive property of the dice, however, also means that D will beat A two-thirds of the time.

So the trick lies in making your opponent choose first. If she chooses A, you must choose D. If she chooses B, you must choose A. If she chooses C, you must choose B, and if she chooses D, you must choose C.

Similar analysis will work with the lower deck of dice.

The general lesson from the wager is that blind faith in "first mover advantage" is often misplaced. Sometimes, the odds of the business game are such, that it is advantageous for you to allow your opponents to make the first move and then decide your own move (including whether you want to move at all or not).


Subsequent to my posting of the above blog post, Futile France wrote to me and clarified that Buffett-Gates wager involved four dice and not three.

I had relied on an old paper version of the HBR article (which mentioned three dice - I reconfirmed), whereas, Futile not only looked up his (corrected) electronic copy of the same article, he also checked out the Fortune article. He also gave two links which I'd like to share here:

Thanks Futile!