Kelly's Heroes

by Jake Jacobs
10 July 2014

Jake Jacobs

Have you heard of "The Incredible Chabot Paradoxical Proposition?" Probably not. A recent post at BG Online reminded me of some obscure backgammon books I read many years ago.

The author, Michelin Chabot was a French-Canadian player with a mathematical bent. His books addressed the question: How much should you bet? Danny Kleinman reviewed Michelin's first guide, and gave it zero stars. (As Chabot is forgotten today I fear the author read his own books.)

In the second of his books, How To Maximize Your Profit Using the Kelly Criterion he introduced his "Paradoxical Proposition." Here is the position, the terms follow.

Let's suppose we are betting $1000 on this prop. You will start off playing Black, playing this position 360 times. You start with $1000 "capital" on paper. You must bet 5% of your capital, however much it is, on each trial. You must double, and I must beaver. Besides tracking your current capital we will track your roll each time. As an example, you start with $1000, so you must bet $50; then you double, I beaver, and you roll 63. We record the fact that you rolled 63, and that you now have $1200. You bet 5% of that, which is $60; you double, I beaver, and you roll 54. We record that, and the fact that you now have $1440. Having fun so far?

After your 360 trials we note your final capital. Now it is my turn. I take over as Black, starting with $1000 as you did. Instead of rolling I get whatever sequence of 360 rolls you just had, so you cannot complain that I got lucky rolls. I also must bet 5% of my capital each roll, but unlike you, I do not double. After 360 trials we compare my final capital with yours. Whichever of us ends up with more capital wins $1000 from the other player.

Care to play?

The proposition sounds crazy, but the side not doubling is nearly a 10 to 1 favorite. To understand why, we need to talk about risk of ruin. Suppose you have a favorite fish that you play three times week for $25 a point. Some weeks you lose, but mostly you win. You have never lost more than fifty points, nor won more than one hundred. You have been averaging nearly $1000 a week, which is good, because you have been out of work for a year, and the $12,000 cash you have saved up is all that stands between you and dumpster diving. One night you walk into the club, and your fish is waiting for you, and he is excited. "I won! I won! I won Powerball!" And then he says: "I feel like playing for more money. Let's play for $3,000 a point." Why not, you think. His winning the lottery didn't make him a better player. If you were making a thousand a week at quarters, now you should be able to make $120,000 a week! So you roll a 41, and play 24/23, 13/9. He rolls 55, you fan, and he doubles. You take, and three minutes later you are broke and out of action until you find a job and save some money. You took a risk, and you were ruined.
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