Can gambling theory stop overtraining?

The Kelly criterion calculates the best size bet in a game of chance. A key principle of the criterion is that over-betting will lead to ruin. Under-betting only slows the growth rate.

When selecting training loads, “over-betting” will always lead to over-training. “Under-betting” may slow progress, but consistent growth is more important.

The Formula

The Kelly criterion was derived in the 1950s. It allows a gambler to choose the best bet based on the probabilities of the hand they’re dealt. The bankroll will grow at the greatest compound growth rate, while eliminating financial ruin as a possible outcome.

The formula is: %K = %W – ( %L / R ), where

  • %K is the portion of the bankroll that should be bet;
  • %W is the chance of a winning outcome;
  • %L is the chance of a losing outcome; and
  • R is the ratio of winnings to the bet (aka. the odds)

In a coin toss with odds of 1:1 (%W = 50%, %L = 50%, R = 1), you’ll double your money if you win. But the Kelly criterion suggests betting nothing, because if you play the game long enough, you’ll be ruined.

%K = 50% – ( 50% / 1 )
%K = 50% – 50%
%K = 0%

But what if it’s a biased coin? Let’s say %W = 60% and %L = 40%. Then bet 20% of your bankroll–$20 if you’re starting with a bankroll of $100.

%K = 60% – ( 40% / 1 )
%K = 60% – 40%
%K = 20%

And if the odds are 2:1 for the same game, then you should bet 40% of your roll.

%K = 60% – ( 40% / 2 )
%K = 60% – 20%
%K = 40%

By using the Kelly criterion, you’ll never bet 100% of your roll, and you’ll never bet anything for a negative expected profit. You can’t be ruined.

But if you chronically bet greater than the prescribed portion, the eventual outcome is $0.

In contrast, betting less than the Kelly portion will slow the growth of the bankroll, but it also reduces the volatility.

What does this have to do with athletes?

The Kelly criterion reminded me of the daily bets that we make with training loads. Every day, we choose how long and how hard to train. That’s our bet.

If we always over-bet, then on a long enough timeline, the value of our fitness is going to zero. We’ll be over-trained and sitting at the bottom of a hole we can’t climb out of.

If we always under-bet, we’ll sacrifice some gains in the short-term, but our fitness will continuously grow. (That is, until our volume bumps into constraints of time, lifestyle, age, or expertise.)

“Do as little as needed, not as much as possible.”
~ Henk Kraaijenhof

How to apply it

You’ve got a big day planned tomorrow. It’s going to be hard, but you’ve been looking forward to it. It’s getting close to race season, so the fun begins. The plan for tomorrow is 10x one minute at 130% of race pace.

But when you wake up, your recovery test is throwing red flags. Your youngest son has had a hacking cough for a few days. Your legs feel tired going up stairs.

In effect, your bankroll the night before was $100, but this morning it’s only $20.

Do you adjust your bet to be a smaller portion of your roll? Or, screw it, just bet the whole thing?

“Note that in some instances there is not enough of a good bet or investment to allow betting the full [portion], so one is forced to underbet, reducing somewhat the overall growth rate and the risk.” (p.22, emphasis mine)

“The reason is that using too large [a portion] and overbetting is much more severely penalized than using too small [a portion] and underbetting.” (p.18)