The Kelly Criterion

Epistemic Status: Reference Post / Introduction

The Kelly Criterion is a formula to determine how big one should wager on a given proposition when given the opportunity.

It is elegant, important and highly useful. When considering sizing wagers or investments, if you don’t understand Kelly, you don’t know how to think about the problem.

In almost every situation, reasonable attempts to use it will be somewhat wrong, but superior to ignoring the criterion.

What Is The Kelly Criterion?

The Kelly Criterion is defined as (from Wikipedia):

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:

{\displaystyle f^{*}={\frac {bp-q}{b}}={\frac {p(b+1)-1}{b}},}

where:

  • f * is the fraction of the current bankroll to wager, i.e. how much to bet;
  • b is the net odds received on the wager (“b to 1″); that is, you could win $b (on top of getting back your $1 wagered) for a $1 bet
  • p is the probability of winning;
  • q is the probability of losing, which is 1 − p.

As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), and the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of the bankroll at each opportunity (f* = 0.20), in order to maximize the long-run growth rate of the bankroll.

(A bankroll is the amount of money available for a gambling operation or series of wagers, and represents what you are trying to grow and preserve in such examples.)

For quick calculation, you can use this rule: bet such that you are trying to win a percentage of your bankroll equal to your percent edge. In the above case, you win 60% of the time and lose 40% on a 1:1 bet, so you on average make 20%, so try to win 20% of your bankroll by betting 20% of your bankroll.

Also worth remembering is if you bet twice the Kelly amount, on average the geometric size of your bankroll will not grow at all, and anything larger than that will on average cause it to shrink.

If you are trying to grow a bankroll that cannot be replenished, Kelly wagers are an upper bound on what you can ever reasonably wager, and 25%-50% of that amount is the sane range. You should be highly suspicious if you are considering wagering anything above half that amount.

(Almost) never go full Kelly.

Kelly betting, or betting full Kelly, is correct if all of the following are true:

  1. You care only about the long-term geometric growth of your bankroll.
  2. Losing your entire bankroll would indeed be infinitely bad.
  3. You do not have to worry about fixed costs.
  4. When opportunities to wager arise, you never have a size minimum or maximum.
  5. There will be an unlimited number of future opportunities to bet with an edge.
  6. You have no way to meaningfully interact with your bankroll other than wagers.
  7. You can handle the swings.
  8. You have full knowledge of your edge.

At least seven of these eight things are almost never true.

In most situations:

  1. Marginal utility is decreasing, but in practice falls off far less than geometrically.
  2. Losing your entire bankroll would end the game, but that’s life. You’d live.
  3. Fixed costs, including time, make tiny bankrolls only worthwhile for the data and experience.
  4. There is always a maximum, even if you’ll probably never hit it. Long before that, costs go up and people start adjusting the odds based on your behavior. If you’re a small fish, smaller ponds open up that are easier to win in.
  5. There are only so many opportunities. Eventually we are all dead.
  6. At some cost you can usually earn money and move money into the bankroll.
  7. You can’t handle the swings.
  8. You don’t know your edge.

There are two reasons to preserve one’s bankroll. A bankroll provides opportunity to get data and experience. One can use the bankroll to make money.

Executing real trades is necessary to get worthwhile data and experience. Tiny quantities work. A small bankroll with this goal must be preserved and variance minimized. Kelly is far too aggressive.

If your goal is profit, $0.01 isn’t much better than $0.00. You’ll need to double your stake seven times to even have a dollar. That will take a long time with ‘responsible’ wagering. The best thing you can do is bet it all long before things get that bad. If you lose, you can walk away. Stop wasting time.

Often you should do both simultaneously. Take a small amount and grow it. Success justifies putting the new larger amount at risk, failure justifies moving on. One can say that this can’t possibly be optimal, but it is simple, and psychologically beneficial, and a limit that is easy to justify to oneself and others. This is often more important.

The last reason, #8, is the most important reason to limit your size. If you often have less edge than you think, but still have some edge, reliably betting too much will often turn you from a winner into a loser. Whereas if you have more edge than you think, and you end up betting too little, that’s all right. You’re gonna be rich anyway.

For compactness, I’ll stop here for now.

 

 

 

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4 Responses to The Kelly Criterion

  1. Dan says:

    It looks like some of the ways in which the 8 criteria don’t hold are reasons to bet less than Kelly (e.g. 7, 8) and others are reasons to bet more than Kelly (e.g. 1, 2, 6). I’m not sure why you come down so strongly on not betting more than Kelly (or even than half Kelly).

    • TheZvi says:

      It’s the more important mistake to avoid making. Given the conditional where you’re trying to grow a bankroll you can’t replace, the reasons the other way don’t apply, and betting too big becomes *very* bad, turning winners into losers. I do think the other mistake is important too, but wanted to get the foundational post out of the way here and follow up later with a ‘reasons to risk it all’ later.

  2. Pingback: Rational Feed – deluks917

  3. Pingback: The Art of the Overbet | Don't Worry About the Vase

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