The Kelly Criterion

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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:

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.