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:

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.