Why Kelly Criteria Don't Work for Individual Players

Fellow short-term investors:

For years I've read, and learned through sorry experience, that betting the full Kelly amount is risky, and that most competent players bet less...or, amounting to the same thing, bring a bigger bankroll than Kelly would suggest, just to keep the risk of ruin low.

How is it, though, that the formula for obtaining the maximum rate of return in a betting situation would also frequently produce such a high risk of ruin?

For some wierd reason, I was thinking about this today while driving into work. I'm not a mathematicain and am a pretty literal guy, so I came up with this example which demonstrates the problem:

First of all, as we all recall, the Kelly formula stipulates that the best bet is "The edge over the odds." So, if you've got $100,000 in your betting bank account, and if you're betting in a lottery, for instance, where a ticket costs $1 and the odds of winning are one in 100,000, and the payoff is $1,000,000 you have a ten to one advantage (great odds!) Thus the Kelly Criteria would be met if you bet 10 (the edge) divided by 100,000 (the odds) times your bankroll, which would be 1/10,000th of of your $100,000...or $10. This seems reasonable.

On the other hand, let's assume that you're playing mini-roulette with a wheel that has only 10 slots, so the odds are 10 to 1 that you'd win any single bet. Let's assume that the payoff is $100 on a $1 bet, again a 10 to 1 edge. Kelly would tell you to bet 10 (the edge) divided by 10 (the odds) = 1 times your kitty, or ALL $100,000 on a single spin!

Now you can see that if a team of 10 or 100 players used this strategy, they'd win money pretty fast, but 9 out of 10 individuals would be bankrupted just so that the 1 out of 10 could win big for everybody.

Anyhow, I don't play on a blackjack team, and so I don't bet the Kelly amount.

In its purest form, Kelly applies only to win-or-lose bets (i.e. you bet a dollar and you either lose your dollar or you win two dollars). In this case:

Kelly Bet = Expectation x Bankroll
(where Expectation = expected long term dollars won per dollar bet)

For bets with more complicated outcomes than a simple win-or-lose bet (including both your lottery example and your roulette example), you must make a correction for the variability of your return. The link below (to an earlier post) explains how this is handled, and provides clarification about Kelly betting.

you have a 1/10 chance of winning with a 100:1 payoff...first, no such game would exist. Second you would just bet every number on the game you described to lock in a guaranteed win. duh! You would bet $10,000 on each of 10 numbers and win 100x10,000 on the number that hits ($1,000,000 per spin guaranteed win)

Fat Freddie:

I read your post and it was great. Just to clarify my language a bit, the "edge over the odds" really is the "expectation" per bet.

Please explain, though, if you can...maybe with an example...how to divide this by the variance, the standard deviation squared, in order to get the correctred Kelly amount for an actual game. I know that the standard deviation for the average blackjack game...at least the ones I simulate...is around 14 per 100 hands. How would you calculate the S.D. for a single bet, which would be needed to figure the size of that one bet?

I realize this is probably obvious to the math guys, but I'm just a player.

When you say that standard deviation is "14 per 100 hands". I assume this means when you play 100 hands, (using predetermined casino conditions, strategy and betting ramp), then the standard deviation of your total dollar outcome will be 14 of your minimum bet units.

In practical terms here's what to expect: If you use a minimum bet of $15 (an arbitrary value chosen because it corresponds to the sample calculation below), and you play sessions of exactly 100 hands, then (a) 68% of your sessions will be within $210 (1 standard deviation of 14 x $15) of your expectation, and (b) 95% of your sessions will be within $420 (2 standard deviations of 14 x $15) of your expectation. (Note that the values of 68% and 95% come from normal probability tables.)

As you play more and more hands, the good and bad hands (and/or good and bad sessions) tend to average each other out, and your AVERAGE outcome (outcome per hand) becomes less variable. However at the same time your DOLLAR outcome (total outcome for all hands) becomes more variable. The distinction between average outcome and dollar outcome is important when discussing the standard deviation of multiple hands (this distinction is not necessary when discussing the standard deviation of a single hand). The standard deviation you mentioned (14 per 100 hands) is almost certainly based on dollar outcome rather than average outcome.

It is relatively simple to calculate the effect that the number of hands has on standard deviation. If we define S as the standard deviation for a single hand, and N as the number of hands, then:

Standard Deviation of AVERAGE Outcome (Bet Units/Hand) = S / (Square Root of N)
Standard Deviation of DOLLAR Outcome (Bet Units) = S x (Square Root of N)

Using the convenient fact that the square roots of 1, 100 and 10,000 are 1, 10 and 100 respectively, we can use the above equations to come up with the following series of equivalent statements (based on the example of standard deviation = 14 per 100 hands):

For 1 Hand, Standard Deviation of AVERAGE Outcome = 1.4 Bet Units/Hand
For 100 Hands, Standard Deviation of AVERAGE Outcome = 0.14 Bet Units/Hand
For 10,000 Hands, Standard Deviation of AVERAGE Outcome = 0.014 Bet Units/Hand
For 1 Hand, Standard Deviation of DOLLAR Outcome = 1.4 Bet Units
For 100 Hands, Standard Deviation of DOLLAR Outcome = 14 Bet Units
For 10,000 Hands, Standard Deviation of DOLLAR Outcome = 140 Bet Units

Now for a sample calculation. Remember that the formula for a full Kelly bet is:

Full Kelly Bet = Expectation x Bankroll / Variance
(where Variance is Standard Deviation squared)

IF
Bankroll = $10,000
Expectation (expected gain) = 1% (expressed as a decimal this is 0.01)
Standard Deviation = 14 bet units per 100 hands

THEN
Standard Deviation for a single hand = 14/(square root of 100) = 1.4
Variance = Standard Deviation Squared = 1.4 x 1.4 = 1.96
Full Kelly Bet = 0.01 x 10,000 /1.96 = $51

As you mentioned, it is not prudent to bet full Kelly, so let's assume you bet 30% of Kelly, which is a reasonable value. In this case your bet unit should be 0.3 x $51 = $15. Remember this is your minimum bet unit, and is based on the bet spread and bet ramp you used when determining your standard deviation. If you decide to change your bet spread or bet ramp, you should re-estimate your standard deviation, and recalculate your optimal bet size.

Freddie,
Thanks for taking the time to provide that nice explanation, it clarifies the situation quite a bit for me.

Your Kelly math was off; Kelly never risks everything. On the mini-roulette the fair odds are 9:1, the actual odds are 100:1. The expectation is .1*100 + .9*(-1) = 9.1. The Kelly-optimal bet is $100,000*9.1/100 = $9,100. That doesn't seem entirely unreasonable for a decent shot of becoming a millionaire.