Applying Kelly Criterion to a session

How would you apply the Kelly criterion to a session as opposed to individual bets? Let's say for example a casino is offering a promotion -- if you do $1,000,000 coin in, they'll give you $10,000 cash. If you don't complete the $1M coin in, you get nothing back. You think about it for a second and find they have $1 9/6 JOB (99.54% return, 19.51 variance, 4.417 SD), giving you a 0.54% edge including the $10k bonus.

You play at half kelly and want to make sure your BR is big enough to play this. Normally, you'd just use the formula (Edge/Variance)*Bankroll which is (0.0054/19.51)*BR. You determine you need 7,225 bets or $36,125 worth of BR to play this game at half kelly ROR (~1.8%?). You have $40,000 bankroll, so you think you're safe.

Of course, that's not quite right since you aren't getting that 1% back on each play, but only at the very end. You could run really bad at the start, pushing your bankroll below the required amount to make a single bet. Alternatively (far-fetched scenario) it could be a situation where your coin-in EV is going to eat your entire bankroll before you can even finish the play (EG: $10M coin in for $100k cash bonus would cost you $46k in EV, but you only have $40k BR...so obviously this wouldn't be playable under those circumstances, but the "regular kelly" formula would say to play it).



I know you can calculate the standard deviation of the $1M coin in (about $9,876) and go from there, but I'm not particularly interested in knowing what that number is, since that still wouldn't tell you at what ROR you'd be playing to. (Perhaps you can play this promo over and over again.)

Instead, I'd like to figure out how (if possible) you can treat it as one big $1,000,000 wager with a tiny amount of variance. The tiny amount of variance is relative to the $1M wager. This way you can determine if your bankroll is large enough to support making this type of play. Or, if there is another way of going about it to determine BR requirements.

BTW -- This isn't a real play AFAIK, just using an example. Therefore, I'm not interested in just hearing the answer like "playing that with 40k roll would be playing at X% ROR". I'd like to know how to come up with the solution. Teaching a man how to fish kinda thing.

I'd like to hear people's thoughts on this. Or perhaps I'm way overthinking it and it's really as simple as "if you have $36,125, then go play it, otherwise don't" sorta thing, but that logic just feels intrinsically wrong to me.

As created by Kelly, the formula is for individual bets with only two possible outcomes. In this example, we're looking at a huge number of bets, with millions of different possible results. When looked upon like that, the best we would do is come up with the likelihood of going bust compared to the expected growth. I would then use those numbers to create something I could enter into a Kelly calculator and see if whatever bankroll I had was sufficient.

Quote: RS

How would you apply the Kelly criterion to a session as opposed to individual bets? Let's say for example a casino is offering a promotion -- if you do $1,000,000 coin in, they'll give you $10,000 cash. If you don't complete the $1M coin in, you get nothing back..

I'll be interested in the solution to do this, as it applied to a small online promo that I took last night. It was a 3 tier bonus where...

Wager 1,000 they gave me £30
Wager another £4,000 they gave me another £120
Wager another £5,000 they gave me another £150

I know: Small potatoes!

So, with a 0.5% Blackjack available ( allowing for slightly imperfect play ) I had a 2.5%, or £250 eventual player advantage.

I wanted to have a reasonable risk of making a profit, a reasonably short length of play and a reasonably small session bankroll in play. Since the bonus was only coming at the ends of those three late stages, i was unsure how to proceed....

... So, I stuck my finger in the air and went for...
Initially, wagering 1 spot at £5 per hand.
It was painfully slow, so switched up to 5 simultaneous seats of multiplayer Blackjack, each flat betting £4 per hand, so including splits and doubles, that would be about 10,000/(5*4) = 500 wagers

Initial buy-in was £100

I don't have the first clue about how to factor in covariance of multiple seats, but the swings seemed acceptable.

Anyhow, It passed a few hours away and ran pretty damned close to expectation.
The first tier ran very well initially betting just one spot.

After I'd hit the 1,000 tier, my bankroll was £210 ... Became £240 with first tier bonus applied.
After I'd hit the 1,000+4,000 tier, my bankroll was £193 ... Became £313 with second tier bonus applied.*
Then it ran a bit poorer, but not exactly bad.
After I'd hit the 1,000+4,000+5,000 final tier, my bankroll was £236... Became £386 with third tier bonus

I still had 1x150 wagering requirements so I did some trivial 1,2,3,3,3 progressive and soon cashed out £400

So I'd made £300 profit: £50 more than my initial estimated EV of £250. Albeit, a little of that was from fortunate progressive wagering at the end.

Because the initial tier ran well, I never needed to buy in for any extra beyond the first £100. If at any time it had run bad, i'd have maybe thrown a couple more hundred at it.

I'm hopeful that someone can explain the co-variance aspect, and advise me whether I was just very lucky, or whether my strategy was something like sensible.

* rounded numbers.

As to the original, the Kelly formula doesn't directly address situations where you may or may not get rewarded down the road for play now. If there is a chance of blowing the whole bankroll on this play, I would not count the cash back or any bonus you might get in determining bet size, just to be conservative. If you are confident of reaching the end goal, then I would. This is just an educated opinion as opposed to the result of doing any math.

Quote: TomG

As created by Kelly, the formula is for individual bets with only two possible outcomes. In this example, we're looking at a huge number of bets, with millions of different possible results. When looked upon like that, the best we would do is come up with the likelihood of going bust compared to the expected growth. I would then use those numbers to create something I could enter into a Kelly calculator and see if whatever bankroll I had was sufficient.

This is not true. It is true, I believe, for the dividing by the variance short cut, but the original Kelly formula will work for any number of possible outcomes, as in video poker. The optimal bet size is whatever will maximize the expected log of the bankroll after the bet.

More information.

For this scenario (1% bonus after wager threshold met):

Find all games with <1% edge. There shouldn’t be many. For each you’ll need it’s variance, return and minimum bet. Your bankroll in units is obviously your cash divided by the minimum.

Use a Markov chain to calculate your risk of ruin for each game. You know your starting unit bankroll and it needs to survive (threshold / minimum) bets. All else equal, the lower the variance and minimum, the better.

For each game, multiply the chance of survival by 1% and add to the base return. Play the game with the highest adjusted return IF it’s above 100%

The optimal Kelly bet is zero for all negative expectation games so you can’t apply it here.

Wiz,

On your "The Kelly Criterion" page, you wrote:

Producti [(1+wixi)^(n*pi)] - 1, where
wi is the net payout for the ith outcome
xi the stake for the ith outcome
pi the probability of the ith outcome.

What is the variable "n"?

Dog Hand

Quote: DogHand

Wiz,

On your "The Kelly Criterion" page, you wrote:...

Thank you. You're right, that was quite a mess. That article has been through a few site overhauls and the formatting didn't survive as intended. However, it's probably on me that I didn't say what n is. Please have a new look and see if it makes more sense. I couldn't get the formula to display properly in HTML so just did a screen capture from Word.

Kelly article

Wiz,

In your updated article you wrote this equation:

(1 + (1/36)*0.01*3 + (14/36)*0.01*1 + (20/30)*0.01*-1 + (1/36)* 0.01*4) - 1 = 0.00025381

First of all, I assume the "30" is a typo for "36".

However, the calculation is still unclear to me. If we use the formula you had earlier in your article, shouldn't the calculation instead be this:

{[1+0.01*3]^(1/36)}*{[1+0.01*1]^(14/36)}*{[1+0.01*(-1)]^(20/36)}*{[1+0.01*4]^(1/36)}-1 = 0.000196

Dog Hand

You're absolutely right. Thank you for the correction.

Quote: Ace2

For this scenario (1% bonus after wager threshold met):

Find all games with <1% edge. There shouldn’t be many. For each you’ll need it’s variance, return and minimum bet. Your bankroll in units is obviously your cash divided by the minimum.

Use a Markov chain to calculate your risk of ruin for each game. You know your starting unit bankroll and it needs to survive (threshold / minimum) bets. All else equal, the lower the variance and minimum, the better.

For each game, multiply the chance of survival by 1% and add to the base return. Play the game with the highest adjusted return IF it’s above 100%

The optimal Kelly bet is zero for all negative expectation games so you can’t apply it here.

Thanks, that makes sense, although most of it all kinda goes over my head as far as how to calculate it, particularly markov chains.

Can you clarify why you’d multiply chance of survival by 1%? I assume that figure represents the return portion for the 1% bonus? EG: 90% survival rate means the bonus is worth 0.9% return, not 1%?

If possible, can you describe how to do a Markov chain? I’ve tried figuring it out before and didn’t find much regarding gambling. I know how to do it for (I guess a basic example where) you have an option to run an ad for your company and you figure you have a 80% chance to retain a customer and 20% chance to lose them, and a 40% chance to gain a customer and a 60% chance that customer doesn’t come over to you from a competitor....then you determine if running the ad will increase or decrease your base. But I have no idea how to do that with gambling. :(

Quote: Wizard

You're absolutely right. Thank you for the correction.

Wiz,

You're missing the second right parenthesis just before the caret in this term:

(1 + (0.01*1)^(14/36)

Dog Hand

From the Kelly page:

“Example 2: A casino in town is offering a 5X points promotion in video poker. Normally the slot club pays 2/9 of 1% in free play. So at 5X, the slot club pays 1.11%. The best game is 9/6 Jacks or Better at a return of 99.54%. After the slot club points, the return is 99.54% + 1.11% = 100.65%, or a 0.65% advantage. The Game Comparison Guide shows the standard deviation of 9/6 Jacks or Better is 4.42, so the variance is 19.5364. The portion of bankroll to bet is 0.0065 / 19.5364 = 0.033%”

I got a slightly different answer using the simple Kelly formula.

Take (100.65 / 99.54)^2 * 19.5364 to get 19.8520 which is the variance (V) of the game including the promotion. Its return (R) is 1.0065. Its imaginary payoff on a “to 1” basis is V/R + R - 1 = 19.73034. So the optimal Kelly bet is the advantage of R - 1 divided by the payoff equals 0.000327422 of bankroll.

The whole reasoning behind the Kelly formula is based on three assumptions :
- independent trials.
- same game repeatedly.
- Infinite repetition .

If the repetition is finite, the formula includes a N factor (N=nb of repeats).
The calculus takes advantage of infinity, but also exact similaritiy of the trials, in order to progress towards the formula. In other cases, there is no demonstration. Only some unfounded confidence that it is « more or less » valid.

In the situation described here, it looks like a one-shot opportunity (if you consider the individual bets, each but the last have negative EV, so Kelly says Don’t Play). The Kelly criterion is designed to optimize long-term growth of your bankroll over time. Definitely not your objective: your goal is to not bankrupt before the last bet. So, better forget that formula and develop an optimization that is relevant to your objective.