Kelly Betting Optimization

Can I get a verification here?

One of the The Wiz Kelly Betting examples shows:

"A blackjack card counter perceives a 1% advantage at the given count. From my Game Comparison Guide, we see the standard deviation of blackjack is 1.15 (which can vary according to the both the rules and the count). If the standard deviation is 1.15, then the variance is 1.15^2 = 1.3225. The portion of bankroll to bet is 0.01/1.3225 = 0.76%."

Which makes sense. But it seems to me also that the Kelly optimization is instantaneous, varying with perceived expectation and grubstake at the time. Thus Kelly should theoretically be recalculated every time either bankroll and(or) expectation changes. Seems common sense to me. Of course, one wouldn't likely recalculate it each time in a real game, just use a common sense Kelly bet size increases or reductions as ongoing circumstances dictate.

Does anyone agree?

Thx.

https://wizardofodds.com/gambling/kelly-criterion/

Deleted.

Quote: APEppink

Can I get a verification here?

Of course, one wouldn't likely recalculate it each time in a real game, just use a common sense Kelly bet size increases or reductions as ongoing circumstances dictate.

Does anyone agree?

Thx.

https://wizardofodds.com/gambling/kelly-criterion/

I agree with your interpretation.
Let's say that you are counting and estimate the Player advantage is 1% at that moment and that because your bankroll is sufficient, Kelly says bet $100. A few hands later, your edge is an estimated 2.5%. Then Kelly says bet $250... 0% bet zero. etc.
Of course, when you have no edge, you might want to play table minimum to stay in the game. You need to be betting much lower values than your Kelly bet at those times, because here you are depleting your bankroll as a cost for playing.
I understand that in reality, many players decide their max bet or unit size based on a kelly assessment of their bankroll and work backwards from that to decide their betting ramp..

It's pretty much impossible or impractical at least to update your bets to kelly in real time. Rather, it makes a lot more sense to do something like this -- say you have a $100K bankroll. You'd determine what your optimal spread/ramp is for an $80k bankroll, even though you have $100k. Once your bankroll gets down to $80k (if it does), then you re-evaluate and figure out what the optimal spread/ramp is for a $70k or $60k bankroll. If you go from $100k to $120k, then re-evaluate and change your spread, as if you had a $100k bankroll --- IF you want/need to.

In other words, give yourself some slack when calculating your spread/ramp, that way you don't have to continuously re-adjust.

Another thing to note, at least for many casinos, your spread/ramp/betting amounts do not and can not increase at the same rate as your bankroll, because once you get to a certain level, you really can't (or shouldn't) be betting more, due to the casino's threshold/sweatiness. Maybe you can bet 2x$300 with a 50K bankroll....you can't necessarily just bet 2x$600 with a 100K bankroll.


Also, you probably should not be betting at full kelly, since you're going to have some wild swings. It's perfectly fine to under-bet. Full kelly basically means the amount to wager to MAXIMIZE BANKROLL GROWTH. Not sure what kind of BR you have or what stage you're in. If you're starting out and really want to grow your bankroll but take risk, then bet full kelly, hope you get a nice upswing, then let it smooth out. If you are (or plan to be) a professional AP / main-source of income is AP and already have a bankroll.....you don't need to maximize your bankroll's growth -- rather -- "live off the interest" while growing it a bit.

Thx to those replying.

One last observation. Think of 'Kelly in reverse' (or something). For tourists, ploppies etc. games ALWAYS have a negative expectation and customers play all kinds of crazy bets (probably effectively random), thus the house, from its perspective, can never bet Kelly. Yet they make money consistently hand over fist, presumably due to their infinite bankroll (in practical terms), to which Kelly would be inapplicable. Still, even as large as their bankrolls are, you'd think there'd be infrequent occasions where customers, in aggregate, would come close to bringing the house near to 'short term' bankers ruin, due to variance.
Thoughts? Statistics?
Thx.

Fascinating thread.

Seems to me, even if Lady Luck had prepared the extremely improbable run of luck that could "ruin" a casino, the casino could take some kind of preventive action in a timely manner to thwart Lady Luck. Casinos don't put all their eggs in one basket. They have lotsa baskets, and they watch each constantly.

However, I'm not sure exactly what options might be available to a casino if lotsa slot machines suddenly began giving max wins, and dealers at almost every table couldn't win more than one hand an hour. Hopefully, others who are more knowledgeable have some ideas about preventive actions available to a casino.

Wouldn't it be fun to be there, though? "George, isn't that the second straight flush you got in the last 30 minutes? And, Mary, how did you just get two four-of-a-kind hands in a row? Yeowza!!"

Quote: APEppink

Thx to those replying.

One last observation. Think of 'Kelly in reverse' (or something). For tourists, ploppies etc. games ALWAYS have a negative expectation and customers play all kinds of crazy bets (probably effectively random), thus the house, from its perspective, can never bet Kelly. Yet they make money consistently hand over fist, presumably due to their infinite bankroll (in practical terms), to which Kelly would be inapplicable. Still, even as large as their bankrolls are, you'd think there'd be infrequent occasions where customers, in aggregate, would come close to bringing the house near to 'short term' bankers ruin, due to variance.
Thoughts? Statistics?
Thx.

I like how you're trying to think in reverse... That's a very good mentality to take to a lot of things ;-).

You're on the right track with the fact that the house has a "infinite bankroll" which doesn't apply to Kelly. Also, the variables (house edge, average bet, etc) are ever changing with every single player that comes in the door. One blackjack player might flat bet the house minimum and play perfect basic strategy, while his table mate bets between $10 and $500 and is betting randomly and playing horribly. The first guys average bet and house edge are both tiny, where as the 2nd guy is betting way more and has a completely different house edge. This is why Kelly is not applicable from the casinos side. From the players side we use it because based off our betting ramps that we've created for our given bankroll, we know what our STATIC average bets and advantages are.

Quote: APEppink

Yet they make money consistently hand over fist, presumably due to their infinite bankroll (in practical terms), to which Kelly would be inapplicable.

I like the idea of looking at Kelly from the casino side. They don't have an infinite bankroll. (Some have negative amounts of money, some have lost money hand over fist).

If a casino has $100 million and I go and bet red or black on their roulette wheel, the correct Kelly bet for them would be a little over $5 million. But I don't have that much money, so if the bet was made for that price they would risking $5 million to only win something in the thousands. It would be a horrible bet for them. The correct bet for them according to Kelly Criterion is whatever I post up, even if only $5.

So how is the casino industry able to earn billions every year? They just keep making good bets over and over. Which is what we can do. Kelly Criterion should be considered 'maximally aggressive,' not optimal. (otherwise we would have to say the most successful gamblers only make sub-optimal bets). Go over Kelly and the risk of ruin hurts profits too severely. Spend too much time re-evaluating bet or size and you'll also hurt profits by losing opportunities to make more good bets.

Makes me think of an interesting question: on a 9-5-3-2-2-1 Deuces Wild machine, what's the correct bet size according to Kelly Criterion?

Quote: APEppink

Still, even as large as their bankrolls are, you'd think there'd be infrequent occasions where customers, in aggregate, would come close to bringing the house near to 'short term' bankers ruin, due to variance.

Gaming Commission requires them to have enough cash on hand to ensure there is no possibility of this.

While casinos are limited I guess my response was more that for any of us posting on these forums, in respect to us, the casino does have an infinite bankroll. My guess would be with 95%+ of the patrons in a casino, to them, the casino has an infinite bankroll. It's only the whales that have as much or more money than the casino that could actually hurt the casino.

Interesting idea though that the casino could use Kelly to determine what a safe maximum bet would be to take FROM a whale... Wonder if any of them have figured that out =P.

Can the Kelly Criterion be usefully applied to sizing craps bets when laying odds from the don't? Odds are zero-sum, sure, but they are also by definition favored to win from the don't. If the goal might be to maximize profit during a winning streak by wagering increasing percentages or minimizing losses by wagering decreasing percentages of the bankroll, I think the Kelly could help guide the amount of lay odds bets.

Let's say I walk to the rail with $1,000, and put aside $333 to be used for the flat bets on the don't (chip stack 1), and the other $667 will be used for laying odds (chip stack 2). Respective winnings are returned to the appropriate chip stack.

I choose to lay odds as a proportion of my odds bankroll (chip stack 2) according to the point. On 4 / 10, I'll lay 33% of the chip stack. On 5 / 9, I'll lay 20% of the chip stack. On 6 / 8, I'll lay 9.09% of the chip stack. The math could be easy at the table, divide by 3 or 5 or 11.

Is this overbetting a bankroll? Probably. But I favor playing for the shortest amount of time & hoping to sneak into a short win streak, which isn't too far-fetched if I'm just trying to win a majority of my don't pass points, say 6 out of 10.

I'm just learning about the Kelly recently & hoping to apply it as a money management tool next time I hit the tables. Is there more math to consider?

Input appreciated!
Dan

I think you are misusing the, 'Kelly Criterion,' since Kelly is meant for betting (or investing) at an advantage, the optimal Kelly bet on this proposition is $0 overall and $0 on the odds because you do not have an advantage.

The best system to take advantage of a short winning streak on the Don't Pass is to bet everything you have with you on one Don't Pass wager. If you win, you'll have doubled your money and are probably willing to leave. All bets on Craps (as far as line bets) have a negative expectation, no system is mathematically better than any other.

Quote: APEppink

Can I get a verification here?


/gambling/kelly-criterion/

re-up please: this page :"404 Page Not Found"
.........

Kelly Criterion tells you how much to bet relative to your bankroll, variance, and ADVANTAGE. Frequency of winning has no part in the Kelly Criterion.

Quote: Mission146

I think you are misusing the, 'Kelly Criterion,' since Kelly is meant for betting (or investing) at an advantage, the optimal Kelly bet on this proposition is $0 overall and $0 on the odds because you do not have an advantage....

You're correct, no advantage on the Don't Pass Odds due to the payout.

But I still think that using a Kelly optimization can be useful when sizing DP odds bets. If you consider that the player is a 2:1 favorite behind the 4&10, 3:2 on 5&9, and 6:5 on 6&8, you could translate these into percentage points of 2%, 1.5%, and 1.2% then develop a usable chart to dictate the amount of odds.

Let's say I walk up to the rail with $1,000 and play the DP, laying odds using those percentages of my bankroll as threshold indicators. My lay odds would be $20 behind the 4&10, $15 behind the 5&9, and $12 behind the 6&8. As you win or lose, you would adjust your bets in the following way:


4&10: for each $100 won/lost, adjust lay odds by $2.

Bankroll	Lay Odds
$750-$849	$16
$850-$949	$18
$950-$1049	$20
$1050-$1149	$22
$1150-$1249	$24

5&9: for each $200 won/lost, adjust lay odds by $3.

Bankroll	Lay Odds
$700-$899	$12
$900-$1099	$15
$1100-$1299	$18

6&8: for each $500 won/lost, adjust lay odds by $6.

Bankroll	Lay Odds
$750-$1249	$12
$1250-$1749	$18

Just something to consider if you are looking for a logical, bankroll-connected way of raising or lowering your DP odds as you play. Again, I know that there is not as advantage per se on the DP odds, but it seems to me a good a way as any to adjust your bets based on wins/losses.

Dan