Atlantic City and he attributed his success to negotiating favorable terms with those

casinos, terms that he believes gave him a player's edge. He faults the casinos for

accepting his terms without doing the math.

For the game itself he negotiated the dealer standing on a soft 17, a maximum bet of $25,000

to $100,000, no restrictions on doubling down, and resplitting for up to four hands. Best of

all, he negotiated a 20% rebate on any loss he incurred in a single day for a loss of

$500,000 or more.

Apart from the effect of the rebate, he calculated that the game had a house edge of 0.253%.

Casino management probably thought that the house still had the advantage and would merely

be giving up a percentage of their profit. Proper analysis would have shown that this rebate

would shift the odds in Don's favor.

The house edge is defined as the ratio of the average loss to the initial bet, For an

average win or loss of one betting unit it can be converted to the probability of winning by

p = (1 - He)/2. We will need this for what follows.

For an He of 0.00253, p = 0.498735 and the probability of losing is q = 0.501265. We shall

use this to calculate Don's expected winnings taking into account the loss rebate.

At what point should he have decided to quit for the day? A loss of $500,000 would be a good

quitting point because playing further would involve the risk either of increasing the loss

or decreasing it and losing the rebate.

The optimal quitting point for a win can be calculated using the gambler's ruin formula for

an unfair game. For a house advantage of 0.00253 p = 0.49873, q = 0.501265 and

r = q/p = 1.00508291.

The probability of winning n units before losing 5 units is

w = (1 - r^5)/(1 - r^(n + 5)).

The player is betting on winning n units with probability w against losing 4 units, taking

into account the rebate, with a probability 1 - w and the player's expectation is

nw - 4(1- w)

This has a maximum of 0.57093 betting units at n = 15 so the player should

stop if he reaches a win of 1,500,000. and his expected average return per day is $57,093.

Teliot has done a similar calculation using simulation and has gotten comparable, though not

identical results.

The strategy works because the house edge in properly played blackjack is small and r is

very close to 1. At r = 1.1 the strategy is unprofitable for all stopping points. This

corresponds to a house advantage of 4.76%. It follows that this strategy is useful only for

blackjack and baccarat and that the house may safely offer this rebate for slots or roulette

without fear of being taken advantage of.

Calculations were done using Derive 6. Maxima were found using calculus.

Quote:PerditionHe also gave us this

A television actor: not the same person.

More along the lines of your approach is this:Quote:puzzlenutTeliot has done a similar calculation using simulation and has gotten comparable, though not

identical results.

the loss rebate theorem

Here are the results of this theorem, when applied to Don Johnson with a $100,000 wager and 20% rebate:

Here is the full theorem:

Quote:IbeatyouracesIn my book, he manipulated casino managers to give him an edge on the game. That makes him a cheater, not an AP. Same goes for Ivey.

Manipulate? Sure, if you accept the definition of "to use or change (numbers, information, etc.) in a skillful way or for a particular purpose". But that's a far cry from 'cheating'!?!?

People in all industry's negotiate deals every day, some are to their advantage and other that are not. This is no different, in both cases the individuals who negotiated and agreed to terms with Johnson and then with Ivey just made bad deals.

Quote:teliot

Here are the results of this theorem, when applied to Don Johnson with a $100,000 wager and 20% rebate:

Thank you for your attention; I think this is a very worthwhile topic. As I read your image, the loss exit point is $2,600,000, the win exit point is $2,400,000, and the expected win is $124,999.

The result of your simulation is that the loss exit point should be $500,000. the win exit point should be $1,600,000, and the expected win is $61,876.

The result of my calculation, which does not take into account the standard deviation of the game, is that the loss exit point should be $500,000, the win exit point should be $1,500,000 and that the expected win is $57,093. This is in pretty good agreement with your simulation but not with your calculation.

I think the idea of using a formula is a good one. I was able to find the optimal quitting point for a gain merely by differentiating the expression for the player's expectation, setting to zero, and solving. Derive and other computer algebra systems can do such things quickly and accurately. Simulation takes time and the error is not always easy to estimate.

A cheater?? He negitoiated a deal, caino accepted it, how is that cheating? Same with Ivey, casio didn't mind turning the cards, after all he was losing, is it only okay to do things while the player is losing and not while they are winning? Ever exprienced dealer trying to upset players, either through fast dealing, section spinning, they don't always offer a fair game despite the HE, you reall do sound jealous.Quote:IbeatyouracesIn my book, he manipulated casino managers to give him an edge on the game. That makes him a cheater, not an AP. Same goes for Ivey.

False. See this post:Quote:puzzlenutThe result of your simulation is that the loss exit point should be $250,000. the win exit point should be $1,600,000, and the expected win is $61,876.

don johnson 3 could he have won more

My simulation gave a quit loss of 2,750,000 and a quit win of 2,200,000 and an average win of $125,209.

Quote:puzzlenutThe result of my calculation, which does not take into account the standard deviation of the game, is that the loss exit point should be $250,000, the win exit point should be $1,500,000 and that the expected win is $57,093. This is in pretty good agreement with your simulation but not with your calculation.

Your method gives incorrect results. You misquote my results. The standard deviation is everything. You are using the wrong house edge.

Quote:Perdition

He also gave us thisQuote:puzzlenutA television actor: not the same person.

As sodawater would say: In other news…the pope is still catholic

don johnson 2 how he beat blackjack

you now are quoting from the tables on the following page. Could you explain the difference?

Yes, for the house edge you have to take into consideration that Don Johnson is playing from a shoe with a cut card. That moves the edge towards the house. I actually state that in the second article very clearly: "It is a bit of arcane science, but when the dealer uses a cut card, the edge moves slightly towards the house side (the so-called “cut card effect”). In the case of this game, the edge is about 0.29% using a cut card."Quote:puzzlenutTeliot: I am using Don Johnson's calculated house edge, which is 0.253%. Also I am quoting from your tables on this page

don johnson 2 how he beat blackjack

you now are quoting from the tables on the following page. Could you explain the difference?

Also, in the article above (#2), I fixed his loss limit at its smallest possible value, $500k. I make this restriction very clear in my analysis. The three articles, together with the article on the Loss Rebate Theorem, work together. Read them all and you will understand.

I really think your academic research here is shoddy. Read everything, understand it, and then see if you can reproduce it using your method. Don't call it wrong.

How is anything cheating when a casino agrees to it? Never heard of casinos cheating players? Such as getting high rollers so pissed they have to be propped up in seats, short stacked decks, players restricted to flat betting even though they weren't counting, you take what you can, it's dog eat dog and sometimes open war-fare, quit being naive like everything has to be above board, they don't give a monkeys about social damage or impacts, so you take it anyway you can, because that is what they would do.Quote:IbeatyouracesJealous? No. I'm not knocking the guy. it's just how I see it.

If I were to convince a dealer and a floor supervisor to flash two community cards at Mississippi Stud, you wouldn't call that cheating?

Don Johnson cheated? Give me a break, he took advantage of there GREEDY nature, only an idiot would not do the same, if they had the chance

Quote:IbeatyouracesJealous? No. I'm not knocking the guy. it's just how I see it.

If I were to convince a dealer and a floor supervisor to flash two community cards at Mississippi Stud, you wouldn't call that cheating?

If this happened, and

1) it was clearly demonstrated that the dealer and supervisor okay'd the flash. Preferably in writing but if not, then in actions and verbal agreements that others witnessed, and

2) if they weren't colluding with the player for their own gain, and

3) if it didn't directly negatively impact any other player financially, then

No, not cheating in my opinion.

Of course this situation wouldn't happen because they would be knowingly giving the player a HUGE advantage.

I think it's shrewd the way a whale can get them to agree to a rebate (in writing) that if played correctly can result in positive EV. Given some of the rebates that whales negotiate, and given the way they can be manipulated by the player, I'm surprised they still exist. But it sounds like they still do.

All this would be your real gamble, in fact it would take a lot of balls, pardon the expression. Enough so that I could believe that Johnson to some degree or other stumbled into the situation. Johnson was already known as a high roller, and I know he said it must have been clear that he was no AP at BJ ; he could point to not varying his bet size.

All this I shall say is IMHO, the H in there this time for sure.

I used Don Johnson's figure of a house advantage of 0.253% out of respect for him, and I quoted your figures on your page 2 out of respect for you. I don't expect any respect for myself.

I shall be happy to recalculate using whatever parameters you suggest.

Quote:odiousgambitThe possible [probable?] element of sleaziness involved in getting a casino to agree to unwise loss rebates would involve role-playing. This role-playing would convince them you were just a degenerate gambler, but of the well-heeled and demanding whale type. You'd go through an initial period of hopelessly losing money without ever showing signs of smartening up. It would seem your average bet was increasing all the time while simultaneously pouting about this and that, especially how you're thinking about taking your action elsewhere.

Role playing? Like, the way casinos always show people laughing, clapping, and winning? They never show the long lines at the Cage, or the overflowing ashtray, or the cost of a simple soda, or a steadily depleting bank account. Just happy, smiling winners.

What's good for the goose...

It is not what makes a lot of sense, it is what maximizes win per session. A stop loss of $500k does not maximize win per session. Don't argue with me about what makes sense, learn from what I did and see if you can find an easier way. I was contacted by an AP who showed me an iterative solution that got the same results I got, it blew me away how smart the guy was. So, blow me away or learn something, either way, just don't misstate my work.Quote:puzzlenutTeliot: I think that a loss stopping point of $500,000 makes a lot of sense. Whichever way it goes from there he loses, either from losing more or from forfeiting the rebate.

Quote:IbeatyouracesIn my book, he manipulated casino managers to give him an edge on the game. That makes him a cheater, not an AP. Same goes for Ivey.

Ivey is a different story. There is far more to that story and poker then what some people think, including blatant dealer manipulation, just use your imagination on the poker. Everything together MIGHT make him a cheat. But, as far as normal edge sorting goes I don't consider that cheating.

What Don Johnson did, was not cheating, that's INSANE, He made a deal the casinos, The casinos got greedy and got what they deserved. I wish he would have kept his mouth shut is all, but after making 15 mill I guess he dose not care at this point.

People make deals with casinos all the time are they all cheaters? If your not a sucker your a cheater?

How is putting on an act of cover play when counting cards any different? You are manipulating casino employees to help gain an advantage.

Quote:puzzlenutThe optimal quitting point for a win can be calculated using the gambler's ruin formula for

an unfair game. For a house advantage of 0.00253 p = 0.49873, q = 0.501265 and

r = q/p = 1.00508291.

The probability of winning n units before losing 5 units is

w = (1 - r^5)/(1 - r^(n + 5)).

The player is betting on winning n units with probability w against losing 4 units, taking

into account the rebate, with a probability 1 - w and the player's expectation is

nw - 4(1- w)

This has a maximum of 0.57093 betting units at n = 15 so the player should

stop if he reaches a win of 1,500,000. and his expected average return per day is $57,093.

This is a great post, and I hate to pick on just one point, but...

Does this ignore variance? ie, doesn't it assume that each decision results in either a win or a loss of one unit? There are other possibilities too. The player may get a blackjack for a return of 1.5 units. The player may double and split, possibly several times, for a result anywhere between -8 and +8 units. (Note that house edge is determined as a fraction of one unit, not as a fraction of the total amount of money put in play).

Having said that, the variance is pretty low (most hands either win or lose one unit) so this is probably pretty close to correct.

A related question:

Since variance helps him in this situation, which deviations from basic strategy should he make (assume that he is not counting). I'd assume that he will want to make a lot of close doubles where the EV of standing is fractionally higher than that of doubling (I think a lot of soft doubling opportunities against small dealer cards fall into this category). He may also want to forgo surrendering in very close situations. I'm not sure if any pair splitting situations are close enough for this -- although with DAS and resplitting, splitting pairs can be very high variance.

Quote:FaceWhat's good for the goose...

Absolutely!

To be clear, I can't see what Johnson was able to do as cheating. Saying there could be an element of sleaze is my way of playing that down; you could also say it was gamesmanship.

................http://www.richardmarcusbooks.com/casinoScamOfTheMonth.php

Quote:Buzzardbaccarat edge sorting used by Phil Ivey

I'd like to pull off what Johnson did, but not Ivey ... not sure if my qualms get in the way, but they refused to pay Ivey.

It's not cheating to play a game well enough within the rules to have an advantage. There were no rules against giving high-rollers negotiated loss rebates; that's why the casinos did it. If Johnson is smart enough to negotiate a loss rebate big enough to give him an edge, and the casinos are dumb and greedy enough to negotiate the same, that's just business. Not cheating.

There were no rules, apparently, against dealers turning cards for superstitious players in mini-bacc in London -- that's why the casino allowed it. Do you think no one was watching Ivey play? Every suit and every camera in the place was sweating that game. They had no problem with it until after he won.

False. It is precisely because of the variance that it is nowhere close to correct.Quote:AxiomOfChoiceHaving said that, the variance is pretty low (most hands either win or lose one unit) so this is probably pretty close to correct.

Quote:Ibeatyouraces

How is that cheating? The rules were clearly stated, agreed to, and signed by both parties. That seems about as honest as it gets.

Quote:teliotFalse. It is precisely because of the variance that it is nowhere close to correct.

Which part is false? The variance of blackjack is fairly low. With the rules mentioned, I'd suspect that the standard deviation is around 1.2?

Personally, I think that the question of optimizing his playing strategy to take the loss rebates into account is far more interesting.

The part that is false is any assumption that the poster's results are even close to correct. They are not.Quote:AxiomOfChoiceWhich part is false?

Quote:teliotThe part that is false is any assumption that the poster's results are even close to correct. They are not.

Oh, I see. Although, I think that the source of the large error is not the standard deviation (you mention that it's 1.15, even lower than I guessed), but the assumption that the optimal quitting point (on the losing end) is -$500,000. I didn't question it when I first read it, but now I see why that's wrong (at least, I think I do)

Dam I thought I was re reading my post. I was going to open with the same I usually agree with Ibeat, as wellQuote:sodawaterI usually agree with a lot of what ibeatyouraces posts, but calling Johnson and Ivey cheaters is insane. In both those cases, the casinos decided for themselves to do what they thought would be profitable. In both those cases, the player was smarter and was able to legally beat the game.

It's not cheating to play a game well enough within the rules to have an advantage. There were no rules against giving high-rollers negotiated loss rebates; that's why the casinos did it. If Johnson is smart enough to negotiate a loss rebate big enough to give him an edge, and the casinos are dumb and greedy enough to negotiate the same, that's just business. Not cheating.

There were no rules, apparently, against dealers turning cards for superstitious players in mini-bacc in London -- that's why the casino allowed it. Do you think no one was watching Ivey play? Every suit and every camera in the place was sweating that game. They had no problem with it until after he won.

I do not know how many days he got away with this offer but his win of $15 million must be a lot bigger than his expected win.

ie he got really really lucky. This is something that the media and especially himself did not say.

Probably the expected win was in the $1 million range.

And this is something that the average guy would not understand anyway.

The other point I saw in this thread is how the casinos could not figure this out etc.

I do not think there a lot of people on this board that have played rebates.

I have played loss rebates extensively. A lot of the casinos that I have played offered rebates (not in the US).

And I have played rebates together with counting. ie the game had normal Ev for a counter and on top I got rebates usually from 10%-20%.

The maximum rebate I got offered was 35%. Some people would say that I am just full of it and no way a casino would offer 35% rebate, but they have.

And based on this experience comparing Counting and say 20% rebate (with no Counting and say 0.3% HE) it is not even close.

Counting is more profitable by 3-4 times at least.

Quote:AceTwoThe maximum rebate I got offered was 35%. Some people would say that I am just full of it and no way a casino would offer 35% rebate, but they have.

A 35% loss on theoretical is one thing, 35% on actual another. The term 'loss rebate' I believe should mean on actual. The difference is huge and somewhat counter-intuitive. I'm not saying you used it wrong either. You might clarify.

Explain why you think this please. I think, I have to disagree, if your bet sizes are large enough. Loss rebates are far better.Quote:AceTwo

Counting is more profitable by 3-4 times at least.

A 20% loss rebate, would not be as good, only if you are making a lot of bets during a rebate session. 1 enormous bet with a 20% rebate would be a far better advantage then counting. Also the reaction from the casinos when your simply flat betting is going to be meet with open arms and encouragement. Counting will get you more heat then you can imagine.

I see I did misquote your Don Johnson page No. 2 with respect to your loss stopping limit of

$500,000 and I have made a correction. I feel very comfortable with your page No. 2 but less

comfortable with your page No. 3 and your theorem.

I think that the variance or standard deviation of the game of blackjack is relevant only to

the question of whether or not either stopping limit is likely to be reached in a single

day. Please do not forget that the results of my calculation are quite close to the results

of your simulation on page 2. That should be comforting to both of us.

I have recaculated Don Johnson's strategy using a house edge of 0.29%. Using this,

p = 0.49855, q = 0.50145 and r = q/p = 1.005816869.

I'll now use your loss exit point of $2,600,000 and your win exit point of $2,400,000. For

these values n = 24. The probability of winning 24 units before losing 26 units is

w = (1 - r^26)/(1 - r^50) = 0.4838012753.

The player's expectation is

24W - 20.8(1 - w) = 0.8742971334 betting units, which is $87,429. This is better than before

but somewhat less than your calculated value of $124,999.

I'll now try to optimize the player's expectation with respect to both stopping points. I'll

call the win stopping point a for "ahead" and the lose stopping point b for "behind."

For a win stopping point of 15 the probability of winning 15 units before losing b units is

w = (1 - r^b)/(1 - r^(15 + b))

and the player's expectation is

E = 15w - 0.8b(1 - w)

which has a maximum at b = 20, so E(15,20) = 0.9406394916

I'm doing this by differentiating E with respect to b, setting the result equal to zero and

solving for b to the nearest integer.

Using b = 20, I'll optimize for a

w = (1 - r^20)/(1 - r^(20 + a)

E = aw - 16(1 - w) and the optimal value for a is 19 and E(19,20) = 0.9598383989

Optimizing for b again using a = 19

w = (1 - r^b)/(1 - r^(19 + b))

E = 19w - 0.8b(1 - w)

and the optimal value of b turns out to be 20, so our optimal stopping points are

+$1,900,000, -$2,000,000 and the expected win is $95,983.

Don has said that one of the casinos at which he has won would be happy to have him back but

would give him a 20% rebate only on losses exceeding $2,000,000. I think we can safely

advise him not to be deterred.

Is it possible that a casino such as the Wynn would offer discounts on losses?

As the other member correctly stated, it isn't really a "rebate", it's a discount on a paid loss. I think that matters from the players perspective more than a statistical one.

How the tax man keeps up with all this is another question. Strange that a casino could simply write off such a discount without the player being issued some sort of income form.

Quote:puzzlenutTeliot:

Here are the results of the LRT using ev = -0.29%, sd = 1. They match yours exactly.

2000000, 1900000, 95984.12, 380.0, 0.4846

380.0 = expected hands played.

0.4846 = probability of hitting win goal.

Essentially, your work is reproducing the proof of the LRT in the single case when sd = 1. It is very nice.

Here are the results of the LRT using ev = -0.29%, sd = 1.15.

2300000, 2100000, 110407.52, 420.3, 0.4950

That's how much variance is worth. Higher moments also have an effect on profitability, which is why a simulation is the ultimate answer.

Quote:puzzlenutI think that the variance or standard deviation of the game of blackjack is relevant only to

the question of whether or not either stopping limit is likely to be reached in a single

day.

No, variance is incredibly important for loss rebates. High variance is your friend.

Consider this: if the variance were 0, you could not possibly win.

variance is best for the loss rebateQuote:AxiomOfChoiceNo, variance is incredibly important for loss rebates. High variance is your friend.

Consider this: if the variance were 0, you could not possibly win.

Quote:AxiomOfChoiceNo, variance is incredibly important for loss rebates. High variance is your friend.

Consider this: if the variance were 0, you could not possibly win.

I have changed my position on the importance of variance in calculating probabilities. Teliot has convinced me that it is necessary to consider variance in computing the probability and I am looking into his methods. The formula I have been using was derived for a coin tossing game with a variance of 1. There probably is not too much error with respect to blackjack with its low variance of 1.32 but it might be inaccurate to use it for slots with a variance of 76.38.

Quote:puzzlenutThere are no games with zero variance. The games with the lowest variance are the banker and player bets in baccarat, yet it is one of the best games for a loss rebate because of the low house edge. The games with the highest variance are the slot machine and the single-number bet in roulette, however I have calculated that there is no advantage to a loss rebate for a game with a house edge of 4.76% or greater, which is the case with roulette and the majority of slot machines. The variation in how much can be won depends not only on the variance of the game but the amount bet. In this respect a small bet on a high variance game is equivalent to a large bet on a low variance game.

First, there are absolutely games with a lower variance than baccarat.

Second, you understand that variance is measured in dollars squared, right? In other words, "variance" already contains the information about the amount bet.

Quote:puzzlenuthowever I have calculated that there is no advantage to a loss rebate for a game with a house edge of 4.76% or greater

Also, I am not sure how you calculated that, but it is clearly wrong.

And, for the record, there are single-zero roulette wheels.

I should have qualified my statement to say that my calculation was for Don Johnson's rebate offer, which was 20% on a loss of $500,000 or more and for the game of blackjack. My method of calculation is given below. There have been many other rebate offers, including 100% rebate offers, and they must be analyzed according to their own conditions and the games for which they are offered.

As to standard deviation (square root of variance) of various games, I am relying on the Wizard's table at

https://wizardofodds.com/gambling/house-edge/

Such a table would make no sense if it were not normalized for a unit bet. The house edge of single-zero roulette is given as 2.70%. The Wizard does not give variances for roulette because there are many types of bet with different variances but with the same house edge.

According to the classical gambler's ruin formula for an unfair game, the probability of

winning n units before losing 5 units is

w = (1 - r^5)/(1 - r^(n + 5))

where r is the ratio of the probability of losing to the probability of winning, and the player's expectation is

E = nw - 4(1- w)

which is the number of betting units won times the probability of winning them minus the number of betting units lost times the probability of losing them.

If r >= 1.1 E is zero or negative for all values of n. This corresponds to a house edge of 4.76%. For 1.1 > r > 1.0, E can be positive and will have a maximum value. In Don's case this occurs at n = 15.

If r < 1 E increases indefinitely with n and there is no optimal quitting point for gains. Don probably was able to achieve this using a card counting strategy.

As given above, w is for a predetermined loss limit of 5 betting units. This would be changed for a different loss quitting point.

This classical gambler's ruin formula is quite old. Teliot has a more sophisticated one that deserves further study and he has developed a "loss rebate theorem" based on it.

puzzlenut makes some bad assumptions while doing decent work

eliot jacobson continues his quest to be world's most loathsome man, now pulling ahead of Mullah Omar into the #6 spot

stockholm syndrome ploppies call for APs in handcuffs

You joined just 5 days ago, yet you have strong opinions about me, as if you've been a member of this board for years. How many long time members have taken personal shots at me for no other reason than they feel like being nasty and can't sit on their hands? I can only think of 1. Maybe the moderator will check some IP's. Maybe the moderator will start paying attention to some of the stuff I put up with on this site. I'm more than happy to be called wrong on substance. Is that too hard for you?Quote:RicardoEsteban

eliot jacobson continues his quest to be world's most loathsome man, now pulling ahead of Mullah Omar into the #6 spot

Quote:teliotYou joined just 5 days ago, yet you have strong opinions about me, as if you've been a member of this board for years. How many long time members have taken personal shots at me for no other reason than they feel like being nasty and can't sit on their hands? Maybe the moderator will check some IP's.

It's a little disingenuous for someone so desperate for recognition and publicity as you to feign indignation that a random person is familiar with your "work."

You really should sit on your hands.Quote:RicardoEstebanIt's a little disingenuous for someone so desperate for recognition and publicity as you to feign indignation that a random person is familiar with your "work."

"When you see a good move, sit on your hands and see if you can find a better one."

-- Siegbert Tarrasch (Chess grandmaster)