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.

http://apheat.net/2013/07/02/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.

http://apheat.net/2013/05/03/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.