puzzlenut
Joined: Sep 19, 2013
• Posts: 71
February 11th, 2014 at 6:48:16 AM permalink
Don Johnson was a recent winner of \$15 million playing blackjack at three casinos in
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.
Perdition
Joined: Sep 3, 2013
• Posts: 610
February 11th, 2014 at 6:53:50 AM permalink
He also gave us this

puzzlenut
Joined: Sep 19, 2013
• Posts: 71
February 11th, 2014 at 7:02:30 AM permalink
Quote: Perdition

He also gave us this

A television actor: not the same person.
teliot
Joined: Oct 19, 2009
• Posts: 2189
February 11th, 2014 at 7:15:49 AM permalink
Quote: puzzlenut

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

More along the lines of your approach is this:

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:

Poetry website: www.totallydisconnected.com
Ibeatyouraces
Joined: Jan 12, 2010
• Posts: 11933
February 11th, 2014 at 7:26:30 AM permalink
deleted
DUHHIIIIIIIII HEARD THAT!
MidwestAP
Joined: Feb 19, 2012
• Posts: 1262
February 11th, 2014 at 7:36:09 AM permalink
Quote: Ibeatyouraces

In 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.
puzzlenut
Joined: Sep 19, 2013
• Posts: 71
February 11th, 2014 at 8:11:53 AM permalink
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.
RolexWatch
Joined: Jan 20, 2014
• Posts: 35
February 11th, 2014 at 8:13:53 AM permalink
Quote: Ibeatyouraces

In 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.

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.
Ibeatyouraces
Joined: Jan 12, 2010
• Posts: 11933
February 11th, 2014 at 8:25:13 AM permalink
deleted
DUHHIIIIIIIII HEARD THAT!
teliot
Joined: Oct 19, 2009
• Posts: 2189
February 11th, 2014 at 9:05:58 AM permalink
Quote: puzzlenut

The 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.

False. See this post:

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: puzzlenut

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 \$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.
Poetry website: www.totallydisconnected.com