Hold Percentages On Twenty One

HOLD PERCENTAGE FOR TWENTY-ONE GAMES

A RESEARCH PAPER

SHAWN McGHIE

CASINO MANAGEMENT

UNIVERSITY OF NEVADA

The following information was abstracted from the above mentioned research paper written in 1984. The statistical information contained therein was taken from John Scarne's New Complete Guide to Gambling, Edward Thorpe's Beat the Dealer, Rouge et Noir Winning at Casino Gaming and the American Institute of Certified Public Accountants Audit and Accounting Guide. The information concerning the Jim Kirby study was added in 1987.

In Scarne's Guide to Gambling he discusses a method of computing the theoretical advantage to the house for twenty-one and notes, "Since the rules of play give the player more freedom in his decisions and since players don't conform to any single strategy (many aren't even consistent, playing one way one time, and another way a few minutes later) how do we arrive at a figure for the number of busts the player may expect? First, we must consider a hypothetical player is consistent in his play and follows the same rules as does the dealer. If we assume this, we can calculate an exact percentage by finding out how many times the probability theory says the dealer and player will bust on the same round". Using this "mimic the dealer" strategy, he computes a house advantage of +5.9%. This computation was made using liberal rules for doubling down, splitting cards, and without any consideration for the arrangement of the cards in a deck.

Recent studies in casinos have developed information that indicates the average house advantage enjoyed by casinos for twenty-one games is much lower than that developed by Scarne. The improvement in play has increased because of the attention to counting systems and basic strategies readily available in gambling books.

The gaming win of twenty-one is therefore the function of the volume of money bet applied against the statistical advantage of the game.

It is not practical to measure the volume of money bet on a twenty-one game in a continuing casino environment. Therefore, the most common analytic tool for twenty-one is the win to drop percentage, more commonly called the "hold percentage". However, the hold percentage does not reflect the total game's wagers won by the casino. Drop is not the total of all wagers, but the amount of chips exchanged for cash or cash equivalent (markers). Win or loss is the change in a game's net assets as a result of gaming transactions. The win or loss can be determined for each game on each shift as shown in the following illustration from the audit and accounting guide:

Cash in the drop box		$6,000
Credit issued		3,000
Total Drop		$9,000

Less: Beginning table inventory	$14,000
Chip transfers
Fills	5,000
Credits	-1,000
	$18,000
Ending table inventory	-11,000	7,000

Win		$2,000

Percent of Win to Drop (Hold Percent)		22%

Since the hold percentage is the only information available, the question becomes one of understanding how it works and its relationship to the house game advantage. There are five factors, each independent of the other, that work simultaneously to determine hold percentage on a twenty-one game. They are: 1) the buy-in, 2) average bet, 3) house advantage, 4) hands per hour, and 5) hours played. This information can be simulated with a computer model and then varied to understand the effects of each factor. The following examples were generated using such a computer model.

To illustrate this approach, a model was developed based on the following assumptions:

Basic Model

Buy-in	$100.00
Average bet	$5.00
House advantage	2.5%
Hands per hour	50
Hours played	2.5

Hold percentage	15.63%

To explore the model and the effects of a single factor on the hold percentage, all factors remain constant except the average bet. If the average bet is raised in increments of $5.00, the hold percent can be expected to increase as follows:

Increasing Average Bet

Average Bet	Hold Percentage

$5.00 (original model)	15.63
10.00	31.25
15.00	46.88
20.00	62.50
25.00	78.13

Therefore, increasing the average bet from $5.00 to $10.00 and having the player on the game for 2.5 hours, will increase the hold percentage from 15.63% to 31.25% on a buy-in of $100.00.

To explore the model further, the original factors are constant except hands per hour. Making this factor variable and increasing the hands per hour by five, the hold percentage will increase by the following amount:

Game Speed

Hands Per Hour	Hold Percentage

50 (original model)	15.63
55	17.19
60	18.75
65	20.31
70	21.88
75	23.44
80	25.00

If the preceding two examples are compared, it is obvious that increasing the average bet is more powerful than the game speed assuming the other factors are constant.

The effect of hours played can also be measured. If this factor is varied and the original factors held constant, the model shows the following:

Time on the Game

Hours Played	Hold Percentage
2.5 (original model)	15.63
3.0	18.75
3.5	21.88
4.0	25.00
4.5	28.13
5.0	31.25

The other two factors, buy-in and house percentage, can also be varied. The control over both is somewhat limited as explained later, but special circumstances could exist that would warrant their analysis in a similar fashion. With the above information available, the twenty-one manager can focus on techniques to increase the games hold percentage and, therefore, the win. Each of the five factors can be addressed in terms of management actions.

Buy-In:

The twenty-one manager generally has little control over the buy-in of players. He may be in a position to affect it through credit policies, but for the most part, this factor is the responsibility of the casino marketing department. That is, to market the property to guests who like to gamble and have the where-with-all to spend money. The emphasis on buy-in that results in drop can be monitored by comparing drop to activity, advertising, entertainment, special events, junkets, conventions, and even the bus program.

Average Bet:

Increases in average bet can be encouraged by a number of means. The breakdown of chips for the buy-in is the first opportunity a casino has for the encouragement. For example, if the player in our model is given $1 tokens for his $100 buy-in, he is probably less likely to maintain an average bet as high as another player who is given $5 chips. The same concept probably applies to how winning bets are paid. If a player wins a $50 bet that consisted of $5 chips and he is paid in $25 chips, he is more likely to be upgraded in average play than a player who is paid with $5 chips.

The limits on a game can also affect the average bet. The minimum and maximum limit is generally intended to prevent the player from doubling his bet continually until he wins. However, the casino might agree to take the limit off if a player agrees to play at least a minimum amount for a specific period of time. Another technique is to increase the table minimum as the play on that table increases.

House Advantage:

The sources quoted earlier gave a range of -.1% to +15.0% for single deck twenty-one play. The range depended upon the strategy of play used by the player. Rouge et Noir issued a staff report in 1983 that included rather sophisticated mathematical approaches to hold percentages and concluded that the use of two decks would increase the hold percentage by +.2%, and four decks would increase it by +.4%.

The twenty-one manager can consider the rules under which the games are to be played. He needs to encourage play, but must consider the house advantage as rules are established. The surrender rule in Atlantic City was popular with the players, but not with the twenty-one managers. Rule restrictions that favor the house are as follows:

Rules that Increase the House Percentage

Rule Restrictions	Increase in House Advantage
Dealer hits soft 17	+0.2
No soft doubling down	+0.1
No doubling down on 11	+0.8
No doubling down on 10	+0.5
No doubling down on 9	+0.1
No doubling down except 9, 10 or 11	+0.2
No doubling down after splitting	+0.2
No splitting of aces	+0.2
Tie hands are won by the dealer	+9.0

Hands Per Hour:

Hands per hour is important, but has numerous ramifications for the twenty-one manager to consider. Casino philosophies differ widely on this subject because of player attitude. The faster the game the less time the dealer has to talk to the players. Some players enjoy fast games while others, particularly those just learning to play, become irritated by game speed and, therefore, unhappy.

The single deck twenty-one creates a problem for both the dealer and the player if the hands per hour is increased too much. Game speed can be accomplished without detrimental effects by using a multiple deck shoe. A new invention is being tested in Atlantic City that automatically shuffles the cards. With multiple deck shoes and automatic shuffling, hands per hour can be greatly increased. The twenty-one manager must consider all these factors as well as his players, adjust the game speed, and monitor the hold percentage to develop the proper balance for his games.

Hours Played:

This is the most difficult problem the twenty-one manager faces. It ranges from the attitude of a player who wants to quit when he is ahead, to the personal egos of those not getting enough attention. In between are the problems of :

Player Comfort:

Are the stools comfortable to sit at for a long period of time? Does the player get enough to drink to keep him happy? Are the lights too bright or too dark? Is the music too loud or too soft?

Mandatory Playing Time:

If he is on a junket and has agreed to playing time, how do you monitor his activity?

Grading of Player:

Are good players spotted and given personal attention, such as a complimentary dinner or overnight lodging to encourage them to stay in the game?

Even though the hold percentage of a twenty-one game is not the same statistically as the hold percentage of a slot machine, it is still a very valuable management tool. The Nevada Gaming Control Board reported hold percentages as shown in the following table:

Twenty-one Hold Percentage

Downtown Las Vegas	12.11
Strip Las Vegas	14.69
Reno	17.55
Lake Tahoe	14.63
Laughlin	14.86

The above schedule indicates a considerable range between the best and the worst. The twenty-one model described herein demonstrates a simple method of analyzing the effects of the five factors, and seems to indicate that hold percentage on games can be increased by management's attention to these factors. If the hold percentage is increased and the drop remains the same, the casino will win more money.

Other:

In 1985, Jim Kilby of the University of Nevada, Las Vegas developed various computer models based on different algorithms. He then tested the computer against actual observations that included $40,000,000 in bets. All of his models predicted house win higher than his actual observations. In conclusion, he stated that the house wins at a rate equal to the average bets per hour.

Another consideration of the twenty-one manager is to maximize the house advantage by eliminating errors, or allowing procedures that players can use to their advantage. This involves training programs to ensure proper payoffs and standardization of shuffling, holding and dealing the cards. It also includes establishing internal control procedures to ensure that the prescribed procedures are being followed.

Shawn McGhie was a consultant with McGhie Consulting in Reno, Nevada and is presently employed by Fair Isaacs, Co. You can reach McGhie Consulting at (775) 329-2999 or find more information on the Web at www.200.com/consulting.

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