•MULTI-HAND BLACKJACK
Summary
Some casino softwares give the option to play several hands of blackjack against a single dealer hand. Splitting your bet among several hands reduces variance, but also increases the time required to complete a wagering requirement. The data that follows suggests that the reduction in variance is not worth the increased time investment under typical circumstances. For the same time investment, a player can reduce variance to a greater extent by playing a single hand and adjusting bet size. You can compare the time and variance changes using the return & variance with hours of play calc.
Analysis
With Playtech software, it takes me ~4.3 seconds to play a single hand of BJ or ~15 seconds to play 5 hands. For n hands, it takes me approximately 1.6 + 2.7n seconds. The 1.6 second delay relates to the time between deals and the dealer plays. The 2.7 second delay relates to the time it takes to play a hand. I play hands blackjack quickly. Many players would take more time to play an individual hand, increasing the 2nd number (2.7).
The table below shows the variance and time required for a $100 20xB bonus with this rate of play. The betting options are listed in order from lowest variance to highest variance. The first row of the table and lowest variance option is betting $1 per hand and playing a single hand. The 1 standard deviation range with the $100 20xB bonus is $52, and playing with this style would take 2.11 hours at the listed rate of play. Note that the single hand options have a disproportionately low number of hours required. For example, playing a 1 hand and betting $2 per hand takes significantly less time and has a lower variance than playing 5 hands and betting $1 per hand. By these measures, it is better to complete a wagering requirement by playing 1 hand with $2 bets than 5 hands with $1 bets.
Bet Size per Hand |
Number of Player Hands |
Standard Deviation per Hand |
1 SD with $100 20xB Bonus |
Rounds per Hour |
Hours Required |
1 |
1 |
1.16 |
$52 |
840 |
2.11 |
1 |
2 |
1.35 |
$60 |
510 |
1.74 |
1 |
3 |
1.52 |
$68 |
370 |
1.59 |
2 |
1 |
1.64 |
$73 |
840 |
1.05 |
1 |
4 |
1.67 |
$75 |
290 |
1.53 |
1 |
5 |
1.81 |
$81 |
240 |
1.47 |
2 |
2 |
1.91 |
$86 |
510 |
0.87 |
3 |
1 |
2.01 |
$90 |
840 |
0.70 |
2 |
3 |
2.15 |
$96 |
370 |
0.80 |
4 |
1 |
2.32 |
$104 |
840 |
0.53 |
3 |
2 |
2.34 |
$105 |
510 |
0.58 |
2 |
4 |
2.36 |
$106 |
290 |
0.76 |
2 |
5 |
2.56 |
$114 |
240 |
0.74 |
5 |
1 |
2.59 |
$116 |
840 |
0.42 |
The next table keeps the required time constant. Bet size is adjusted so that the required time is 1.00 hours. With 1 hand, the bet size is 2.11. With 5 hands, the bet size per hand is 1.49, and the bet size per round is 5*1.49 = 7.45. The results show that as number of hands increases, standard deviation also increases. When time is kept constant, variance is smaller with single-hand than multi-hand.
Bet Size per Hand |
Number of Player Hands |
Standard Deviation per Hand |
1 SD with $100 20xB Bonus |
Rounds per Hour |
Hours Required |
2.11 |
1 |
1.68 |
$71 |
840 |
1.00 |
1.72 |
2 |
1.77 |
$75 |
510 |
1.00 |
1.60 |
3 |
1.92 |
$81 |
370 |
1.00 |
1.53 |
4 |
2.06 |
$87 |
290 |
1.00 |
1.49 |
5 |
2.21 |
$93 |
240 |
1.00 |
The decrease in variance by splitting bets between hands may become worth the increased time with certain other softwares, where there is a longer delay between rounds. Earlier I listed the rate of play as 1.6 + 2.7n seconds. The player hand to common delay ratio is 2.7 / 1.6 = 1.69. 2-hand play becomes worthwhile when this ratio is less than 0.83. For example, a delay of 3 + 2.4n . 3-hand play becomes worthwhile when the ratio is less than 0.55, and 5-hand play becomes worthwhile when the ratio is less than 0.34.
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