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Basic Strategy in blackjack sometimes differs little from next best

22 March 2010

Surprisingly few blackjack buffs actually know what's behind Basic Strategy. Ask around. Representative answers range from polling the experts about their preferences in every instance to watching lots of games and seeing which decisions win most often.

In reality, the rules are based on the "expectation" for each triad of two player cards and one dealer up-card. Expectation is a statistical average that accounts for probabilities of winning and losing various amounts. In blackjack, it's expressed as a percentage of the initial wager, before the deal. It's like an edge on each hand. Positive or negative values show players or the house favored, respectively.

Expectation calculations are only practical with a computer. One standard method is to simulate each three-card set separately, millions of times, then tally results and generate corresponding statistics. Another approach uses a program that systematically cycles through all possible draws, in a "combinatorial" analysis. Both techniques have been applied to different size shoes and house rules, with bettors standing, hitting, doubling, or splitting. All credible gurus get the same answers either way, and there's no question as to the validity of their findings.

Since expectation for each set of cards – executed in the alternate ways – is known, it's easy to identify the decisions yielding the most promising expectations. An example for a strong hand is 10-10 versus six-up. In an eight-deck shoe, expectation is +70.31 percent standing, +43.18 percent splitting, -85.26 percent hitting, and -170.52 percent doubling. Basic Strategy is to stand, for the greatest average win. A typical weak hand is nine-eight versus eight-up. Players are underdogs no matter how they choose to go. With eight decks, expectations are -38.61 percent standing, -50.24 percent hitting, and -100.48 percent doubling. Basic Strategy is to stand because 38.61 cents on the dollar gives the least average loss of any of the options.

Some cases are close enough that sacrificing expectation by ignoring Basic Strategy and making the next best pick might be warranted by other factors. The narrowest margin is on ace-two versus five-up. Basic Strategy, to double, has an expectation of +13.636 percent. However, the value when hitting is +13.633 percent. With a starting bet of \$100, the difference would be three tenths of a cent. You could hardly be faulted for hitting if, say, you were nervous about putting \$200 up for grabs.

The next closest call is 10-two versus four-up. The rule in an eight-deck game is to stand, with an expectation of -21.110 percent. Hitting has an expectation of -21.116 percent. You're praying for a miracle either way. The projection is to average a mere six tenths of a cent per \$100 bet worse loss by hitting, if the table is rich in 10s and you're convinced a low card is due.

The difference in expectation for the Basic Strategy decision and the next best alternative is less than 1 percent, a penny per dollar or a dollar per \$100 bet, in 14 situations. These are shown in the accompanying table, with data for eight-deck shoes.

Expectations for Basic Strategy and the next best decision for the 14 cases in which differences are under penny per dollar bet

```Hand	Basic Strategy	  Next Best         Difference
A-2v5	double:	+13.636%  hit:    +13.633%  0.003%
10-2v4	stand:	-21.110%  hit:    -21.116%  0.006%
9-7v10	hit:	-53.651%  stand:  -53.772%  0.121%
A-4v4	double:	+6.138%	  hit:    +5.936%   0.202%
A-7v2	stand:	+12.343%  double: +12.067%  0.276%
T-6v10	hit:	-53.597%  stand:  -54.083%  0.486%
A-6v2	hit:	+0.039%	  double: -0.474%   0.513%
A-7vA	hit:	-9.474%	  stand:  -10.026%  0.552%
8-4v4	stand:	-20.787%  hit:    -21.364%  0.577%
9-3v4	stand:	-20.880%  hit:    -21.492%  0.612%
7-5v4	stand:	-20.466%  hit:    -21.111%  0.645%
7-2v2	hit:	+7.569%	  double: +6.791%   0.778%
5-4v2	hit:	+7.704%	  double: +6.793%   0.911%
6-3v2	hit:	+7.674%	  double: +6.751%   0.923%```

Arguably, the only useful entries in the table are the seven involving choices between doubling and hitting or standing. There, solid citizens give up little in edge by holding back instead of doubling, or going whole hog rather than hitting or standing. These are conditions under which players may believe reducing or increasing immediate bankroll swings trumps the long-term impact of expectation. As the poet, Sumner A Ingmark, noted:

Those who adhere to rules too blindly,
May find that luck behaves unkindly.