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Best of Donald Catlin

Gaming Guru

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JackBlack

9 July 2000

Here, as promised last month, is an analysis of a new casino game called JackBlack.  This was one of four new games presented by Prime Table Games at last Fall's World Gaming Congress convention in Las Vegas.  Though not yet in the casinos, I believe it will make it to the casino floor successfully.  To see why I think so requires a bit of history.

    In his benchmark text The Theory of Gambling and Statistical Logic, Academic Press 1977, Richard Epstein proposed a variation of Blackjack known as "Zweikartenspiel" in which both of the dealer's cards were dealt face up but the dealer won ties.  In October of 1979 the casino/hotel Vegas World introduced the game as Double Exposure, although they added a soft 17 hit for the dealer and paid only even money on a player Blackjack.  The player's Blackjack did, however, beat a dealer's Blackjack.  With the advertising hype by Vegas World touting the game, the game gained some popularity and during the 80s it appeared in other Las Vegas casinos, generally some of the smaller ones downtown and along the Strip.  As many of you know, Vegas World went bankrupt while building what is now called the Stratosphere Tower, and the Stratosphere Hotel and Casino now sits on the site of the old Vegas World.  It is my impression that the Double Exposure game sort of died along with Vegas World.  Oh, I still see it around and a new, rather inferior version in which the player can only double on 9, 10, and 11 is gaining some ground in Atlantic City; I can't imagine why.  It will be interesting to see if the Atlantic City game eventually goes the way of the Vegas game.

    What is the problem with Double Exposure?  Well, maybe nothing, but in my opinion the fact that the dealer wins ties makes the game frustrating for the player.  I remember, in the early 80s, playing the game at the little Morocco casino on the Strip.  My delight at being able to see both of the dealer's cards soon ebbed when I started losing those ties.  I lasted about 30 minutes.  Of course, maybe it's a great game and I'm just an old stick-in-the-mud, but my impression is that most players share my view.

    So what does all this have to do with JackBlack?  JackBlack is a Blackjack type game where not only the dealer's initial two cards are dealt face up, but the dealer plays first and his entire hand is face up.  What is more, the dealer doesn't win ties.  Sound good?  Here's how it works.

    JackBlack is played on a standard 21 table.  The JackBlack game begins with each player making a unit bet on a space in front of them.  The dealer then deals two cards face up, which are common cards shared by all players, and two cards face up to himself.  The dealer then completes his hand using standard hit/stand rules from Blackjack.  The dealer does not hit any soft hand over 16 and the entire dealer's hand is face up.  If the dealer busts, the hand is set aside and the dealer deals himself a new two-card hand and plays it.  If the dealer busts this second hand then all players win even money on their bets and a new game begins.  Otherwise, the dealer's hand is this second dealt hand.

    All players play a common hand that starts with the two face-up player cards mentioned above.  This means, naturally, that the players' common hand must be played according to a fixed set of decisions.  Of course, players could surrender (yes, it is offered) or double without affecting other players, but splitting is another matter.  So the games creator, Derek Webb, hired me and another analyst (the very talented Stanley Ko) to independently derive the optimum strategy for playing this game.  His plan was that the optimum splitting strategy so derived would be that employed in the game's player decisions.  Here are the details.

    To begin with, a tie in this game (with the exception of 21 ties) is a push; no money changes hands.  This means that hard hit/stand strategy is obvious.  The players' hard hand should be hit until it equals or exceeds the dealer's hand.  Although it is probably the case that most soft ties should stand, it is conceivable that certain of these might be better if hit (or doubled).  If this turns out to be the case, then the player would have the option of taking the tie before the hand is hit; essentially surrendering for the full bet.  In other words, no player is forced to risk money hitting a tied soft hand even though it is the correct play.  Similarly, players not wanting to risk additional money in a splitting situation have the option of surrendering for 1/2 of their bet.  Obviously, if splits are only allowed in situations that are favorable to the players' hand, then such a surrender option should never be taken.

    Here are the rules under which the optimum strategy was derived:

  • Player can double on any first two cards.
  • In optimal splitting situations, players' hand can only be split once.
  • Player can double after splitting.
  • Player can hit after doubling, i.e., a doubled hand will, consistent with optimum hit/stand strategy, receive more than the mandatory one card.
  • Tie, hit, or double if soft tie is to be hit.
  • In optimal splitting situations, player can choose to surrender for half of bet.
  • Two card 21s (Blackjack) have no special significance, they are just 21.
  • Two successive dealer busts is a win for the player.

In addition to the above, here are the payoffs for the game.  PH will represent players' hand and DH will represent the dealer's hand.
 

Players' Hand Dealer's Hand Payoff to Player
First two Two Busts +1
Bust DH<22 -1
PH = 17 - 20 PH -1
PH = 17 - 20 PH>DH +1
PH = 17 - 20 PH = DH 0
2- or 3-card 21 DH<22 +1
4-card 21 DH<22 +3/2
5-card 21 DH<22 +2
6 or more card 21 DH<22 +3
Figure 1
JackBlack Payoffs

Note that players win all 21 ties with the dealer.

    To analyze this game I used an infinite deck approximation.  Close calls were settled by later using a finite deck simulation.  As mentioned earlier, the hard hit/stand decisions are obvious.  With these in hand, I first calculated the expected returns by hitting the players' particular hard total versus each of the dealer's 17 through 21 hands.  Any of these that came out positive are clearly hands that should be doubled when possible.  I then calculated expected returns by hitting soft hands.  Obviously, if the soft hand beats the dealer the correct strategy is to stand but, as I mentioned earlier, it might be the case that a soft tie should be hit or doubled.  In fact, since all hands that carry a positive expected return by hitting should be doubled, if possible, and the expected return for a tie is zero, if a soft tie has a positive expected return by hitting, it should be doubled if it is a two-card soft hand.  As it turned out, the players' soft 17 versus the dealer's 17 is such a hand.  Here are the results for hard doubling.
 
 

Dealer's Total
Players' Total 17 18 19 20 21
21 STD STD STD STD STD
20 STD STD STD STD HIT
19 STD STD STD HIT HIT
18 STD STD HIT HIT HIT
17 STD HIT HIT HIT HIT
16 HIT HIT HIT HIT HIT
15 HIT HIT HIT HIT HIT
14 HIT HIT HIT HIT HIT
13 HIT HIT HIT HIT HIT
12 HIT HIT HIT HIT HIT
11 DBL DBL DBL HIT HIT
10 DBL DBL DBL HIT HIT
9 DBL DBL HIT HIT HIT
8 DBL DBL HIT HIT HIT
7 DBL HIT HIT HIT HIT
6 DBL HIT HIT HIT HIT
5 DBL HIT HIT HIT HIT
4 DBL HIT HIT HIT HIT
Figure 2
JackBlack Hard Hit/Stand/Double Table

Some of the doubles look strange until you realize that the rules allow the players' hand to be hit more than once after doubling.  This same remark applies to the following soft double table.
 

Dealer's Total
Players' Total 17 18 19 20 21
21 STD STD STD STD STD
20 STD STD STD STD HIT
19 STD STD STD HIT HIT
18 STD STD HIT HIT HIT
17 DBL HIT HIT HIT HIT
16 DBL DBL HIT HIT HIT
15 DBL DBL HIT HIT HIT
14 DBL DBL HIT HIT HIT
13 DBL DBL HIT HIT HIT
12 DBL DBL HIT HIT HIT
Figure 3
JackBlack Soft Hit/Stand/Double Table

Naturally, in any situation covered by Figures 2 and 3, if three or more cards are involved, the DBL should be replaced by HIT.

    Splitting strategy is determined as follows.  A separate calculation is done to determine the conditional expectation of a single card being hit with an Ace through 10 and then played optimally according to the tables in Figures 2 and 3.  This number is doubled and then stored in an array sp(j,k), j being the split card and k being the dealer's total.  This number is compared with the players' conditional expectation exp(2j,k) of optimally playing a hand of value 2j versus the dealer's up card of k.  If sp(j,k) > exp(2j,k) then the player should split; otherwise the player should play according to the strategies in Figures 1 and 2.  Here is the resulting splitting strategy (SPL indicates a split).
 

Dealer's Total
Players' Pair 17 18 19 20 21
10 - 10  STD STD STD STD STD
9 - 9 STD SPL SPL STD HIT
8 - 8 SPL SPL HIT HIT HIT
7 - 7 SPL SPL HIT HIT HIT
6 - 6 SPL SPL HIT HIT HIT
5 - 5 DBL DBL HIT HIT HIT
4 - 4 DBL DBL HIT HIT HIT
3 - 3 SPL SPL HIT HIT HIT
2 - 2 SPL SPL HIT HIT HIT
Ace - Ace SPL SPL SPL SPL SPL

Notice that the 9 - 9 hand is split versus the 18, even though this would be a push for the player.  If some players feel uncomfortable with this decision, they can always surrender for half of their bet, but it would be foolish of them to do so.  Even splitting the Ace - Ace hand versus the dealer's 21 is a good play and the decision isn't even close.

    With the above finished, the rules for JackBlack can be spelled out precisely:

  1. Player can double on any first two cards.
  2. Players' hand will be hit after doubling according to optimum strategy.
  3. If the dealer's total is 17, 18, 19, 20 , or 21, the players' Ace pair will be split.
  4. If the dealer's total is 17 or 18 the players' 2-2, 3-3, 6-6, 7-7, and 8-8 will be split.
  5. If the dealer's total is 18 or 19, the players' 9-9 will be split.
  6. Players not wishing to participate in splitting can surrender for half their bet.
  7. Player can double after splitting.
  8. For players' 2 card soft 17 versus dealer's 17, the player can take the tie, hit or double
  9. For players' 3 or more card soft 17 versus dealers 17, the player can hit or take the tie.
  10. Payoffs are as in Figure 1; Blackjacks are simply 21 totals.

There you have it.  I think this is an interesting game with lots of nice features.  It involves player strategy, which I think players enjoy, but the strategy is much easier to learn than Basic Strategy for either Blackjack or Double Exposure.  Also, with the players sharing a common hand, I believe that there will be a camaraderie developed between them that will add a social aspect to the game.  We'll see.

    What about the house edge?  Well, in case you were wondering, the edge occurs because the dealer has to bust twice for the players to win.  I ran two 200,000,000 simulations using the above rules, one for six decks with a cut card at 78 from the bottom and one with four decks with a cut card 52 from the bottom.  The results were 1.651% for the six-deck case and 1.616% for the four-deck case.

    Here is an interesting bit.  In order to do the analysis, I first had to determine the probability of obtaining a 21 total with 2, 3, 4, 5, 6 or more cards .  It turns out that there are many more multiple-card hard 21s than soft 21s.  There are more than 10 times more hard four-card 21s than soft four-card 21s, for five-card hands it's more than 30 times, and for six-card hands it's around 85 times as many hard as soft.  But here's a strange one for you.  In 20,000,000 hands, I got two 10-card 21s!  I didn't think I'd see any 10-card 21s; I've never see that hand at a Blackjack table.  On the other hand, there's probably a lot of stuff in this life that I haven't seen (including 20,000,000 Blackjack hands).

    See you next month.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers