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Heads-up play, or playing with more players… which is better in blackjack?
In this article, we explore the effect of number of players at the blackjack table (equally applicable to online and offline play). Does your expectation improve (profit or loss divided by the total of all wagers)? Does it depend on the game and rules? What happens to my hourly win rate?
It is a common belief that the less players playing at the blackjack table the better. This belief, especially if you are counting cards, if is based on the premise that you will get to play more hands when the count is favorable with less players at the table.
This sounds reasonable, but does it hold up in real play? The game of blackjack hides many surprising mathematical and statistical oddities, despite its simple rules. To test this premise, we used our blackjack simulator called Blackjack Audit from DeepNet Technologies. Each session covering a different set of rules played a minimum of 300 million rounds of blackjack.
The tables below show the results with different blackjack rules and number of players. The statistics for the multi-player sessions include only the hands played at the first position in the blackjack table. The 'Avg. bet' column shows the average total bets per round, across the four player sessions (the average bet per round varies very little with number of players). The unit bet size for all games was one dollar. All references to 'shoe' in this article mean the unplayed cards that the dealer deals from. The 'Pen.' Column shows the shoe penetration that was used for that row's simulations. The penetration represents the minimum number of cards played from the shoe before the cards are reshuffled (rounded down if necessary). We used the full High-Low count system as published by Stanford Wong in his popular book, "Professional Blackjack".
| 1 |
2 |
4 |
6 |
| DAS, S17 |
8 |
1-$10 |
$2.20 |
75% |
0.52% |
0.51% |
0.51% |
0.53% |
| DAS, H17 |
6 |
1-$8 |
$2.02 |
75% |
0.43% |
0.43% |
0.43% |
0.43% |
| NoDAS, H17 |
2 |
1-$4 |
$2.46 |
67% |
0.46% |
0.47% |
0.47% |
0.49% |
| NoDAS, H17 |
1 |
1-$3 |
$2.20 |
67% |
0.92% |
0.95% |
1.07% |
0.79% |
Table 1: Full High-Low count system: indices, insurance index, and betting levels
The first three rows in Table 1 immediately shatter the belief that your odds of winning improve with less players, with two or more decks. The expectation (the percentage of each bet you should expect to win or lose, on average) is effectively constant from one to six players.
The most shocking result in this data is that the four player expectation in the single deck game is better than all other number of player variations. This is not an error: it is caused by the fact that with four players, the shoe is played much deeper than the other games. In effect, the cut card is ending up on average a few cards in at the start of the third round in the four-player game, resulting in more cards dealt per shoe. With four players, an average of 3.01 rounds are played until the shuffle. With six players, an average of 2.05 rounds are played until the shuffle. In the head-to-head game, we fail to get any deeper since at most two hands are dealt. This extra round makes a tremendous difference, given the added advantage that the High-Low count system delivers as penetration increases. The best performance is surprisingly achieved with four players (not head-to-head), where the cards are dealt deeper by the time you get to take cards on your last hand, providing deeper penetration effectiveness of the true count for index plays.
To help understand this single deck anomaly, consider the following analysis. In the four-player game, we can expect 28 cards to be dealt out after two rounds (2.7 cards per average blackjack hand, times 5.1 average hands/round, times 2 rounds. The average hands/round is more than 5 due to hand splitting. Both the 2.7 and 5.1 numbers are from exhaustive simulations). The cut card will be 66.67% * 52 = 34 cards into the shoe. Since approximately 14 cards will be dealt per round (2.7 * 5.1 hands), an extra 8 cards will be dealt after the cut card on average. This means the shoe will be dealt deeper by the time you have to play your last hand of the shoe. In the six-player game, we can expect 38 cards to be dealt after two rounds (2.7 cards per average blackjack hand, times 7.1 average hands/round, times two rounds). This means the cut card comes in the last four cards dealt in the second round on average, after you've played your hand, with no extra third round. In the single player game, the occurrence of the cut card in the round makes little difference given the smaller number of cards per round.
Readers may ask why extra depth improves the expectation. For the card counter, depth is the greatest factor in leveraging a positive edge, since the true count will change in greater proportion, providing more information for play deviations, and bet spreading. With greater depth, each exposed card that happens to further unbalance the count up, means a proportionately higher count, which means more play deviations and higher bet.
Players should be cautioned though that these computer simulation results are not likely to transpose into equal effect when playing live at a casino. Casinos rarely deal as deep as two thirds of the shoe in single deck blackjack. Also, players joining and leaving the table will affect the number of cards dealt, altering the cut card position. The better lesson to take from this study is that playing with fewer players does not necessarily provide a better expectation, and that the reverse (playing with more players is better) is also not necessarily true. Generally, casinos will deal a minimal number of rounds in single deck blackjack, usually equal to 6 minus the number of players (i.e. two rounds with four players). This limits the positive advantage of using more players at the table to control the position of the cut card.
| 1 |
2 |
4 |
6 |
| DAS, S17 |
8 |
1-$10 |
$2.16 |
75% |
0.40% |
0.40% |
0.38% |
0.40% |
| DAS, H17 |
6 |
1-$8 |
$1.99 |
75% |
0.28% |
0.27% |
0.27% |
0.26% |
| NoDAS, H17 |
2 |
1-$4 |
$2.43 |
67% |
0.29% |
0.28% |
0.26% |
0.27% |
| NoDAS, H17 |
1 |
1-$3 |
$2.17 |
67% |
0.62% |
0.58% |
0.63% |
0.44% |
Table 2: High-Low count system, no indices (betting levels only)
In Table 2, we used the High-Low Count System for betting only (no index plays). We see identical trends as Table 1, with the exception that the single deck game has equal performance in one and four players, unlike Table 1. A possible explanation is that without play deviations from indices, the extra depth in the four player game does not provide an extra edge.
| 1 |
2 |
4 |
6 |
| DAS, S17 |
8 |
$1 |
$1.10 |
75% |
-.35% |
-.34% |
-.35% |
-.35% |
| DAS, H17 |
6 |
$1 |
$1.10 |
75% |
-.52% |
-.52% |
-.51% |
-.53% |
| NoDAS, H17 |
2 |
$1 |
$1.10 |
67% |
-.50% |
-.49% |
-.50% |
-.45% |
| NoDAS, H17 |
1 |
$1 |
$1.11 |
67% |
-.24% |
-.26% |
-.16% |
-.21% |
Table 3: Basic Strategy only (flat bets)
With basic strategy only (flat bets), we see the same multi-deck trend to flat expectation. But there is a glaring anomaly in the single deck row, with four players that is difficult to explain. At first, it may seem tempting to justify the improved expectation for the same reason as table 1, when card counting (greater depth). The hypothesis being that the extra penetration (which is still true in the basic strategy case with four players) is the cause for the improved performance. But penetration is not a plausible factor when a player is using only basic strategy. A simple justification would be to consider a shuffled eight-deck shoe, from which we pull the last 52 cards. If we deal from this reduced pack of cards, our expectation is still the same as with the initial eight decks, and is not the single deck expectation. When card counting, especially with index plays, penetration does matter, because of the greater proportionate information for better advantage plays (as shown above).
So why do we see the improved expectation in the four player single deck, basic strategy game? I explored the possibility that it is caused by the infamous cut-card effect. This highly controversial and very complex phenomenon is often used to explain uncertain data in blackjack, so I ran additional simulations to test this theory precisely for our anomaly.
But first… what is the cut-card effect? It happens due to the common practice of using a cut-card (usually a plastic colored insert) to determine when to reshuffle. The important characteristic is that the cut-card is placed at the start of the deal, and then the round in which it is exposed is dealt to completion, after which the cards are reshuffled. The net effect is that in some deep-penetration games (especially in single and double deck games), the last round played can have a statistically high number of low-valued cards played. When there are more low-valued cards, then the player's advantage worsens. The cut-card effect is also dependent on the number of players, but not in any simple or intuitive way. Depending on the penetration and number of players, the cut card may come out near the beginning or end of the last round in a consistent fashion, which impacts the amount of the cut-card effect.
To help explain the anomaly in the Table 3 data above, consider the following simulation results:
| 1 |
2 |
4 |
6 |
| NoDAS, H17, 1 deck, basic strategy |
Shuffle after every round |
-.16% |
-.14% |
-.14% |
.15% |
| NoDAS, H17, 1 deck, basic strategy |
2/3 penetration cut-card |
-.24% |
-.26% |
-.17% |
-.22% |
Table 4: Cut-card effect analysis
The cut-card effect can be seen in the much better expectations in games where we shuffle after each round. But, notice the far smaller improvement in the four-player game. The only variable we modified was the cut-card versus the shuffle, so the smoking gun seems to be in the hands of the cut-card effect!
To further explore this, let's look at some more simulation data for the above single deck game with the cut card, and analyze the player expectation by round played:
| 1 player |
|
|
|
| |
1 |
15.04% |
-0.13% |
| |
2 |
15.04% |
-0.16% |
| |
3 |
15.04% |
-0.15% |
| |
4 |
15.04% |
-0.12% |
| |
5 |
15.04% |
-0.10% |
| |
6 |
14.97% |
-0.18% |
| |
7 |
9.59% |
-1.03% |
| |
8 |
0.20% |
-6.71% |
| 2 player |
|
|
|
| |
1 |
21.82% |
-0.12% |
| |
2 |
21.82% |
-0.14% |
| |
3 |
21.82% |
-0.13% |
| |
4 |
21.81% |
-0.16% |
| |
5 |
12.70% |
-1.09% |
| |
6 |
0.00% |
-6.59% |
| 4 player |
|
|
|
| |
1 |
33.41% |
-0.15% |
| |
2 |
33.41% |
-0.16% |
| |
3 |
33.00% |
-0.19% |
| |
4 |
0.16% |
-4.22% |
| 6 players |
|
|
|
| |
1 |
49.00% |
-0.15% |
| |
2 |
49.00% |
-0.17% |
| |
3 |
1.98% |
-3.26% |
Table 5: Cut-card effect by round played
Look carefully at the last round or two for each of the number of player games. The expectation decreases dramatically as expected with the cut-card effect. But in the four-deck game, the penetration and number of players results in exactly 3 rounds played, with virtually zero occurrences of four rounds, since the cut-card is coming out consistently near the start of the third round. The result is less deviation in the player expectation, and a reduction in the cut-card effect.
It should be noted once more that this anomaly is highly dependent on the exact simulation parameters. A real casino game will not have four 'robotic' players, each playing exactly the same strategy, nor will the dealer place the cut-card exactly two thirds in the pack every time, or even any time! The cut-card effect varies unpredictably based on the penetration and number of players. No one should take this data to literally to mean that they should avoid exactly four player single-deck games.
Technical detailsSome readers may argue that your win rate improves with less players, despite the constant expectation. Your win rate per hour is the expectation times the number of hands per hour, times the average total bets per round. Of these three factors, only the number of hands per hour decreases as more players join the table (the average bets per round were checked in the above simulations, and did not vary significantly). But the increase in hands per hour is only a beneficial factor in a positive edge game! When using basic strategy only, you will lose more money per hour playing head-to-head blackjack since you are going to lose on every hand in the long run. If you are a basic strategy player, enjoy the company of more players and your bankroll will last longer.
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