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This article was originally published in the Blackjack Insider, a Blackjack e-mail newsletter edited and published by Henry Tamburin, author of "Blackjack: Take the Money and Run". The article was written by Dan Pronovost, President of DeepNet Technologies.
Number of Players in Blackjack
In this article, we explore the effect of number of players at the blackjack table. 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 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 calledBlackjack Audit from DeepNet Technologies (www.HandheldBlackjack.com). 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".
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. 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 is better than all other games. 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 per shoe. With six players, an average of 2.05 rounds are played per shoe. In the head-to-head game, we fail to get any deeper in the shoe 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 shoe is 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 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.
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.
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. The extra penetration by the time we play our hand does not provide an advantage since we are using basic strategy only. But the large drop at six players continues due to one round less being played on average.
Table 3: Basic Strategy only (flat bets)
With basic strategy only (flat bets), we see the same six and eight deck trend to flat expectation, with mostly the same in single and double deck. Even without card counting, penetration does have a valid and significant effect as shown by the slightly improved expectation in the single deck four-player game (3.0 rounds per shoe, providing extra penetration). Hence, the four-player single deck game just manages to cause an extra round to be dealt on average, resulting in deeper penetration. Expectation improves with deeper penetration, which is why single deck games have a lower house advantage than multi-deck games. The mathematical logic behind this is explained with great clarity in Henry Tamburin's book "Take the Money and Run", page 27.
To justify why your expectation improves as we deal deeper into the shoe, even when only playing basic strategy, start by looking at the probability of getting a blackjack. At the start of a single deck game, the odds of getting a blackjack (a ten and ace combination in your first two cards) is: 16/52 x 4/51 + 4/52 x 16/51 = 0.04827. Now, consider the same question on an average shoe with only 1/4 of the deck left. Assume an average distribution of the dealt cards, leaving one of each rank: 4/13 x 1/12 + 1/13 x 4/12 = 0.05128. Notice that the odds are better of getting a blackjack when there are less cards to be dealt. Hence, with 3 to 2 blackjack payout, we can assume a better expectation when there are less cards to be dealt, even when only playing basic strategy.
To double-check the above results from Blackjack Audit, we ran some identical simulations in another popular blackjack simulation program CVData (www.qfit.com). For the equivalent game in the first row of Table 1, the single player expectation was 0.501%, and the six player expectation was 0.492%. As another example, CVData generated -0.23% and -0.15% for the one and four player expectations in the fourth row of Table 3. The same trends discovered in this article surfaced in other identical simulations.
Some 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.