The Small Effects of Multiple Losses
by MathProfIn blackjack, the results of one round are negatively correlated with the results on the next hand. The players expectation increases after a loss, and decreases after a win and a push. Research done by Gwynn and Griffin suggest that the player's expectation increases roughly .1% after a loss in a single deck game. Now what happens after multiple losses I believe that the traditional theory suggests that these effects are quite minimal. Recently this idea has been challenged; one gambling author has asserted that the player attains a 1% advantage after 5 consecutive losses. The purpose of this post is to investigate these claims. I have carried out a number of sims to study the effect of previous losses on expected results.
I have run a series of sims to study the effects of consecutive losses. The sim deals rounds off the top of a single deck. If there is a win or a push, we exit (i.e. reshuffle). If we have a loss, we play on. If win or push on the second round, we reshuffle; otherwise we play a third round. We keep track of what happens on the first round, on the second round (after 1 loss), on the third round (after 2 consecutive losses), etc. Each sim involves 1 billion rounds.
First we study the effects under ideal conditions. We employ what in the good old days were called "Strip Rules" (S17). We deal as deeply as possible. When my computer ran out of cards, I had it recycle the cards in the discard tray. Obviously this is undesirable, so I tried to prevent this by refusing to deal a round if there were 6 or fewer undealt cards remaining. Even so, we ran out of cards on 7691 rounds. The following are the results that were obtained in our 1 billion round simulation
Sim 1: Deeply dealt S17 game PrevLosses N EV SE Tot Gain 0 521,953,986 -0.002% 0.005% -11661 1 250,035,410 0.089% 0.007% 222759 2 119,690,280 0.161% 0.010% 192963 3 57,255,615 0.260% 0.015% 149055 4 27,370,214 0.302% 0.022% 82741 5 13,077,100 0.398% 0.032% 52049 6 6,242,841 0.349% 0.046% 21772 7 2,980,048 0.500% 0.066% 14905 8 1,296,592 0.217% 0.100% 2815 9 97,905 -2.707% 0.376% -2651 10 9 N/A Ran of cards on 7691 RoundsN is the number of rounds that are dealt, EV is the Expected Value, and SE is the Standard Error. The last column is N*EV, which tells you the total number of bets won (or lost) immediately after that many consecutive losses. The player's advantage does increase slightly after each successive loss. However the biggest edge obtained was .5%, and to achieve that we had to assume that the dealer went so deep that we always had 7 rounds dealt. After 7, we run into the dreaded "cut-card" effect, which significantly decreases our expectation. Note that if we ever lose 8 consecutive rounds in the deck, we get a ninth round only when we have used many high cards in the first 8. Hence our expectation here is very negative.
Note after 5 consecutive losses, our approximate edge is only .4%. (between .33 and .46%) This is significantly below the 1% edge that had been claimed-- a claim which I had characterized as a wild exaggeration.
Unfortunately we have a large range of possible values, which is due to the relatively large sampling error (SE). There are two reasons for this. First, is the size of the samples themselves. We do not often lose the first 5 rounds off the deck. Hence, even though we dealt 1 billion rounds, we had only 13 million rounds that immediately followed 5 losses. Second, and perhaps more important, our expectations are relatively small numbers, so modest standard errors are large in comparison. A .03% standard error is fairly low, but if you edge is only .4%, your sampling error is almost 10%.
The above sim was under a classical game of blackjack dealt down to the end of the deck. Such a game cannot be found. So now let us examine some real games. To get the needed deep penetration, we go to Northern Nevada, where we also have to contend with unfavorable rules (D10-H17). This is our next sim
Sim 2: Nothern Nevada (D10-H17, shuffle point 13 cards) PrevLosses N EV SE Tot Gain 0 522,861,057 -0.467% 0.005% -2439221 1 250,606,813 -0.383% 0.007% -959489 2 120,030,352 -0.317% 0.010% -380582 3 57,452,862 -0.211% 0.014% -121268.5 4 27,476,708 -0.185% 0.021% -50802 5 13,135,301 -0.100% 0.030% -13150.5 6 6,254,288 -0.133% 0.044% -8297.5 7 2,153,758 -0.709% 0.075% -15271.5 8 28,861 -7.120% 0.665% -2055 Ran out of cards on 14 RoundsNote that we never obtain an edge in this game! The house edge declines with each successive loss, but just when it gets down toward 0, the dreaded cut-card effect kicks in and starts to increase it. And after 5 consecutive losses, instead of having a 1% advantage, we have only a .1% disadvantage.
So now let us look at a Vegas game. In the next sim, the dealer hits Soft 17, and uses a fixed shuffle point of 26 cards.
Sim 3: Vegas (H17, shuffle point 26 cards) PrevLosses N EV SE Tot Gain 0 531,962,725 -0.192% 0.005% -1023285 1 255,457,198 -0.098% 0.007% -250861.5 2 122,581,096 -0.023% 0.010% -27973.5=20 3 58,773,085 0.087% 0.015% 50843.5 4 27,694,762 0.090% 0.022% 24958.5 5 3,530,937 -1.907% 0.062% -67329.5 6 197 -20.558% 8.205% -40.5
Here we can get a slight edge of .1% after we lose 2 rounds off the top. But again, the cut card effect kills us as we get deeper into the deck. And here, after losing 5 consecutive rounds, instead of having a 1% advantage we have 1.97% disadvantage! If you were betting 1% of your Kelly b= ankroll here, you would lose your shirt!
Now in fairness, I must point out that my use of a fixed shuffle-point at 26 cards was what has led to the poor performance here. My impression is that this reflects current typical conditions, but I will let others comment on that. In case you are wondering what happens with deeper "cuts", I did another sim with my "cut card" back 4 cards. This essentially eliminated the House edge on the 5 row; it produced a statistical tie there. Of course, if you assume that the dealer will always give you a fifth round, no matter how deep it forces her to deal, then you will have an edge on the 5 row. This is seen in the final sim.
Now in the final sim we try to examine the power of a betting system based upon counting losses. You start your count at 0. With every loss, you add 1. With every win/push, you subtract 1. I assume that you have to "play-all", and that you cannot force a reshuffle after a win. However, I will give you a dealer who always deals 5 rounds, so we will have no cut card effect. The rules are H17:
Sim 4: Using a Count based on losses. (H17, deal 5 rounds) Count N EV SE Tot Gain -4 14,544,488 -0.525% 0.030% -76380 -3 28,034,992 -0.437% 0.022% -122629.5 -2 107,956,862 -0.346% 0.011% -373323 -1 181,856,443 -0.267% 0.009% -486132 0 374,774,165 -0.183% 0.006% -687046.5=20 1 168,003,867 -0.106% 0.009% -177590.5 2 92,135,160 -0.034% 0.012% -30887.5 3 22,104,698 0.105% 0.024% 23199.5 4 10,589,325 0.181% 0.035% 19182 Bets won in positive situations: 42,381.5=20 Bets won in negative situations -1,953,989.0 Min Spread to break even 46.1We have an edge only when we have incurred 3 net losses. We would make big bets here. We make minimum bets at all other times. We won 42K big bets, and we lost 1.9M small bets. Now I will caution you that there is a fair amount of sampling error in the above sim, because the edges are so small. But please observe that we had been playing these 1 billion rounds, we would have needed a spread of 46.1 to 1, just to break even!
Gives a whole new meaning to the expression "Czechs Play!"