Blackjack Bet Size for Recreational Card Counters
FROM ET FAN:
Blackjack Betting: The Leib Criterion vs. the Kelly Criterion and Bet Sizing for Recreational PlayersBy John Leib
© 1993 Blackjack Forum
[From Blackjack Forum Vol. XIII #3, September 1993]
Just when you thought it was safe to buy in,
[Note from Arnold Snyder: In this article, our favorite heretic, mathematician John Leib, continues his arguments against the universally-accepted Kelly method of bet-sizing for blackjack betting.
Leib in fact attempts to answer some questions Iíve had about optimal blackjack betting for many years. The problem was articulated to me in the early Ď80s by a player who said he went to Vegas five or six times per year, always with $4000 to play with. He was a card counter, but did not entertain fantasies of becoming a pro. He was happy spreading his bets from $25 to a couple of hands of $100, had been playing this way for years, was well ahead of the game and was also collecting a lot of valuable comps.
From a Kelly perspective, he was grossly overbetting his blackjack bankroll, but he argued this point with me. "If I lose all the money I brought to play with, itís not like Iím broke," he said. "Iíll be back a few months later with the same bankrollófour thousand bucks. And if I win a couple thousand on a trip or even double my bank, which Iíve done on occasion, I donít add it to my bankroll for the next trip. When I come to Vegas, I bring four thousand. That is my stop loss, I guess, not my bankroll."
The problem I had was that this player wanted my advice on the optimal counts at which to raise his bets and/or spread to two hands. There isnít an easy answer to his question, no matter what anyone says.
The playerís outlook on his bankroll is probably not all that unusual for a casual player, and in fact may be closer to the way in which the majority of players think about their gambling bankrolls. The Kelly method of bet-sizing fails to address this type of playerís concerns. Where does this leave us? John Leib proposed The Leib Criterion.]
John Leib on the Leib Criterion for Blackjack Bet Sizing
Simulation work supports discrediting the Kelly Criterion as optimal for any blackjack bettor, regardless of bankroll and trip goals, on any particular gambling trip. Its replacement is a thing of personal decision. I shall provide you with the information necessary to make that decision.
This material may not apply to professionals, those readers who do not have another source of income in addition to what they glean from the playersí side of the blackjack table. These are a special class to which (I am guessing) you do not belong, and to which I shall not presume to give advice. But it is unlikely that they approach a full Kelly betting fraction.
For purposes of the rest of us, I have assumed trips to Las Vegas or other gambling Meccas, be they occasional or frequent, are made with a total budget determined at the moment of leaving home. This budget is typically composed of two distinct partitions: an "expenses" partition for room, food and incidentals; and a "gambling" partition for action at the tables. I shall suggest a slightly different partitioning which may make more sense to you.
Proportional betting seems rational. Make the most of those good, positive expectation situations. I would not try to discourage proportional betting. But, what proportion?
We try to do our bet sizing as some fraction of the current state of the "gambling" partition and our understanding of the expectation on the next hand. This is a somewhat flawed approach because it doesnít recognize the majority of the situations: those for which the expectation is zero or negative. Refusing to play in negative situations may be impossible, or bad for camouflage. Occasionally you can change tables or go to the rest room when things look bad, but usually you have to play through the bad hands.
So you may wish to accommodate these zero and negative expectation hands as a separate type of "expense" item in your trip budget, perhaps further partitioning the "gambling" partition into "gambling expense," which pays for your seat at the table during hands which are even or negative, and "gambling investment," from which your positive expectation wagers are taken and to which your positive expectation wager wins are credited. "Investment" wagers are then fractions of the current state of the "gambling investment" partition. When the term "bankroll" is used here, it refers to the "gambling investment" partition.
The "gambling expense" partition is a logical way to look at the rent for the "office" from which you work, much like the office of a lawyer or accountant, and is no less a real expense than room and food. It should be estimated from the game(s) you plan to play, how long you plan to play, and how you plan to bet in negative and zero expectation situations. Some conservatism may be in order in making this estimate.
A word about "risk of ruin." Were you able to bet fractionally and perfectly, you would never go broke. But this is not the real world. Two possibilities (not including "barring") exist that may cut short your gambling activity: you run out of money or you run out of courage.
We all (well, most of us) have, upon occasion, become so discouraged by poor results in potentially lucrative situations that, with no probabilistic justification for our decision, we stop playing, perhaps even for the remainder of the trip. So, to deal with the discouragement factor, I have, through simulation, looked at the effect of "chickening out" at levels of loss of 50%, 70%, 80%, 90%, and 95% of the initial bankroll. Such "chickening out" would be, for those of us who are not full-time professionals, the meaning of "ruin" for that trip, and shall be the meaning of "ruin" for the remainder of this article.
Now, some technical discussion about the simulation. Iím sorry, but this is necessary so you can judge its validity and applicability to your gambling proclivities.
Simulation of Blackjack Bet Size Strategies
The variance for the outcome of a hand of blackjack is about 1.26 times the bet squared, due to doubling, splitting and 3/2 payoffs for naturals (see Griffin, The Theory of Blackjack, Revised and Expanded edition, page 167). But this includes ties, which add essentially nothing to the variance while appearing to reduce the "risk of ruin" that a particular betting strategy incurs.
Since about 10% of the hands result in ties (see reference above), proper "risk of ruin" calculations require this variance to be divided by 0.9. This adjustment yields a variance of 1.40 for non-tied hands, with its corresponding standard deviation of 1.18 (as compared to the usually-quoted 1.12).
For calculating time in a casino, the number of non-tied hands must be increased by the factor of 1/0.9 to get the expected positive hands. The hand expectation must also be increased by this same factor, as the expected win comes in the fewer number of hands.
Since less than half of the hands will provide the positive expectation opportunity to be included in our computations (the rest of the hands are taken care of by our "betting expense" partition of the trip budget), I have chosen to consider one dayís hard play to yield approximately 500 such situations, and only 450 of these will be non-ties. For a weekend, I used 1000/900, and for a week I used 2500/2250.
The simulation emulates the toss of a biased coin on which we get to wager on the better side. Payoffs are even money plus an amount to increase the standard deviation to 1.18 times the bet. (This is not the same as betting 1.18 times as much with an even money payoff because you donít collect the expectation on the extra fraction of a bet.) Perfect fractional betting is employed and no casino limits exist. The betting fraction is always some fixed multiplier times the instantaneous expectation for all wagers.
Positive expectations do not come like that so some composite distribution of the positive continuum must be used. At a loss to tell you what your style of game selection and play will yield, I referred to Dr. Edward O. Thorpís classic, Beat the Dealer, 1966 Edition. Using tables 8.1 (page 104) and 8.5 (page 117), I constructed a rough approximation of the distributions shown on those pages.
The results of this part of the simulation process are shown in the tables under the heading of "Composite Projections." (When other distributions become of interest, I will be happy to run them.) The composite projections give you a good approximation of what to expect of our "investment" budget in head-to-head play at single-deck.
The results of the simulation are tabulated in the manner I felt was most easily understood and used. You probably already have some idea of what kind of bettor you are, assuming you are (more or less) a proportional bettor. Do you typically make a half-Kelly bet? A full-Kelly bet? A two-Kelly bet? Are you going for a day? A weekend? A week? And do you "chicken out" before you lose it all?
Now I am going to ask you to perform that ever-popular task: "Meet me half-way." Before you look at the tables, decide on an objective way to select a betting fraction which is just right for you, a "Figure of Merit" (FOM) by which you can judge the appropriateness for you of the competing Kelly fraction table entries. Such an FOM will include the positive value to you of average win (in terms of your initial bankroll) and the negative value to you of the probability that you will reach your "chickening out" level during the trip.
It is this old "risk versus reward" concept revisited. To be proper for you, these must be balanced, one against the other, because more of one necessarily means more of the other. You may be surprised to discover, after you have defined your FOM and then looked at the tables, that no betting fraction can satisfy what you thought were your requirements for projecting a successful trip. A successful trip projection is an acceptable combination of average win and the probability of disappointing results.
There is a table for bettors who like to bet half the Kelly fraction, one for full-Kelly bettors and additional tables for bettors with persuasions to bet one and one-half, two, two and one-half, and three times the one-Kelly fraction.
The tables contain the results for approximations of one day, one weekend, and one week of play, with a column for hand expectancies of 1%, 2%, 3%, 4% and 5%, and a row for each "chickening out" level. The results shown in each entry presume the entire time will be spent in that positive expectation situation, or in non-positive situations covered by your "betting expense" budget partition.
Finally, the composite results are presented, reflecting the more typical situation of varying hand-to-hand expectations for head-to-head single-deck play.
Find whom you think you are in the tables, and you can see what perfect proportional betting means, both in average trip win and the risk of suffering your definition of ruin.
A few comments about the tables:
I propose The Leib Criterion as a replacement for The Kelly Criterion. This new criterion is based on the unquestionable correctness of a fractional bettor wishing to maximize his or her expected (average) win over a finite number of fixed-expectation wagers, but with the added requirement of limiting the likelihood of finishing the series of wagers "in the red," to no more than about one-half.
The conversion factor in going from Kelly to Leib is simple and wasy to remember: One Leib Criterion equals Two Kelly Criteria. In other words, to bet according to The Leib Criterion you must bet twice what you would in following The Kelly Criterion.
This has the pleasant result of more than doubling your expected win while holding the likelihood of loss to approximately one-half, no matter how long a series of wagers you plan to make.
Perhaps this 50-50 split between the probabilities of winning and losing has caused some authorities to conclude that "It can be proved mathematically that an over-bettor who consistently bets twice the optimal amount will break even over the long haul" (Wong, Professional Blackjack, page 103.) The fallacy in this conclusion is, of course, that when you win, your average win is much larger than is your average loss when you lose; and you will win as often as you will lose.
Well, one size may not fit all. But if you wish to really understand proportional betting, and to find your comfort zone as a function of expectation, The Leib Criterion and the accompanying tables should give you what you need.
Snyder Comments on the Leib Criterion for Blackjack Bet Sizing
I think Leib has got something here, but before you run out and start betting double Kelly, bear in mind that Leibís methodology would, in most cases, result in a bet that is less than full-Kelly, if you were figuring out Kelly in the traditional fashion.
Normally, we consider our total bankroll when estimating our optimal Kelly bet. Leib suggests first separating our gambling bankroll into two portionsó"gambling expenses" and "gambling investment."
This might provide a truer estimate of the appropriate Kelly bet, since Kelly advises no bet unless there is a player advantage.
But, what portion of your bankroll do you assign to "gambling expenses?" That would depend on the games you play in.
If you play in shoe games, and you do not table hop relentlessly, you will probably play 80+% of your hands without an advantage. So, even if you use a fairly large spread, a sizeable portion of your total playing bankroll should be assigned to the "gambling expenses" portion of your budget. Betting double Kelly with the remaining "gambling investment" portion might call for smaller bets than if you were betting full-Kelly based on your total bankroll. The wisdom of table-hopping shoe games becomes very apparent with a divided bankroll, as your investment dollars increase as your expense dollars decrease.
But even in single-deck games, most of your hands will not be played with the edge in your favor. Since single-deck games also usually require a player to keep his betting spread down, probably close to half of most playersí total bankrolls should be assigned to the "gambling expenses" bank.
One difficulty inherent in Leibís methodology is that your "gambling expenses" bankroll will also be subject to fluctuations. You might "assign" $500 to your "gambling expenses" bankroll, but lose $1000 on negative advantage hands due to bad luck. You do not want to underestimate your potential table "expenses."
A conservative way to overcome this difficulty would be to overestimate your "gambling expenses" from the start, and transfer funds from "gambling expenses" to "gambling investment," or vice versa, according to your actual win/loss results.
In order to employ Leibís methodology with any accuracy, we need to look at the frequency distributions for the various games, as well as the various blackjack betting spreads the player might use. Leib may draw fire for some of his unorthodox conclusions, but I find his basic idea here appealing. Since a blackjack player, out of practical necessity, must constantly violate Kelly, Leibís separation of the playerís bankroll into "expenses" and "investments" is a unique effort to rethink Kelly for the reality of the game, and the variety of its players.
Again, Leib may draw fire for some of his unorthodox conclusions, but I find his basic idea here appealing. Since a blackjack player, out of practical necessity, must constantly violate Kelly in his blackjack betting, Leibís separation of the playerís bankroll into "expenses" and "investments" is a unique effort to rethink Kelly for the reality of the game, and the variety of its players. ♠
For more information on how professional gamblers size their bets to maximize their earnings for bankroll size and game conditions, see Arnold Snyder's Blackbelt in Blackjack.
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This Blackjack Forum article recommends the Leib Criterion for blackjack betting for recreational card counters. Whereas the Kelly Criterion recommends betting proportional to your blackjack bankroll, Leib points out that most recreational blackjack card counters will never, in fact, get into the long run in their blackjack play. For such players, the value of casino comps and other factors may outweigh the value of blackjack betting proportionally to your blackjack bankroll.