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# Difference between revisions of "Harvey Dubner"

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His graph #5 is entitled “Frequency of Favorable Situations for High-Low System, Reshuffle at 6 Cards from End.” This implies that he not only used the terms “Hi-Lo strategy” and “High-Low system” somewhat interchangeably, with comments by his son Robert reproduced below indicating a preference for the former, but also that Thorp had probably borrowed the term “high-low” to use in his 1966 revised edition of Beat the Dealer. | His graph #5 is entitled “Frequency of Favorable Situations for High-Low System, Reshuffle at 6 Cards from End.” This implies that he not only used the terms “Hi-Lo strategy” and “High-Low system” somewhat interchangeably, with comments by his son Robert reproduced below indicating a preference for the former, but also that Thorp had probably borrowed the term “high-low” to use in his 1966 revised edition of Beat the Dealer. | ||

− | + | [[File: Dubner1.JPG|thumb|right|375px|<small> Dubner’s Hi-Lo strategy as provided to the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference. The top image is that of an actual 3 x 5 card that was handed out to attendees. The clamor by attendees to obtain one of the copies caused, in the words of Dubner’s wife, Harriet, a “riot.” The bottom image is taken from an 8 1/2 x 11 photocopy of nine apparent slides most likely presented at the panel discussion. Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion. Thorp remembers that they were. Dubner’s son Robert doubts that they were presented at a later meeting or published after the conference. (Courtesy of Dubner family)</small>]] | |

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+ | [[File: Dubner2.JPG |thumb|right|375px|<small> Dubner’s Hi-Lo strategy. Image is taken from an 8 1/2 x 11 photocopy of nine apparent slides most likely presented at the panel discussion. </small>]] | ||

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Following the presentation, Thorp and Julian Braun embarked on a detailed analysis of the Hi-Lo strategy, Braun using the computer of the IBM facility in Chicago and the programming approach Thorp had used to develop the ten-count strategy he had introduced in the 1962 edition of Beat the Dealer. Thorp published the system in the 1966 revised edition of Beat the Dealer as the “complete point count system.” Some refer to it as the “High-Low” system. Although Jerry Patterson pointedly asks the rhetorical question, “Did he (Thorp) have Dubner's permission?” such a request is not normally needed in scholarly work, and Thorp fully acknowledges Dubner’s work in the 1966 revised edition of Beat the Dealer. Perhaps the more interesting question concerns why Thorp and Braun excluded Dubner from this additional work. | Following the presentation, Thorp and Julian Braun embarked on a detailed analysis of the Hi-Lo strategy, Braun using the computer of the IBM facility in Chicago and the programming approach Thorp had used to develop the ten-count strategy he had introduced in the 1962 edition of Beat the Dealer. Thorp published the system in the 1966 revised edition of Beat the Dealer as the “complete point count system.” Some refer to it as the “High-Low” system. Although Jerry Patterson pointedly asks the rhetorical question, “Did he (Thorp) have Dubner's permission?” such a request is not normally needed in scholarly work, and Thorp fully acknowledges Dubner’s work in the 1966 revised edition of Beat the Dealer. Perhaps the more interesting question concerns why Thorp and Braun excluded Dubner from this additional work. | ||

− | + | [[File: Dubner3.JPG |thumb|left|375px|<small> Dubner’s summary of his consideration of betting in terms of financial theory. The image is taken from a photocopy of nine apparent slides most likely presented at the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference. Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion. (Courtesy of Dubner family)</small>]] | |

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+ | [[File: Dubner4.JPG |thumb|right|375px|<small> Dubner’s comparison of his Hi-Lo strategy and Thorp’s ten-count strategy. The image is taken from a photocopy of nine apparent slides most likely presented at the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference. Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion. This provides the results of the only known rigorous comparison of the two strategies. Thorp, on page 94 of the 1966 revised edition of Beat the Dealer, opines that the two strategies are of “comparable power.” Dubner’s analysis shows that the Hi-Lo is superior. The “Figure of Merit” is defined in Figure 2. (Courtesy of Dubner family)</small>]] | ||

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We can only speculate on the answer. Perhaps Dubner, a private man, simply wanted to return to his professional endeavors, considering the blackjack calculations an interesting diversion now solved to his satisfaction. Perhaps Thorp was annoyed that Dubner “stole the show.” I hope and believe that the former was the case. | We can only speculate on the answer. Perhaps Dubner, a private man, simply wanted to return to his professional endeavors, considering the blackjack calculations an interesting diversion now solved to his satisfaction. Perhaps Thorp was annoyed that Dubner “stole the show.” I hope and believe that the former was the case. |

## Revision as of 13:01, 30 November 2017

**Harvey Dubner** is a retired engineer and mathematician living in New Jersey, noted for his contributions to finding large prime numbers. In 1984, he and his son Robert collaborated in developing the 'Dubner cruncher', a board which used a commercial finite impulse response filter chip to speed up dramatically the multiplication of medium-sized multi-precision numbers, to levels competitive with supercomputers of the time, though nowadays his focus has changed to efficient implementation of FFT-based algorithms on personal computers.

He has found many large prime numbers of special forms: repunits, prime Fibonacci and Lucas numbers, twin primes, Sophie Germain primes, and primes in arithmetic progression.^{[1]} In 1993 he was responsible for more than half the known primes of more than two thousand digits.

Dubner was also the first to publish a paper on the first blackjack point count (The High Low Count) which is used by most blackjack card counters today. This was presented at the Fall Joint Computer Conference held in Las Vegas in 1963 at a panel titled "Using Computers in Games of Chance and Skill".

Dubner was also the first to publish a paper on the first blackjack point count (The High Low Count) which is used by most blackjack card counters today. This was presented at the Fall Joint Computer Conference held in Las Vegas in 1963 at a panel titled "Using Computers in Games of Chance and Skill".

## Contents

- 1 Role in Blackjack Systems Development
- 2 Harvey Dubner and the Development of the “Hi-Lo” System
- 2.1 Early Years
- 2.2 Uproar at the 1963 Fall Joint Computer Conference
- 2.3 Examination of Family Claim
- 2.4 Dubner’s Professional Career
- 2.5 Early Interest in Blackjack
- 2.6 The Computer Used in the Blackjack Simulations
- 2.7 Dubner’s Approach to the Blackjack Problem
- 2.8 Independent Derivation of Optimum Bet Size Criterion
- 2.9 Family Claim that Dubner Was Not Aware of Thorp’s Early Work
- 2.10 Presentation of Hi-Lo at the 1963 Fall Joint Computer Conference
- 2.11 Casino Reaction, Dubner’s Personal Blackjack Winnings, and His Personal Reflections

- 3 References
- 4 External links

## Role in Blackjack Systems Development

An extensive biography of Dubner, based on interviews with him and his family and highlighting the importance of his development of the "Hi-Lo" system for blackjack, is presented in an abridged, highly-edited form in an appendix to the novel *Never Split Tens* by Les Golden. The complete report based on Professor Golden's scholarly research follows.

## Harvey Dubner and the Development of the “Hi-Lo” System

### Early Years

Harvey Alan Dubner was born on July 14, 1928, the son of Reuben and Frances Dubner, Jews of Eastern European descent who lived in New York. He had a younger brother, Neil Peter Dubner. He and his wife Harriet (Weiss) had four children, in order of age, Robert Joseph, Emily Rachel, Terry Ann, and Douglas David. Although his entire family was Jewish, his son Robert’s recollections below indicate they did celebrate the exchange of gift ritual of Christmas with Hanukkah. He attended Brooklyn Polytechnic University in New York, from which he earned the Bachelor’s of Science in electrical engineering in 1949, summa cum laude, and the Master’s of electrical engineering in 1951. He played flippantly with his middle name. Although his birth certificate states his middle name as “Alan,” his Master’s degree diploma from Brooklyn Polytech in 1951 provides it as “Allen.” His son Robert notes that “it is clear that he never knew how his middle name was spelled. He would use his middle initial when forced to. But asked about his middle name, he would shrug and spell out Allen. But it was always clear to me that he was just coming up with something to make the questioner go away; he neither knew nor cared how it was spelled.”

Although educated and employed as an electrical engineer, he was a proficient mathematician and as early as the 1950’s a skilled computer programmer. He found great pleasure and joy in attacking mathematical problems working alone. He would first try to solve the problem analytically, using, that is, only equations. If the problem was not amenable to such solution, he would write a computer program to simulate the effect. His career brought him to in 1960 to the Curtiss-Wright Corporation in Paterson, New Jersey. There he used an IBM 1620 computer to run blackjack simulations, leading to the Hi-Lo strategy. In so doing, he solved the problem of an optimum bet, indpendently recreating the 1956 work of John L. Kelly, now known as the Kelly criterion. He considered the 2’s through 6’s as “lows” and the ten-value cards and Aces as “highs.” His analysis then showed that an index calculated by subtracting the number of low-value cards left in the deck from the number of high-value cards left in the deck and dividing by the number of cards remaining to be played provided an excellent sensitivity to the quality of the cards remaining to be played.

### Uproar at the 1963 Fall Joint Computer Conference

Dubner, then thirty-five years old, presented his blackjack results during a panel discussion on games of chance and skill, entitled “Using Computers in Games of Chance and Skill,” moderated by Edward O. Thorp at the 1963 Fall Joint Computer Conference held in November in Las Vegas. He had learned of the panel discussion, most likely in the trade magazine, Electrical Engineering, and had contacted the conference and asked to be included as one of the presenters. A “call for papers” had been published on page 51A of the April, 1963, issue of Electrical Engineering magazine, the trade magazine of the IEEE, the Institute of Electrical and Electronics Engineers. (I am indebted to Ms. Jennifer Hart of the Crerar Library of the University of Chicago for her aid in locating this announcement.) It was general in nature, and did not mention the panel discussion on games of skill and chance. The actual program, a listing of session topics, speakers, and their topics, was published in the September, 1963, issue. It mentioned the panel discussion, but did not mention Thorp’s name in that context. The call for papers and the program, including the panel discussion, may well have been published elsewhere. Gambling writer and player Jerry Patterson, who writes that he was a speaker at the panel discussion, in his column “Harvey Dubner: The Forgotten Man of Blackjack,” relates that the conference organizers not realizing that most “computerniks” (as they were called at that time borrowing the suffix from the Russian Sputnik satellite) were also inveterate blackjack players, they scheduled the Panel Discussion in one of their smaller meeting rooms. The room filled up and overflowed 45 minutes before the session was scheduled to start! Hundreds of conference attendees were pushing and shoving to get into the room. The crowd, of course, had been drawn by Thorp.

Newspaper coverage of the conference provides insight into Dubner’s sense of play. He asked the panel, “Can you make money playing blackjack?” After the panel emphatically replied, “Yes,” Dubner said, “Then I wish we’d get this meeting over with. I’m losing $25 an hour while I’m sitting here.”

Dubner had his results for his betting and playing strategies printed on 3 x 5 index cards. The clamor by those in the overflow, standing-room only crowd to obtain a copy of the “Basic HI-LO Strategy” card caused, in the words of his wife, Harriet, a “riot.” Jerry Patterson writes that Dubner “stole the show” and was “mobbed” at the conclusion of the panel discussion by the crowd “all wanting copies of his handout.” Patterson recalls, “Here at last, many were saying, is a system that is practical, that can actually be used in the real world of casino play.”

An image of that card is reproduced in Figure 1. Figures 2 and 3 reproduce additional material found by the Dubner family folded neatly in Dubner’s copies of the 1962 and 1966 editions of Beat the Dealer. A total of nine graphs, tables, and summary of analyses were found copied onto the back and front of 8 1/2 x 11 sheets of paper. These were most likely copies of slides he presented at the panel discussion or handouts for attendees at the panel discussion. Of course, the material could have both been presented as slides with copies reproduced for handouts. Thorp remembers they were presented at the panel. Dubner’s son Robert doubts that they were presented at a later meeting or published after the conference. The similarity in font and styles of the various materials conference. implies they were created at about the same time, for presentation together.

His graph #5 is entitled “Frequency of Favorable Situations for High-Low System, Reshuffle at 6 Cards from End.” This implies that he not only used the terms “Hi-Lo strategy” and “High-Low system” somewhat interchangeably, with comments by his son Robert reproduced below indicating a preference for the former, but also that Thorp had probably borrowed the term “high-low” to use in his 1966 revised edition of Beat the Dealer.

Following the presentation, Thorp and Julian Braun embarked on a detailed analysis of the Hi-Lo strategy, Braun using the computer of the IBM facility in Chicago and the programming approach Thorp had used to develop the ten-count strategy he had introduced in the 1962 edition of Beat the Dealer. Thorp published the system in the 1966 revised edition of Beat the Dealer as the “complete point count system.” Some refer to it as the “High-Low” system. Although Jerry Patterson pointedly asks the rhetorical question, “Did he (Thorp) have Dubner's permission?” such a request is not normally needed in scholarly work, and Thorp fully acknowledges Dubner’s work in the 1966 revised edition of Beat the Dealer. Perhaps the more interesting question concerns why Thorp and Braun excluded Dubner from this additional work.

We can only speculate on the answer. Perhaps Dubner, a private man, simply wanted to return to his professional endeavors, considering the blackjack calculations an interesting diversion now solved to his satisfaction. Perhaps Thorp was annoyed that Dubner “stole the show.” I hope and believe that the former was the case.

It was natural for Thorp to compare the power of the complete point count, based on the Hi-Lo strategy of Dubner, to his ten-count strategy. Thorp, on page 94 of the 1966 revised edition of Beat the Dealer, opines that the two systems are of “comparable power.” Dubner created a comparison of the Hi-Lo strategy to the ten-count strategy, which may have been presented at the panel discussion. Thorp recalls that it most likely was presented. The results are summarized in Figure 3, the only known rigorous comparison of the two strategies. Dubner’s analysis shows that the Hi-Lo is superior. The complete point count system, more detailed than the Hi-Lo, would have been more powerful.

Jerry Patterson suggests that without Dubner’s development blackjack would not have attained its popularity as the most popular casino game. The ten-count strategy presented by Thorp in the 1962 edition of Beat the Dealer and in the 1966 revised edition require the player to calculate an index including the tenths and hundredths digits. This cumbersome necessity would limit the number of players able to play blackjack profitably. In Patterson’s view, the development of the easier to use Hi-Lo strategy by Dubner is responsible for the wide popularity of the field of card counting. Inexplicably, although my conversations with the Dubner family make it clear that his joy resulted from solving the problem, Dubner has not been elected to the Blackjack Hall of Fame. To me this is an egregious oversight and I hope it is rectified during his lifetime. Without Dubner, no Blackjack Hall of Fame would exist.

### Examination of Family Claim

Working with the Dubner family, we can discuss the question of whether Dubner indeed developed the Hi-Lo strategy without, as they claim, ever hearing of Thorp. The recollection of events more than fifty years in the past by a man near 90 years of age is challenging.

Most, if not all, blackjack historians have assumed that Dubner read the 1962 edition of Thorp’s Beat the Dealer and was thereby stimulated to examine the game himself, leading to the his development of the Hi-Lo strategy. Dubner’s son, Robert Dubner, in our early correspondences made the same assumption. It was natural to do so. Thorp distinguished tens and non-tens in his ten-count strategy published in 1962 and Dubner distinguishing “highs” vs. “lows” in the “Hi-Lo” strategy seems a natural extension rather than a complete re-invention. Yet, further discussion with those members of the Dubner family involved, Harvey, his wife Harriet, and his eldest son Robert, has led them to change their opinion. Dubner firmly states he did not know of Thorp until the 1963 conference. That he worked independently of anyone else on the project is well-known, confirmed by Thorp himself.

An absolute validation of this thesis would be had if the Dubner family retained some of the paper tape printout of Dubner’s computer runs. In the early days of computers, output was provided on paper tape and punched cards. Later computers provided output on magnetic tape and within the computer memory. Those paper tape printouts would have either had the date of the run encoded or would have been identified by Dubner marking them with a marking pen, stick-on label, or a note taped to the paper tape. A date of any run pre-dating the November, 1962, publication of Beat the Dealer would establish with certainty that Dubner developed or had begun to develop the Hi-Lo strategy without any external influence. As we see below, however, the tapes were discarded by the family.

Much evidence suggests that Dubner did know of Thorp’s work. As noted, the documents discovered by the Dubner family folded in his copies of Beat the Dealer seem to be copies of nine slides created for presentation at the conference or publication and/or handouts of such for the conference. They include Figure 3, a comparison of his work with Thorp’s. Dubner possesses copies of both the 1962 and 1966 editions of Beat the Dealer. He used the term “strategy” to refer to his system, the same term that Thorp uses in the 1962 edition to describe his five-count strategy and ten-count strategy. Dubner calculates a “Hi-Lo index,” shown in Figure 1, and Thorp calculated a “ten richness” index by calculating the ratio of “others/tens,” although he doesn’t use the term “index,” referring to it simply as the “ratio.” Both men provided not only a guide to playing strategy but also a guide to betting. Thorp based his on the Kelly criterion (see later). Dubner apparently did so as well, but determined the criterion by himself. No previous system, including that of the Four Horsemen of Aberdeen, had provided a betting guide. On the other hand, the work was apparently performed while Dubner was working at Curtiss-Wright in 1962-1963. If the work was begun prior to the November, 1962, publication of the 1962 edition of Beat the Dealer, then Thorp was not a seminal influence. If it was begun in late 1962 or 1963, then whether or not Dubner had become aware of Thorp’s work remains in question. The evidence, accordingly, seems to be that Dubner’s recollection is incorrect.

The alternative possibility, however, remains, that he developed the Hi-Lo strategy before learning of Thorp. As will be discussed later, Dubner had an interest in playing blackjack in Las Vegas long before the 1963 conference. With the development of computers, Dubner became a skilled programmer and it is conceivable, as the family states, that he created software simulations of the game for diversion. He had access to computers from his professional work. As an electrical engineer, he either subscribed to the trade magazine Electrical Engineering or read it at work. A call for papers for the 1963 Fall Joint Computer Conference was published in that journal in April, 1963, and the program was published in that journal in September, 1963, including a mention of the panel discussion on “Using Computers in Games of Chance and Skill.” Perhaps the call and program were also published elsewhere.

We need to remember that no journals dedicated to “computer science” existed at that time and indeed many early computer science curricula were part of university departments in “electrical engineering and computer science.” Computer science in its early years concerned the electronics and electrical components more than the software. I remember as an undergraduate at Cornell University in the 1960’s courses being designated by “EECS.” It could have been that Dubner developed or began to develop the Hi-Lo strategy and then read of a forum in Electrical Engineering at which to provide his results, without ever hearing of Thorp and his work.

Although neither the call for papers or program as published in Electrical Engineering mentioned Thorp’s name, that Thorp was to moderate the panel had to have been publicized somewhere. As Jerry Patterson pointed out, Thorp was “the draw” and the conference organizers must have utilized the renown he had achieved resulting from the 1962 edition of Beat the Dealer to arouse interest in the panel discussion. The overflow crowd attests to their success.

At the date of this writing, Thorp has not been able to provide information from his files as to when he was invited to moderate the panel, when and how that development was publicized, and if he selected the members of the panel. It seems highly likely that he did have a major impact on selection of the panel. Of the participants noted on pages 93-94 of the 1966 revised edition of Beat the Dealer, he had worked with Julian Braun, William E. Walden was his student at New Mexico State and later a collaborator, and Allan Wilson apparently knew Julian Braun from San Diego State College.

Dubner either then learned of Thorp or did not learn of Thorp. In the former case, Dubner, now committed to participating in the panel, would have learned of Thorp and would have purchased Beat the Dealer. He could have then run simulations of the ten-count strategy using the same software that he had used in analyzing the Hi-Lo strategy and then prepared what is apparently a slide, reproduced in Figure 3. In this scenario, he developed or had begun to develop the Hi-Lo strategy before becoming aware of Thorp. He may have added the Hi-Lo index and betting guide afterwards.

On the other hand, it may be that Dubner remained unaware of Thorp’s work until the conference itself. Crucially, the September, 1963, issue of Electrical Engineering which presented the program for the November conference provided “R. A. Kudlich” as the chairman of the panel discussion, without mentioning Thorp’s name in that context. The call for papers did not even mention the panel discussion. Perhaps the conference organizers publicized Thorp as moderating the panel in other journals which Dubner had not read and/or by poster flyers mailed to academic departments. In this scenario, Dubner learned of Thorp at the conference, bought Beat the Dealer after the conference, performed the analysis that led to the results of Figure 3, and then either presented the nine apparent slides at a later meeting or published them. He may have simply held onto them himself.

This scenario is unlikely. The Dubner family states that Dubner did not present the information in the nine apparent slides at a later date, and Thorp recalls that Dubner presented the information comparing the Hi-Lo to the ten-count strategy at the panel discussion. In fact, on page 94 of the 1966 revised edition of Beat the Dealer, Thorp writes that Dubner made “great claims” for his strategy and that “(H)is calculations supported his claims.” This implies that Thorp’s recollection is correct, that Dubner presented the results of his computations.

One piece of intrigue concerns the comment by Thorp on page 94 of the 1966 revised edition of Beat the Dealer. As noted above, there Thorp states that the Hi-Lo and ten-count strategies are of “comparable power.” Figure 3 shows that the Hi-Lo, even in Dubner’s formulation, is superior. The more detailed calculations subsequently made by Julian Braun would have produced an improvement, in particular, providing strategy decisions for all player strategies, not only the hard standing guide provided by Dubner. Thorp’s statement implies either an emotionally-motivated defense for his ten-count strategy, which is understandable but which I hope is not the case, or that Dubner did not in fact produce the comparison slide at the conference (and that Julian Braun did not later compare the two). The latter case would imply Dubner bought Beat the Dealer after the conference, supporting Dubner’s recollection. Thorp, as noted, does recall that the comparison information was in fact presented.

As noted, Thorp based his betting guide on the optimum bet analysis of John L. Kelly of Bell Telephone Laboratories, commonly referred to as Bell Labs, published in 1956. Robert Dubner believes that his father independently derived the result. I do believe that belief to be plausible, Dubner’s mode of scholarly work being to work independently of others. (In the sciences, the first step taken by researchers is to go to the library, now usually digital but in previous generations the physical library, and determine if the work has been done by others. In mathematics, an incredibly splintered field, perhaps this is not always the case.)

The Kelly criterion, which can be rigorously proven to provide the optimum betting size, is simple and intuitive and can be expressed for the case in which only two outcomes from a $1 bet are possible as

f = p - q/b ,

where f is the optimum fraction of your current bankroll to wager, p is the probability of success, q is the probability of failure, and b is the net bet or payoff, the amount of money you win if successful above and beyond your $1 bet. (If, for example, you are playing a game with a 0.60 probability of winning, therefore a 0.40 probability of losing, and a payout of $2 for every $1 wagered, then the optimum fraction of the bankroll to wager is 0.40. Go for it!!) From the formula, we see that the optimum bet increases with increasing probability of success, decreases with increasing probability of failure, and increases with increasing size of the payout. All dependences make sense. It is often equivalently expressed as

f = (bp - q)/b,

with the numerator then being the expectation value of a single $1 bet and the denominator being the payoff for a winning $1 bet. I’m sorry, although in this form we can refer to the criterion in pithy terms as being “expected winnings/winnings from successful bet,” I prefer my algebraically simpler version.

In addition to the rigorous proof, numerous “heuristic,” that is, reasonable, practical demonstrations, of this simple formula exist. It is reasonable that Dubner, a gifted mathematician, was able to prove the formula himself without knowledge of Kelly’s work, or to simply derive it by trial and error and computer simulation using the three parameters, p, q, and b, in different combinations. After all, the above form, bp – q, and (p – q)/b are some of the only few simple algebraic forms in which p, q, and b behave in the sensible manner noted above. Indeed, mathematician and physicist Daniel Bernoulli (1700–1782) had effectively arrived at the formula based on an analysis of geometric means of possible outcomes centuries earlier, and his result was commonly known among mathematicians as well as economists. Deriving it required a statement of the problem, a basic knowledge of probability theory, and use of the calculus, within the skill set of most mathematicians. After considering these various possibilities, my feeling is that Dubner began and performed the majority of his investigations without hearing of Thorp. At some time, perhaps after the call for papers or after reading the program in the September, 1963, issue of Electrical Engineering, he learned of Thorp. He then created the comparison diagram and presented it to the panel discussion along with his other material. The evidence does not support Dubner learning of Thorp only at the conference itself.

In research, investigators build upon the work of their predecessors. Thorp was initially influenced by the work of Baldwin, Cantey, Maisel, and McDermott. Dubner’s work was at some stage influenced by that of Thorp. Thorp’s work subsequent to the 1963 Fall Joint Computer Conference was influenced by the work of Dubner. That one researcher builds upon the work of others is how scientific knowledge progresses. No one’s work is diminished in importance as a result. As Sir Isaac Newton wrote (in a phrase the first documented form of which was stated by the 12th century scholar Bernard of Chartres), “If I have seen further it is by standing on the shoulders of Giants.”

Dubner was always and remains a quiet man, not interested in fame or fortune. His interest was mainly in the joy of solving mathematical and engineering problems. Like many of us interested in the mathematics of the game, he was not at all a gambler. It is appropriate that the lack of biographical material on his life, personal philosophy, research approach, and contribution to the game of blackjack be documented. That is done for the first time here.

The following is a moderately edited version of communications between me and the Dubner family in January, February, and March of 2017. The first person refers to Dubner’s son, Robert. Comments by his wife Harriet, who I will note had a career as a speech pathologist, are enclosed in square brackets. Comments by me relating the content of phone and mail conversations with the family or elaborating upon or clarifying concepts are enclosed in curly brackets.

### Dubner’s Professional Career

He started working in 1949, just before graduating from college. His first jobs out of college were with the Arma Corporation (1949–1951) and with the Avion Corporation (1951–1960). Those jobs involved defense work, including fire control strategies for U.S. Navy surface vessels at Arma and the design of the infrared seeker head for the Navy’s Sidewinder missile, his main project at Avion.

Professionally, in the 1950’s my father worked on missile guidance systems, in particular infrared homing systems. He was part of the team that developed the AIM-9 Sidewinder. He designed the optics, which meant ray tracing, which meant a number of people, mainly women, doing arduous calculations with mechanical calculators. That led him to learning the capabilities of electronic computers. He got hold of his first computer, a 1956 Royal-McBee LGP-30, in part to do those calculations. {LGP is the acronym for “Librascope General Purpose” and, later, “Librascope General Precision.”}

He also became fascinated by catadioptric telescope design as a result of his optics work. In particular, he was much taken with the elegance of the Maksutov design. In 1976, having been interested in astronomy and star-gazing all his life — when I was about nine years old he got me an Edmund Scientific three-inch Newtonian as a Christmas present, and he messed with it at least as much as I did — and having lusted after one for years, he bought a Questar Standard 3.5-inch telescope. I have it now; I recently had it serviced and converted to the Duplex model, where it can be removed from its alt-azimuth mount and placed directly on a tripod for field work or as a telephoto lens.

[We moved to Ridgewood, New Jersey, in 1963 and he worked at Simmonds {Precision Products}. I believe he went to work at Curtiss-Wright in Paterson, New Jersey, in 1960 for a couple of years and then Simmonds also for a couple of years. In late 1963 he went to work at Computer Applications until that company went bankrupt and then he formed his own company, Dubner Computer Systems.] {Robert Dubner provides corrected dates.} He went to Simmonds in early 1963, and from Simmonds to Computer Applications sometime around 1967. Dubner Computer Systems was formed in mid-1970.

Although by the end of 1964 my father was at Simmonds, I am pretty sure that he was at Curtiss-Wright in the 1962-1963 time frame. We moved from Washington Township, New Jersey, to Ridgewood in June of 1963. Curtiss-Wright had many locations. I seem to recall that my father worked at the Curtiss-Wright location in Paterson. We used to go as a family to a restaurant named Steve’s which I believe was near where Dad worked. A dim memory has something on the wall of the restaurant relating to the Kennedy assassination, which took place in November of 1963.

Curtiss-Wright was one of the country’s leading aerospace engineering companies. The Paterson plant was a major one. My father reported directly to the “big boss,” the plant manager. My father’s title was “Chief of Advanced Design,” and at the time he was in charge of the biggest project going on there, a huge flight simulator. There were many dozens of engineers involved: mechanical, electrical, production, test, you name it. They all reported, directly or indirectly, to my father on that project. As far as I know, this was the first major flight simulator based on a digital computer, rather than primarily based on analog computers. I regret to report I can’t recall what aircraft it was. I don’t think I ever knew, although I can describe in detail how he and his team decided to use a digital computer at its heart. My father can’t remember the aircraft either.

I became interested in my father’s career and engineering in general when I was in the sixth grade, which was the 1964–1965 school year, my having been born in 1953. I was eleven years old that Christmas, and I remember a class project involving a mural depicting a winter scene. I augmented that mural by incorporating two NE1 neon bulbs as the eyes of the snowman. Powered by a 90-volt radio “B” battery, a simple RC driving circuit caused the snowman’s eyes to blink.

(My teacher, who I realize in retrospect was a very sweet woman in her twenties, was utterly bewildered by me. I mean, c’mon. I was a scrawny eleven-year-old kid, not even five feet tall, and I show up with a handful of electronic components and a 90-volt battery, connect them together with some alligator clip jumper cables — without a drawing — and the lights start blinking. She must’ve felt like she was falling down the rabbit hole.)

The key thing is that my dad brought those components home from his job at Simmonds Precision Products in Tarrytown, New York. The other key thing is that I was, at around that age, starting to pay attention to my father’s work and career. I relate the anecdote about the snowman’s eyes to establish my bona fides and to mark that point in time.

A few years later when I was in the ninth grade, the 1967-1968 school year, I learned to program computers. I worked with my father for the next thirty years, so I can lay some claim to being able to come up with some good guesses about what happened in the times before I really started paying attention.

Curtiss-Wright still exists, as does Simmonds Precision Products. It is an 85-year-old aviation company. The “Wright” goes back to the Wright brothers and the “Curtiss” goes back to Glenn Curtiss. Curtiss and the Wright brothers were vigorous competitors in the early days of aviation but eventually their companies merged. Either Curtiss-Wright or Simmonds Precision Products would have had, in 1963, a print shop capable of creating those 3 x 5 cards.

### Early Interest in Blackjack

Starting around 1950 when he was twenty-two years old, the business and professional work at both Arma and Avion occasionally took him to the West Coast and he developed a habit of stopping overnight in Las Vegas on his way home to New York probably both for conferences and for convenience in travel. Because traveling from California to New York by turboprop aircraft was noisy, lengthy, and tiring, it was much more civilized to take a short hop to Las Vegas, spend the night there, and then take an early flight back to the East Coast than it was to travel by airplane at night. There, other engineers introduced him to Las Vegas and casino gambling.

Although he wasn’t a gambler, he loved games and puzzles. He also loved the hustle and bustle and sheer weirdness of Las Vegas; we talked about that many times. He was a mathematician and an innovative thinker, and, at some point after learning to play blackjack, he figured out that unlike craps or roulette (which are pure games of chance) or poker (which is mainly a game of skill) blackjack offered the opportunity of informed advantage play. It was during these visits to the casinos of Las Vegas that he developed his interest in possible advantage play in blackjack.

It was obvious to him that knowing what cards had been played could lead to an advantage over the house, if only the task of recalling those cards and the resulting calculations could be made manageable. But it was after he started using computers in the late 1950’s that he began thinking about using computers to develop a method that could be used by a gambler in a casino.

### The Computer Used in the Blackjack Simulations

He easily could have started the {blackjack computer simulation} work at Curtiss-Wright and kept it up at Simmonds. I am pretty sure the computer my father used at Curtiss-Wright was the IBM 1620. The computer work was probably done off-hours at Curtiss-Wright. [I agree. It was at Curtiss Wright on off hours.] My father had some big spools of black eight-track punched paper tape in his files. (I later worked with such tape for several years, so I have clear retrospective memories that it was the eight-track ASCII type, and not the earlier six-track Baudot coded tape used in Flexowriters.) As I type this, I am having a visceral memory of how that heavy oily paper looked, felt, and smelled, as if were in my hand right now. I remember my father telling me that those tapes were the saved output of blackjack simulation runs. I am pretty sure the computer my father used at Curtiss-Wright was the IBM 1620. I can’t swear to it, but the timing works. And the 1620 had the 1621 paper tape reader and the 1624 tape punch.

Although his memory is weak on this point, the work had to have been at Curtiss-Wright. He might have done some preliminary work at Avion on the LGP-30, but the LGP-30 used six-level punched tape. My memories are as clear as they can be after over fifty years, and although six-level and eight-level tape are each one inch in width, the hole patterns are different, and dang it all, the blackjack results were punched on the eight-level tape used on the IBM 1620. And, again, there doesn’t seem to have been enough time after starting at Simmonds for him to have done the necessary non-work-related research there.

I know Harvey worked with the 1620. I have no memory of him working with the IBM 704 or 709. That doesn’t mean much because until I started programming in 1968 I didn’t pay attention to the “which computer” question. I know his first machine was the Royal McBee LGP-30, and I know about the 1620. But I don’t know much about other machines he used at work until I really got involved around 1971. Online articles describing the IBM 704/709 don’t mention punched paper tape. Those machines apparently used magnetic tape along with card punchers and readers for data storage. And I “know” he punched the output of his blackjack simulations on paper tape. This all points to the 1620, and away from the 704/709. Without being sure, my guess is that he did the blackjack simulations on an IBM 1620; the timing is about right. [We moved to Ridgewood in June 1963 soon after Harvey went to work for Simmonds. But I think he did his simulations when he worked for Curtiss-Wright.]

{Only mild uncertainty remains in the family memory whether the blackjack simulations were performed at Simmonds or Curtiss-Wright. The family moved in June, 1963, after which he began to work for Simmonds, and the 1963 Las Vegas conference was held in November, 1963. This probably allowed insufficient time to prepare the paper and submit it for presentation, implying the work was performed before the move, at Curtiss-Wright.}

I don’t believe he “worked at night” on the computer. He would have written the code at night at home. He would have gotten it working during the work day. We always mixed in extracurricular stuff in with the paying work. It was fun, it kept things interesting and exciting, you learned things that way, and the paying work “always” got top priority, with the extra stuff filling in the cracks. Then he would do the long runs for statistically significant results overnight, gathering the results the next morning. {Because of the relative slowness of computer processing, this was the mode of computer analysis of time-consuming programs in the early days of computing.}

A “long run” for him, given the power of the computers of the time, was on the order of ten thousand hands.

The output paper tapes from his computer runs are long gone, along with boxes of IBM cards, 8-inch and 5.25-inch and 3.5 inch floppy disks, and piles of green-striped computer printer paper. When I talk about seeing the paper tape, I remember seeing it in the house we lived in before we moved to Ridgewood in 1963. I was ten years old when we moved. I used to play with those tapes when he was done with them. By the late 1960s he wouldn’t have had access to an IBM 1620 and the tape would have been so much scrap; they would have been discarded by then.

The sad thing is, he almost certainly had notebooks from that time, and he likely would have recorded his work in them, too. I had a look around the house a few weeks ago, but the earliest of his notebooks seem to be from around 1980 or so, which is when we started the real push into large-precision arithmetical number theory. The older notebooks don’t seem to be around. Back around 2006 or so my parents had a flood in their basement, and it’s very possible those earlier notebooks were damaged beyond recovery at that time.

### Dubner’s Approach to the Blackjack Problem

As to what happened during the development of his counting rules, my knowledge of my father and the way he liked to work tells me it went like this. He played blackjack from time to time in Las Vegas and got interested in the possibility of advantage play. He would not have attempted to find anybody else’s work on the subject. He and I talked about blackjack a lot over the years, and he never mentioned anybody but Thorp, but he never indicated that he learned about counting from Thorp.

Having gotten interested, he would have done a mathematical analysis. Taking the formal math as far as he could, he would then have started writing computer code. He would have investigated random number generators and created his own. He would have figured out how to use an RNG to shuffle a simulated deck of cards — which you may know isn’t actually as straightforward as it sounds; there are a lot of ways of screwing up both random number generation and shuffling — and then he would have started playing simulated games.

I don’t know the details of blackjack, but I believe there are some straightforward rules for when to ask for additional cards based on your hand and the dealer’s up cards. And once you have rules of play, you can simulate multitudes of hands and experiment with methods of counting and calculating your edge.

To elaborate on this concept, in one of your questions, you wonder if “intuition” had something to do with it. I can assure you that intuition was involved in the process of developing the hard theory, but it was the hard theory and the results of simulations that led to his results. This is how mathematical and scientific research works: an investigator starts with a suspicion, an intuition, a hunch, maybe even a wish. But those feelings have to survive rigorous analysis. Otherwise they are just guesses.

I speak as one familiar with this kind of thing. I know how I would respond if I were to get interested the way I believe that my father got interested.

First, I would learn how blackjack is played. I would have learned the rules of the game, and I would have investigated optimal play in the case where I have no knowledge about the constitution of the deck. I am talking about things like, if I am holding two tens, it’s kind of stupid to draw. And if I am holding a ten and a two, it’s kind of stupid to stand. (At, least, I think that’s the case; I am not a blackjack player.)

And I would have verified mathematically that when the deck is rich in high cards that I am more likely to win, and when the deck is rich in low cards I am more likely to lose.

Operating on that omniscient level I doubt any simulation is necessary; if you know exactly what cards are left in the deck, then the probability of winning or losing can be calculated precisely.

I believe that for a full deck, this has long been worked out and represents the “basic strategy,” which is how to best play your hand against each possible dealer upcard. That procedure is also the foundation behind casino rules and payouts; it’s what makes blackjack the game it is. But all of those conclusions are based on averages; they would have assumed that you are playing against an infinite deck of cards.

Knowing my father, he would have built a blackjack engine and run it some thousands of times in order to verify the “basic strategy.” He would have done it for two reasons: First, to verify the published basic strategy. Second, in order to test his blackjack engine. If for any reason his results differed from the published versions, he would know that he had to check his implementation of the random number generator and his “basic strategy” decision making. He wouldn’t expect to discover the published techniques to be wrong, but you never know. And it’s very easy to make implementation mistakes, so he wouldn’t have moved forward on a belief that the published techniques were incorrect until he had really verified everything else. The basic principle is that it’s necessary to make sure your initial conditions match what came before you.

Now we get into “informed play”: If you know the composition of the remaining cards in the deck because you kept track of every card that was played, you can, in theory, calculate precisely the odds of winning your next hand. It’s a combination, of course, of every possible hand the dealer can deal both you and himself, but it is in principle calculable.

That’s where artistry and simulation come into play. Perfect knowledge is too costly for a human to work with; he wanted to come up with something simpler, something that could be done without equipment at a casino table. He would have started with various scenarios: What if I know exactly how many aces there are left, and how many face cards, and how many twos, threes and fours? Something like that. Then he would have started simulating: The computer would play thousands of games. He would have grouped each hand into buckets based on the strategy he was testing at that time. He would have kept track of how many times each bucket resulted in a win. I feel confident that he iterated through that kind of process until settling on “count the high cards, count the low cards, subtract the lows from the highs, and divide by the number of cards left.”

So the process would have started with the conviction that an edge was possible even with “chunky” knowledge of the composition of the remaining deck, rather than precise knowledge. Informed intuition would have led to the creation of various scenarios. Simulation would allow each scenario to be tested, and the results of the tests would lead to one that seemed optimal: That is, it got “good enough” results without requiring superhuman memory or arithmetical abilities.

What I don’t know is precisely what he was testing in those simulations. I do know the results he arrived at. And I worked with him often enough on other similar investigations to be morally convinced that he arrived at the hi-lo index by testing a number of other, similar, ways of counting the cards, and keeping track of which method seemed to work best in terms of being feasible at a casino table and effective in generating returns.

You ask about the term “basic” on the Hi-Lo card. I have no direct knowledge of why he used it. But knowing my father, I am positive the word means, “There is more to the story.” As I mentioned earlier, the method on the card wasn’t the best possible method. It was my father’s attempt to balance practicality with effectiveness.

### Independent Derivation of Optimum Bet Size Criterion

It was at that point that Harvey ran up against the question of given that you have an edge, how much do you bet? Too little, and your winnings grow slower than they need to. Too much, and you’ll go broke if there is a run of hands against you. The simulations would have shown: “When the hi-lo index is “this,” then the probability of winning is “that.” The next question becomes: “If I am playing blackjack and my probability of winning the next hand is 52%, how much should I bet?”

Imagine a coin flipping game: Tails means you win your bet, heads means the casino takes your money. You somehow know the coin is biased to come up tails 52% of the time, and you know that if you play long enough you’ll come out ahead. How much should you bet? If you have $100 and place it all on the table, you have a 48% chance of going broke on the first flip. Bet nothing and you might as well have stayed home.

You can infer that there is some optimal amount to bet to maximize the rate at which you make money, given the magnitude of the edge and the amount of money in your wallet. That’s what my father figured out, and he did the analysis, and he came up with the optimal amount to bet once the card counting rules gave him the size of the edge.

So, given a known edge in the game, there clearly is some point between betting nothing and betting everything you have that will maximize your winnings per hand over many hands. That can be plotted and analyzed as winnings-per-hand against size-of-bet (with both axes expressed as a fraction of your stake); it is an upwardly convex curve {imagine an opened umbrella held overhead} with the optimum at a peak.

That point can be analytically found using simple calculus. Finding good numerical values would be trivial using computer simulation. I asked my dad about it once, and he told me that he had done the math.

This is exactly what John Kelly did and published in 1956 and which became known as the Kelly criterion, or a Kelly bet. He learned sometime later that he had independently recapitulated the work of John Kelly. I am making educated inferences about some of this, but he told me he learned of the Kelly criterion after he had recreated it himself. {He told me as early as 2006} that he had rederived the Kelly criterion and only later learned of Kelly’s work.

So, there it is. Some of that I know my father did because he told me: He independently duplicated Kelly’s math, and I know he simulated tens of thousands of blackjack hands.

### Family Claim that Dubner Was Not Aware of Thorp’s Early Work

Based on my recent long conversation with my parents, I hadn’t fully realized how much casino blackjack he’d been playing during the 1950s. It’s my understanding that has grown over the last few weeks. I hadn’t realized just how much blackjack he played in Las Vegas during the 1950s.

My father, through the years, and during recent discussions inspired by your questions where we have walked through his memories, has always said that he never even knew who Thorp was until the 1963 conference. We’ve been talking to him about Thorp, and he says definitely that he didn’t know about Thorp’s work prior to 1963. He and my mother are in agreement that he played blackjack in Las Vegas starting as early as 1950 or so, and that his interest in advantage play grew steadily throughout that decade, culminating in his developing his method through computer simulation after the computers became available. My belief now is that the work was being done in parallel by Thorp and my father; neither knew about the other. {The independence of effort in developing the Hi-Lo strategy is confirmed by blackjack writer Peter Ruchman, who quoted Thorp as writing: “At the 1963 conference Harvey Dubner presented the . . . (high-low system), which he had thought up himself.}

As I noted, his work taking him to Las Vegas in the early 1950s led him to start playing blackjack. Being mathematically inclined, he would have quickly figured out the theoretical possibility of advantage play. After he started using computers in 1956, he would have automatically started thinking about using them to develop workable techniques. He remembers doing just that.

One thing my father is adamant about: He worked out the possibilities of what became his hi-lo strategy on his own. It grew out of playing casino blackjack from time to time during the 1950s and the early 1960s. I have repeatedly revisited with him the question of “Where did you get the idea that counting the cards could lead to an edge over the house?” His answer is unwavering: “I worked it out myself.” When asked if he learned of the possibility that he learned of it from Edward Thorp, his unwavering response: “I never even heard of Thorp.”

{That his work was at least begun without knowledge of Thorp’s 1962 book} makes sense. There just wasn’t enough time between publication of Thorp’s book in November, 1962, and the November, 1963, conference for my father to have developed the expertise to play casino blackjack, work out the math, write the programs, do the simulations, and develop what he referred to as “his method.” Between Beat the Dealer coming out in the fall {November} of 1962 and my father starting a new job in 1963 there just wasn’t enough time to do the work the way he did it. He had to have been working on it before Thorp published.

He couldn’t possibly have learned about counting from Thorp’s book in 1962, then become competent at casino play, figured out the math, worked out the concept, written the code, and developed his method all between the book coming out and then starting a new job in early 1963. If nothing else, he wouldn’t have been in a casino often enough. And this was all extracurricular work. He wouldn’t have let it interfere with his job, so his time commitment wouldn’t have been great. I had previously thought that he learned about it from Thorp but that can’t be right; there just isn’t enough room in the timeline. And when I was doing that speculation, I didn’t know he’d been regularly going to Las Vegas during the 1950s. I figure also that card counting research and development in the early 1960s was a case of “when it’s time to railroad, railroads pop up all over.” I’ve been told that various gamblers had been trying to count cards for years. They kept quiet about it, so as not to lose what little edge they had, but word was spreading. J.L. Kelly published his work on optimal betting in 1956. Computers were becoming more generally available. So it’s not surprising to me that a lot of activity was happening in parallel.

[Harvey says he was not working with anybody when he created his Hi-Lo system. And no he did not have a software collaborator. He was not in contact with Julian Braun. He says he might have been in contact with him after the {1963 Las Vegas computer} conference, but not for any reason. Harvey is sure he never heard of Thorp until the 1963 joint computer conference. He is sure he worked independently. Harvey said he created the {3 x 5} card before he even knew who Thorp was.] So, putting together everything I know now, it’s clear to me that Harvey was one of a small number of people at that time who had the interest, the inclination, the mathematical training, the innovative insight, the computer programming ability, and — not insignificantly in that time frame — the access to a computer necessary to conceive and build the simulations needed to create and test useable blackjack counting strategies. But it is clear to me that he started his work because of the possibilities that became apparent to him while playing and thinking about blackjack, and not because he learned about card counting from anybody else.

Concerning the comparison of the Hi-Lo and Ten-Count strategies {displayed in the apparent slide}, does it demonstrate that Harvey knew about Thorp and the Ten-Count Strategy at the time of the conference? Sure does. That leaves the question you find most interesting: Does it show that Harvey learned about counting from Thorp? Harvey is adamant that he developed the idea on his own. My belief, knowing him, is that he did develop it on his own. My belief is that when he learned of the Ten-Count strategy he compared it to his method, found his to be superior, and that may well have led to him deciding to present the Hi-Lo strategy at the Fall 1963 conference. {Or, he may have already decided to present at the conference before reading Beat the Dealer and simply added the comparison to the material he had already planned on presenting.}

Once he learned of the ten-count strategy he would very naturally have compared it to his. And upon finding his Hi-Lo strategy to be superior, and then learning of the panel session at the Fall Joint Computer Conference, it’s natural that he would have used it as an opportunity to go public. It’s distinctly possible that he would have gone to that conference even without the “games of chance” session; he was a computer professional at a time when it was still a new field. I note that he had a paper published at the 1970 Spring Joint Computer Conference (http://dl.acm.org/citation.cfm?id=1476964). I am speculating that he was planning on that trip to Las Vegas anyway, and the panel discussion was a chance to have some fun and show off his work. It seems clear that by mid-1963 my father had read the 1962 Beat the Dealer, because it appears from that handout {Figure 3} that he tested Thorp’s ten-count system using the same simulation techniques he used for developing his Hi-Lo. {The format employed by Dubner in the handout does not allow immediate comparison with Thorp’s published results, and this comparison would likely have required the additional simulation to which Robert Dubner refers.} But there is no question in my mind that he did the bulk of his work before he read Beat the Dealer. He started his work and he developed his methods, and then read Beat the Dealer. Remember, he wasn’t a professional gambler. His interest was theoretical and not because he wanted to win at blackjack. He wouldn’t have bought the book except for the fact that he was already doing research and development on card counting.

How much of his final work was done after reading the book versus before nobody will ever know. My guess is “not much”; the essence of my dad’s method is that it is simple; it would have coalesced fairly early.

{The 1962 copy in their family library is the hardcover first edition from Blaisdell Publishing Company.} I see one rare book dealer offering one in fine condition for $500. My mother later commented that maybe we should sell it.

{The findings of the family of the copies of apparent slides which may also have been used as handouts at the 1963 conference led to a further consideration.} There are a number of interesting things in that handout. He apparently did some simulations that involved hundreds of thousands, even millions of hands. Pure speculation, of the type that I really should stop making: I know he has told me in the past that his runs involved tens of thousands of hands. This leads me to wonder if he started his work on the slower Librascope LGP-30 {while he worked at Avion, from 1951–1960}, and later refined it on the faster IBM 1620 {at Curtiss-Wright}. But that’s absolutely pure speculation. {If true, Dubner clearly began his investigations before the publication of the 1962 first edition of Beat the Dealer.} The letter-sized sheet of paper was prepared at the same time as the 3x5 card, which means at some point in late 1962 or early 1963 when he still had access to the 1620 computers at Curtiss-Wright.

If Thorp’s account in the 1966 edition of Beat the Dealer is accurate, Harvey didn’t present those sheets at the meeting. Thorp says on page 94 of the paperback 1966 edition {of Beat the Dealer} that “Exactly how much better or worse (the Hi-Lo system) is than the Ten-count method is not known.” It sure doesn’t sound like Thorp saw those sheets of paper. My dad was a natural showman; he may have decided that the sheets were too technical and he should just stick with the “We’re in Las Vegas, and with this little card you can beat the house!” fireworks, and leave presenting the math for another time…a time that never came. {I’d be shocked if Harvey didn't present the sheets, as handouts or slides/handouts. This was a technical meeting and he wouldn't just provide the slide of the 3 x 5 card. Thorp, as stated, remembers that the material was presented. His comment in Beat the Dealer may resulted from either not at the time of its writing remembering the details of Dubner’s presentation or not wishing to quote results from a study the details of which he had not examined or verified for himself.}

I can’t comment on any possible competition in the development of the theory of card counting, or who might have appropriated work from somebody else. I am certain Harvey did his work independently. My understanding is that Mr. Thorp, in the later edition of Beat The Dealer, acknowledged Harvey’s work and credited him with putting card counting onto a solid mathematical footing.

I very strongly doubt that he even knew of anybody else’s work, much less was taking anybody else’s work into account. I don’t know if he had published any papers at that point in his career, but he shortly would start to do so. If anybody else’s work had informed his, I am certain he would have credited them. And I’m pretty sure it would have come up in our conversations over the years.

For complex problems, especially work related, sure, he would research, and collaborate, and build on the work of others. He did that a lot. But he was doing the blackjack thing for fun. He paid a lot of attention to blackjack over the years, and many of his memories are pretty clear. He doesn’t remember where he did the work, but he does remember doing it, and he remembers why he did, which was for the fun of it and to prove that you could get an edge over the house in casino blackjack.

He wouldn’t have found it to be fun to start with somebody else’s work. It would have been far more characteristic of him to start from scratch. He was not surprised to learn that he was among a group of people doing similar work. To some extent it was the availability of computers that led both Thorp and my father to do the work they did. When it comes time to railroad, railroads pop up all over the place.

In addition to his current assertions, I had many conversations with dad about blackjack and counting over the years. He never mentioned Thorp except in the context of what happened after the 1963 conference. He never once said that he learned about the possibility of getting an edge from anybody else. He always spoke of it in terms of figuring out that it could be done and working out how on his own. He never spoke about building on anybody else’s work, or working with anybody. He always spoke about working out the possibility himself, and working the math and the computer simulations to come up with his method. He never referred to it as anything but “his method.” For example, he would talk about how “theoretically there are better methods than ‘my method,’ but they are much harder to use, and the gains are small.” He didn’t even use the name “Hi-Lo” when talking to me, and he didn’t even call it “card counting” much, except when talking to other people. When he and I discussed it, it was always just, “my method.”

I have always been interested in the technical aspects of his work on blackjack, and we talked about it many, many times. He always — always! — spoke in terms of what he had figured out. He never once said that he had learned about the possibilities of card counting from somebody else.

And if he had, he would have acknowledged it. I worked with him for years doing number theory research. He learned a lot from other investigators, like Richard Brent, and Peter Montgomery, and Hugh Williams, to name just a few. He often spoke in terms of what he had learned from those other people. He was always careful about what he learned from other people, and the original work he did himself.

If the idea that Harvey did the work all on his own is controversial, well, it’s only because nobody ever asked him. He’s been in the phone book all this time, and for many years his e-mail address has the same and available. If anybody asked, as you are asking, how, he would have said, as he is saying now, that he did it all on his own. If somebody can demonstrate that they were in touch with my father about blackjack before he gave that presentation, I would be both astonished and incredibly interested in talking to them.

If somebody thinks they can prove my father read Thorp’s book {before he began his own investigations}, or somehow got the idea for card counting other than on his own, well, I’m pretty sure that his reaction over the years would have been to ignore them. He was always like that. He knew what he’d done and, although he liked being acknowledged for it, he didn’t go out of his way to correct misapprehensions. Heck, I happened to pick up a copy of {William Poundstone’s} Fortune’s Formula a few days ago. Lo and behold, my father’s counting method is described in there, but without attribution, except vaguely in Thorp’s direction.

So it’s not surprising that dad’s method evolved into being described as “an improvement over Thorp’s method.”

It’s incontrovertible that Thorp did publish before my father gave his presentation in 1963, so it’s perfectly reasonable for the world to have concluded that Dubner’s method was an improvement over Thorp’s. But Harvey is clear that he didn’t know about Thorp’s work, and his doing the work de novo fits every discussion he and I ever had about it. {At this point, after discovery of the apparent slide comparing the Hi-Lo strategy to Thorp’s ten-count strategy, Robert Dubner would state that his father did not know about Thorp’s work at least when he began his investigations and possibly until most of it was completed.}

### Presentation of Hi-Lo at the 1963 Fall Joint Computer Conference

According to the program as published in the {September, 1963, issue of the} trade journal Electrical Engineering the conference was to take place in the Las Vegas Convention Center. You’ll note at the bottom right of page 20A in the program {see bibliography} that on Wednesday, November 13, at 8:00 p.m., “Computers Applied to Games of Skill and Chance” was scheduled, chaired, inexplicably, by “R.A. Kudlich, General Motors Corp.”

In the “backmatter” {of the published proceedings; see bibliography}, “E. O. Thorp” is listed as one of the Session Chairmen. {This information, however, was published in the proceedings of the conference, and not before it. Dubner first encountered Thorp at this conference, according to his recollection.} In the second big box on the page {see bibliography} there is a set of tabs, labeled Abstract, Source Materials, Authors . . .. The Source Materials tab has a link to the Back Matter. {The proceedings were published both by the Association of Computing Machinery and Spartan Books of Baltimore, Maryland. Neither, however, provides excerpts from the games of skill and chance panel discussion, only papers formally presented to the various conference sessions. The 3 x 5 index card on which Dubner printed the “Basic HI-LO Strategy” and, if they were presented, the nine slide/handout material, are the few, if only, artifacts remaining from the panel discussion on games of skill and chance.}

Dr. Kudlich was a long-standing and reliable administrator of the IEEE, and he apparently organized the 1963 “Computers in Games…” panel. It seems equally clear to me that Thorp was brought in as the pre-eminent popular figure, because of Beat the Dealer, to chair the actual session. (An inspired move, if you ask me!) My educated guess is that Kudlich was probably there, probably introduced Thorp to the enthusiastic crowd, and then Thorp moderated the discussion. In short, it really looks to me like Thorp was rightly brought in as a celebrity figure to moderate the panel. {At the date of this writing, Thorp has not been able to provide information from his files as to when he was invited to moderate the panel, when and how that development was publicized, if he selected the members of the panel, and if Kudlich in fact introduced him.}

I asked him about how he got to present at the 1963 conference. He said that he learned about the panel discussion on computers and games of chance — the topics are published in advance — and he contacted the conference and asked for a slot. {That is, his participation was not based on personally knowing Thorp.} (My father is a lifetime member of the IEEE, and Electrical Engineering Magazine was an organ of the IEEE.) And he simply doesn’t remember. The answer {to exactly how he learned of the panel discussion}, again, is that nobody knows; nobody can know. But you can be absolutely sure that Curtiss-Wright had a library that subscribed to all the publications of all those groups, as well as the publications of the American Federation of Information Processing Societies. At that time a company like Curtiss-Wright would have had all of them available, and my father and his coworkers would have paid attention to them. The astonishing thing would have been if a panel on “Computers Applied to Games of Skill and Chance” had gone unnoticed by my father, given his interest at that time in blackjack. I don’t know who R. A. Kudlich was, but it’s easy to think that when that September 1963 magazine was put to bed Edward Thorp hadn’t yet agreed to chair the panel.

### Casino Reaction, Dubner’s Personal Blackjack Winnings, and His Personal Reflections

I know people who have made tens of thousands of dollars using his method. Hell, I know a guy who claimed he bought a house with card-counting winnings. But not my father. As my mother and I have said, he wasn’t a gambler. He is, in fact, remarkably risk averse. He has told me that when playing at a one dollar table when the deck got hot, his hand would shake when he put down the twenty dollar bet.

The pickings were easy in the beginning. The casinos didn’t know what was going on, and particularly downtown in Vegas the dealers would play a single or double deck right down to the bottom.

As time went on, of course, you know what happened. Two, four, even six decks. The yellow “stop here and shuffle” card halfway down. Lately the casinos, I am told, have deployed their Doomsday Machine: continuous mechanical shuffling, wiping out any hope of an edge.

And, of course, the dealers themselves had nothing better to do than count the cards, and when they saw the counting betting pattern they’d signal the pit boss, who’d intervene. That led to the team play concept made noisily public by the MIT crew, where one person would quietly count and signal a cowboy to come play a few hands when the deck got hot. But you know all about that.

People have a tendency to gleefully assume that my father was banned from casinos all over town. Not so. Nobody knew who he was by sight. He kept his head low when he was playing blackjack. The one and only time my father was asked to leave — it was at a downtown casino, I think he once told me — he was actually behind at the time. “But I’m losing!” he told me he told the pit boss. The polite reply was something like, “We don’t want your money.”

[He never stayed long at any casino. He didn’t want them to know he was counting. Even so, at one point we were asked to leave although he was betting very little. I don’t remember the name of the casino. We stayed at the Riviera so I guess we played more there than anywhere else. We also played in town.]

My father wasn’t there to win. He was there for the satisfaction and fun of experiencing his results at work. He kept careful records; I am sure you will start slavering when I tell you those records still exist, as far as I know. Among other things there is a graph, covering at least a decade or two of once or twice a year visits to Las Vegas, showing the amount of money earned per hour of play. It slopes steadily down to the right, both because of some outsized returns early on — a case of “beginner’s luck” — combined with the effects of the steadily escalating countermeasures by the casinos over the years. [He says what started out as an interesting mathematical problem became a big thing. And as time went on, it became bigger and bigger. At the beginning, he had no idea of the ramifications of his discovery. Harvey says suddenly he was involved in a situation where he was making money and had a good chance of making a lot of money some time in the future. But he was doing other things professionally that were even more interesting. He was doing more and more work that was much more significant than card counting. Remember always he was a mathematician, not a gambler. He never benefited financially from his blackjack creations. He was a mathematician, never a gambler.]

I will reiterate the point: My father wasn’t a gambler, and he had no particular interest in blackjack. But having learned that it might be possible to get an edge on the casino by tracking what cards had been played and thus developing some knowledge of what was left in the deck, and having his interest piqued by that, he would have — based on many discussions about this over the years — done the analysis, done the math, written the code, done the simulations, and formulated his method, completely on his own.

He was proud of how easy his method was to use. Perfect counting could do a little better than his rules, but it required perfect memory and a complex series of responses, and it could only do a few percent better. At least, that’s what he told me.

So he twisted the tail of the casinos with that 1963 paper. I am absolutely sure he did it on purpose; his sense of humor is extremely puckish.

{He did make enough money on one trip to buy his wife a $200 diamond pin, which they still possess.} [We bought the pin in Ridgewood, New Jersey. I believe the name was Webber Jewelers. That’s just a vague memory.] The jeweler in Ridgewood was Weber Jewelers.

You wonder if those sheets of paper {containing the nine materials} were created after 1963 for a second presentation. I tell you that they were not. There is no evidence anywhere that my father ever talked publicly about blackjack counting ever again. He talked about it plenty privately. He made no secret about it. He relished having created the first practical blackjack counting system, and would tell anybody about it at the drop of a hat. But he wasn’t part of the gambling world, and didn’t expend any energy pursuing it. He never did any subsequent research or simulations.

After 1963 he was done, except to go to Las Vegas and occasionally Atlantic City and count at blackjack, keeping track of his winnings. His long-maintained gently downward sloping graph of $/hour over time appears to be lost, but he was still winning when, around 1977, not long after the sole time he was asked to leave a casino because he was caught counting, he gave it up. I asked why he stopped. He said it was because it had become boring.

But my father had a unique opportunity. A mathematician at heart, he liked to prove things formally. As an engineer, he would switch to pragmatic methods when the formal math fell short. Having embraced computers for solving problems starting in 1956, he knew when to switch from formal methods to computational ones. He had the computer programming skills to implement the methods, and he had access to the computer needed to perform the calculations. He also had the wry sense of humor needed to publish his results in Las Vegas.

## References

- ↑ Caldwell, Chris. "Harvey Dubner".
*The Prime Pages*. http://primes.utm.edu/bios/page.php?id=5. Retrieved 29 April 2013.