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Gambling

Yang Li

University of Washington Bothell

STMATH 420: History of Math

Dr. Robin Angotti

October 18th, 2021

Gambling

Development of Gambling Games

What is gambling? Slots, Blackjack, Russian roulette, whether you have been to Casino or not, there will be some names jumps out when you talk about it. But these are not gambling itself, you can only call them a manifestation. The core of gambling is betbetting, without a bet, those activities are only games. That is, if you have a bet, everything can be a proper gambling. That makes gambling become one of the most popular and ancient “entertainment”, in most case. Comment by Haojie Chen: Nice introduction, it make the readers more simply understand the main point of this paper

According to The History of Gambling website, the earliest evidence of gambling can be found in 2300bc, China. But furthermore, a pair of dice had been uncovered from an Egyptian tomb even earlier, from 3000bc. It may be an impossible mission to know what and when was the earliest gamblingThe earliest gambling was born, since it could be done even without any tool, left us no evidence.

According to this article, most scholars agree that the first playing cards appeared in China in the 9th century, although there is no exact rule recording left for us to study. But one thing for sure is, it had little relation to the classic 52 card decks system we know today.

Another popular gambling game – Baccarat, was first mentioned in 1400s when it migrated from Italy to France. Although just like the card game, it spent another hundredstook hundreds of years to develop and finally became the game played in casino nowadays.

Blackjack born inwas born somewhere around 16ththe 16th to 17th century. According to the article, one inference is that blackjack came from a Spanish game called ventiuna as this game appeared in a book written by the author of Don Quixote in 1601. Other assumptions were: It was the game of trente-un from 1570; Or it was quinze from France decades earlier. Since there were many similar games in different cultures from all over the world, even if we tried to focus inon Europe, there were still too many possibilities, it willwould be almost impossible to judge where and how blackjack was born.

1638 is a big year for gamblers. In that year, the first casino, the Ridotto, was established in Venice, Italy. “It provided a controlled gambling environment amidst the chaos of the annual carnival season.” (The History of Gambling website) According to it, during the 19th century, casinos started to spread out widely, while at the same time in US, much more informal gambling houses started to appear.

After this drastic change in gamblingthe gambling world, gambling games were starting to be unified and becoming more formal. Between 18th century to 19th century, some famous gambling games finally “born” with a more formal rule. In 1796, roulette bornwas born in Paris. It then spent 50 years to change from the European version to the form we know today, spreading and becamebecoming popular in 19th century. Although poker itself may have many ancestors from all over the world, a formal “date of birth” would be 1829. According to the website, in that year, an English actor named Joseph Crowell reported that a recognizable form of game was being played in New Orleans, which symbolized the birth of poker.

Although those weren’t gambling itself, only games used for gambling, from the history of development of those games we can still see some trajectory of the development of gambling itself as well.

Martingale Betting System

With the bornbirth and spread of casino in 17th century, gambling was changing with each passing day. Some strategies were created in this environment by gamblers who wanted to win a game for sure or those bookmakers who wanted to squeeze more money from the gamblers. One of them is Martingale, a general and maybe most popular gambling system. In case you never interested in those “Hormone-stimulating activities”, here is a brief explanation of Martingale:

Let’s say you have a clear target today – to win $1, so you start with a $1 bet. If we win, then congratulationcongratulations, it’s time to go home; If you lose, the next bet becomes $2. You can keep this loop until you win a game, or you don’t want to continue your lostloss. This strategy’s subject is to “hit and run”, as long as you are able to win one round, target fulfills and it’s time to go home.

“Its first documented use was well before the 18th century. It is rumored to have been started by the greedy casino owner John Martindale.” The name of Martingale system is also a variation on his surname (Colon, N, 2017). According to the author, Martingale’s popularity is mainly because it has a “straight forward and seemingly rational approach”..” To be brief, it’sit is easy for playerplayers to play this way, without any complex strategy, with a simple target and reasonable gains and losses balance. From the sample above, we can see in each loop, the result of winning is always the same — $1. That makes the player have this feeling: “I’m just repeating this game repeatedly. The more I lose, I will have largera larger chance to win. And at last, I will finally get my one dollar.” From a “mathematical” point of view, let’slet us say the chance for playera player to win this game is 1/10, then some players will believe that they will be most likely to succeed after simply playplaying this game 10 rounds. This simple strategy seems to have promised outcome for players, but actually “the system opens up the player to an unlikely but very high-risk approach and for a very slight reward” (Colon, N, 2017).

Even if we get rid of those complex probability calculations, only think of calculating the cost of this strategy. Think it in “real math”,,” it’s easy to notice and calculate that even if the player starts with $1, maybe the lowest bet ever, even if the story goes to a happy end as the previous player though – ten rounds before winning, the final bet will be $512. And the total bet will be $1023. This reminds me of a classic quiz – fold your A4 paper 10 times, see what you will get. Even it’sit is so clear that n power of 2 can bring little number to a crazy place, most people still don’t realize this “power of power”.

Some may still say: “Even so, we still can win $1 for sure if we prepare enough money.” That’s obviously wrong due to Probability. If we still use the previous example, the chance for player to lose all 10 rounds will be about 35%, which simply means that banker will be able to win $1023 from 35% players, lose $1 to 65% players, hopefully. Although this example can be an extreme situation, which means for game having higher chance to win Martingale can work a little better, this high-risk low-payment strategy has squeezed tons of money from the gamblers.

Anti-Martingale

Those players who lost their money by using the Martingale strategy were finally starting to question this “convenient” strategy as well as thinking one thing – Since the Martingale strategy didn’t give us any benefit, what if we do the opposite way? That’s how Anti-Martingale, some call it Reverse Martingale, bornwas born. The basic idea of “doing the opposite way” is: When you win a game, instead of running away, you double the bet; but when you lose a game, halving the last bet. Now the question is if this opposite strategy is going to work, at least better than the origin one. The main concept is –

Suppose the first bet to be X. Martingale promise a X profit with X*2^(N-1) (Nth round), increasing risk for every round; Anti-Martingale has no promised profit, but only X/2^(N-1), decreasing risk for every round if you keep losing. What’s more, if you win a round N, you will have X/2^(N-1) income, together with X/2^(N-1) principal, while the bet for round N+1 will be X/2^N, equal to your income + principal, which makes the risk to be X/2^(N-1). In the same way, if you win round N+1, the risk for round N+2 will still be your round N bet, which is X/2^(N-1).

In brief, Anti-Martingale can keep making the risk half when player is in a losing trend; keep the risk unchanged, doubling possible income when player is in a winning trend. This provides much reasonable risk than the origin one. However, since there won’t be any promised profit, the key ofto using this system will be whether you will be able to choose the right time to stop. But even if you are noy a person who’s good at seizing the opportunity, mathematically speaking, you won’t be able to lose more than twice the first bet if using Anti-MartingaleMartingale, which can be a huge advantage.

The Labouchere System

Some called cancellation system or split martingale. According to The Labouchere Betting System – how to use it and when website, “this system was developed by keen roulette player Henry Labouchere (1821-1912). Although it was designed to be used at the roulette table, it can actually be used with any even money proposition though, including other casino games such as blackjack and baccarat. Sports betting is an option as well.” The general idea of this strategy is like Martingale, it can guarantee a winningwin if you are able to finish the whole process. Not like the Anti-Martingale, there will be no need to think the “when to stop” question, unless you decide to interrupt it. And it won’t be a simple hit and run strategy as well, due to its slight complexity.

FirstThe first step of the system is to prepare a sequence of numbers of your choice. The bet will be the sum of firstthe first number and last number of this sequence. Then if you win this round, those used first and last numbernumbers will be removed; If you lose, you will add the number of bets to the end of the sequence. When all numbers from this sequence are deleted, the whole process is finished. For example, let’s use 1 2 3 as the initial sequence. Bet 1 will be 1+3 = $4, suppose I lose, that will make sequence become 1 2 3 4. Bet 2 is now 1+4 = $5, suppose I win, sequence becomes 2 3. Bet 3 is 2+3 = $5, suppose I win again, sequence now contains nothing, process finishes. Calculating the result, I lose $4 while winning $10, the gain is $6, which equals tothe sum of the origin sequence.

As you can see, Labouchere System tries to take a balance between risk and gain. While guaranteeguaranteeing a modifiable profit, it decreases the risk to a pretty low level. There is no complex math calculation in it, which makes it easier to be understood and operated than Anti-Martingale. However, it still shares samethe same problem of Martingale – since this guaranteed profit is based on walking through the whole process, luck is eventually required. As the sequence gets longer and numbers get larger, risk will grow fast.

Kelly Criterion

Entering the 20th century, with the development of math system, gamblers started to realize that it would be irrational to try to pursue a “100% winning strategy”, since gambling is always related to individuals’ luck. So, they were starting to pay attention onto mathematical concepts based on probability and data analysis instead of simple guarantee of winning. Kelly criterion could be the most famous and popular one among this type of strategies.

Kelly Criterion, some called it Kelly Strategy or Kelly bet, “was developed in 1956 by Bell Labs scientist John Kelly, the formula applied the newly created field of Information Theory to gambling and investment.” (Yoder, N, 2021) According to the author, this formula calculates the proportion of one’s net worth to wager in order to maximize the expected logarithm of wealth increase. Also, a huge advantage of this formula is that it can theoretically avoid the possibility of bankruptcy. The formula is: f* = (bp – q)/b, where f* is the percentage of our total wealth that we should put at risk, p is the probability of a win, q is the probability of a loss, and b is the odds. (For example, if bet is $1, profit is $2, then b = 2/1 = 2.)

Due to the complexity of this system, it will be difficult to show the result by a simple example. A detailed one is provided by Yoder, N, showing relationship between bet size and profit in certain condition: Let’s say “there’s a 50/50 chance of gain or loss, with a gain of 6% and a loss of only 5%.” After one win and one loss, we can then calculate the profit due to different bet sizesizes from 0.2 to 5.0 and see how the bet size can influence our profit. Detail of this example is presented in Figure 1.

Figure 1

Table of relationship between Bet Size and Profit

Although the percentage of win and loss grows at the same rate as we expected, we can observe a high peak of profit at 0.83% profit with 1.50x bet size. Before this peak, profit generally grows in a one-to-one relationship. Between 1.5 and 3.0 bet size, the profit is still positive, but it’sit keepkeeps decreasing with the increasing bet size. When betthe bet size is larger or equal to 3.5, profit becomes negative which means we are starting to lose money.

What if we use Kelly Criterion to try to calculate the proper bet size? In this case, p = q = 0.5, while b = 6% / 5% = 1.2. Then f* = (1.2 x 0.5 – 0.5)/1.2 ≈ 0.0833 = 8.33%. Since the base loss is 5%, bet size will be 8.33% / 5% = 1.66. This result is very close to the peak we find in the table which gives us the best bet size.

The last thing to remember is, what the formula gives us is only estimate, we should only use it as a reference. “9 times out of 10, reality will be less profitable than our estimates.” (Yoder, N, 2021) So please don’t totally rely on this strategy. Although it seems to be so logical and reliable, nobody can tell what will happen in reality. There’s a famous story of gambling, happenedwhich happened in game of roulette at the Monte Carlo Casino on August 18, 1913. The ball of the roulette fell in black 26 times in a row which didn’t make sense at all. “The probability of a sequence of either red or black occurring 26 times in a row is (18/37) ^ 25 or around 1 in 66.6 million, assuming the mechanism is unbiased.” (Gambler’s fallacy Wiki) No strategy or system can save you from this situation for sure.

Card Counting

The last strategy I want to introduce is card counting. It’s another strategy born in the 20th century, maybe the most famous card games strategy ever. Many people may know this specific technique from the famous movie Rain Man, used by Dustin Hoffman’s character Raymond Rabbit. People with such special talent isare quite persuasive to be able to use such special technique in real-time card games, however, card counting isn’t just super power, can be used by certain genius, it’s a fancy real gambling technique.

American mathematician Edward O. Thorp, some called the father of card counting. “He used early computers to examine whether keeping track of the cards could be lend itself to gaining an advantage over the game.” (Ofton, L, 2020) According to Ofton, he was able to gain a clear advantage over the game, which leads to the theory of card counting. “His conclusions, published in the book Beat the Dealer in 1962, marked the birth of card counting.

According to Card Counting Wiki, basicthe basic idea of card counting is to set up a system, assigning a positive, negative, and zero value to each card. Lower cards increase the count, increasing the percentage of high cards in the deck while high cards do the opposite thing. Due to this feature, different systems can be created for different levellevels of users. The system created by Edward O. Thorp, published in Beat the Dealer, the 10-count, is harder to use and less profitable than later systems.” (Card Counting Wiki) It cost him huge energy and calculation to use this “prototype” of card counting system

The later systems were mainly classified by the level of count. Briefly, levelthe level of count is decided by how many different types of numbers (counted by the absolute number, ignore 0) are using in this system. With highera higher level of count, the system supposesis supposed to be more efficient, but also harder to use.

For example, the level-1 count system, Hi-Lo system, only uses -1, 1 and 0 to do the counting, which means Hi-Lo system is easy for card counting beginners to use, but less precise. While level-2 count system, Hi-Opt II system, uses 2, 1, 0, -2 to do the counting, more difficult to use, but more accurate in calculating the situation than Hi-Lo system. A detailed table of some main counting systems, provided by card counting wiki, is presented in Figure 2.

Figure 2

Table of Card Counting systems

Card Strategy

2

3

4

5

6

7

8

9

10, J, Q, K

A

Level of count

Hi-Lo

+1

+1

+1

+1

+1

0

0

0

−1

−1

1

Hi-Opt I

0

+1

+1

+1

+1

0

0

0

−1

0

1

Hi-Opt II

+1

+1

+2

+2

+1

+1

0

0

−2

0

2

KO

+1

+1

+1

+1

+1

+1

0

0

−1

−1

1

Omega II

+1

+1

+2

+2

+2

+1

0

−1

−2

0

2

Red 7

+1

+1

+1

+1

+1

0 or +1

0

0

−1

−1

1

Halves

+0.5

+1

+1

+1.5

+1

+0.5

0

-0.5

−1

−1

3

Zen Count

+1

+1

+2

+2

+2

+1

0

0

−2

−1

2

As long as understandingunderstand the system and concept of card counting, everyone can use it, of course after plenty of practice. But before you startingstart to practice using them, users should always remember that result given by this system provides an incomplete analysis to situation only. It doesn’t exactly tell you how much you should bet or what card you should play next. Also, it doesn’t give you the exact analysis due to the limitation of mental arithmetic. What it gives is a balance between detailed situation analysis and controllable amount of calculation. So, playerplayers still need to judge their next action by analysinganalyzing the given data again.

Conclusion

Math concepts used in gambling strategies are mostly simple enough and straightforward, especially before 20th century. The nature of gambling and gamblers decides that strategies should be immediate, or at least easy enough for users to use without extra help. Although gambling itself usually gives people negative feelings about it, strategy itself can have multi-purpose, symbolizing humans’ mathematical development as well as mental growth. It’s interesting to do research on it using mathematicala mathematical perspective only.

References

The history of gambling. Gambling.net. (n.d.). Retrieved October 24th, 2021, from https://www.gambling.net/history/.

Colon, N. (2017, December 12). All you need to know about the martingale betting system. Maintenance. Retrieved October 27th, 2021, from https://www.888casino.com/blog/casino-tips/martingale-betting-system.

The Labouchere Betting System – how to use it and when. (n.d.). Retrieved November 5th, 2021, from https://www.gamblingsites.com/systems-strategies/labouchere/.

Yoder, N. (2021, April 30). The kelly criterion – quantitative trading. Nick Yoder. Retrieved November 10th, 2021, from https://nickyoder.com/kelly-criterion/.

Ofton, L. (2020, June 15). The history of Blackjack and card counting. Blackjack Apprenticeship. Retrieved November 20th, 2021, from https://www.blackjackapprenticeship.com/the-history-of-blackjack-and-card-counting/.

Wikimedia Foundation. (2021, September 29). Card counting. Wikipedia. Retrieved November 21, 2021, from https://en.wikipedia.org/wiki/Card_counting.

Wikimedia Foundation. (2021, October 28). Gambler’s fallacy. Wikipedia. Retrieved November 21, 2021, from https://en.wikipedia.org/wiki/Gambler%27s_fallacy.

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