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Searching for viable betting strategies in gambling, with consideration to Roulette Joseph Toninato UMD University Honors Capstone Project Introduction Gambling is one of the world’s oldest past times. Dice have been found in Egyptian tombs that are dated to be roughly 8000 years old. Since then, gambling has taken many different forms. Cards, dice, slots, wagers on sporting events, all of these fall under the umbrella of gambling. Whatever form it takes, all methods of gambling share the same structure. There is a game of chance, where people place wagers of money or assets or any multitude of things in the hopes of winning some amount of goods that are greater than their wager. Approximately 1.6 billion people worldwide reported gambling in some fashion in 2010. While many of these stated that they played the lottery, a large portion ventured into casinos. One of the most popular and iconic games in a casino is Roulette. This game is thought to originate from a perpetual motion machine invented in the 17th century, and played in its current form since before 1800. Roulette is a simple game in which a wheel is spun with numbered, colored, slots ranging from 0-36, and a double 0, or 00, in the American version. A ball is then allowed to spin around the outside of the wheel and fall into a random slot. This is the winning number. In addition, the numbers are colored either red or black, and the zero or zeros are colored green. A player may make one, or multiple, of many bets. These include betting on a color, a single number, a range of numbers, or odd or even numbers. Every type of bet has a payout equal to the probability the ball will land on the group of numbers that correspond to it. However, these probabilities are calculated out of 36 possible slots instead of 37 or 38. This allows the house to have a small advantage over the players. To many, this is simply a game of chance, as each roll is independent of all others. However, some believe there exist strategies one can implement on the way they bet in order to “beat” the system. One of the most common strategies implemented for Roulette is the Martingale system. This system is based off of odds of 1:2. Under this system, every time the gambler loses a round, he or she doubles his or her bet. The theory behind this is the fact that no matter how many times the gambler loses in a row, the first win will result in a positive profit of the original bet. Mathematically, we can say that the amount of money won from a win on round n is , where () = 2 0 with being the initial bet. In addition to= this, 2 we, can conceptualize the total amount of 0 money loss from a losing streak of length n as −1 () = � 02 = 02 − 0 = − 0. =1 Therefore, the overall profit of this strategy will be . 0 0 There have been( )numerous= () studies− ( )on= this −strategy,( − mostly) = testing its effectiveness in real world settings. the effectiveness is called into question because the loss factor increases exponentially as the game progresses. According to a study by Nigel Turner, doubling a bet after every loss, resulted in an overall greater loss of money than if the player had used a constant bet. This was based off of mean and median profits. While Turner states that using the Martingale system could be beneficial when playing for only a few wins, the longer one plays, the more one will typically lose using this method. Another paper, written by J. Laurie Snell, talked much about gambling, probability and if it is possible to find a system that always wins using mathematics. His paper ended with a few comments on the Martingale system, including a brief history on the term. Like Turner, his calculations showed that, while the Martingale system is often used and can produce some profit n the short term, it is overall not a system that can be counted on by gamblers to make sure they beat the casino. In fact, Snell calculated that if someone sat down at a Roulette table with $10 and wanted to walk away with $100, that gambler had a 99.995% of going broke before they reached their goal. As an endnote, Snell pointed out that Blackjack, with the proper strategy, can be a “favorable” game to the gambler. The last paper I would like to acknowledge is one by David K. Neal and Michael D. Russell. This paper went into detail about a Martingale system where the player does not double his/her bet, but multiplies it by a factor of . The authors went on to create functions that calculated the expected number of bets ≥ 1 needed to win once, the expected bet amount on a given turn, and the expected fortune of the player after k turns and after a single win. In this way, they found that, like the two previous papers, the Martingale system does not provide a high probability of winning, whether or not However, the probability can be raised by finding the optimal starting fortune, optimal = m 2., and optimal starting bet. The results from these paper begs the question, “Are there strategies that can be used to produce a positive overall profit in Roulette and other games of chance?” That is what this project set out to find out. The methodology was similar to Turner’s but added in more strategies that seemed, at least theoretically, plausible. Strategies The following strategies compromised the majority of concentration during this project. We start by noting a few things that were used for every strategy including the constant k, called the betting factor, defined as: , 1 where p is the probability of winning. For each= strategy, a win function W(n), a loss function L(n), and a profit function P(n) = W(n) – L(n), was made, where n is the round number. These functions define exactly what they are called. The win function is used to calculate the amount of money won from any bet at turn n, while the loss function calculates the total amount of money lost at turn n. The expected value of the profit function was also found using the formula , −1 or [] = ∑=1 ()(1 − ) = , 1 −1 1 1 −1 −1 [] = ∑=1 ()(1 − ) () ∑=1 () ( ) The first of the strategies used in this project was a constant bet. This means that throughout the simulation, the bet remained at $1. This was used as a control strategy, as it is often the most common bet for gamblers at a roulette table. Under this system, , , . Clearly, the profit( will) = only be positive (when) = . The expected() = value of − this profit was found to be > . −1 1 −1 −1 Taking the limit as n approaches[] = ∑infinity=1( it becomes −) � � obvious= �that � the expected profit of this betting strategy is zero. The second strategy tested was a modified Martingale system where, instead of maintaining odds of 0.5, the odds were any . Thus, instead of doubling the bet 1 every time a lose occurred, the bet would be multiplied= by k. In this way, we could analyze the behavior of the system for all bets, rather than sticking to just one betting factor, where much research has been done. For this reason, , . −1 −1 From this, we see( )that= the, expected value() is= −1 () = − −1 = . −1 1 1 −1 1 This result is highly[ surprising.] = ∑=1 This (means) �1 − that� under (the modified − 1) − Martingale betting system is expected to produce an exponentially increasing profit. The third strategy explored was one based off of the modified Martingale system. This experimental method increased based off of the betting factor, but instead of increasing after every loss, the bet would only increase after k consecutive losses. In this way, the amount of money one is losing is not drastically increasing every turn if the bettor is on a losing streak. The win, loss, and profit functions were found to be , , �� �� �� () = () = −1 � − 1� − � �� − � � � � � � � The expected value (of) which= is − � − 1� + � � � − �, − 1 � � � �+1 2 − 1 2 − 1 2 − 1 2 (( ) ) − + (( ) ) + ( ) − 1 − [] = � � � 2 � − 1 ( ) − 1 − � � � � 2 2 − 1 − 1 − 1 − ( ) (( ) ) − � � (( ) ) + + − − 1 2 − 1 − 1 (( ) − 1) ( ) − 1 3 − 1 −1 2 2 − �( ) (2 − ) − − − 1 � � − 1 2 − 1 −2 (( ) ) − 1 + ( ) � � � − 1 ( ) − 1 � � � � 3 −1 − 1 − 1 − 1 (( ) ) − 1 � � (( ) ) 2 − � �1 − � � �� � − � − 1 − 1 − 1 �( ) − 1� ( ) − 1 Methods The three strategies described above were used in two types of simulations, wherein every single simulation started with an initial bet, . The first group of trials, referred to as Win In One simulations, used each strategy0 = until 1 the “player” “won” once. After this event, the turn number and amount of money earned, or lost, was recorded. The other simulations, referred to as Turn Based, gave the “player” an initial amount of money, denoted S, and then allowed the systems to play out until 100 rounds had been played, or the “player” was unable to place the prescribed bet. Three S-values were used: $100, $1000, and infinite money. The first two were done in an attempt to simulate an average player, an excessive player. The third S-value was used to see what the potential profit could be if every simulation was allowed to run for 100 turns. After each simulation, the round number, amount of money gained or lost, not the amount of money the “player” had left, and the number of “wins” was recorded. In addition to this, three more strategies were tested using the second type of simulation.

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