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

,

𝑛𝑛 where 𝑊𝑊(𝑛𝑛) = 2𝐵𝐵

𝑛𝑛 𝑛𝑛 0 with being the initial bet. In addition𝐵𝐵 to= this, 2 𝐵𝐵 we, can conceptualize the total amount of

0 money𝐵𝐵 loss from a losing streak of length n as

𝑛𝑛 𝑖𝑖−1 𝑛𝑛 𝐿𝐿(𝑛𝑛) = � 𝐵𝐵02 = 𝐵𝐵02 − 𝐵𝐵0 = 𝐵𝐵𝑛𝑛 − 𝐵𝐵0. 𝑖𝑖=1 Therefore, the overall profit of this strategy will be

.

𝑛𝑛 𝑛𝑛 0 0 There have been𝑃𝑃(𝑛𝑛 )numerous= 𝑊𝑊(𝑛𝑛) studies− 𝐿𝐿(𝑛𝑛 )on= this𝐵𝐵 −strategy,(𝐵𝐵 − 𝐵𝐵mostly) = testing𝐵𝐵 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 of𝑚𝑚 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, optimal𝑚𝑚 = 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. For𝑘𝑘 each= 𝑝𝑝 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 , 𝑛𝑛 𝑖𝑖−1 or 𝐸𝐸[𝑃𝑃] = ∑𝑖𝑖=1 𝑃𝑃(𝑖𝑖)(1 − 𝑝𝑝) 𝑝𝑝 = , 𝑛𝑛 1 𝑖𝑖−1 1 1 𝑛𝑛 𝑘𝑘−1 𝑖𝑖−1 𝐸𝐸[𝑃𝑃] = ∑𝑖𝑖=1 𝑃𝑃(𝑖𝑖)(1 − 𝑘𝑘) (𝑘𝑘) 𝑘𝑘 ∑𝑖𝑖=1 𝑃𝑃(𝑖𝑖) ( 𝑘𝑘 ) 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 profit𝑊𝑊𝑐𝑐( 𝑛𝑛will) = only 𝑘𝑘 be positive𝐿𝐿𝑐𝑐 (when𝑛𝑛) = 𝑛𝑛 . The expected𝑃𝑃𝑐𝑐(𝑛𝑛) = value 𝑘𝑘 of −𝑛𝑛 this profit was found to be 𝑘𝑘 > 𝑛𝑛 . 𝑖𝑖−1 𝑛𝑛 1 𝑛𝑛 𝑘𝑘−1 𝑘𝑘−1 Taking the limit as n approaches𝐸𝐸[𝑃𝑃𝑐𝑐] = 𝑘𝑘 ∑infinity𝑖𝑖=1(𝑘𝑘 it becomes −𝑖𝑖) � 𝑘𝑘 � obvious= 𝑛𝑛 �that𝑘𝑘 � 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 be𝑝𝑝 multiplied= 𝑘𝑘 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,

𝑛𝑛 , 𝑛𝑛 . 𝑛𝑛 𝑘𝑘 −1 𝑛𝑛 𝑘𝑘 −1 From this, we𝑊𝑊𝑚𝑚 see(𝑛𝑛 )that= 𝑘𝑘the, expected𝑊𝑊 value𝑚𝑚(𝑛𝑛) is= 𝑘𝑘−1 𝑃𝑃𝑚𝑚(𝑛𝑛) = 𝑘𝑘 − 𝑘𝑘−1

= . 𝑖𝑖−1 1 𝑛𝑛 1 𝑛𝑛−1 1 𝑛𝑛 This result is highly𝐸𝐸[ 𝑃𝑃surprising.𝑚𝑚] = 𝑘𝑘 ∑𝑖𝑖=1 This𝑃𝑃𝑚𝑚 (means𝑖𝑖) �1 − that𝑘𝑘� under (the𝑘𝑘 modified − 1) − Martingale𝑘𝑘 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

𝑛𝑛 , 𝑛𝑛 𝑛𝑛 , �𝑘𝑘� 𝑘𝑘 �𝑘𝑘� �𝑘𝑘� 𝑛𝑛 𝑊𝑊𝑒𝑒(𝑛𝑛) = 𝑘𝑘 𝐿𝐿𝑒𝑒(𝑛𝑛) = 𝑘𝑘−1 �𝑘𝑘 − 1� − 𝑘𝑘 �𝑘𝑘 �𝑘𝑘� − 𝑛𝑛�

𝑛𝑛 𝑛𝑛 𝑛𝑛 � � � � � � 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑛𝑛 The expected value𝑃𝑃𝑒𝑒 (of𝑛𝑛) which= 𝑘𝑘 is − �𝑘𝑘 − 1� + 𝑘𝑘 �𝑘𝑘 � � − 𝑛𝑛�, 𝑘𝑘 − 1 𝑘𝑘

𝑛𝑛 𝑛𝑛 � � � �+1 2 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 2 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 2 𝑘𝑘 − 1 𝑘𝑘 2 𝑘𝑘 (𝑘𝑘( ) ) − 𝑘𝑘 + 𝑘𝑘(𝑘𝑘( ) ) + 𝑘𝑘 ( ) 𝑘𝑘 − 1 − 𝑘𝑘 𝐸𝐸[𝑃𝑃𝑒𝑒] = � 𝑘𝑘 𝑘𝑘 𝑘𝑘 � � 2 � 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘( ) − 1 𝑘𝑘 − 𝑘𝑘 𝑘𝑘 𝑛𝑛 𝑛𝑛 � � � � 2 2 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 𝑛𝑛 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 𝑘𝑘 − 𝑘𝑘 ( ) 𝑘𝑘(𝑘𝑘( ) ) − 𝑘𝑘 𝑘𝑘 � � (𝑘𝑘( ) ) + 𝑘𝑘 + 𝑘𝑘 − 𝑘𝑘 𝑘𝑘 𝑘𝑘 − 1 𝑘𝑘 2 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 − 1 (𝑘𝑘( ) − 1) 𝑘𝑘( ) − 1 𝑘𝑘 𝑘𝑘 3 𝑘𝑘 − 1 𝑘𝑘−1 2 2 𝑘𝑘 − �( ) (2𝑘𝑘 − 𝑘𝑘) − 𝑘𝑘 − 𝑘𝑘 𝑘𝑘 − 1 𝑛𝑛 � � 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 2 𝑘𝑘 − 1 𝑘𝑘−2 (𝑘𝑘( ) ) − 1 + 𝑘𝑘 ( ) � � 𝑘𝑘 � 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 𝑘𝑘( ) − 1 𝑘𝑘 𝑛𝑛 𝑛𝑛 � � � � 3 𝑘𝑘−1 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 𝑛𝑛 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 − 1 (𝑘𝑘( ) ) − 1 � � (𝑘𝑘( ) ) 𝑘𝑘 2 𝑘𝑘 𝑘𝑘 − � �1 − � � �� � − 𝑘𝑘 � 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 − 1 𝑘𝑘 𝑘𝑘 − 1 �𝑘𝑘( ) − 1� 𝑘𝑘( ) − 1 𝑘𝑘 𝑘𝑘 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 strategy𝐵𝐵0 = 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. The first additional strategy used the modified Martingale system until such a time that the “player” was unable to make the next bet, then switched to a constant bet of $1. The next strategy was similar, but started with the experimental strategy instead of the modified Martingale before switching to a constant bet. The last strategy again started with the experimental method. However, when the “player” could not place the next necessary bet according to the system, it switched to a bet of $1 and continued playing using the modified Martingale system. For these three methods, simulations were run only with S = $100 and S = $1000, as the “player” would never swap strategies if it started with an unlimited amount of money due to always being able to make the next necessary bet. For each combination of simulation type, strategy, and S-value, simulations were run with k = 2, 6, 18, and 38 to determine if a higher or lower betting factor is more favorable. Every simulation was run 1000 times, and an average, standard deviation, and a median was calculated for all outputted information. In addition to this, the proportion of trials within the 1000 that produced a positive profit was also calculated for each simulation type. It should be noted that neither a house limit to how high a bet can be, nor a house advantage was taken into account. This was done in an attempt to generalize the results to fit more games other than just Roulette. Results

Table 1 summarizes the results of the Turn Based simulations. As you can see, in general none of the strategies were able to produce a positive median profit consistently over varying k-values. The most prominent trend seen in the data is that of change in profits between betting factors. It appears that as the betting factor increases from 2 to

38, the median profit decreases dramatically when a limit is placed on how much money the “player” starts with. When k = 2, we see that the “player” ends with a positive or zero profit for each strategy. However, even a jump from k = 2 to k =6 produces mostly negative profits.

Another trend that is apparent exists when there is no limit on the starting money.

In these unlimited simulations, both the modified Martingale system and the experimental one produced positive profits for every k-value. In addition, the constant bet strategy only ended with the “player” losing money when k =18 and 38. Again, we make a note that no systems that switched between two strategies were tested under this unlimited scenario. This is due to the fact that the switch was implemented only when the “player” was unable to make the next necessary bet. With unlimited funds, the next bet could always be made, and thus, no switch would occur.

When looking at the Win In One simulations, summarized by Table 2, we see that there is, surprisingly, a general increase in median profit and positive profit proportion as k increases. As a reminder, these simulations ran until the “player” had won a single round and then stopped and reported the turn on which the win occurred and the amount of money lost or won over the course of the trial. The fact that we see mostly positive profits amongst all the trials for all the strategies is not surprising, as many times the

“player” wins before the round number reaches k, allowing the “player” to still be within

the realm where the profit from one win exceeds the total losses. The fact that we see

increasing proportions of positive profits corresponding to higher k-values is, however,

very surprising, as the chance of winning any particular round decreases. This being the

case, it is expected that the total money lost would increase more rapidly, leading to a

higher chance of negative profits.

In the Turn Based simulations, the constant betting system generally produced

profits very close to zero. In almost every situation, the median profit was either 0 or

under 10. When the betting factor was 38, in every situation, this system produced an

overall median loss of money. In addition to this, a loss was seen when S was infinite

and k = 18. In every situation, a positive profit was seen in roughly 50% of trials. When

looking at the Win In One trials, this trend of near zero profits held. The proportion of

trials with positive profits was slightly higher

The modified Martingale system had very different results. In the TB trials,

median non-negative profits were only seen when k = 2, or S was infinite. In all

situations where S was finite, negative median profits that decreased as k increased were seen. This trend was also present in the percentage of positive profits within the trials.

With increasing betting factors, the proportion of positive profits decreased, quite rapidly.

An interesting result was that when S was infinite, not only was the profit positive for each k-value, but when k was greater than 2, we saw exponential high profits.

However, when we look at the percent of positive profits in these simulations, we see a marked decrease as k increases. This means that even though the amount of money won

increases with k, the chance of winning money does the opposite.

The WIO simulations showed another interesting, although expected, result. In

the introduction to the modified Martingale system, we saw that the expected value of the

profit function was always positive. When this system was tested, we see almost all

positive profits. The lowest proportion of positive profits is 92.9%, for . This

percent decreases as k increase, but it starts at 100% for , and only𝑘𝑘 drops = 38 to 92.9%.

The median profit for this strategy also increases as k does.𝑘𝑘 = 2

A similar trend in TB simulations was seen in the experimental system. Here we

also saw mostly positive profits. The only negative median profits occurred when S was

finite was in place, and k was greater than 2. The proportion of positive profits followed

the same trend when the money limit was $100. However, when , fluctuating proportions of positive profits were seen in addition to fluctuating𝑆𝑆 median= 1000 profits as k varied. When S was infinite, a general increase in median profit and percent of positive profits was seen as k increased. Only when did this not hold.

If one was to leave after a single win 𝑘𝑘using= 38 this strategy, according to the WIO data, a higher k-value would prove to produce a higher median profit and a higher proportion of positive profit. Referring back to Table 2, we see that when , the

“player” was able to make money about half of the time. When k was increased,𝑘𝑘 = 2 not only did the median profit increased from 1 to 14 for and 30 for , but the percent of trials that produced a positive profit increased𝑘𝑘 = 18 as well to 𝑘𝑘96%= 38and 91.1% respectively. For each of the strategies that switched between betting systems, the general

pattern was a decrease in profit and proportion of positive profits as k increased until k

was high enough to allow for a quick switch between the two betting systems used. In all

three strategies, a general decrease was seen, followed by a quick increase, then another

decrease. For example, using the Martingale to Constant strategy, when , the

median profit decreased from 49 to -348 as k changed from 2 to 18, 92.9%𝑆𝑆 =and1000 1.80% positive profits respectively. However, when , the median profit then increased to

-61 with a positive profit occurring in 25.5% of𝑘𝑘 trials.= 38

Discussion

The results section, and the tables showing the data may make people believe that

using a particular strategy is guaranteed to make them rich, as it appears with the

modified Martingale system with . However, before everyone runs to the nearest

roulette table, let us analyze what 𝑘𝑘these= 18 numbers mean in more depth.

We start with the strategy that seemed to produce the highest median profit for the

TB simulations, the modified Martingale system. The greatest median profits for these

were 1.58 x 1013, 4.5 x 1042, and 2.18 x 1066, for respectively.

However, these results were found when S was unlimited.𝑘𝑘 = 6, 18 ,Thisand 38means that for a real person to replicate these results, they would also have to have unlimited funds with which to gamble. This is obviously not possible for anyone. In addition to this, the average, and median, number of wins for these strategies was, 17, 5, and 2 respectively. Since this system forces the player to multiply his or her bet by the betting factor after every loss, someone using this strategy would most likely have to be able to consistently bet multiple millions or even billions. In addition to this, positive profits were only seen in 95%, 84.1% and 65.8% of trials respectively. Thus, this strategy, while appearing to work,

would not be plausible for any person.

We might think that using this system and leaving after a single win would be better, since just under 100% those trials came out with positive profits, when k was less than 38. However, these results also state that the average win came at about turn k, with the median at slightly less. This means that the bettor would have to typically front

21 $( 𝑘𝑘 before being able to win. This comes out to be $9,331, or $2.31 x10 for 𝑘𝑘 −1 𝑘𝑘−1 and 18) respectively. When , the strategy is just the normal Martingale system,𝑘𝑘 = 6

which produces a profit of $1𝑘𝑘 no= matter 2 when the player wins. Therefore, this system is

also not very plausible, or exciting, for the average person. While some may have the

thousands of dollars necessary for this system when , it should be reiterated that

these results are the averages and medians of 1000 trials.𝑘𝑘 = 6 Any individual could have a very different experience where they lose much later, losing far more money, simply based off of probability.

Another strategy that produced a high percent of positive profits was the modified

Martingale to Constant strategy from the TB simulations where S = 1000. This strategy

allowed for 92.9% of the trials to have a positive profit when . While, this may

seem to be the strategy to use at the casino, we must remember𝑘𝑘 tha= 2t this was based off of

a probability of 1 in 2. By this, it is meant that a player will typically win half of the

time, giving them a very likely scenario to win only small amounts of money. This is

fine, however, it must be remembered that these results are based off of averages, which

can very drastically from individual experiences. Overall, this strategy may be plausible to what casinos refer to as “high-rollers”, or those people who spend thousands of dollars.

It is not very plausible for the average person, though.

In general, it can be stated that the strategies that produced low to moderate proportions of positive profits are the most plausible for the average gambler. These strategies, a constant bet with for example with , are quite typical actually for those who frequent casinos.𝑘𝑘 =They 6 usually represent𝑆𝑆 the= 100“low-risk, low reward” mentality. By making sure not to lose money too quickly, gamblers are able to continue gambling for extended periods of time. This proves to lead to losing money more often than not. On the opposite end, the strategies tested that have high median profits and/ or proportions of positive profits, are almost exclusively the “high risk, high reward” type. these strategies, like the ones discussed above, are not plausible for the average person because they typically require an exorbitant amount of money in order to fund them.

This holds for both the TB trials and the WIO trials. In both situations we see that when a large amount of money is won, a large amount of money must first be bet, and most likely lost. The higher the end profit, the higher the total loss gets to be before that successful round.

These tests, while significant in their own right, are not to be confused for conclusive evidence a certain strategy should or should not be implemented. More study is needed on the statistical error of the trials, in addition to real world application. As stated in the introduction to the strategies, no house advantage was taken into consideration. This means that these results would be very skewed if used on an actual roulette table because the odds are not as succinct as in they were here. Future tests would account for this, as well as test other strategies. We saw that the strategies that switched between betting systems seemed to allow for a higher chance, although still

low, of producing a positive profit. This could be expanded upon to where the player

switches back to the original system if he or she accumulates a certain amount of money.

Perhaps the player switches after losing a certain amount, but before they cannot place

the next bet. These situation change how the money would flow, and maybe how much

money the player could win.

Conclusion

We saw that, while some strategies did produce a very high profit at a high rate,

they would not be very popular amongst the average gambler because of the excessively

high amount of funds one would need to finance them. The strategies that could be used

by normal people, were shown to barely turn a profit, and have a low success rate. We

can conclude that, while some strategies and implementations look promising, the ones

tested here will not guarantee fortune. On an ending note, while gambling in general is

not encouraged, if anyone is looking to try a system used here, leaving the table after one

win and using the experimental system with a moderate k-value is recommended.

Acknowledgements

I would like to my advisor, thank Marshall Hampton, for all the help he has given me throughout this project. He was always willing to meet and give suggestions as to

how to better my project, explain how to write a bit of code, and teach me how to more

easily find my functions. Thank you very much!!

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Tables

Strategy S k Wins Profit Turns Proportion 50.027 0.054 100 2 4.88632246 9.77264491 0 46.30% 50 0 100 16.807 0.842 100 6 3.74001855 22.4401113 0 52% 17 2 100 100 5.612 1.016 100 18 2.33857327 42.0943188 0 51.20% 6 8 100 2.663 1.194 100 38 1.6291363 61.9071794 0 50% 2.5 -5 100 49.944 -0.112 100 2 4.9312454 9.8624908 0 44.40% 50 0 100 16.822 0.932 100 6 3.64796157 21.8877694 0 51.10% 17 2 100 Constant 1000 5.659 1.862 100 18 2.25827784 40.6490012 0 51% 6 8 100 2.613 -0.706 100 38 1.63214426 62.021482 0 48% 2 -24 100 50.017 0.034 100 2 5.08532905 10.1706581 0 47% 50 0 100 16.83 0.98 100 6 3.73457455 22.4074473 0 53.40% 17 2 100 Unlimited 5.43 -2.26 100 18 2.31440197 41.6592354 0 44.10% 5 -10 100 2.602 -1.124 100 38 1.54595795 58.7464021 0 49.50% 2 -24 100 Table 1 Results of Turn Based simulations. Columns feature strategy used, S-value, k-value, number of wins, profit, number of turns played and proportion of trials with positive profits within subset. Under Wins, Profit, and Turns, numbers within subset are mean, standard deviation, and median (top to bottom). Strategy S k Wins Profit Turns Proportion 36.75 0.426 74.095 2 18.9407564 56.0843648 33.3560279 55% 47 42 100 0.818 -13.47 5.723 6 1.2823886 152.216862 3.00154631 7.80% 0 -43 4 100 0.116 98.316 3.236 18 0.39076425 3136.39932 0.91931994 1.30% 0 -19 3 0.054 -38.522 3.094 38 0.24319363 8.41952519 0.47475192 0.10% 0 -39 3 47.85 -9.27 95.913 2 10.1742329 202.404255 15.686475 90.60% 49.5 48 100 1.16 226.104 7.891 6 1.84765089 8523.08442 5.33193666 6.90% 0 -259 5 Mod Mart 1000 0.174 -27.994 4.382 18 0.49997397 5400.86896 1.22702143 0.90% 0 -343 4 0.043 -39.005 3.085 38 0.2029586 7.88169373 0.45164032 0% 0 -39 3 49.907 38.168 100 2 4.77547444 260.105584 0 98.80% 50 49 100 16.765 6.47E+35 100 6 3.7796906 2.04E+37 0 95% 17 1.58E+13 100 Unlimited 5.553 -9.81E+121 100 18 2.31009254 1.40E+123 0 84.10% 5 4.50E+42 100 2.622 -6.46E+154 100 38 1.54515485 0 65.80% 2 2.18E+66 100

Strategy S k Wins Profit Turns Propotion 48.74 -0.544 98.056 2 7.94629118 26.0330133 10.4377109 57.50% 50 4 100 8.668 -0.852 53.13 6 7.56710229 143.231188 33.4947001 25.10% 6 -73 43 100 2.837 4.944 50.488 18 2.99690047 216.21748 29.1817311 18.90% 2 -90 39 1.59 12.546 63.144 38 1.7259429 327.418923 23.510551 17.50% 1 -76 57 49.851 -0.451 99.92 2 5.08218729 36.3442775 2.52982213 61% 50 5 100 14.76 11.112 87.384 6 6.35931135 644.466147 23.6757489 67.70% 16 171.5 100 Exp 1000 4.545 20.807 80.952 18 3.09670221 1514.44551 23.8458449 47% 5 -273 100 2.397 -41.233 90.678 38 1.78734037 889.552718 14.5507402 52% 2 14 100 50.39 1.656 100 2 5.00589242 27.4959941 0 61.40% 50 7 100 16.686 47297.536 100 6 3.73943677 1222504.84 0 88.70% 17 268 100 Unlimited 5.418 -94565.775 100 18 2.28834274 1921398.22 0 82.10% 5 449.5 100 2.642 -228.745 100 38 1.61134491 15854.0739 0 71.40% 2 365 100 Table 1(cont) Results of Turn Based simulations. Columns feature strategy used, S-value, k-value, number of wins, profit, number of turns played and proportion of trials with positive profits within subset. Under Wins, Profit, and Turns, numbers within subset are mean, standard deviation, and median (top to bottom).

Strategy S k Wins Profit Turns Proportion 49.177 -0.22 98.331 2 6.43701065 58.7525954 7.54588493 57.10% 50 43 100 16.453 3.945 98.143 6 4.50600656 316.011578 10.8886691 13.10% 17 -38 100 100 5.634 21.724 99.765 18 2.31547088 460.00312 2.10765681 32.80% 5 -27 100 2.432 -35.873 92.029 38 1.75797226 83.959896 16.619337 26.30% 2 -61 100 Mart-Const 49.882 -0.717 99.787 2 5.20222444 195.152688 2.80204969 92.90% 50 49 100 16.736 -72.667 99.833 6 3.78240804 1184.24668 3.54226745 5.60% 16 -253 100 1000 5.659 -113.823 100 18 2.22073201 4422.7969 0 1.80% 6 -348 100 2.656 15.802 100 38 1.63595243 1687.66492 0 25.50% 2 -61 100 49.607 -0.183 99.344 2 6.2345345 25.8763811 6.06881327 58% 50 5 100 13.868 -1.773 82.826 6 6.00888365 148.547332 24.854204 29.50% 15 -88.5 100 100 3.253 2.446 57.961 18 2.96779359 219.079426 27.9588121 19.80% 2 -100 51 Exp-Const 2.082 8.025 78.787 38 1.72750116 346.50044 17.7070938 19.90% 2 -100 65 49.834 -1.681 100 2 5.18201672 35.20353 0 55.60% 50 3 100 1000 16.655 -13.817 99.757 6 3.75420685 646.970332 3.00582534 65.80% 16.5 155.5 100 Strategy S k Wins Profit Turns Propotion 5.425 -12.834 94.674 18 2.76695338 1490.69428 14.4335738 46.80% 6 -144.5 100 Exp-Const 1000 2.449 -44.633 93.517 38 1.80406587 923.684937 10.8082621 52.90% 2 35 100 49.403 0.162 98.484 2 7.1907091 26.9571223 8.95119366 60.90% 50 5 100 8.897 2.564 54.6 6 7.43974064 159.359089 33.0440298 26.90% 6 -80 43 100 2.767 9.486 51.117 18 2.90846526 280.878099 28.9054811 18.50% 2 -91 39 1.665 -16.297 64.252 38 1.79443445 257.135819 23.5004844 0% 1 -77 58 Exp-Mart 50.115 2.399 100 2 4.99406805 21.9867965 0 60.50% 50 5 100 14.514 -8.26 87.505 6 6.1930734 658.957418 23.0074622 68.30% 16 173.5 100 1000 4.61 -33.338 82.236 18 3.01645471 1452.89852 23.201565 46.30% 5 -342 100 2.435 9.121 90.93 38 1.80583389 932.121559 14.0852936 55% 2 52 100 Table 2 Results of Turn Based simulations. Columns feature strategy used, S-value, k-value, number of wins, profit, number of turns played and proportion of trials with positive profits within subset. Under Wins, Profit, and Turns, numbers within subset are mean, standard deviation, and median (top to bottom).

Strategy K Profit Turns Proportion -0.012 2.012 2 1.37758984 1.37758984 0 2 49.20% 0.283 5.717 6 5.05949696 5.05949696 2 4 61.20% Constant -0.497 18.425 18 17.640555 17.266688 5 13 60.30% 1.025 34.353 38 35.6953293 28.9258551 12 16 63.70% -0.029 1.968 2 4.48355453 1.42232169 1 1 51.50% 110.643 5.817 6 1134.35336 5.30996297 4 4 86.90% Exp 73773.012 17.609 18 1562561.92 17.112889 14 12 96% 696.293 37.052 38 14591.3862 30.4827902 30 28 91.10% 1 2.2028 2 0 1.43850106 1 2 100% 5.94E+26 6.245 6 1.33E+28 5.64223447 1.04E+03 4 100% Mart Mod 2.18E+122 18.264 18 1.01E+124 17.6217715 1.09E+15 13 99.50% -1.78E+155 35.671 38 29.6463094 5.54E+34 27 92.90% Table 3 Results of Win In One simulations. Columns feature strategy used, k-value, profit, number of turns played and proportion of trials with positive profits within subset. Under Profit and Turns, numbers within subset are mean, standard deviation, and median (top to bottom).