Kevin Harrigan a, Vance MacLaren b, Dan Brown c, Mike J. Dixon d & Charles Livingstone e a Gambling Research Lab, University of Waterloo, Canada

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1 This article was downloaded by: [Monash University Library] On: 03 June 2014, At: 22:19 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Gambling Studies Publication details, including instructions for authors and subscription information: Games of chance or masters of illusion: multiline slots design may promote cognitive distortions Kevin Harrigan a, Vance MacLaren b, Dan Brown c, Mike J. Dixon d & Charles Livingstone e a Gambling Research Lab, University of Waterloo, Canada b Department of Psychology, Brandon University, Canada c School of Computer Science, University of Waterloo, Canada d Department of Psychology, University of Waterloo, Canada e School of Public Health and Preventive Medicine, Monash University, Melbourne, Australia Published online: 21 May To cite this article: Kevin Harrigan, Vance MacLaren, Dan Brown, Mike J. Dixon & Charles Livingstone (2014): Games of chance or masters of illusion: multiline slots design may promote cognitive distortions, International Gambling Studies, DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

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3 International Gambling Studies, Games of chance or masters of illusion: multiline slots design may promote cognitive distortions Kevin Harrigan a *, Vance MacLaren b, Dan Brown c, Mike J. Dixon d and Charles Livingstone e a Gambling Research Lab, University of Waterloo, Canada; b Department of Psychology, Brandon University, Canada; c School of Computer Science, University of Waterloo, Canada; d Department of Psychology, University of Waterloo, Canada; e School of Public Health and Preventive Medicine, Monash University, Melbourne, Australia (Received 13 December 2013; accepted 22 April 2014) Problem gamblers often have distorted beliefs about gambling, including illusion of control and gambler s fallacy. Most multiline slots games allow players to adjust the number of wagered paylines and the amount bet per line, and over time this control may support incorrect conclusions and promote distorted gambling beliefs. We created software to run simulations of a popular multiline slots game and examined the effects of betting on single versus multiple paylines. Simultaneous multiline betting tends to produce a less varied gambling experience because it increases the frequency of legitimate wins and losses disguised as wins, while decreasing the occurrence of big wins. It also shortens consecutive series of losing spins and it prolongs the time a typical player takes to exhaust funds. Indirect control over losing streaks may give some players the false impression that they can play skilfully and predict the occurrence of wins. However, applying five different wagering strategies in our simulations showed that none had any real effect on the average percentage of wagers that would be paid back to players as prizes. Player control over multiline slots games may lead frequent gamblers to incorrect conclusions that sustain excessive play despite recurring losses. Keywords: gambling; problem gambling; electronic gaming machines; losses; irrational beliefs Introduction Problem gamblers often hold beliefs about gambling that are incorrect or distorted, and it is widely believed that such erroneous cognitions may be critical in the development and maintenance of problem gambling (Blaszczynski & Nower, 2002; Ladouceur & Walker, 1996). The two cognitive distortions that have been most consistently shown to be higher in problem gamblers than non-problem gamblers are the illusion of control and the gambler s fallacy (Goodie & Fortune, 2013). Illusions of control may foster player behaviour that is intended to alter the odds of winning at the game. The gambler s fallacy is the belief that after a number of consecutive losses a win is due. The focus of the present article is the source of these two cognitive distortions in relation to modern multiline slot machine games. Although some authors and treatment providers attribute problem gamblers faulty thinking about control and their belief in the gambler s fallacy to their misunderstanding of probability and random events, problem gamblers who play slot machines do not have an inherent tendency toward faulty thinking about these *Corresponding author. kevinh@uwaterloo.ca q 2014 Taylor & Francis

4 2 K. Harrigan et al. mathematical subjects that is any different from non-problem players (MacLaren, Fugelsang, Harrigan, & Dixon, 2012). An alternative view is that modern multiline slot machine games contain structural characteristics that mislead frequent players to reach conclusions about the games that are incorrect and that may encourage continued play even as losses mount. Most modern multiline slots games contain at least two features that suggest to players that they can exert more control over game outcomes than is really the case. The two features that help foster this illusion of control (Langer, 1975) are playeradjustable reinforcement schedules and player-adjustable win sizes. Player control over losing streaks and volatility of outcomes In this article we use the term celebratory feedback rate to report what is typically reported in the gambling literature as hit frequency. Like hit frequency, celebratory feedback rate includes any spin that results in audiovisual feedback that indicates that a prize has been won. It is a more inclusive term because it includes winning outcomes where a prize is earned on the spin that is greater than the total amount wagered on all paylines for the spin, but also includes instances where the total prize is less than the total wager per spin. We report the legitimate win rate separately as the rate at which legitimate wins occur with total prizes exceeding total wagers. This legitimate win rate will always be less than or equal to the celebratory feedback rate. As has been shown previously (Harrigan, Dixon, MacLaren, Collins, & Fugelsang, 2011), the number of paylines that a player places bets upon determines the schedule of reinforcement (i.e. the average frequency with which winning, or seemingly winning, outcomes are delivered). Counterintuitively, the number of lines played has no effect whatsoever on the hold or house-edge (i.e. the average proportion of the wager that is kept by the casino and not returned as prizes). Although the outcome of each individual gamble is randomly determined and unpredictable, when players place bets on many paylines on a single spin, they are effectively betting on multiple events simultaneously (Harrigan & Dixon, 2009). The frequency of hits can increase dramatically because the chances of winning a prize on at least one of the simultaneous bets increases with the number of bets that are placed. Thus a player can experience more frequent hits simply by placing simultaneous bets on multiple paylines. When wagering on multiple paylines there is a tendency for many of the hits to be small credit gains that are obtained on one or a few paylines, but with the total number of credits gained being less than the total amount wagered on all of the paylines. These hits are still presented with audiovisual feedback to celebrate any prize that is won, even when the prize amounts are trivial. On such fake wins (Wilkes, Gonsalvez, & Blaszczynski, 2010), the sights and sounds that are presented are typically brief but have content similar to what is given following legitimate wins (i.e. when the amount won is larger than the total amount wagered, generating a net profit for the player). We consider this arrangement to be misleading to players and we prefer the term losses disguised as wins (LDW) to denote this game design element that masks net monetary losses by presenting winning sights and winning sounds in order to blur the distinction between legitimate wins and LDWs. In Pavlovian conditioning terms, the audiovisual feedback that accompanies emotionally evocative legitimate wins also occurs on LDWs so that the two become subjectively equivalent. Conflating these two types of outcomes in this way allows LDWs to serve as secondary operant reinforcers. The frequent signalling of hits in multiline gambling may promote a subjective impression of very frequent winning that is illusory because many of the hits are LDWs. This creates an anomalous situation where players

5 International Gambling Studies 3 feel as if they are winning consistently, yet end up with no money in their wallet. The emotional equivalence of LDWs and legitimate wins has been demonstrated experimentally both LDWs and legitimate wins trigger psychophysiological arousal responses that are significantly greater than full losses (Dixon, Harrigan, Sandhu, Collins, & Fugelsang, 2010). Indeed, much of the increased reinforcement that is experienced when players opt to place bets on multiple paylines is directly attributable to the high frequency of LDWs. The second way that players can exert some control over their gambling experience is to adjust the size of their wagers per line and, in so doing, adjust the size of the winnings that are realized. Slot machine games allow players to adjust the amount wagered per payline on each spin, so if they feel lucky on a particular spin they can bet more and potentially win more, or if they are feeling more cautious they can bet less and potentially minimize their losses. In multiline slot machines, players use one set of buttons to select the number of lines they wish to play, and a separate set of buttons to choose the number of credits they wish to bet on each line. Playing more paylines and wagering more credits per payline means that monetary outlay increases multiplicatively. Experienced slots players typically use a mini-max strategy (Livingstone, Woolley, Zazryn, Bakacs, & Shami, 2008), in which the amount wagered per payline is usually kept to a minimum but bets are placed on most or all available paylines which results in the maximum celebratory feedback rate for that game, and compared to max bet this wagering strategy delays the depletion of funds and prolongs their time on device. Players may also strive to win larger prizes by betting larger amounts (Templeton, Dixon, Harrigan, & Fugelsang, 2014), but this max bet strategy depletes funds quickly if the hoped-for large prizes do not materialize. Importantly, players always have the option to bet larger amounts whenever they feel that a win is imminent, and this interactivity may foster an illusory belief that slot machines can be played strategically so that with practice and knowledge one can develop into a skilled slots gambler (Myrseth, Brunborg, & Eidem, 2010). The problem with this faulty reasoning is that any effective strategy would have to capitalize on some predictable patterning of wins and losses. Such predictability is simply not possible when all spins occur at random and are independent of one another. Nevertheless, to discover an effective strategy of timing the size of one s bets to coincide with winning outcomes is the holy grail for problem gamblers. When wagering on a single line, players will either lose their spin wager, or get a legitimate win accompanied by celebratory feedback. There are no LDWs. During single line play, gamblers will tend to experience chains of full losses interspersed with occasional legitimate wins. Players subjectively refer to the chains of consecutive losses as losing streaks. When wagering on multiple paylines, the length of these losing streaks tends to be reduced as players encounter far more outcomes that have celebratory feedback. People have a natural tendency to try to understand their surroundings and to be able to predict future events on the basis of salient current and past events. In the gambler s fallacy, there is a belief that imminent game outcomes can be predicted on the basis of recent outcomes, though in fact all events are independent of one another (Tversky & Kahneman, 1971). Since the celebratory feedback rate is controllable as a function of the number of simultaneous wagers placed on multiple paylines, a player can reduce the average length of losing streak by betting on as many paylines as possible. Furthermore, a high celebratory feedback rate means lower volatility in the length of losing streaks. With single line play, the volatility of losing streak lengths can be high (occasionally one will encounter a very short losing streak, whereas most of the time streak length will be long). In multiline play very long losing streaks are quite rare a fact that leads to lower

6 4 K. Harrigan et al. volatility of losing streak length. This may support the subjective impression that outcomes can be predicted. After playing a multiline game extensively, an experienced gambler might acquire an informal heuristic that a chain of losses rarely exceeds a certain length, and when that length is reached a spin with celebratory feedback is likely to occur (i.e. the gambler s fallacy that a win is due ). Increasing the celebratory feedback rate does not negate the fact that all outcomes are independent. However, having extensive a posteriori experience and knowledge of the celebratory feedback rate of a particular game might give experienced players a false sense of having a priori knowledge of the game, and hence misguided confidence in their skill at predicting outcomes. The scenario that would lead players to fall into this trap involves reducing the apparent randomness of the game by consistently playing all paylines in order to get the highest celebratory reinforcement rate, the shortest average length of losing streaks, and the lowest losing streak volatility. In this situation, encountering a long losing streak might entice players to increase their bets since in their minds they are due for a win. Thus, a player could effectively exert a very powerful form of control over the game that seems to make the pattern of losses and spins with celebratory feedback more predictable. Although this would be yet another instance of illusory control that has no real effect on the hold or payback percentage, the subjective effects on players might lead them to misinterpret any long series of losses as support for their hope that a spin with celebratory reinforcement is imminent and stimulate their appetite for the anticipated reward, thereby reinforcing continued play despite recurring losses (often in the form of LDWs). Increasing the number of lines played not only reduces the volatility of the length of losing streaks, but also reduces the outcome-to-outcome variability. Consider a player who bets five credits on one line versus a player who distributes his or her wager across five lines (one credit per line). If each player encounters the same win of 10 times his or her wager on one of the paylines, the single-line player would win 50 credits, whereas the multiline player would win 10 credits. Thus the 50 credit win would markedly stand out from the chain of losses in single line play, leading to a perception of high volatility in terms of the size of the amounts won. By contrast, the small win of 10 credits would blend in amongst the other small wins and LDWs in multiline play. This smoothing of the experience in multiline play is therefore due to both a reduction in losing streak lengths and to a reduction in outcome-to-outcome variability. Player control of time on device The higher variability with one line wagered results in many players losing their bankroll quickly, while some players have a big win and can either play for a long period of time or leave with their big win. A consequence of the variability is that the median time on device with one line wagered is less than the median playing time with nine lines wagered (we consider the median to be the statistic most of interest here because the frequency distribution of wins is highly skewed, so occasional large wins have less effect on the median than the mean). Dow Schull provides insights into why players may choose to wager on multiple lines: Whether they actually give longer time-on-device or just the impression of it, the trick to multiline slots is this: while mechanical slots either pay nothing on a given spin or significantly more than the amount of the initial bet, multiliners pay something frequently but usually for less than the amount of the initial bet. By creating wins where players receive less than their wager, writes one game designer and consultant, we give them a sense of winning but also continue to accrue [their] credits. This sense of winning is communicated

7 International Gambling Studies 5 by presenting gamblers with the same audiovisual feedback colorful blinking lines, sounds, a musical score that occurs during actual winning. (Dow Schull, 2012, p. 121). Turner (2011) studied the prize structure of three realistic, but fictitious, single-line electronic gaming machine (EGM) games and various other gambling games to show the volatility and the percentage of winners after a certain number of spins. Two of the EGM games had mostly small and middle-sized prizes, but no jackpot. For these games there were 26% and 35% winners after 1 hour but only 2.3% and 5% winners after 50 hours. The other simulated game had small prizes and a jackpot. In that game, 15% were ahead after 1 hour and 17% were ahead after 50 hours. In this latter game with the jackpot, many of the 17% of simulated players who were ahead after 50 hours probably won at least 1 jackpot. Harrigan, Dixon, and Brown (2014) conducted a similar study in which they simulated a commercially available 20-line game called Money Storm. Their results were similar to Turner s results when the simulated players wagered on 1 line there were 24% winners after 1 hour and 13% winners after 50 hours. For the simulated gamblers who wagered on 20 lines there were 22% winners after 1 hour but 0.0% winners after 50 hours (only 3 of the 10,000 simulated players who wagered on 20 lines were ahead after 50 hours). In a separate simulation, Harrigan et al. (2014) simulated 10,000 players who arrived with $100 and made $1 wagers until broke. The percentage of players who played for more than 1 hour before going broke was 17.7% when wagering on 1 line and rose dramatically to 49.0% when wagering on 20 lines. Although most players who wagered on one-line lost their money faster, their maximum possible time on the machine is actually longer due to the long tail of the distribution. The slot machine game Dolphin Treasure To investigate the effects of wagering on one line versus multiple lines we wrote a Java computer program to simulate a popular multiline slot machine game called Dolphin Treasure, which was developed in Australia (Artistocrat Leisure Limited, North Ryde, NSW). It has been popular since the late 1990s in various versions. In the Office of the Liquor and Gambling Commissioner (OLGC, 2006) in South Australia released a list of the 250 top performing slot machine games in terms of net gaming revenue. Dolphin Treasure was the fourth most prominent game on the list. The popularity of Dolphin Treasure has also been identified in qualitative and quantitative research conducted in South Australia (Livingstone et al., 2008) involving interviews with 64 selfidentified recovering slot machine problem gamblers and a telephone survey of 180 nontreatment seeking gamblers. Dolphin Treasure figured prominently with both groups of gamblers, and Dolphin Treasure was identified as the most popular game by 22 of the problem gamblers. Design of the game Dolphin Treasure is a typical multiline slots game, with five animated reels that spin independently of one another, with their final resting positions of the middle row determined by five randomly generated numbers. The order of the symbols on each reel is fixed so, for any given reel, there is a dependency between the image on the middle row and the image on the row above and the row below. Not all combinations of the images on the top, middle and bottom rows are possible. Also, in multiline slots there is a dependency between winning outcomes on the various lines due to the overlapping nature of the lines. For example, 2 occurrences of the 9 symbol starting from the left on a played line is a win

8 6 K. Harrigan et al. that pays 2x the wager. Assuming the player wagered on all lines (i.e. the common minimax strategy), if the 9 symbols occurs on the top row of both reel 1 and reel 2 then that is a win of 2x the line wager on line 2 and 2x the line wager on line 6 for a total win of 4x the line wager. At the end of a spin there are 3 symbols seen on each reel, forming a 3x5 grid of symbols across the reels. Most of the winning outcomes require matching symbols (e.g., 3 seahorses) to appear on paylines that span the reels running left to right. Prize amounts increase as the number of matching symbols on a payline increases, and if the matching symbols are rare or have special bonus features associated with them. Players can bet on single or multiple paylines, with up to 9, 15, or 20 paylines being typical. Players can also adjust the size of their wagers, typically from 1 to 5 or 1 to 10 credits per payline. Multiple prizes can be won on a single spin when bets are placed on more than one payline, and wins are proportionally greater when more credits are wagered per payline. We have access to a Dolphin Treasure slot machine in our research lab. It is configured so that players can wager 1 5 credits on 1 9 paylines, and thus the minimum wager per spin is 1 credit (i.e. 1 credit bet on 1 payline) and the maximum wager is 45 credits (i.e. 5 credits on each of 9 paylines). Each credit is valued at two cents. The nine paylines are shown schematically in Figure 1 and show that line one is the middle horizontal row, line two is the top horizontal row, line three is the bottom horizontal row, line four is V shaped, line five is an upside down V, and so on for all nine lines. The paytable of a slots game states the various prizes that can be won for different outcomes. The panel entitled # Winning Symbols in Table 1 shows the Dolphin Treasure paytable. As an example from the # Winning Symbols panel of Table 1, 3 occurrences of the low-paying J symbol pays 5x the line wager, 4 pays 25x the line wager, and 5 pays 100x the line wager. J is a line win meaning that the winning J symbols must be consecutive on a played line and start with the leftmost reel. The W symbol pays the highest, with 5 W symbols paying 9000x the line wager (i.e. a jackpot ). To know how likely each of the outcomes is to occur requires not just the paytable, but also a map of each of the five reels. We were able to generate a reel map for the version of Dolphin Treasure in our research lab, and this is shown in the Reels panel of Table 1. For instance, the low-paying J symbol occurs four times on reel one, seven times on reel two, one time on reel three, two times on reel four, and five times on reel five, while the high-paying W symbol only occurs once per reel. Reel 5 has 44 symbols while the other reels each have 30 symbols. Dolphin Treasure also has two symbols with special features. The Treasure Chest is a scatter win, which means that the Treasure Chest symbols can be anywhere on the reels and do not need to be on a payline going from left to right. For Treasure Chest wins the player wins both credits and free spins. The amount of credits won is shown in Table 1, two occurrences of the Treasure Chest pays 2 credits, 3 pays 5, 4 pays 50, and 5 pays 400. Additionally, for Treasure Chest wins in which there are 3, 4, or 5 Treasure Chest symbols Figure 1. The nine lines in Dolphin Treasure.

9 International Gambling Studies 7 Table 1. Paytable and number of occurrences of each symbol on each reel for Dolphin Treasure. # Winning symbols Reels Symbol Prizes Symbols per reel Jack J Queen Q King K Ace A Octopus O Fish school FS Starfish SF Treasure Chest S Turtle T Seahorse SH Sunrise W TOTAL the player also wins 15 free spins. All prizes in free spins are tripled. The Sunrise symbol acts as a wild symbol, meaning that it can be substituted for any symbol on a payline to form a win, but it cannot contribute to a Treasure Chest scatter win. Any win containing the wild symbol pays double. Figure 2 shows 4 examples of Dolphin Treasure outcomes with the winning positions shown within the 3x5 grid of symbols that are visible to the player when the reels stop. Figure 2 contains a regular line win, a line wins with the wild symbol, a scatter win, and multiple wins on one spin. Method We wrote a computer program in Java to simulate the outcomes for the version of Dolphin Treasure we have in our research lab. Using the fixed layout of the 5 reels and paytable from Dolphin Treasure, our program was able to analyse all 35,640,000 possible outcomes in base mode and simulate the additional Treasure Chest free spins. This version of Dolphin Treasure has a hold of 12.13%, meaning that for every dollar wagered approximately 12 cents is retained by the slot machine, on average. The payback percentage is the percentage that the player is paid back, on average, on each spin. The payback percentage for this Regular Line Win Wild Line Win Scatter Win Multiline Win A W S S J A W Q S T W A Q S J 3xA on line 4, pays 10x the line wager. 2xQ + 2xW on line 5, pays50x the line wager. 4xS, pays 50x the total wager. 1xT + 1xW on line 1 and 2xJ + 1xW on line 4, pays 14x the line wager. Figure 2. Example wins in Dolphin Treasure. The leftmost example is a regular line win on line 4. The next example has a line win on line 5 with 2 wild symbols with the prize doubled. The third example has a scatter win with prize multiplied by the total wager. The fourth example has multiple wins on the same spin.

10 8 K. Harrigan et al. version of Dolphin Treasure is 87.87% (i.e. 100% 12.13%). Each outcome is completely independent of other outcomes and is unpredictable, but the layout of the reels and paytable ensures that the hold is adhered to across a very large number of plays, with the Law of Large Numbers ensuring close approximation of the pre-set hold as the number of individual plays grows without bound. Using our program we simulated all possible outcomes in Dolphin Treasure with one and nine lines wagered and analysed the results. Each time we encountered a win with 3, 4 or 5 Treasure Chests we randomly generated 15 free spins (and sometimes these free spins generated additional free spins). The results of our simulation were compared with the game s hold and hit frequency to confirm that our simulation results matched the actual game in our lab. From the results of this simulation, we generated descriptive statistical information about the game including the number of spins in base mode (by base mode spins we mean all spins except the free spins that are won with 3, 4 or 5 Treasure Chest symbols), the number of free spins, the amount of time between free spin initializations (i.e. the time between wins with 3, 4, or 5 Treasure Chest symbols), the payback percentage broken down by base mode spins and free spins, the legitimate win rate, and the celebratory feedback rate. Next, we simulated millions of spins and produced results for a Gambler s Ruin scenario whereby players arrive with $100 and make wagers of 90 cents until broke, on either 1 line or 9 lines. We reported the percentage of players who had a peak balance above 2x and 10x their initial bankroll players who, if they had quit gambling during the session, had the potential to leave having doubled or made 10 times their initial bankroll. We chose the upper limit of 10x because Turner, Zangeneh, and Littman-Sharp (2006) surveyed 96 gamblers and determined that $1000 was the median amount they would have to win to consider the win to be a big win. We also reported the median playing time for each group and the percentage of players who played for less than 1 hour on their $100 bankroll. To investigate the gambler s fallacy whereby gamblers intuit that after a number of losses a win is due, we simulated millions of spins and reported the length of losing streaks with wagers placed on one line and nine lines. To determine the effect of LDWs on streaks, in our nine-line analysis we first considered losing streaks in which consecutive spins resulted in no celebratory feedback (i.e. an LDW or legitimate win ended the losing streak), and then considered losing streaks in which consecutive spins contained no legitimate wins (i.e. LDWs were included in losing streaks, and thus only legitimate wins ended the losing streak). For each condition, we reported the number of losing spins needed before 90% of the losing streaks ended. Finally, based on the length of the streaks we generated one million spins and simulated players having different wagering strategies whereby the amount of the wager (i.e. the number of lines wagered and the credits wagered per line) varied with each strategy. We then reported the payback percentage and amount lost using each strategy. Results Table 2 shows the results of our calculations of various descriptive statistics about Dolphin Treasure. The cycle of a slot machine game is the number of possible outcomes. Dolphin Treasure has 35,640,000 unique outcomes in base mode, calculated by multiplying the number of symbols on each reel (i.e. 30 * 30 * 30 * 30 * 44). Of these outcomes, 249,966 are scatter wins that reward the player with 15 free spins. We generated 249,966 * 15 free spins. Some of these free spins generated more free

11 International Gambling Studies 9 Table 2. Descriptive statistics for Dolphin Treasure. 1 line 9 lines Base mode spins 35,640,000 35,640,000 Free spins 4,190,265 4,185,720 Total spins 39,830,265 39,825,720 Number of free spin initializations 249, ,966 Spins between free spin initializations Minutes between free spin initializations Payback % from base mode spins 64.96% 64.96% Payback % from free spins 22.85% 22.93% Total payback % 87.81% 87.89% Legitimate Win rate (base mode only) 11.28% 16.69% LDWs (base mode only) 0% 14.38% Celebratory feedback rate (base mode only) 11.28% 31.07% spins for a total of approximately 4,100,000 free spins. The player gets a scatter win that initiates 15 free spins, on average, every 14.3 minutes assuming a rate of 10 spins per minute. The 87.87% payback percentage of the game is composed of 64.96% from base mode spins and approximately 22.93% from free spins (this 22.93% varies slightly each timewerunoursimulationasthefreespinsare generated randomly). With 1 line wagered the legitimate win rate is 11.28% and there are no LDWs. With 9 lines wagered, 16.69% of spins result in a legitimate win and 14.38% result in an LDW, resulting in a celebratory feedback rate of 31.07%. To determine whether wagering on multiple lines yields a smoother game experience (less outcome-to-outcome variability) we generated all 35,640,000 spin outcomes with 1 line and 9 lines wagered. We then measured the variability of these outcomes around the expected outcome determined by the hold of the game (see Harrigan & Dixon, 2009, p. 89). In the 1-line game the wager per spin was $0.90 on a single line. In the 9-line game the wager was $0.10 on each of 9 lines, for a total spin wager of $0.90. In both cases the hold in each game was $.11 ($0.90 wager times the hold percentage 12.3%), Thus the expected outcome was $0.90 minus $0.11 or $0.79. When wagering on 1 line the standard deviation around the expected outcome in Dolphin Treasure was and it dropped to 5.25 with 9 lines wagered. Figure 3 shows another way of capturing the smoother playing experience in the nine-line game. Two sample blocks of 250 spins are shown in Figure 3,1 block with one line wagered and 1 with 9 lines wagered. The solid black line depicts the estimated balance based on the hold (showing that the more spins, the more they can expect to be down). The dotted line is the running balance (how much they are up or down) with one line wagered while the solid grey line represents nine lines wagered. The graph highlights that compared to the nine-line game, the actual running balance in the one-line game deviates much more from the expected running balance. We considered peak balances of 2x and 10x the original bankroll as big wins and showed that 21.69% of the 1-line players reached 2x their original bankroll compared with 20.46% of the 9-line players. At this point they could have quit gambling and doubled their money. As for achieving a balance of 10x the original stake at some point in the scenario, just 2.55% of the 1-line players and 0.27% of the 9-line players did this. We noted a dramatic difference in the length of time taken to completely extinguish the funds of the 1- line and 9-line players, with 9-line play resulting in a median time to depletion of 49.7 minutes versus 31.2 minutes when playing 1 line. More of the 1-line players depleted their funds in less than an hour (73%) compared to those playing 9 lines (58%).

12 10 K. Harrigan et al. Figure 3. A sample graph showing the accumulated balances (how much the player was up or down) over 250 spins of 1-line play and 9-line play. The black line shows the expected accumulation determined by the hold. Figure 4 shows the results of our streak analysis. As shown by the solid lines in Figure 4, the prior likelihood of losses does not vary with the number of preceding wins or losses, but the dotted lines indicate that as streaks lengthen, the posterior likelihood that the streak will end soon increases. This is because the number of completed streaks (i.e. a sequence of losses followed by a win) accumulates as the number of spins in the sequences gets longer. Thus the dotted lines indicating streaks completed at different lengths are cumulative. Figure 4a shows that with 1 million spins and 1 line wagered, over 90% of the losing streaks end with between 1 and 20 losses in a row. In Figure 4b we considered losing streaks to end when the result of a spin has celebratory feedback (i.e. the losing streak ends with either an LDW or a legitimate win). Figure 4b shows that with 9 lines wagered 90% of losing streaks end in 7 spins or fewer. Figure 4c shows the effect of LDWs on the losing streak lengths. In Figure 4c we considered losing streaks to end when the result of a spin is a legitimate win. Figure 4c shows that this increases the length at which 90% of losing streaks end from 7 to 13. In each of these 3 graphs, approximately 99% of all streaks generated by the simulations are plotted, with anomalous streaks longer than the horizontal axes omitted. To some, the Figure 4 may suggest that there may be an optimal strategy for betting in slot machine play simply by increasing the bet size after long losing streaks. To reveal that such an optimal strategy is an illusion that can end up costing the player money, we analysed 1,000,000 spins while simulating gamblers using 1 of 5 different strategies to play the slot machine. The min strategy involved placing the minimum bet (one credit worth two cents) on a single line. The max strategy involved playing the maximum

13 International Gambling Studies 11 Figure 4. One million simulated players observations of the percentage of losing streaks that end with a given length (non-cumulative; solid lines), and percentage of losing streaks that end with a given length or shorter (cumulative; dotted lines). Vertical lines indicate the streak length at which 90% or more streaks end. Figure 4a is for one line wagered. Figure 4b is for nine lines wagered and a losing streak ends when a spin results in celebratory feedback. Figure 4c is for nine lines wagered and a losing streak ends when a spin results in a legitimate win. number of lines (9) and betting the maximum amount (5 credits) on each of these lines (for a total spin bet of 90 cents). The mini-max strategy involved playing the maximum number of lines (9), but betting the minimum bet (1 credit/2 cents) on each line (for a spin bet of 18 cents). The so called optimal strategy involved a mini-max bet (nine lines, one credit per line) until the player accrued seven losses in a row. They then switched to a maxbet strategy (nine lines, five credits per line) until a win occurred, whereupon they switched back to mini-max wagers. This optimal strategy was based on Figure 4b which showed that with nine lines wagered, losing streaks rarely go beyond seven spins. Finally we simulated a gambler playing a random strategy in which he or she randomly chose the number of lines wagered to be between one and nine and the number of credits wagered per line to be between one and five. Thus the wagers were randomly between 2 cents and 90 cents. As can be seen in Table 3, none of these so called strategies, including the so Table 3. Expenditures and payback percentage across one million spins using various wagering strategies with each credit valued at two cents. Strategy Total wager Win Loss Payback % Min $20,000 $17,429 $ % Max $900,000 $793,856 $106, % Mini-max $180,000 $158,771 $21, % Optimal $232,823 $204,920 $27, % Random $300,139 $265,868 $34, %

14 12 K. Harrigan et al. called optimal strategy, changed the payback percentage substantially from the pre set 87.87%. There were, however, marked differences in the amounts of money that would be lost over the course of one million spins. The lowest losses were incurred by playing the min strategy (one credit on one line) followed by the popular mini-max strategy. Importantly, the seemingly optimal strategy failed to change the payback percentage, but cost the player more than the mini-max strategy, and 10x more than the min strategy. The max-bet strategy was of course even worse, since the amount bet directly influences the amount lost. Discussion In this study we examined Dolphin Treasure as a typical example of a modern multiline electronic gaming machine, and we produced simulations of the game in order to assess the degree to which players can exert control over the outcomes of the game. We simulated five playing strategies and showed that none of these strategies affected the odds of winning money over an extended period of play. All outcomes are randomly determined and the hold or payback percentage remains constant regardless of the number of lines wagered or the number of credits wagered per line, so Dolphin Treasure remains a pure game of chance. Even though there is no possible way to play the game and gain any real control over the payback percentage, players are given ample opportunity to produce tangible effects on aspects of the game that may (misleadingly) appear to be correlated with monetary gain. Safety in numbers Placing simultaneous bets on multiple paylines does exert real influence over the celebratory feedback rate. As shown in Table 2, playing multiple paylines has no effect on the overall frequency of free spins or payback percentage, but it creates a nearly threefold increase in the celebratory feedback rate. Placing simultaneous bets on multiple paylines reduces the volatility of game outcomes because more bets are placed and resolved per unit of time. As long as the payback percentage is less than 100%, players will tend to lose money at a rate per gamble that is highly volatile in the short term, but as the number of individual gambles increases, the statistical Law of Large Numbers ensures closer and closer approximations to the hold. Likewise, when fewer gambles are observed the outcomes will have greater variability. Multiline betting accelerates the number of gambles resolved per spin, and therefore per unit of time. For example, after approximately 1 hour of play a gambler wagering on 1 line would have placed 600 bets on 600 spins, but a 9 line player would have placed a total of 5400 bets on 600 spins. Consequently, there is far greater consistency in the amounts lost by the 9-line players after one hour than among the one-line players. What this may mean for a multiline player is that the game outcomes have the appearance of being more predictable betting on nine lines than they would be after playing one line for the same number of spins over the same duration. As shown in Figure 3, actual running balances are closer to expected running balances in the nine-line game than the one-line game. Similarly, statistically there is less variability between actual spin outcomes and hold-determined expected spin outcomes in the nine-line games than the one-line games. This smoothing of the game experience with nine lines wagered may make Dolphin Treasure feel less like a game of chance and more predictable, even though there is no effect whatsoever on the payback percentage. Although a few players betting on 1 line may benefit from the greater volatility and

15 International Gambling Studies 13 experience large wins that increase their initial bankroll to over 10 times its initial value, these events are still quite rare (only 2.55% of 1-line players), so the majority of players may exercise their option to bet on all paylines in order to avoid missing out on any potential wins. Betting on more lines means risking more money per spin, but the odds of winning something increase and the player may be left with the false impression that this style of playing is less risky because the odds of winning nothing at all are reduced. Dixon et al. (2010) have shown that players may derive the same excitement from legitimate wins and LDWs, and winning nothing is obviously not very rewarding. Paradoxically, if a player is afraid of winning nothing for many spins in a row, he or she may avoid the aversive losing streaks if willing to risk more money per spin by betting on more lines. The higher celebratory feedback rate that is obtained by doing this may give a false sense of security, but it comes at the price of betting larger amounts of money with losses that increase proportionally because the hold does not change. Multiline betting not only reduces the volatility of rewards and losses; it also extends the time-on-device that the median player experiences. Our gambler s ruin scenarios show an increase from approximately 31 minutes of play to expend a $100 bankroll when betting on 1 line, to approximately 50 minutes when betting on 9 lines. Also, fewer players run out of money in less than an hour when betting on 9 lines. Prolonged time-on-device might give players the impression of gaining more opportunities to encounter free spin bonus rounds and win large prizes, but it ultimately has no real bearing on their overall expenditures. As shown in Table 3, the payback percentage is the same when playing one line (i.e. the minimal bet strategy) or nine lines (i.e. the mini-max strategy). Indeed, all five strategies that we tested produced essentially identical payback percentages. Another possible effect may stem from the fact that the prospect of actually winning money might not be what motivates some players. For non-problem recreational gamblers, getting to play longer translates into greater entertainment value for the money they spend on gambling. They essentially get a better deal if they can stretch out the time it takes to lose their stake and participate longer for that expenditure. More ominously, there is a growing body of evidence to suggest that problem gamblers are often motivated to gamble because it gives them a means of temporarily escaping aversive emotional states (e.g. MacLaren, Harrigan, & Dixon, 2012; Stewart & Zack, 2008; Stewart, Zack, Collins, Klein, & Fragopoulos, 2008; Thomas, Allen, & Phillips, 2009). If these so-called escape gamblers can use their knowledge of how to prolong their play time by multiline betting, then problematic gambling behaviour might be reinforced by an extended period of negative reinforcement (i.e. unpleasant thoughts and feelings are avoided for a longer time as the gambling sessions in which they are engrossed have longer durations). For problem and non-problem players alike, the illusory protection against loss or against being unable to continue playing as an escape may be nullified if they also bet large numbers of credits per line and deplete their funds quickly. This might account for the phenomenon of some players preferring the mini-max strategy to prolong play even though the size of potential wins is minimal if the wagers per line are minimal. Skilled slots players Experienced players may learn to use their control over the frequency and size of wins to effectively prolong their play, and come to the erroneous conclusion that the game involves an element of skill that can be mastered. This illusion of control is further facilitated by the fact that multiple line betting reduces the volatility of the losing streak

16 14 K. Harrigan et al. lengths and thus the end of a losing streak may appear predictable and promote decisionmaking that is slanted by the gambler s fallacy. As shown in Figure 4, theapriori likelihood of having a spin with celebratory reinforcement does not change depending on the number of winning or losing outcomes preceding any spin, but the a posteriori conclusion that an experienced gambler may reach is that the longer the chain of losses, the more imminent is a win. A very experienced player may have encountered enough spins to approximate the distribution of losing streaks within an entire cycle of the game. The gambler s fallacy comes from applying such knowledge of the game to make guesses about the distribution of wins and losses that should occur in a very small set of spins. When playing with 1 line, 90% of losing streaks end with 20 or fewer losses in a row so a gambler might conclude that any time he or she has witnessed 20 losses in a row, there is approximately a 9 in 10 chance that the next spin will be a hit. Likewise, the same9in10oddscouldbeassumedafterjust7spinswhenplaying9lines.aplayer might opt to start betting larger amounts at this point in order to maximize the size of the supposedly imminent win. Memorable instances where players do encounter a large win after following this strategy may sometimes occur and produce proportionately large wins that reinforce continued use of the strategy and support the gambler s fallacy upon which it stands. However, the illusory control that this fallacy affords also ignores the fact that all outcomes are random and independent, and that the outcomes in a small sample of spins often do not reflect the population distribution of outcomes across the game s cycle. We have shown that none of our strategies can shift the base rate of hitting on any spin from fluctuating around an average of 11.28% when playing 1 line, or 31.07% of spins with celebratory feedback when playing 9 lines, and no strategy can reduce the hold from 12.13%. Indeed, Table 3 shows that applying a seemingly optimal strategy based on our detailed knowledge of the distribution of streak lengths had no effect on the hold, and ultimately cost the player more money relative to players using the min strategy or the mini-max strategy. These behaviours might seem irrational but in some sense they are really quite reasonable, given the feedback that the game provides to players who may understand the impacts of their playing style on the frequency and size of wins, but who do not understand that they have no possible way of shifting the hold in their favour. Conclusion The design of modern multiline electronic gaming machines of the type described here does a masterful job of conveying the false impression that players can control salient features of the game that seem as if they should be related to hold or payback percentage. If a gambler learns that he or she can control the frequency and size of wins, it would seem rational to assume that the odds of profiting are likewise controllable. This belief seems perfectly rational and not what we normally think of as distorted thinking. However, what is probably not so obvious to players is that any increase in the frequency or magnitude of wins that they can effect by adjusting the number of lines and the wager per line is equally offset by a proportional increase in expenditure so there is no net change to the hold. It is true that players can exert real control over the frequency and size of wins and these impacts can be compelling, but those effects are completely independent of the outcome that should really be of most interest to gamblers whether they win money or lose money over the long haul. Indeed, players will almost certainly lose money over the long haul if the payback percentage is anything less than 100%, and it always is.