International Journal of Mental Health and Addiction

, Volume 9, Issue 1, pp 60–71

What Does a Random Line Look Like: An Experimental Study

Authors

    • Centre for Addiction and Mental Health
    • Department of Public Health SciencesUniversity of Toronto
  • Eleanor Liu
    • Centre for Addiction and Mental Health
  • Tony Toneatto
    • Centre for Addiction and Mental Health
    • Public Health Sciences and PsychiatryUniversity of Toronto
Article

DOI: 10.1007/s11469-009-9251-z

Cite this article as:
Turner, N.E., Liu, E. & Toneatto, T. Int J Ment Health Addiction (2011) 9: 60. doi:10.1007/s11469-009-9251-z

Abstract

The study examined the perception of random lines by people with gambling problems compared to people without gambling problems. The sample consisted of 67 probable pathological gamblers and 46 people without gambling problems. Participants completed a number of questionnaires about their gambling and were then presented with a series of random and non-random lines. The participants rated lines as random if the pattern stayed near zero (the middle of the screen) and did not form anything that resembled waves. The probable pathological gamblers rated 2 of the patterns (jumps, and multi-wave) as significantly less random than non-problem gamblers. They also rated random lines significantly less random than the non-problem gamblers. That is, they seem to be able to find patterns both when they are really there and when they only appear to be there as in the case of random drift.

Keywords

Problem gamblingRandom chanceErroneous beliefs

It is well known that in situations of uncertainty, people tend not to reason very well (Kahneman and Tversky 1982). One case of such a ‘situation of uncertainty’ is the occurrence of random events. When experiencing random events, people often impose patterns on the experience. The purpose of this study was to explore the naïve human concept of random chance by asking people to rate graphs of various lines in terms of how random they look. The graphs depicted a random walk where the changes (up or down) in the graph were either driven entirely by random chance, or were driven in part by one of 8 non-random routines. In addition, this study compared the perception of these random lines by people with gambling problems compared to people without gambling problems.

People have a reasonably good intuitive sense of the nature of random events. Most people expect random events to be erratic and unpredictable. That is people expect random events to “look random” (Kahneman and Tversky 1982). But many aspects of the nature of random events are counterintuitive. For example, if you toss an unbiased coin, heads will come up roughly half the time. Most people would accept that as true. But suppose that the coin landed with the head side up 5 times in a row, the chance that the next flip is another head is still 50%. This is counterintuitive, but is fundamental to understanding the concept of random chance. Data reported by Turner et al. (2005; see also Turner et al. 2006) indicates that nearly half (41.8%) of the general population do not understand this basic part of random chance.

Problem and pathological gamblers often exhibit errors in reasoning. Considerable research has suggested that gambling behaviour is associated with a wide variety of beliefs and attitudes about gambling behaviour (e.g., Langer 1983; Ladouceur et al. 1991; Toneatto et al. 1997; Turner et al. 2006). For example, gambling behaviour is often accompanied by beliefs about the ability to control luck and the efficacy of superstitious behaviour to influence gambling outcomes even when such outcomes are chance-determined. Kahneman and Tversky (1982) and numerous others have shown that people in general, frequently make errors in reasoning by the over application of rules of thumb or heuristics such as representativeness heuristic and the availability heuristic. The representative heuristic for example, involves judging something to be more likely if it is representative or stereotypical of that category. For example, many people believe that ticket number “12-5-23-7” is more likely to be the winning ticket than “1 2 3 4” because it is more representative of what people think a series of random numbers should look like a (see Turner et al. 2005). Although many non-problem gamblers make this error, it is even more common amongst pathological gamblers (see Turner et al. 2006).

As Turner (2000) has noted, part of the problem lies with the random chance itself. Random events most often do actually look ‘random’. The heuristics of representativeness and availability are more often than not helpful strategies. It is not surprising that people sometimes interpret a random pattern as meaningful if it appears to form a predictable pattern. Moreover, there is a strong survival advantage in being able to detect real patterns. A pattern of dark and light stripes may be random, or it could be a tiger, hiding in the shadows. To interpret the pattern as a tiger and remove oneself to safety is likely to be a safer policy than to test out the hypothesis and walk toward the stripes. So erring on the side of over interpretation may, in general, be a good survival strategy.

One of the inspirations for this particular study was the way in which probable pathological gamblers describe their experience of gambling. During interviews conducted as background research for this and other studies (e.g., Turner and Liu 1999; Turner et al. 2006), several problem gamblers have described the roller coaster-like effect of winning. Some even used their hand to illustrate the ups and downs of their fortunes during a gambling session. That is gamblers seem to believe that there were patterns in the ups and downs of their gambling. One person described how when he was losing he kept at it because he knew that he was on the down side of the wave and would follow it up the other side. Probable pathological gamblers may be more likely to read patterns into a random walk than non-problem gamblers?

The topic of this study was to explore the naïve human concept of random chance. It examined the extent to which people can differentiate patterns generated by pure random chance, from patterns that were not random and the types of non-random patterns that participants are most likely to confuse with random chance In addition, this study compared the perception of these random lines by participants with gambling problems compared to participants without gambling problems. Participants were shown a series of graphs of various lines and asked to rate the graphs in terms of how random they looked. The graphs depicted a particular type of random event called a random drift pattern or a random walk. The graphs depicted a random walk where the changes (up or down) in the graph were either driven entirely by random chance, or were driven by one of 8 non-random routines In a random drift the value of something at time t is dependent to some extent on the previous value at t-1, but is modified randomly. The minute to minute changes on the stock market are an example of a drift pattern that is mostly random (a random walk). The effect of the previous value is called an autocorrelation. In time series analysis the autocorrelation effect is controlled for before the analysis begins (see McCleary and Hay 1980) usually by differencing the time series (Δxt = xt – xt-1). The current paper however will not employ time series analysis, but will examine differences between probable pathological and non-problem gamblers in terms of how they rate the randomness of various random and partially random walk patterns.

Method

Participants

The sample consisted of 113 participants consisting of 67 people with gambling problems as defined by the South Oaks Gambling Screen (SOGS, Lesieur and Blume 1987, 1993) (SOGS >= 5; 80% male) and people 46 without gambling problems (SOGS =< 1; 52% male). People who scored 2 or more on the SOGS but less than 5 on the SOGS were excluded from the study (n = 10). All non-problem gamblers were recruited through newspaper advertisements asking for “social gamblers”. The probable pathological gamblers were either recruited from an advertisement placed in a treatment centre (n = 36) or through newspaper advertisements (n = 29). Analysis of the probable pathological gamblers in treatment (M = 12.3 SD = 4.7) scored significantly higher on the SOGS than probable pathological gamblers not in treatment (M = 9.0, SD = 3.2), t(55) = 3.2 = , p < .01, d = .30, but not differ significantly in terms of gender composition, gambling frequency, or past year losses.

Table 1 presents a breakdown of participants’ characteristics. On average the participants were in their mid to early forties (M = 42.5, SD = 13.6). Most of the respondents were male (67.6%). There were some gender imbalances in the different groups. Our probable pathological gambler sample had more male participants in them (54 male vs. 13 female). This is in fact a reflection of reality, in that the majority of probable pathological gamblers are male, however, this gender imbalance may distort some of the findings.
Table 1

Participant Characteristics: Means and Standard Deviations (in brackets)

 

Non-problem (n = 46)

Problem (n = 67)

GENDER (% male)

52%

80%

AGE

42.9 (13.8)

42.7 (12.8)

SOGS past year

0.35 (0.5)

11.1 (4.3)

Money risked in past year (in Canadian dollars)

307 (799)

8206 (12081)

Gambling frequency (estimated days per year)

114 (170)

289 (389)

For gambling frequency, the participants listed how often they play each of 16 different types of games (e.g., once a month, 5 times per year). This was converted into the number of times in the past year and then aggregated across games.

Procedure

Participants who responded to the advertisements were invited to the Centre for Addiction and Mental Health for a two-hour interview. Participants who were recruited through the advertisement were paid $10 in gift certificates for participating in the study. Participants were given a questionnaire regarding their gambling behaviours (e.g., types, frequency, amount wagered and reasons for gambling. Participants also completed several psychometric measures including the SOGS. For the random lines task, the participants were given the following instructions:

Background: Suppose you were tracking the value of a stock on the TSX or changes in Ontario’s annual rainfall and you wanted to know if the changes were predictable or purely random. Would you be able to tell if the changes were random or followed a pattern?

Instructions: In the next task you will be presented with a series of lines. These lines were created using a combination of a random number generator and one of several different predictable mathematical formulas. Your job is to rate the predictability of each line.”

The participants were then shown a graph with 3 completely random lines on it (similar to Fig. 1), and a second figure with three completely non -random lines on it (a smooth sine wave, a straight line and a zig-zagging line). The participant viewed each pattern and rated the pattern on a 5 point scale: 1—more than 95% random; 2–75% random, 25% predictable, 3–50% random, 50% predictable, 4–25% random, 75% predictable, 5—more than 95% predictable. The participant was allowed to look at the figures for as long as they needed. Four random figures and two of each type of non-random figure were presented to the participants so that the participant rated a total of 20 line patterns. The order of items was varied to counterbalance any order effects.
https://static-content.springer.com/image/art%3A10.1007%2Fs11469-009-9251-z/MediaObjects/11469_2009_9251_Fig1_HTML.gif
Fig. 1

Examples of random drift patterns: lines generated using only the random number generator

Measures

Gambling and Pathological Gambling

The assessment package for pathological gambling was similar to that used by Turner et al. (2006), which found that each of these measures had a high degree of internal consistency and convergent validity. To measure gambling behaviour we used a gambling behaviour questionnaire that asked the participants about frequency (gambling occasions in past year), and amounts wagered for 16 different forms of gambling. Total scores for gambling occasions in the past year and money risked were computed based on their answers (see Table 1).

Pathological gambling was assessed using the SOGS. Numerous studies have shown that the SOGS is a valid and reliable instrument for assessing pathological gambling (e.g., Abbott and Volberg 1996; Lesieur and Blume 1987, 1993). The SOGS was framed in terms of ‘past 12 months’ (alpha = .92).

Random Lines

To determine the nature of the participants’ concept of randomness we presented them with a series of lines. The lines were generated by the computer’s random number generator or by one of several non-random algorithms. Each line was made up of 500 numbers that was then shrunk to a 17 cm long graph line. Each graph included a horizontal line across the middle of the graph to indicate an ordinate (y) value of zero and a vertical line at the far left to indicate the ordinate (y – scale) that had values ranging from −100 to +100. No values were given on the graphs for the abscissa. These items were shown to the participants as graphs and the subjects were asked to rate the lines on a scale from 1 (more than 95% random) to 5 (more than 95% predictable). The patterns used here were cumulative random patterns, often called a random walk or random drift pattern (McCleary and Hay 1980), where each new value is made up of the previous value, plus a “change” value ranging from negative (line goes down) to positive (line goes up). The change value was either a random number or was computed using one of 8 different mathematical algorithms. The random numbers were generated using the random number generator (RNG) available in Quick Basic 4.5 which relies on Lehmer’s congruential iteration (see Brysbaert 1991; Onghena 1993) to produce a pseudo random sequence of numbers. Although not technically random, studies by Brysbaert (1991) and Onghena (1993) have shown that this iteration produces a sequence that is indistinguishable from a random sequence of numbers when seeded with the computer’s clock. The repeat cycle of this RNG is approximately 17.7 million.

Random Lines

The general formula for the random lines was as follows:
$$ x_t = x_{{t - 1}} + we $$
(1)

Where xt is the value at a particular time, xt−1 is the previous value of x, e is the random disturbance (or error) that can range from +.5 to −.5, and w is a weight to determine the impact of the random disturbance. The value of w varied from item to item, but remained constant during the creation of a particular line item. The size of change per iteration for both random and non-random patterns was also varied from item to item to make it difficult for the participant to focus on any one aspect of the patterns to determine if they were random.

Four random graphs were shown to each participant. Figure 1 gives 2 examples of random lines patterns.

Non-random Lines

We used 8 different types of non-random line patterns that followed the general formula of,
$$ x_t = x_{{t - 1}} + we + b_1 y_1 \left[ { + {\text{ }}b_2 y_2 + b_3 y_3 } \right] $$
(2)
Which is essentially the same as formula 1, with the addition of one or more y values that represent non-random factors that produced either a linear trend or a wave pattern, and b is a weight for the non-random element in the formula. To create the angular waves (e.g., big wave & small wave), the near zero, and the jumps pattern, the y1 value was determined using a logical if-then statement. Some patterns (e.g., jumps & multi-waves) had more than one y term and their respective weights are included in the formula as y2 and y3. The values of w and b varied from item to item, but remained constant within a particular item. The weights were varied so that the two items of each type given to the participants would look different from each other. The eight non-random patterns were as follows: Near zero pattern (see Fig. 2) drifted up or down but was forced to stay near the zero line. Jumps (see Fig. 3) were essentially the same as the near zero patterns, however, after exactly 20 iterations (0.68 cm) the pattern would either jump up or down. The size and direction of the jump varied. If the value at time t-1 was below zero it would jump up at t + 1, if above zero, it would jump down. The resulting pattern looks somewhat like a cardiogram of a patient with heart disease. The noisy wave (see Fig. 3) produced a pattern of waves but the waves were somewhat obscured by random noise. The skewed wave (see Figure ) was the same as the noisy wave, but the upward and downward parts of the wave were different (e.g., the downward slope might be steeper than the upward slope or vice versa). The multi-wave patterns (see Fig. 4) were created by combining 4 different wave patterns with different amplitudes and wavelengths. The result looked random, but was mostly non-random. The linear pattern (see Fig. 4) consisted of a line that increased or decreased gradually away from zero, but also included some random noise. The big wave pattern (see Fig. 5) consisted of a line that would increase almost linearly to a fixed point and then decrease almost linearly. The small wave (see Fig. 5) was the same as the big wave, but had a smaller amplitude and higher frequency. The waves were somewhat triangular in shape. We could have used the computer’s sine-wave function to produce a very smooth wave form, but the pointy waves are more obviously non-random. These two wave patterns were obviously non-random and were included in the study to encourage the participant to use the entire range of the 5-point scale. It is important to note that with the possible exception of the big and small waves all of these lines patterns could have been created randomly; what differentiates the random and the non random is the fact that patterns created by random chance do not repeat.
https://static-content.springer.com/image/art%3A10.1007%2Fs11469-009-9251-z/MediaObjects/11469_2009_9251_Fig2_HTML.gif
Fig. 2

Near zero (thick line) and skewed wave patterns (thin line)

https://static-content.springer.com/image/art%3A10.1007%2Fs11469-009-9251-z/MediaObjects/11469_2009_9251_Fig3_HTML.gif
Fig. 3

Jumps (thick line) and noisy wave patterns (thin line)

https://static-content.springer.com/image/art%3A10.1007%2Fs11469-009-9251-z/MediaObjects/11469_2009_9251_Fig4_HTML.gif
Fig. 4

Linear (thin line) and multi-wave patterns (think line)

https://static-content.springer.com/image/art%3A10.1007%2Fs11469-009-9251-z/MediaObjects/11469_2009_9251_Fig5_HTML.gif
Fig. 5

Big wave (thin line) and small wave patterns (thick line)

Results

Random Lines

Each participant rated a total of 20 lines that included 4 random lines and 2 items for each of the 8 types of non-random lines. The average rating for the 9 line types is given in Table 2. Each participant rated 4 random lines that varied in terms of the change weight (w). A repeated measures analysis found a significant linear relationship between the size of the change value and the participants rating of the random lines, F(1, 112) = 10.4, p < .01, eta2 = 085. The random lines with smaller change weights (w) were rated as being significantly less random than those lines with larger change weights. There was no quadratic or cubic effect, F < 1. The ratings for the 4 random lines were averaged into a single score. Similarly, the average score was computed for each of the 8 non-random patterns by averaging the responses to the 2 items of each type.
Table 2

Randomness Rating of the 9 Types of Items by Problem Gambling Status

 

Non-problem gamblers (n = 46)

Problem gamblers (n = 67)

Full sample (n = 113)

NP<PG

M

SD

M

SD

M

SD

Random

2.30

0.74

2.61

0.72

2.49

0.74

*

Near Zero

2.28

1.09

2.51

0.99

2.42

1.03

 

Jumps

2.18

1.06

2.60

1.07

2.43

1.08

*

Multi-Wave

2.25

0.88

2.85*

0.85

2.61

0.91

***

Skewed Wave

2.57

0.97

2.90*

0.91

2.76**

0.95

 

Noisy Wave

3.33***

1.09

3.19***

1.00

3.25***

1.04

 

Linear

3.70***

1.10

3.77***

1.12

3.74***

1.11

 

Big Wave

4.39***

1.03

4.14***

1.03

4.24***

1.03

 

Small Wave

4.41***

1.10

4.27***

0.97

4.33***

1.02

 

* < p < .05; **p < .01; *** P < .001

Note that asterisks placed after a mean score indicate that it is significantly different from the average score for random lines. The last column indicates if the average score differed between the problem and non-problem groups.

The average rating for each type of line pattern was computed for each participant. The overall difference between probable pathological and non-problem gamblers did not reach significance, F(1, 111) = 3.2, p = .08, eta2 = .028, however the repeated measures analysis of variance revealed that there were some very large differences in the ratings of different lines F(8,888) = 82.7, p < .001, eta2 = .427). More importantly there was a significant interaction between type of line and problem group, F (8,888) = 2.7, p ≤ .01, eta2 = .024. When sex group (male vs. female) was also entered into the analysis there was no main effect or interaction involving sex group. We found no correlation between gender and the rating of any of the line patterns.

As shown in Table 1, probable pathological gamblers tended to rate the big wave and small wave patterns as less predictable then the non-problem gamblers, and they rated the jumps, near zero, multi-wave, and the random lines as more predictable than the non-problem gamblers. Significant pairwise differences were found between probable pathological and non-problem gamblers for the jumps, t(111) = −2.1, p < .05, d = .38, multi-wave, t(111) = −3.6, p < .001, d = .66, and random figures, t(111) = −2.2, p < .05, d = .42. In each case, the probable pathological gamblers rated the figure as less random / more predictable compared to the non-problem gamblers.

The average rating across the random lines was 2.49, which is mid-way between a score of 2 (75% random/25% predictable) and a score of 3 (50% random/50% predictable). This suggests that a substantial number of people felt that the random patterns were not completely random. Some people even gave some of the random lines a rating of 5 (more than 95% predictable).

As shown in Table 2, the random lines were rated as significantly less predictable than the skewed waves t = −3.0, p < .01, noisy waves, t = −6.5, p < .001, linear t = −11.3, p < .001, big wave, t = −13.6, p < .001, and small wave patterns, t = −14.8, p < .001. Probable pathological gamblers rated the multi-wave patterns as more predictable than the random patterns, t = −2.1, p < .05 and the skewed wave pattern was only significant for probable pathological gamblers.

As shown in Table 2, the random lines (M = .2.49), the near-zero (M = 2.42) and the jumps patterns (M = 2.43) were rated as the least predictable of the patterns. The finding for the jumps patterns was somewhat surprising since the jumps in the patterns were, in fact, regularly spaced at 20 iterations apart (0.68 cm). Another interesting finding is the difference between the ratings of the skewed wave patterns (M = 2.75) and the noisy wave pattern (M = 3.23), t(112) = 4.4, p < .001, d = .42. This difference was significant for both groups.

Discussion

According to the present study, the state of understanding of random events is very poor in both probable pathological and non-problem groups. People can tell the difference between the random patterns and some of the non-random pattern (small wave and big wave), but not at all for some other patterns (jumps & near zero). In general, both groups of participants rated lines as random if the patterns stayed near the zero line (in the middle of the screen) and did not form anything that resembled waves. This expectation regarding staying near zero is consistent with the gamblers fallacy and is consistent with numerous other studies that have shown frequent errors in human reasoning about random chance (Kahneman and Tversky 1982; Toneatto et al. 1997; Turner et al. 2005, 2006). If a person has lost too many times, they believe they are due to win and if they double their bets, they can make a profit (see Turner 1998; Turner and Horbay 2003). But at the same time, they expect the return to zero to be erratic—not wave like. Both probable pathological and non-problem gamblers showed evidence of this fallacy.

We found a small number of differences between the probable pathological gamblers and non-problem gamblers. The probable pathological gamblers rated the random lines as significantly more predictable than non-problem gamblers. In addition, probable pathological gamblers gave significantly higher predictability ratings to the jumps and multi-wave patterns. These findings indicate that probable pathological gamblers may be more apt to impose order on random and non-random stimuli and find patterns that suggest predictability weather there is any predictability or not. However it is noteworthy that probable pathological gamblers and non-problem gamblers did not differ significantly on the pattern that received the lowest predictability rating, (near zero), or on those that received the highest predictability ratings (big wave and small wave).

Neither probable pathological gamblers nor non-problem gamblers could tell the difference between the truly random patterns and the near zero and jumps patterns. The low predictability rating of the jumps pattern is one of the more surprising results of the study. The jumps in the patterns were in fact regularly spaced at exactly 20 iterations apart (0.68 cm). The failure to find the regularly spaced jumps suggests a huge problem in the human ability to detect certain types of regularities. Although probable pathological gamblers rated this item as significantly more predictable, than the non-problem gamblers, they did not rate it as more predictable than a random line.

The significant difference between the skewed waves and the noisy waves is interesting because these two patterns were generated by very similar program routines. The only difference between them was that the up and down slope for the skewed wave was asymmetrical. Apparently, participants view patterns with symmetrical waves as more predictable.

Looking at the items that were rated as the least predictable reveals some interesting facts about how people judge randomness. The low predictability rating of the near zero lines is not surprising because returning to zero is exactly what most people expect random numbers to do (e.g., the belief that they are “due to win their money back”). The multi-wave and the jumps patterns also stay near zero. The noisy wave and skewed wave stray away from zero but return in a relatively regular wave-like manner and were judged to be significantly more random than the random lines. The difference between the near zero and the noisy wave was that the noisy wave was allowed to drift further away from zero and drifted back to the zero line in a more wave like manner. The linear pattern was also judged as non-random by most participants probably because it drifted away from zero. Ironically, a pattern that drifts away from zero is actually a fairly common random drift pattern. If a line drifts up by random chance, there is nothing to force it to return to the zero line producing the illusion of a trend away from zero. Taken together these items suggest that the participants expect random chance to remain near zero or return to zero in an erratic manner. In addition they also rated lines with systematic jumps in value (jumps) or regular changes in direction (near zero) as random as long as the line stayed near zero. A line that strays away from zero or returns to zero in a regular wave-like manner will be perceived as mostly predictable.

The results for the multi-wave pattern are a bit of a puzzle. The probable pathological gamblers rated it significantly more predictable than the random lines. In addition, their ratings for the multi-wave pattern were significantly higher (more predictable) than the non-problem gamblers. Given that the participants rated 9 different types of lines, it could be that this effect is a type 1 error; however, the high level of significance for this particular effect (p < .001) challenges this assertion, but does not rule it out. One possibility is that because probable pathological gamblers have spent so much time watching actual random numbers going up and down (e.g., their credits) the multi-wave pattern might have seemed strange to them (compare the multi-wave in Fig. 5 with the random lines in Fig. 1).

Overall the participants were not very accurate at identifying the random patterns as random. These data permit us to speculate on the nature of the participants’ concept of random events. Items that were most likely to be rated as random appear to be those that stay near zero (near zero, multi-wave), or jump up and down around zero (jumps, skew wave). The non-random patterns that were most easily identified as being non-random are those patterns that form symmetrical waves (big wave, small wave, noisy wave) or progressively deviate from zero in a linear fashion (linear). When random events do form a linear pattern or an apparent wave pattern, they will be interpreted as a linear or wave pattern. Consequently, we argue that when apparent waves or linear trends show up in real life at the roulette table, or on the Stock market, people will tend to interpret them as predictable patterns. Probable pathological gamblers appear to be more likely to see these patterns, and interpret them as predictable. Probable pathological gamblers in treatment will frequently remark that they would be ahead, if they could only learn to stop when at the peaks of those waves. The differences found in the present study were relatively small and need to be replicated to determine if there is a general difference in the interpretation of random pattern between probable pathological and non-problem gamblers.

The study also illustrates some of the challenges in educating the general public about the nature of random chance. Although people understand the erratic nature of random chance, they erroneously believe that random chance would produce an outcome that corrects itself. Turner (1998) has shown that if this belief was correct, a doubling strategy after a loss would actually be financially successful. However, in reality the doubling after a loss strategy can result in financial disaster (Turner 1998). This study also has implications for stock market investments. People have difficulty telling the difference between a graph produced by a random sequence and one produced by a predictable sequence. One probable pathological gambler, who told us he was a stock market investor, said that he could “make money out of any of those graphs”.

Acknowledgements

We would like to thank Keith Stanovich, Roger Horbay, and Geoff Noonan for their help in designing and carrying out this study.

The research was supported by a grant from the National Centre for Responsible Gaming. In addition, support to CAMH for salary of scientists and infrastructure has been provided by the Ontario Ministry of Health and Long Term Care. The views expressed here do not necessarily reflect those of the Ministry of Health and Long Term Care or the National Centre for Responsible Gaming.

Copyright information

© Springer Science+Business Media, LLC 2009