Using Neural Networks to Model the Behavior and Decisions of Gamblers in Particular Cyber Gamblers

8/8/2019 Using Neural Networks to Model the Behavior and Decisions of Gamblers in Particular Cyber Gamblers

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O R I G I N A L P A P E R

Using Neural Networks to Model the Behavior

and Decisions of Gamblers, in Particular,Cyber-Gamblers

Victor K. Y. Chan

Springer Science+Business Media, LLC 2009

Abstract This article describes the use of neural networks (a type of artificial intelligence)

and an empirical data sample of, inter alia, the amounts of bets laid and the winnings/losses

made in successive games by a number of cyber-gamblers to longitudinally model

gamblers behavior and decisions as to such bet amounts and the temporal trajectory of

winnings/losses. The data was collected by videoing Texas Holdem gamblers at a cyber-

gambling website. Six persistent gamblers were identified, totaling 675 games. The

neural networks on average were able to predict bet amounts and cumulative winnings/losses in successive games accurately to three decimal places of the dollar. A more

important conclusion is that the influence of a gamblers skills, strategies, and personality on

his/her successive bet amounts and cumulative winnings/losses is almost totally reflected by

the pattern(s) of his/her winnings/losses in the few initial games and his/her gambling

account balance. This partially invalidates gamblers illusions and fallacies that they can

outperform others or even bankers. For government policy-makers, gambling industry

operators, economists, sociologists, psychiatrists, and psychologists, this article provides

models for gamblers behavior and decisions. It also explores and exemplifies the usefulness

of neural networks and artificial intelligence at large in the research on gambling.

Keywords Neural network Gamblers/Cyber-gamblers behavior and decisions Successive bet amounts Cumulative winnings/losses Gamblers fallacies

Introduction

Gambling, or gaming as it is also known, is a human activity that can be traced back

through prehistory and a multitude of gambling forms are apparent in many different

cultures (Dewar 2001). Gambling is often claimed to have positive effects, in that not onlyis it a lucrative and profitable industry worldwide in its own right, worth billions of US

V. K. Y. Chan (&)

School of Business, Macao Polytechnic Institute, Rua de Luis Gonzaga Gomes, Macao,

Peoples Republic of China

e-mail: [email protected]

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dollars per annum (American Gaming Association 2008b; Moss et al. 2003), but it also

fuels tourism (MacLaurin and MacLaurin 2003), employs a considerable amount of labor

at different skill levels (Gill et al. 2006; MacLaurin and MacLaurin 2003; Siegel and

Anders 1999), and is a major source of government revenue in many economies (Gill et al.

2006; Layton and Worthington 1999; Siegel and Anders 1999). Unfortunately, gambling iscoupled with a series of negative socio-economic problems. For example, it financially

afflicts low-income individuals (Layton and Worthington 1999). Furthermore, there is

evidence that certain forms of gambling create pathological gambling (alternatively known

as compulsive gambling, addictive gambling, disordered gambling, or problem gambling)

through addiction, lead to suicide, attract criminal elements, foster corruption (Feigelman

et al. 2006; Layton and Worthington 1999; Friedman et al. 1989) and facilitate money

laundering (Hugel and Kelly 2002).

The total impact and incidence of both the positive contributions and the negative

problems have been intensified by the advent of cyber-gambling on the internet in the mid-

1990s, given that cyber-gambling makes gambling substantially more prevalent and per-

vasive in society by essentially relaxing the constraints on gambling times and venues (Gill

et al. 2006). In particular, with regard to alleviating the related problems, whilst traditional

terrestrial gambling is subject to strict legal and governmental regulations enforced within

the associated jurisdictional boundaries, cyber-gambling renders these restrictions rela-

tively unenforceable because of the absence of boundaries in cyberspace. In summary,

cyber-gambling scales up both the contribution and the problems of gambling. As a matter

of fact, cyber-gambling has been growing since its inception in 1995 (American Gaming

Association 2008a; Dewar 2001) and the magnitude of its annual turnover worldwide is

now tens of billions of US dollars (American Gaming Association 2008a; Dewar 2001)with nearly 23 million people having gambled at well over 2,000 cyber-gambling websites

as of 2005 (American Gaming Association 2008a).

Given such a conjunction, namely that the gambling industry is already a multi-billion

dollar industry and still rapidly expanding, in particular, as a result of the speedy prolif-

eration of cyber-gambling websites and cyber-gamblers in the last decade, how to strike a

balance between gamblings contributions and problems has been undeniably controversial

both socio-economically and politically (Smith 2004; Layton and Worthington 1999).

Needless to say, both the contributions and the problems of gambling depend on gamblers

behavior and decisions, in particular, their bet amounts and their temporal trajectory of

cumulative winnings/losses. Specifically, on the one hand, gamblings contributions to thegambling industrys profit, the industrys employment and government revenue are directly

derived from gamblers bets and losses. On the other hand, gamblings problems with

regard to the gamblers financial affliction, pathological gambling, and possible suicide

stem outright from the gamblers cumulative losses. Therefore, gamblers bet amounts and

their temporal trajectory of cumulative winnings/losses become one of the determinants of

the aforementioned gambling-related controversy. In the face of this controversy, models

for gamblers bet amounts and their temporal trajectory of cumulative winnings/losses can

assist the work of related government policy-makers, gambling industry operators, econ-

omists, sociologists, psychiatrists, and psychologists. In particular, such models helpgovernment policy-makers in reformulating related government policies and putting for-

ward new legislation to regulate the gambling industry in a bid to secure sufficient gov-

ernment revenue while at the same time containing gambling-related problems (American

Gaming Association 2008a; Smith 2004; Hugel and Kelly 2002; Dewar 2001; Jones et al.

2000; Siegel and Anders 1999). Such models also enable gambling industry operators and

their employees to reposition and redesign their games so as to optimize the profit and thus

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guarantee employment (Moss et al. 2003). At the same time, economists and sociologists

can better estimate the socio-economic value of the gaming industry based on the infer-

ences provided by such models regarding bet amounts and cumulative winnings/losses,

whereas psychiatrists and psychologists can better chart the behavior and decisions of

gamblers according to such models so as to improve the treatment and rehabilitation ofpathological gamblers (Ladouceur et al. 2007; Gill et al. 2006; Nelson et al. 2006; Slutske

et al. 2003, 2005; Breen and Zimmerman 2002; Layton and Worthington 1999).

This article documents research on aforesaid models based on neural networks or, more

exactly, artificial neural networks, which are a type of artificial intelligence emulating

biological neural networks (Nikolaev and Iba 2006; Ananda Rao and Srinivas 2003; Arbib

2003), used to model the aforementioned bet amounts and temporal trajectories of

cumulative winnings/losses. Such neural networks were and needed to be trained by an

empirical data sample on the amounts of bets laid and the winnings/losses made in suc-

cessive games by each of a random sample of cyber-gamblers. The data sample was

collected from a legally licensed, well-established cyber-gambling website (sometimes

referred to as an e-casino) where international gamblers played Texas Holdem among other

poker games. This research aimed at letting the neural networks construct the following

two models:

M1: the model for successive bet amounts, which longitudinally models and thus

predicts the bet amounts in the successive game laid by each individual Texas Holdem

gambler based on his/her winnings/losses in a number of immediately preceding games

and his/her current gambling account balance.

M2: the model for the temporal trajectory of cumulative winnings/losses, which

longitudinally models the temporal trajectory of cumulative winnings/losses of eachindividual Texas Holdem gambler based on his/her cumulative winnings/losses in a

number of immediately preceding games.

Despite the fact that Texas Holdem games are more than games of chance in that the

outcomes depend on gamblers intellectual decisions, which, believe it or not, in turn are

generally thought to depend on individual gamblers skills and even strategies (Harrington

and Robertie 2004), it was to the authors surprise that the neural network for M1 upon

training turned out to be able to predict a gamblers bet amounts in successive games

accurately to more than three decimal places of the dollar on average for each of the six

gamblers in our data sample across the board. More importantly, the neural network for M2upon training was also able to track the temporal trajectory of a gamblers cumulative

winnings/losses, i.e., successively predict the gamblers cumulative winnings/losses, with a

similar accuracy again for each of the six gamblers in our data sample across the board.

Undoubtedly, considering the structure of the two neural networks, which will be elabo-

rated on later in the article, the above findings empirically strongly suggest that the

influence of a gamblers skills, strategies, and personality on his/her successive bet

amounts and cumulative winnings/losses is almost totally reflected by the pattern(s) of his/

her winnings/losses in the first few initial games and his/her gambling account balance.

These conclusions at least partially invalidate gamblers various illusions and fallacies

(Clotfelter and Cook 1993), as detailed in the subsection Economic/Psychological

Models below, that they can outperform other gamblers or even bankers (e.g., casinos,

cyber-gambling websites, etc.) For gamblers, the implication is that it is not worthwhile to

excessively and endlessly experiment with their skills, strategies, and personalities in an

attempt to outperform other gamblers or even bankers. For casinos and cyber-gambling

websites, the implications are that their profits can be better maximized through

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emphasizing the provision of particular games and redesigning their games based on the

different predicted temporal trajectories of typical gamblers cumulative winnings/losses in

different games and/or games of different designs.

In addition, from a broader point of view, such neural networks can serve related

government policy-makers, gambling industry operators, economists, sociologists, psy-chiatrists, and psychologists in the manners discussed above. Such application of neural

networks also explores and exemplifies the usefulness of neural networks and artificial

intelligence at large (Hu et al. 2007; Kaklauskas and Zavadskas 2007; Schocken and Ariav

1994), apart from traditional parametric statistics, in the context of gambling research and

socio-economically and politically controversial issues in general.

The remainder of this article proceeds as follows. The section Literature discusses the

literature relevant to this articles research. The section Methodology develops the

neural networks underlying this research. The section Data delineates the empirical data

sample to be fed into the neural networks for training and testing, followed by the section

Results which interprets the results. The implications of the results are further elaborated

in the section Discussion and Conclusions.

Literature

Previous research into gamblers behavior and decisions can be subsumed under two

categories, which the author refers to as economic/psychological models and sociological

studies, respectively, and which are briefly outlined in the following subsections Eco-

nomic/Psychological Models and Sociological Studies. The subsequent subsectionWhat Gaps Does This Research Fill highlights the gaps in the foregoing previous

research that this article seeks to fill.

Economic/Psychological Models

Traditional economics assumes that human beings are rational agents with a view to

maximizing their advantage. In view of the huge expenses and profit of (Ladouceur 2004)

as well as government tax on casinos and gambling websites (American Gaming Asso-

ciation 2008c), it is crystal clear that the total losses of all gamblers on gambling at a

casino or a gambling website must substantially exceed their total winnings. In otherwords, gambling does not maximize gamblers advantage, so traditional economics per se

falls short of being able to explain gamblers decision to gamble in the first place.

Behavioral economics, however, as a relatively newer branch of economics (with Nobel

Prize laureate D. Kahneman as one exponent), assumes bounded-rationality of humans

with one consequence being that judgment under uncertainty often relies on heuristics

which are sometimes prone to biases (Kahneman 2003; Kahneman et al. 1982). Gamblers

decisions to gamble may be an exemplification of such judgments. Although not explicitly

articulated, Delfabbro et al. (2006) carried out empirical research to prove the existence of

particular gamblers mentalities that could be regarded as being implied by behavioraleconomists bounded-rationality assumption. Further to heuristics and biases, behavioral

economics theorized decisions under risk by the prospect theory (Kahneman 2003;

Kahneman and Tversky 1979), which critiqued and was an alternative to the rather dated

expected utility theory of traditional economics. The prospect theory introduced the con-

cept of decision weights, which took the place of probabilities in the expected utility

theory. Decision weights are generally lower than the corresponding probabilities, except

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in the range of low probabilities. In other words, low probabilities, say, of wins in gambles

are over-weighted in human decisions under risk, leading to decisions to take gambles or

the like.

A further direct consequence of bounded-rationality, heuristics, and biases is the various

illusions and fallacies of gamblers, which can be categorized as follows:

the misconceived dependence between successive gamble outcomes, which the famous

term gamblers fallacy is coined to describe (Clotfelter and Cook 1993; Ladouceur

2004), e.g., a win being reckoned to be very likely after a series of losses, a certain

recently occurred outcome being believed to be unlikely to recur in the near future,

illusions of control, e.g., opining that the gambler himself/herself has more control over

a gambles outcome than other competing gamblers or even the banker (Ladouceur

2004), and

superstitions, e.g., thinking that some rituals or other unrelated happenings bear on the

outcomes of gambles (Ladouceur 2004).All these three categories of illusions and fallacies of gamblers end up with gamblers

believing that their own skills, strategies, and rituals can beat competing gamblers or even

the bankers.

Sociological Studies

There has been research into the macroscopic, cross-sectional analysis of socio-economic

factors determining gamblers gambling propensity (Gill et al. 2006; Delfabbro et al. 2006;

Layton and Worthington 1999). Although there is also research that aims to take a lon-

gitudinal perspective, their actual contribution concerns comparing and contrasting gam-

blers gambling propensity before and after a certain event, such as the opening of a casino

(Jacques and Ladouceur 2006), in a macroscopic manner. The author is aware of no

research specifically performing longitudinal or time-series analysis to microscopically

model gamblers behavior and decisions as to their successive bet amounts. Again, whilst

there has been research into the macroscopic, cross-sectional analysis of how different

demographic factors or the like of individual gamblers impinge on their trajectories of

developing pathologic gambling (Nelson et al. 2006) and also research into the classifi-

cation of gamblers based on the trajectories of their gambling propensity over age (Vitaro

et al. 2004), the author is aware of no research microscopically modeling temporal tra-jectories of gamblers cumulative winnings/losses.

What Gaps Does This Research Fill?

It is more than indisputable that both the economic/psychological models and the socio-

logical studies depicted above have been substantively contributing to gambling research.

The economic/psychological models are primarily analytical in the sense of explaining the

motivation behind gambling by means of economic/psychological theories. In contrast, the

sociological studies are cross-sectional, macroscopic, or for the purpose of gamblers

classification and are consequently macroscopically modeling and predicting a gamblers

gambling propensity based on some socio-economic and demographic factors or the like.

So far, there is no research directly modeling gamblers behavior and decisions. Nor are

there any models that are quantitatively predictive in this aspect.

In summary, there has been no previous research on longitudinally modeling or pre-

dicting gamblers behavior and decisions microscopically and quantitatively as to their

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successive bet amounts. Nor is there any previous research on longitudinally modeling or

predicting the temporal trajectory of gamblers cumulative winnings/losses. This research

aims to fill this gap by developing such models.

In addition, as there are no grounds to assume linearity or a particular type of non-

linearity (e.g., exponential) when developing these models, traditional parametric model-ing techniques, which make such a presumption, are regarded as insufficient for these

models. As such, this research employed the non-parametric modeling technique of neural

networks, which is a type of artificial intelligence, on the grounds of their black-box

(sometimes referred to as model-free) nature, which does not presume linearity or any

particular type of non-linearity. Therefore, another contribution of this research is its

application of non-parametric modeling and artificial intelligence in this area as opposed to

traditional parametric modeling techniques, such as least squares regression, which are

widely used in the aforesaid previous research.

However, for the author, the most far-reaching contribution of this research is that the

various illusions and fallacies of gamblers, as enumerated in the subsection Economic/

Psychological Models above, can be partially invalidated by the models constructed in

this article. This research also explores and exemplifies the usefulness of neural networks

and artificial intelligence at large (Hu et al. 2007; Kaklauskas and Zavadskas 2007;

Schocken and Ariav 1994), apart from traditional parametric statistics, in the context of

gambling research and socio-economically and politically controversial issues.

Methodology

Like neural networks employed in forecasts for financial product pricing (Bennell and

Sutcliffe 2004), backpropagation neural networks are employed in this research. The

reason for this approach is that backpropagation neural networks are the most popular

neural network paradigm (Walczak 2001) and have long been demonstrated as universal

approximators (Hornik et al. 1989; White 1990). Moreover, past research has indicated that

backpropagation neural networks have superior performance to other neural network

paradigms (Walczak2001).

Shown in Fig. 1 below is the structure of the fully connected backpropagation neural

network adopted in this research for M1: the model for successive bet amounts. Also in

Fig. 1 are the definitions of the related variables. Simply speaking, this network tries topredict the normalized bet amount betik to be laid by a particular gambler k in the next

game i based on his/her normalized winnings/losses wini-1,k, wini-2,k,, wini-n,k in his/

her n immediately preceding games and his/her current gambling account balance bal_sik.

Such a predicted, normalized bet amount is denoted by betik in Fig. 1.

Whereas readers can refer to relevant literature on neural networks for details (Nikolaev

and Iba 2006; Ananda Rao and Srinivas 2003; Arbib 2003), the basic use of the neural

network in Fig. 1 is as in the following steps:

1. Collect and prepare an empirical data sample consisting ofwini-1,k, wini-2,k,, wini-

n,k, bal_sik and betik for game i = n ? 1, n ? 2, and for each gambler k.2. Out of the data sample in step 1, select a sub-sample for training the network. Here, a

sub-sample means wini-1,k, wini-2,k,, wini-n,k, bal_sik, and betik for some particular

set of selected (i, k)s.

3. Train the network by inputting wini-1,k, wini-2,k,, wini-n,k and bal_sik into it for

each (i, k) in the training sub-sample of step 2, calculate the mean square error (MSE)

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between the predicted output betik and betik over all (i, k)s in the training sub-sample,

and adjust the network parameters intrinsic to the nodes in the network (Nikolaev and

Iba 2006; Ananda Rao and Srinivas 2003; Arbib 2003) so as to minimize MSE which

is defined by

MSE 1NTR

X

i;k2trainingsub

-sample

betik betik2 and NTR the size of the training sub-sample:

4. Repeat step 3 above by a large number of episodes until the network converges, i.e.,

until its parameters stabilize.

5. Out of the data sample in step 1, select the remaining sub-sample (i.e., observations

not in the training sub-sample) for testing the network.

6. Test the network by inputting wini-1,k, wini-2,k,, wini-n,k, and bal_sik into it for each

(i, k) in the testing sub-sample of step 5, and calculating the mean magnitude of

relative error (MMRE) (Foss et al. 2003; Conte et al. 1986) between BETik (i.e., the

predicted output betik after denormalization) and BETik (i.e., betik after denormaliza-

tion) over all (i, k)s in the testing sub-sample to measure how accurately the network

predicts BETik where

MMRE 1NTE

X

i;k2testingsub-sample

BETik BETikBETik

and NTE the size of the testing sub-sample: 1

kiwin ,3

kiwin ,2

kiwin ,1

iksbal _

.

.

.

.

.

.

.

ikteb

wheremeans a neuron;

k = the gambler number;i = the game number, starting from n +

1 for each gambler k;

ikwin = the normalized (see later

paragraphs of this Section)ikWIN ;

iksbal _ = the normalized ikSBAL _ ;

ikteb = the normalized ikTEB ;

ikWIN = the winnings ofgambler k in his/her

game i if positive and the loss inhis/her game i if negative;

ikSBAL _ = the gambling account balance of

gambler kat the start of his/her

game i;

ikTEB = an estimate of

ikBET as predicted

by this neural network;

ikBET = the total bet amount of gambler k in

his/her game i, covering all roundsof bets in the game

Note 1; and

n = the number of each gamblerspreceding games winnings/lossesto be included as inputs into thisneural network.

input layer

hidden layer

output layer

kniwin ,

Note 1: A Texas Holdem game consistsof several rounds of betting.

Fig. 1 The structure of the neural network for M1

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The smaller the MMRE, the more accurately the network predicts.

Readers must have noticed the word normalized in Fig. 1. Normalization of the

inputs and outputs of the neural network here is necessary to guarantee that the values of all

the inputs and outputs are confined to a certain range, say, [0, 1], to which the network

would be tuned during training in steps 3 and 4 above, and thus no extrapolation wouldoccur afterwards (Nahvi and Esfahanian 2005; Drew 2004). Normalization of, say, WINikto the range [0, 1] is done by the formula below:

winik WINik WINminWINmax WINmin

where WINmin = the minimum ofWINikover all (i, k)s in the whole data sample of step 1;

and WINmax = the maximum of WINik over all (i, k)s in the whole data sample.

Conversely, denormalization is done by WINik= winik (WINmax - WINmin) ? WINmin.

Likewise, normalization and denormalization are done analogously for other variables.

Also, whilst the numbers of input layer neurons and output layer neurons in Fig. 1 are

obviously n ? 1 and 1, respectively, readers may notice that the number of hidden layer

neurons therein is free to choose. On the advice of Morphet (2004) and Azoff (1994), the

number of hidden layer neurons was set to

5 if the number of input layer neurons is less than 10, and

n1

2

1 otherwise where xb c = the largest integer less than x.Another kind of leeway is that a certain transfer function (Nikolaev and Iba 2006;

Ananda Rao and Srinivas 2003; Arbib 2003) is attached to each hidden and output layer

neuron, and such a transfer function is again free to choose, usually, among the pure-linearfunction, the log-sigmoid function, the sigmoid function, etc. Upon experiments on these

various functions, the sigmoid function was found to give rise to the highest prediction

accuracy or the smallest MMRE (as defined in (1)) for our data sample, so the sigmoid

function was chosen for all the hidden and output layer neurons. Likewise, the value n is

discretionary. Again the criterion for choosing its value was to render the highest pre-

diction accuracy or the smallest MMRE for our data sample. The section Results below

will show the MMREfor each value ofn within a reasonable range. By the same token, the

network inputs wini-1,k, wini-2,k,, wini-n,k, and bal_sik in Fig. 1 were not arbitrarily

determined at the very beginning of the research, but were the result of an extended

searching process of trial and error. In fact, beti-1,k, beti-2,k,, bal_si-1,k, bal_si-2,k,,etc., as network inputs were all tried but wini-1,k, wini-2,k,, wini-n,k, and bal_sik pre-

vailed in the end based on their highest prediction accuracy or the smallest MMREfor our

data sample.

Figure 2 below shows the structure of the fully connected backpropagation neural

network adopted in this research for M2: the model for the temporal trajectory of cumu-

lative winnings/losses. Also there are the definitions of the variables in addition to those

defined in Fig. 1. Simply speaking, this network tries to predict the normalized, cumulative

winnings/loss cwinik of a particular gambler k immediately after the next game i based

on his/her normalized, cumulative winnings/losses cwini-1,k

, cwini-2,k

,, cwini-m,k

immediately after each of his/her m immediately preceding games and his/her random walk

indicator RWIi-1,k immediately after the last game i - 1. Such a predicted, normalized,

cumulative winnings/loss is denoted by cwinik in Fig. 2. Again, of course, the network

inputs cwini-1,k, cwini-2,k,, cwini-m,k, and RWIi-1,k in Fig. 2 were not determined at the

very beginning but were the result of an extended searching process of trial and error

analogous to that for Fig. 1.

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The random walk indicator is recommended as a predictor in the field of finance for

predicting quantities of a cumulative nature. In view of cwinik in Fig. 2 being cumulative

albeit unrelated to finance, the random walk indicator RWIi-1,k of gambler k immediately

after his/her game i - 1 is included as an input to the neural network and is define by

RWIi1;k

Dcwini1;kD

cwinkffiffiffiffiffiffiffiffiffiffii 1p

2

where cwini-1,k= cwini-1,k- cwin0,k; cwin0,k= the normalized CWIN0,k; CWIN0,k=

the cumulative winnings of gambler k immediately after his/her game 0 (i.e., at the very

beginning before his/her game 1) if positive or his/her cumulative loss immediately after

his/her game 0 if negative : 0 by definition; Dcwink PNkp1

cwinp;kcwinp1;kj jNk

; and Nk= the

total number of consecutive games played by gambler k.

The basic use of the neural network in Fig. 2 is analogous to that in Fig. 1 except that

the inputs and outputs are different accordingly. Consequently, the MMREto measure how

accurately the network in Fig. 2 predicts cwinik becomes

MMRE 1NTE

X

i;k2testingsub-sample

cwinik cwinikcwinik

: 3

kicwin ,3

kicwin ,2

kicwin ,1

kiRWI ,1

.

.

.

.

.

.

.

iknicw

wheremeans a neuron;

k = the gambler number;i = the game number, starting from m + 1

for each gambler k;

ikcwin = the normalized ikCWIN ;

kiRWI ,1 = the random walk indicator of gambler

k immediately after his/her gamei - 1, as elaborated in laterparagraphs of this Section;

iknicw = the normalized

ikNICW ;

ikCWIN = =

i

p

pkWIN1

= the cumulative

winnings of gambler k immediately

after his/her game i if positive orhis/her cumulative lossimmediately after his/her game i ifnegative;

ikNICW = an estimate of ikCWIN as predicted

by this neural network; andm = the number of the gamblers

cumulative winnings/loss to beincluded as inputs into this neuralnetwork.

input layer

hidden layer

output layer

kmicwin ,

Fig. 2 The structure of the neural network for M2

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Data

Like all backpropagation neural networks, those used in this research required an empirical

data sample for training and later testing of prediction accuracy. Such a data sample was

collected from a legally licensed, well-established cyber-gambling website www.betxxxx.com, part of the websites name being concealed by the xxxx for anonymity. There,

international gamblers uniquely identified by their registered identifiers played poker games

like Texas Holdem, Omaha High, Omaha Hi-Lo, and Seven Card Stud round the clock. This

website was chosen because it was one of the very few cyber-gambling websites where the

unique identifiers of all gamblers in an ongoing game together with the gamblers origins

(i.e., cities or countries), their successive bet amounts, their decisions (e.g., to fold, raise,

etc.), and their gambling account balances were all displayed. This research focused on

Texas Holdem because of the top popularity of this family of games in major gambling

markets (Clark 2006) so as to make our research the most representative of general gam-

blers behavior and decisions.

The collection of the data sample involved two stages, in the first of which ongoing

Texas Holdem games at the aforementioned website were videoed using a screen capture

software package. In the second stage, such videos were replayed to identify gamblers who

played a long enough series of consecutive games for the purpose of our longitudinal

modeling. It was found that most gamblers played for a short while, whereas only a small

proportion of them played as many as around 100 consecutive games, which seemed to be

the practical maximum. In fact, from the long videos we managed to identify only six such

persistent gamblers, totaling 675 games. For each of these persistent gamblers, his/her

bet amounts, gambling account balances, etc., throughout his/her series of games werenoted during the replays. As an example, Table 1 is an abridged copy of such data for one

of the persistent gamblers whose identifier was Trevor38. Subsequently, Table 2 shows

the data derived from Table 1 for the construction of M1: the model for successive bet

amounts with the number of preceding games winnings/losses n = 6 as an example, the

data here being before normalization (see the section Methodology above) for easier

visualization. Similarly, Table 3 shows the data derived from Table 1 for the construction

of M2: the model for the temporal trajectory of cumulative winnings/losses with the

number of preceding games cumulative winnings/losses m = 6 as an example, again the

data here being before normalization (see the section Methodology above) for easier

visualization.

Results

M1: The Model for Successive Bet Amounts

Table 4 shows the MMREs (as defined in Eq. 1) of M1 with n = 6, 7,, 20. They are all in

the order of 10-2 and thus are rather small, especially when n = 6, 7, 8, 13, or 14,

indicating rather accurate prediction ofBETik by M1. In fact, their small magnitudes imply

that the mean magnitude of the error in the prediction of BETik is only around 13% of the

magnitude of BETik. It is noteworthy that analogous to any other neural networks for

prediction purposes, the neural network for M1 only tries to predict the main trend of a

gamblers normalized bet amount betik by picking up the pattern(s) (presumably non-

linear) as reflected in the predictors wini-1,k, wini-2,k,, wini-n,k, and bal_sik, which

concern the gamblers normalized winnings/losses in his/her n immediately preceding

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games and his/her current gambling account balance. Nevertheless, the network is notcapable of tracking any extraordinary bet amount laid by the gambler in a particular game,

say, due to his/her exceptionally strong hand of cards in that game when he/she would tend

to bet exceptionally heavily or vice versa. Therefore, it is true that occasionally the

magnitude of relative error MRE BETikBETikBETik

for some game i played by gambler kin our

testing sub-sample was as large as over 1 or 2. The foregoing MMREs in the order of 10-2

Table 1 Data collected for gambler Trevor38 whose gambler number k= 1

Game (i) BAL_Si1 BAL_Mi1 BETi1 =

BAL_Si1 - BAL_Mi1

BAL_Ei1 = BAL_Si?1,1 WINi1 =

BAL_Ei1 - BAL_Si1

1 90.16 89.16 1.00 89.16 -1.002 89.16 87.66 1.50 87.66 -1.50

3 87.66 87.66 0.00 87.66 0.00

4 87.66 87.41 0.25 87.41 -0.25

5 87.41 86.66 0.75 88.08 ?0.67

6 88.08 86.83 1.25 86.83 -1.25

7 86.83 86.83 0.00 86.83 0.00

8 86.83 86.83 0.00 86.83 0.00

9 86.83 85.83 1.00 85.83 -1.00

10 85.83 85.33 0.50 85.33 -0.50

11 85.33 83.83 1.50 88.03 ?2.70

12 88.03 87.78 0.25 87.78 -0.25

13 87.78 87.53 0.25 87.53 -0.25

14 87.53 86.78 0.75 86.78 -0.75

15 86.78 86.78 0.00 86.78 0.00

16 86.78 86.53 0.25 86.53 -0.25

17 86.53 85.53 1.00 85.53 -1.00

18 85.53 85.03 0.50 85.03 -0.50

19 85.03 84.78 0.25 84.78 -0.25

20 84.78 82.78 2.00 89.38 ?4.60

: : : : : :

: : : : : :

BAL_Si1 = Trevor38s gambling account balance at the start of his game i

BAL_Mi1 = Trevor38s gambling account balance in the middle of his game i after laying all rounds of

bets in the game

BETi1 = Trevor38s total bet amount in his game i, covering all rounds of bets

BAL_Ei1 = Trevor38s gambling account balance at the end of his game i, which is equal to BAL_Mi1plus the received winnings in game i, if any, noting that here the received winnings refers to the money

received upon winning the gameWINi1 = Trevor38s winnings in his game i if positive and the loss in game i if negative, noting that: (1)

here the winnings differ from the received winnings mentioned in the definition of BAL_Ei1 above in that, in

a game, the winnings (WINi1) is the received winnings net of the total bet (BETi1), and (2) here the loss must

be equal to BETi1 upon losing the game

Note: The currency for the games played by Trevor38 as listed in this table happened to be the Great

Britain pound (GBP), so all monetary variables in this table are in GBP. Nevertheless, the international

gambling website www.betxxxx.com underlying this research has games in various other currencies, e.g.,

US dollars (USD), euros (EUR), etc. It is obvious that the validity of this research is not confined to any

particular currency

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are just the means of the MREs averaged over all the games played by all the gamblers in

our testing sub-sample. In other words, the networks prediction tracks the overall trend of

any gamblers successive bet amounts well even if occasionally there are considerable

prediction errors (or residuals if deliberately using the terminologies of statistical

regression, which is a more popularly used prediction method than neural networks).

Similar things are, for example, true of financial market prediction models which are,

however, regarded as acceptably useful and accurate if they are able to track the overalltrend of the market well, even if they fail severely during periods of financial turmoil.

In addition, it is worthwhile to point out that only one single trained neural network for

M1 was devised by using our training sub-sample covering all the six gamblers, and the

high prediction accuracy above was achieved by applying this identically trained neural

network to our testing sub-sample, which also included all the six gamblers. This implies

that the relationship between the main trend of betik and the pattern(s) in the predictors

wini-1,k, wini-2,k,, wini-n,k, and bal_sik as represented intrinsically by the post-training

parameters inside this trained neural network, is almost the same for all the six gamblers in

our data sample. In other words, if there is any difference in the main trend ofbetik between

the six gamblers, such idiosyncrasy is reflected not predominately by the aforesaid

relationship but almost totally by the difference in the pattern(s) of wini-1,k,

wini-2,k,, wini-n,k, and bal_sik between these gamblers. The influence of a gamblers

skills, strategies, and personality on his/her successive bet amounts is almost totally

reflected by the pattern(s) of his/her winnings/losses in the several immediately preceding

games and his/her gambling account balance.

Table 2 Data derived from Table 1 for the construction of M1 with, say, n = 6

Game (i) Using game i of Trevor38 to either train or test M1

Upon normalization, these are inputs Predicted output,beti1; upon

denormalization,

estimatesWINi-6,1 WINi-5,1 WINi-4,1 WINi-3,1 WINi-2,1 WINi-1,1 BAL_Si1 BETi1

7 -1.00 -1.50 0.00 -0.25 ?0.67 -1.25 86.83 0.00

8 -1.50 0.00 -0.25 ?0.67 -1.25 0.00 86.83 0.00

9 0.00 -0.25 ?0.67 -1.25 0.00 0.00 86.83 1.00

10 -0.25 ?0.67 -1.25 0.00 0.00 -1.00 85.83 0.50

11 ?0.67 -1.25 0.00 0.00 -1.00 -0.50 85.33 1.50

12 -1.25 0.00 0.00 -1.00 -0.50 ?2.70 88.03 0.25

13 0.00 0.00 -1.00 -0.50 ?2.70 -0.25 87.78 0.25

14 0.00 -1.00 -0.50 ?2.70 -0.25 -0.25 87.53 0.75

15 -1.00 -0.50 ?2.70 -0.25 -0.25 -0.75 86.78 0.00

16 -0.50 ?2.70 -0.25 -0.25 -0.75 0.00 86.78 0.25

17 ?2.70 -0.25 -0.25 -0.75 0.00 -0.25 86.53 1.00

18 -0.25 -0.25 -0.75 0.00 -0.25 -1.00 85.53 0.50

19 -0.25 -0.75 0.00 -0.25 -1.00 -0.50 85.03 0.25

20 -0.75 0.00 -0.25 -1.00 -0.50 -0.25 84.78 2.00

: : : : : : : : :

: : : : : : : : :

Note: The variables in the headings of this table are as defined in Table 1

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Table3

Dataderive

dfrom

Table1fortheconstructionofM2with,say,m=

6

Game(i)

UsinggameiofTrevor38toeither

trainortestM2

Uponnormalization,

theseareinputs

Predictedoutput,

CW^INi1;

upondenormalization,estimates

CWIN

i-6,

1

CWIN

i-5,

1

CW

INi-4,

1

CWIN

i-3,

1

C

WIN

i-2,

1

CWIN

i-1,

1

RWIi

-1,

1

CWIN

i1

7

-1.0

0

-2.5

0

-2

.50

-2.7

5

-

2.0

8

-3.3

3

*

-3.3

3

8

-2.5

0

-2.5

0

-2

.75

-2.0

8

-

3.3

3

-3.3

3

*

-3.3

3

9

-2.5

0

-2.7

5

-2

.08

-3.3

3

-

3.3

3

-3.3

3

*

-4.3

3

10

-2.7

5

-2.0

8

-3

.33

-3.3

3

-

3.3

3

-4.3

3

*

-4.8

3

11

-2.0

8

-3.3

3

-3

.33

-3.3

3

-

4.3

3

-4.8

3

*

-2.1

3

12

-3.3

3

-3.3

3

-3

.33

-4.3

3

-

4.8

3

-2.1

3

*

-2.3

8

13

-3.3

3

-3.3

3

-4

.33

-4.8

3

-

2.1

3

-2.3

8

*

-2.6

3

14

-3.3

3

-4.3

3

-4

.83

-2.1

3

-

2.3

8

-2.6

3

*

-3.3

8

15

-4.3

3

-4.8

3

-2

.13

-2.3

8

-

2.6

3

-3.3

8

*

-3.3

8

16

-4.8

3

-2.1

3

-2

.38

-2.6

3

-

3.3

8

-3.3

8

*

-3.6

3

17

-2.1

3

-2.3

8

-2

.63

-3.3

8

-

3.3

8

-3.6

3

*

-4.6

3

18

-2.3

8

-2.6

3

-3

.38

-3.3

8

-

3.6

3

-4.6

3

*

-5.1

3

19

-2.6

3

-3.3

8

-3

.38

-3.6

3

-

4.6

3

-5.1

3

*

-5.3

8

20

-3.3

8

-3.3

8

-3

.63

-4.6

3

-

5.1

3

-5.3

8

*

-0.7

8

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

:

CWINi1

Pi p1

WINp1

=

Trevor38scumulativewinn

ingsimmediatelyafterhisgameiifpositiveorhiscumulativelossimmediatelyafterhisgameiifnegative;and

RWIi

-1,

1

=

Trevor3

8srandomwalkindicator(seethesectionMethodologyabove)immediatelyafterhisgamei-

1.

Itsvalueforeachgameisomittedhereforthe

sakeofsimplicity,giventhatitisnotmeaningfultoinc

ludeithereinthatitcannotbede

rivedtriviallyfromTable1butca

nonlybecalculatedusingthecom

plicatedEq.

2

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M2: The Model for the Temporal Trajectory of Cumulative Winnings/Losses

Table 5 is an M2 counterpart of Table 4 where the MMREs here are as defined in Eq. 3.

The MMREs are all in the order of 10-3

or 10-2

and thus are rather small, especially whenm = 6, 7, 8, or 13, indicating quite an accurate prediction of CWINik by M2. In fact, their

small magnitudes imply that the mean magnitude of the error in the prediction ofCWINik is

only around 0.84% of the magnitude of CWINik. When interpreting such MMREs, readers

are reminded to refer to the noteworthy point stated in the case of M1 above.

In addition, it is worthwhile to point out that only one single trained neural network for

M2 was devised by using our training sub-sample covering all the six gamblers, and the

high prediction accuracy above was achieved by applying this identically trained neural

network to our testing sub-sample, which also included all the six gamblers. This implies

that the relationship between the main trend of CWINik and the pattern(s) in the predictors

cwini-1,k, cwini-2,k,, cwini-m,k, and RWIi-1,k, as represented intrinsically by the post-training parameters inside this trained neural network, is almost the same for all the six

gamblers in our data sample. In other words, if there is any difference in the main trend of

cwinik between the six gamblers, such idiosyncrasy is reflected not predominately by the

aforementioned relationship but almost totally by the difference in the pattern(s) of

cwini-1,k, cwini-2,k,, cwini-m,k, and RWIi-1,k between these gamblers. That is to say, the

influence of a gamblers skills, strategies, and personality on his/her cumulative winnings/

losses is almost totally reflected by the pattern(s) of his/her cumulative winnings/losses in

the several immediately preceding games.

Discussion and Conclusions

Summarizing all its regulations, a Texas Holdem game involves only one type of behavior

or decisions of a participating gambler, which is the amount of bet to be laid in the game

(Harrington and Robertie 2004). Even the decisions to fold and to raise are just particular

Table 5 MMREs for M2 with m = 6, 7,, 20

n 6 7 8 9 10 11 12 13

MMRE 0.0079 0.0082 0.0083 0.0103 0.0116 0.012 0.015 0.0096

14 15 16 17 18 19 20

MMRE 0.0188 0.0191 0.0143 0.0204 0.0414 0.0346 0.0251

Bold values represent the best results

Table 4 MMREs for M1 with n = 6, 7,, 20

n 6 7 8 9 10 11 12 13

MMRE 0.01 0.0103 0.0138 0.0171 0.026 0.0208 0.0174 0.014

14 15 16 17 18 19 20

MMRE 0.013 0.0241 0.0226 0.0227 0.0235 0.0183 0.0322

Bold values represent the best results

J Gambl Stud

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cases to freeze the current bet amount and to increase the current bet amount, respectively.

If two gamblers lay the same amount of bet in each identical game (with the same hand,

etc.), then it can be said that the behavior or decisions of the two gamblers are identical or,

equivalently, the skills, strategies, and personalities of the two gamblers are identical as far

as Texas Holdem games are concerned. The findings from the section Results above forM1: The Model for Successive Bet Amounts are that all gamblers successive bet

amounts in the 675 Texas Holdem games of our data sample rather strictly obey the

longitudinal model M1 constructed in this research. Given that M1 relates a gamblers bet

amount in a game with his/her gambling account balance at the beginning of the game and

the winnings/losses in his/her few (n) immediately preceding games, it can be concluded

that if some gamblers in our data sample have the same initial gambling account balance

and the same winnings/losses in their few (n) initial games, such gamblers will give rise to

almost the same subsequent time-series of bet amounts as predicted by M1. In other words,

such gamblers will have almost identical behavior and decisions, or equivalently, they have

almost identical skills, strategies, and personalities as far as Texam Holdem games are

concerned. Hence, such gamblers will lead to almost the same gambling winnings/losses in

the long run. Even if there is a difference, it will be negligible as attested by the small

MMREin Table 4. In general, in respect of the aforesaid gamblers behavior and decisions,

the only substantial idiosyncrasy of an individual gambler lies in his/her gambling account

balance and the pattern(s) of the winnings/losses in his/her few (n) initial games, through

which his/her skills, strategies, and personality are practically fully reflected.

Even if a more result-oriented viewpoint is taken to focus only on the cumulative

winnings/loss just as most real-world gamblers do, a similar conclusion can be drawn. The

findings from the section Results above for M2: The Model for the Temporal Trajectoryof Cumulative Winnings/Losses are that all gamblers temporal trajectories of cumulative

winnings/losses in the 675 Texas Holdem games of our data sample quite strictly obey the

longitudinal model M2 constructed in this research. Given that M2 relates a gamblers

cumulative winnings/loss immediately after a game with his/her cumulative winnings/loss

immediately after each of his/her few (m) immediately preceding games, it can be con-

cluded that if some gamblers in our data sample have the same initial cumulative winnings/

loss immediately after each of their few (m) initial games and thus have the same random

walk indicators, such gamblers will give rise to almost the same subsequent time-series of

cumulative winnings/losses as predicted by M2. Even if there is a difference, it will be

negligible as attested by the small MMREin Table 5. In general, in respect of the foregoingcumulative winnings/losses, the only substantial idiosyncrasy of an individual gambler lies

in the pattern(s) of the cumulative winnings/losses in his/her few (m) initial games, through

which his/her skills, strategies, and personality are practically fully reflected.

A combined further conclusion from the last two paragraphs is that a gamblers skills,

strategies, and personality are practically fully reflected in the few (n or m) initial games.

Beyond these few initial games, both his/her behavior and decisions (in terms of the bet

amounts in successive games) and his/her cumulative winnings/losses are fairly accurately

predictable by M1 or M2. In other words, on the one hand, gamblers behavior, decisions,

and cumulative winnings/losses have substantive commonality among different gamblersin that the inherent relationship between the inputs and outputs of M1 and that between

those of M2 are almost the same among different gamblers. On the other hand, the only

idiosyncrasies of individual gamblers are exhibited by the pattern(s) of the winnings/losses

in each gamblers few (n or m) initial games, through which the influence of his/her skills,

strategies, and personalities on his/her behavior, decisions, and cumulative winnings/losses

is practically fully reflected. Therefore, this conclusion partially invalidates gamblers

J Gambl Stud

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various illusions and fallacies (Clotfelter and Cook 1993), as detailed in the subsection

Economic/Psychological Models above, that they can outperform other gamblers or

even bankers through specific skills, strategies, and personalities. Although the conclusion

here cannot rule out the possibility of individual gamblers idiosyncratic performance

ascribable to their specific skills, strategies, and personalities, an implication for gamblersis that it is not worthwhile or even relevant to excessively and endlessly experiment beyond

n or m games with the gamblers skills, strategies, and personalities in an attempt to

outperform other gamblers or even bankers.

From casinos and cyber-gambling websites viewpoint, the implications of this research

are that their profits can be better maximized through emphasizing the provision of par-

ticular games and redesigning their games based on the different predicted temporal tra-

jectories of typical gamblers cumulative winnings/losses in different games and/or games

of different designs as empirically verified by M2.

From a broader viewpoint, such models can assist the work of related government

policy-makers, gambling industry operators, economists, sociologists, psychiatrists, and

psychologists. In particular, through these models, they all can realize and even simulate

the temporal trajectory of a general gamblers losing (or, less likely, winning) money in

such a gambling game as Texas Holdem in this paper and thus quantitatively determine

gamblings political, financial, economic, social, psychiatric, and psychological contribu-

tions and problems in respect of the individual gambler himself/herself and society at large.

Since such models can be applied generically to other gambling games by just feeding into

it empirical data samples of gamblers bet amounts in successive games, etc., in such other

gambling games, different trajectories for different gambling games can be derived for

comparison and contrast between their aforesaid contributions and problems.Such models also explore and exemplify the usefulness of neural networks and artificial

intelligence at large (Hu et al. 2007; Kaklauskas and Zavadskas 2007; Schocken and Ariav

1994), apart from traditional parametric statistics, in the context of gambling research and

socio-economically and politically controversial issues in general.

These models together with its predicted gamblers behavior and decisions can also

serve well as an educational tool for gamblers, in particular, pathological gamblers

undergoing rehabilitation, hopefully helping to partially uproot their irrational illusions

and fallacies, as elaborated on in the subsection Economic/Psychological Models

above, that they can outperform other gamblers or even bankers through their specific

skills and strategies.

Acknowledgments The authors thank Macao Polytechnic Institute for generous financial support under

Research Grant RP/ESCE-6/2004 and Dr. Jonathan Fearon-Jones for his proof-reading. Thanks are also due

to Kai Piu Benny Chan and Chin Wa Tam for their assistance in data collection, data analysis, etc.

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Using Neural Networks to Model the Behavior and Decisions of Gamblers in Particular Cyber Gamblers

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