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Introduction. Today, gambling is a huge industry with a huge social impact. According to a report by the American Gaming Association 1 , commercial casinos in the United States alone made total revenue of over 40 billion US dollars in 2017. On the other hand, different studies reported that 0.12%–5.8% of the adults and 0.2–12.3% of the adolescents across different countries in the world are experiencing problematic gambling 2,3 . Studying the gamblers’ behavior patterns not only contributes to the prevention of problematic gambling and adolescent gambling, but also helps to better understand human decision-making processes. Researchers have put a lot of attention on studying gambling-related activities. Economists have proposed many theories about how humans make decisions under different risk conditions. Several of them can also be applied to model gambling behaviors. For example, the prospect theory introduced by Kahneman and Tversky 4 and its variant cumulative prospect theory 5 have been adopted in modeling casino gambling 6 . In parallel to the theoretical approach, numerous studies focus on the empirical analysis of gambling behaviors, aiming at explaining the motivations behind problematic gambling behaviors. However, parametric models that quantitatively describe empirical gambling behaviors are still missing. Such models can contribute to evaluating gambling theories proposed by economists, as well as yield a better understanding of the gamblers’ behaviors. Our goal is to provide such a parametric model for describing human wagering activities and risk attitude during gambling from empirical gambling logs. However, it is very difficult to obtain gambling logs from traditional casinos, and it is hard to collect large amounts of behavior data in a lab-controlled environment. Therefore in this paper we will focus on analyzing online gambling logs collected from online casinos. Whereas historically the development of probability theory, which then became the foundation of statistics, was tied to chance games, today we use statistical tools to analyze gamblers’ behaviors, roulette game roulette. Recent years have seen an increasing trend of online gambling due to its low barriers to entry, high anonymity and instant payout. For researchers of gambling behaviors, online gambling games present two advantages: simple rules and the availability of large amounts of gambling logs. In addition to the usual forms of gambling games that can be found in traditional casinos, many online casinos also offer games that follow very simple rules, which makes analyzing the gambling behavior much easier as there are much fewer degrees of freedom required to be considered. On the other hand, many online casinos have made gambling logs publicly available on their websites, mainly for verification purposes, which provides researchers with abundant data to work on. Due to the high popularity of online gambling, in a dataset provided by an online casino there are often thousands or even hundreds of thousands of gamblers listed. Such a large scale of data can hardly be obtained in a lab environment. Prior research has begun to make use of online gambling logs. For example, Meng’s thesis 7 presented a pattern analysis of typical gamblers in Bitcoin gambling. It is worth arguing that although our work only focuses on the behaviors of online gamblers, there is no reason to think that our conclusions cannot be extended to traditional gamblers. Naturally, we can treat the changing cumulative net income of a player during their gambling activities as a random walk process 8 . We are particularly interested in the diffusive characteristics of the gambler’s net income. This is another reason why we want to analyze the wager distribution and risk attitude of gamblers, since both distributions are closely related to the displacement distribution for the gambler’s random walks. Within this paper, we will mainly focus on the analysis at the population level. Physicists have long been studying diffusion processes in different systems, and recently anomalous diffusive properties have been reported in many human activities, including human spatial movement 9,10,11 , and information foraging 12 . In a previous study of skin gambling 8 , we have shown that in a parimutuel betting game (where players gamble against each other), a gambler’s net income displays a crossover from superdiffusion to normal diffusion. We have reproduced this crossover in simulations by introducing finite and overall conserved gamblers’ wealth (see 13 for a different way of modeling this using kinetic equations of Boltzmann and Fokker-Planck type). However, this explanation cannot be used in other types of gambling games where there is no interaction among gamblers (e.g., fixed-odds betting games, which will be introduced below), as they violate the conservation of gamblers’ overall wealth. In this paper, we want to expand the scope of our study to more general gambling games, check the corresponding diffusive properties, and propose some explanations for the observed behaviors. One of our goals is to uncover the commonalities behind the behavior of online gamblers. To implement this, we analyze the data from different online gambling systems. The first one is skin gambling, where the bettors are mostly video game players and where cosmetic skins from online video games are used as virtual currency for wagering 8,14 . The other system is crypto-currency gambling, where the bettors are mostly crypto-currency users. Different types of crypto-currencies are used for wagering. Commonly used crypto-currencies include Bitcoin, Ethereum, and Bitcoin Cash, whose basic units are BTC, ETH and BCH, respectively. As the overlap of these two communities, video game players and crypto-currency users, is relatively small for now, features of gambling patterns common between these two gambling systems are possibly features common among all online gamblers. Not only do we consider different gambling systems, but we also discuss different types of gambling games. In this paper, we discuss four types of solely probability-based gambling games (Roulette, Crash, Satoshi Dice and Jackpot), whose outcomes in theory will not benefit from the gamblers’ skill or experience when the in-game random number generators are well designed. In general, there are two frameworks of betting in gambling: fixed-odds betting, where the odds is fixed and known before players wager in one round; and parimutuel betting, where the odds can still change after players place the bets until all players finish wagering. In fixed-odds betting, usually players bet against the house/website, and there is no direct interaction among players; and in parimutuel betting, usually players bet against each other. The four types of games we discuss in this paper will cover both betting frameworks (see the Methods section). When a player attends one round in any of those games, there are only two possible outcomes: either win or lose. When losing, the player will lose the wager they placed during that round; whereas when winning, the prize winner receives equals their original wager multiplied by a coefficient. This coefficient is generally larger than 1, and in gambling terminology, it is called odds in decimal format 15,16 . Here we will simply refer to it as odds. Note that the definition of odds in gambling is different than the definition of odds in statistics, and in this paper we follow the former one. When a player attends one round, their chance of winning is usually close to, but less than the inverse of the odds. The difference is caused by the players’ statistical disadvantage in winning compared to the house due to the design of the game rules. In addition, the website usually charges the winner with a site cut (commission fee), which is a fixed percentage of the prize. We further define the payoff , o p , to be the net change of one player’s wealth after they attend one round. Although the four types of games are based on different rules, the payoffs all follow the same expression. where \(b > 0\) is the wager the player places, \(m > 1\) is the odds, \(1 > \eta \ge 0\) corresponds to the site cut, and f m is a non-negative value based on the odds representing the players’ statistical disadvantage in winning, as mentioned earlier. At least either \(\eta \) or f m are non-zero.
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