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In general, the distributions are symmetric with respect to the y-axis, except in games ( D , F ). The x -coordinate \( _( _/ _ )\) is proportional to the parameter ν . Meanwhile, the high positive auto-correlations, along with the higher probability densities at small integers/half-integers and their inverses, provide evidence that gamblers often follow a multiplicative process when wagering. The multiplication process can be explained by the wide adoption of multiplicative betting systems. “Betting system” here refers to the strategy of wagering where the next bet value depends on both the previous bet value and the previous outcome 22,23 . Although betting systems will not provide a long-term benefit, as the expected payoff will always be 0 in a fair game, still they are widely adopted among gamblers. A well-known multiplicative betting system is the Martingale (sometimes called geometric progression) 23 . In Martingale betting, starting with an initial wager, the gambler will double their wager each time they lose one round, and return to the initial wager once they win. Martingale is a negative-progression betting system where the gambler will increase their wager when they lose and/or decrease their wager when they win. Apart from multiplicative betting, there are many other types of betting systems, such as additive betting and linear betting 23 . The reasons why multiplicative betting systems are dominant in our datasets are: 1) Martingale is a well-known betting system among gamblers; 2) Many online gambling websites provide a service for changing the bet value in a multiplicative way. For example, for the Crash games csgofast-Crash (C) and ethCrash (D), both websites provide a simple program for automatically wagering in a multiplicative way. For the Roulette games and Coinroll (F), the websites provide an interface with which the gambler can quickly double or half their wager. However, for Satoshi Dice (E) and csgospeed-Jackpot (G), no such function is provided, yet we still observe similar results, indicating that gamblers will follow a multiplicative betting themselves. Figure 2 provides us with the distribution of ν , however, it will not tell us whether the gamblers adopt the negative/positive-progression betting systems. Therefore we further analyze the effect on the bet values of winning/losing a round. How the gamblers adjust their wager after winning/losing rounds is shown in Table 3. We can see that although there is a high probability for sticking to the same bet values, the most likely outcome after losing a round is that the gambler increases their wager. When winning one round, gamblers are more likely to decrease their wager. This means that negative-progression strategies are more common among gamblers than positive-progression strategies. Table 3 Statistics about how gamblers change their bet values after winning/losing rounds, roulette portomaso बिटकॉइन कैसीनो बिटकॉइन लाइव कैसीनो. Risk attitude. We now turn to the following question: When a player is allowed to choose the odds themselves in a near-fair game, how would they balance the risk and potential return? Higher odds means a lower chance of winning and higher potential return, for example, setting odds of 10 means that the winning chance is only 1/10, but the potential winning payoff equals 9 times the original wager. In our analysis, we can examine such behaviors based on the gambling logs from Crash and Satoshi Dice games. For the Crash game only CSGOFAST.COM provides the player-selected odds even when players lose that round, whereas for the Satoshi Dice game only Coinroll accepts player-selected odds. We will therefore focus on the data collected on these two websites. For the Crash game on CSGOFAST.COM, the odds can only be set as multiples of 0.01, whereas for the Satoshi Dice game on Coinroll the odds can be set to 0.99·65536/ i where i is a positive integer less than 64000. To simplify our modeling work, we will convert the odds on Coinroll to be multiples of 0.01 (same as for the Crash game). It turns out that in both cases the odds can be modeled with a truncated shifted power-law distribution, where \(\zeta (\cdot ,\cdot )\) is the incomplete Zeta function, and m max is the upper truncation. Note that there is a jump at m max , meaning that the players are more likely to place bets on the maximum allowed odds than on a slightly smaller odds. The estimated parameters \(\alpha =1.881\) , \(\delta =0.849\) , and \(_ >>=1.15\) for csgofast-Jackpot on CSGOFAST.COM, whereas for Coinroll the parameters are found to be \(\alpha =1.423\) , \(\delta =2.217\) , and \(_ >>=2.58\) . From the comparison between the CCDFs of empirical data and fitting curves, as shown in Fig. 3, we can see that the truncated shifted power law can capture the overall decaying trends of odds distribution. The stepped behavior results from the gamblers’ preference of simple numbers.
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