discussed separately. We plot the complementary cumulative distribution function (CCDF) of the empirical data and the fitted distribution to check the goodness-of-fit, see Fig. \[fig:wager\_distribution\]. CCDF, sometimes also referred to as the survival function, is given by $\bar{F}(x) = P(X>x) = 1 - P(X\le x)$.

It turns out that when players are allowed to place arbitrary wagers (games A-G in Table \[table:wager-distribution\]), the wager distributions can in general be best-fitted by log-normal distributions. In particular, in games (A, B, C, E, F, G), the wager distribution can be approximated by the following expression $$\label{eq:log-normal}
\displaystyle P(x) =
\frac{\displaystyle\Phi\left(\frac{\ln(x+1)-\mu}
{\sigma}\right)-\Phi\left(\frac{\ln(x)-\mu}{\sigma}\right)}
{\displaystyle 1-\Phi\left(\frac{\ln(x_\text{min})-\mu}{\sigma}\right)},$$ with $x_\text{min}\le x$ and $\sigma>0$. $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution. Meanwhile in game (D), the fitted log-normal distribution is truncated at an upper boundary $x_\text{max}$, which might result from the maximum allowed small bet value and the huge variation of the market price of crypto-currencies.

During model selection, we notice that when we select different $x_\text{min}$, occasionally a power-law distribution with exponential cutoff is reported to be a better fit, but often it does not provide a decent absolute fit on the tail, and overall the log-normal distribution provides smaller Kolmogorov-Smirnov distances, see 