game).

![\[fig:odds\] Odds distributions can be well-fitted by truncated shifted power-law distributions. ](figure3.pdf){width=".75\columnwidth"}

It turns out that in both cases the odds can be modeled with a truncated shifted power-law distribution, $$\displaystyle P(m) = \left\lbrace \begin{split}
\displaystyle &\frac{\left(\ m - \delta \right)^{-\alpha}}{\zeta(\alpha, \ m_\text{min} -\delta)},
\ &\text{for}\ m_\text{min}\le m < m_\text{max}, \\
\displaystyle &\frac{\zeta(\alpha,\ m_\text{max} -\delta)}{\zeta(\alpha, \ m_\text{min} -\delta)},
\ &\text{for}\ m = m_\text{max},
\end{split}\right.$$ where $\zeta(\cdot,\cdot)$ is the incomplete Zeta function, and $m_\text{max}$ is the upper truncation. Note that there is a jump at $m_\text{max}$, 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 $m_\text{min}=1.15$ for csgofast-Jackpot on CSGOFAST.COM, whereas for Coinroll the parameters are found to be $\alpha = 1.423$, $\delta = 2.217$, and $m_\text{min} = 2.58$. From the comparison between the CCDFs of empirical data and fitting curves, as shown in Fig. \[fig:odds\], 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. A distribution that is close to a power law indicates that a gambler’s free choice of odds 