same time, the “shifted” characteristics here lead to more bets on small odds, which also indicates that the gamblers over-estimate the winning chance of high-probability games. As a result, they under-weight the winning chances of mild-probability games. These are clear empirical evidence of probability weighting among gamblers, which is believed to be one of the fundamental mechanisms in economics [@Barberis2012].

Wealth distribution {#wealth-distribution .unnumbered}
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In the previous study of skin gambling [@Wang2018], we pointed out that the wealth distribution of skin gamblers shows a pairwise power-law tail. This time, by considering the players’ deposits to their wallets on a gambling site as the wealth data, we find that the pairwise power-law tails are also observed for bitcoin gambling. We find that on the gambling website Coinroll, starting from $5660$ cents, the players’ wealth distribution follows a pairwise power-law distribution, with the power of the first regime to be $1.585$, and the power of the second regime to be $3.258$, see Fig. \[fig:wealth\]. The crossover happens at $1.221\times 10^5$ cents. As both wealth distributions of skin gambling and bitcoin gambling can be approximated by a pairwise power distribution, we believe that it is a good option for modeling the tails 