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 $$\label{eq:payoff}
\displaystyle o_p = \left\lbrace \begin{split}
\displaystyle &-b, &\text{with\ probability\ } &p= 1- \frac{1}{m} + f_\text{m}, \\
\displaystyle &(1-\eta)(m-1) b, &\text{with\ probability\ } &q=1-p = \frac{1}{m} - f_\text{m},
\end{split}\right.$$ where $b>0$ is the wager the player places, $m>1$ is the odds, $1>\eta\geq 0$ corresponds to the site cut, and $f_\text{m}$ is a non-negative value based on the odds representing the players’ statistical disadvantage in winning, as mentioned earlier. At least either $\eta$ or $f_\text{m}$ are non-zero.

From Eq. (\[eq:payoff\]), we can obtain the expected payoff of attending one round $$\label{eq:expected-payoff}
\displaystyle E\left( o_p|m, b\right)
= \Big(- (1-1/m +f_m) + (1-\eta)(m-1)(1/m - f_m)\Big)b
= - \Big((1-\eta)m f_m + (1-1/m + 