the analysis of the change of the mean net income with the number of rounds played (time), $\left\langle \Delta w(t) \right\rangle = \left\langle w(t) - w_0 \right\rangle =
\left\langle \sum\limits_{i=1}^t o_p(i) \right\rangle,$ where $w_0$ is the player’s initial wealth, $w(t)$ is the player’s wealth after attending $t$ rounds, and $o_p(i)$ is the payoff from the $i_{th}$ round the player attended. $\langle \cdot \rangle$ represents an ensemble average over a population of players placing bets. In the rest of this paper, $\langle \cdot \rangle$ will always be used for representing an ensemble average. In Fig. \[fig:netincome\] we show the change of $\left\langle \Delta w(t) \right\rangle$ over time. In most of the datasets, players’ mean net income decreases over time, which suggests that in general players will lose more as they gamble more. At the same time, in some datasets such as Ethcrash (D) and Coinroll (F), large fluctuations can be observed.

![ \[fig:netincome\] Change of the mean net income with time for the different datasets. Most of the datasets present a decreasing net income as time $t$ increases. Each point is obtained from an average over at least $200$ players.](figure6.pdf){width="\columnwidth"}

An useful tool for studying the diffusive process is the ensemble-averaged 