of the wager distributions at the individual level suggests a diversity of individual betting strategies. Also, it indicates that a gambler may not stick to only one betting strategy. It follows that the log-normal wager distribution observed at the population level is very likely an aggregate result.

Diffusive process {#diffusive-process .unnumbered}
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For an individual player’s gambling sequence we define “time” $t$ as the number of bets one player has placed so far, and define as net income the sum of the payoffs of those bets. In all the games we analyze, there are only two possible outcomes: a win or a loss. The player’s net income will change each time they place a bet in a round, with the step length to be the payoff from that bet. We can treat the change of one player’s net income as a random walk in a one-dimensional space (see Fig. 1 in Ref. [@Wang2018] for an example of such a trajectory). The time $t$ will increase by $1$ when the player places a new bet, therefore the process is a discrete-time random walk.

Now, let us focus on the analysis of the diffusive process of the gamblers’ net incomes, starting with 