Jackpot statistics, a physicist's approach

Abstract

At a first glance lottery is a form of gambling, a game in which the chances of winning is extremely small. But upon a deeper look, considering that the Jackpot prize of lotteries is a result of the active participation of millions of players, we come to the conclusion that the interaction of the simple rules with the high number of players creates an emergent complex system. Such a system is characterized by its time-series that presents some interesting properties. Given the inherent stochastic nature of this game, it can be described within a mean-field type approach, such as the one implemented in the Local Growth and Global Reset (LGGR) model. We argue that the Jackpot time-series behaves ergodic for six lotteries with diverse formats and player pools. Specifying this consideration in the framework of the LGGR model, we model the lotteries with growth rates confirmed by the time-series. The reset rate is deduced mathematically and confirmed by data. Given these parameters we calculate the probability density of the Jackpot prizes, that fits well the empirically observed ones. We propose to use a single w parameter, as the product of the player pools found under the jurisdiction of the lottery and the chance that a single lottery ticket wins.

Figures (5)

• Figure 1: Time series of the Jackpot prize, for the Powerball from 03.2017 to 07.2019. canada_dataset
• Figure 2: The time series of the lotteries studied (general information provided in Table \ref{['table:lotteries']}). The rows represent the following: (A.) The time series plot of the lotteries, on the time range where the rules and the player pools are not changing, (B.) the mean of the time series $\langle x(t)\rangle_{t}$ along with the convergence value, (C.) the autocorrelation function of the time series $ACF(s)$ until $s$ days along with the confidence interval with $p \le 0.05$
• Figure 3: The stationary probability distribution of the Jackpot prizes $\rho_s (\frac{x}{\langle x \rangle_{t}})$ as a function of the mean rescaled Jackpot $\frac{x}{\langle x \rangle_{t}}$, for the six studied lotteries.
• Figure 4: The empirical growth rate $\mu( x / \langle x \rangle_{t})$ presented as the function of $x / \langle x \rangle_{t}$, along with the theoretical fit and the parameters for the studied lotteries. The empirical data averages are presented along with the error bars for the averages (standard deviation in the ensemble) in the bins for both axes. We also present the number of data points, $N_{data}$, used for calculating the averages in order to demonstrate that the red regions in the plots are not reliable for statistical inference.
• Figure 5: The empirical reset rate calculated by Equation \ref{['eq:reset-exp']}, fitted with the theoretical reset rate, for the Canadian lottery and UK lottery. We used the parameters obtained for fitting the probability density functions.