Multiple-interaction kinetic modeling of a virtual-item gambling economy

In recent years, there has been a proliferation of online gambling sites, which has made gambling more accessible with a consequent rise in related problems, such as addiction. Hence, the analysis of the gambling behavior at both the individual and the aggregate levels has become the object of several investigations. In this paper, resorting to classical methods of the kinetic theory, we describe the behavior of a multiagent system of gamblers participating in lottery-type games on a virtual-item gambling market. The comparison with previous, often empirical, results highlights the ability of the kinetic approach to explain how the simple microscopic rules of a gambling-type game produce complex collective trends, which might be difficult to interpret precisely by looking only at the available data.

Figure 1

Function $\mathrm{\Psi }$ given in (44).

Figure 2

Comparison of (50) with the numerically computed large-time solution (at the computational time $T=10$) to the Boltzmann-type equation (46). The value function is (44) with $\mu =0.5$ and $\alpha =1$. Moreover, the binary interaction is (43) with $\lambda =\frac{1}{10}$ and $w_L=e^{\lambda /2\mu }$. We considered the quasi-invariant scaling (48) with (a) $\epsilon ={10}^{-1}$ and (b) $\epsilon ={10}^{-2}$.

Figure 3

Test 1: Without refilling. Evolution at different times (a) $t=0.1$ and (b) $t=2$ of the multiple-interaction Boltzmann-type model with either $N=5$ gamblers (open circles) or $N=100$ gamblers (open triangles) and of its linearized version (closed circles) in the time interval [0,2] for $\delta =0.2$ and $\beta =0$. We considered $\kappa =0.1$.

Figure 4

Test 1: With refilling. Evolution at different times (a) $t=1$, (b) $t=5$, and (c) $t=25$ of the multiple-interaction Boltzmann-type model with either $N=5$ gamblers (open circles) or $N=100$ gamblers (open triangles) and of its linearized version (closed circles) in the time interval [0,25] for $\delta =\beta =0.2$ [log-normal refilling sampled from (52)]. We considered $\kappa =0.1$.

Figure 5

Evolution of the mean number of tickets $m\left(t\right)$ in the time interval [0,25] for $\delta =0.2, \kappa =0.1$, and several choices of $\beta$.

Figure 6

Test 2. (a) and (b) Comparison of the steady distribution of the multiple-interaction Boltzmann-type model (5) with the Fokker-Planck asymptotic distribution (24) (solid line) and its log-log plot [in (b)] for $N={10}^2$ (open circles), $N={10}^3$ (closed circles), and fixed $\kappa =0.1$. (c) and (d) Comparison of the steady distribution of the linearized Boltzmann-type model (15) with the Fokker-Planck asymptotic distribution (24) (solid line) and its log-log plot for $\kappa =0.1$ (open circles) and $\kappa =0.01$ (closed circles).

Figure 7

Test 3. (a) Estimate of the approximate collision invariant $\chi$ [cf. (28)]. (b) A log-log plot of the distributions (24) and (37).

Figure 8

Test 3. Comparison between the time evolutions of the Boltzmann-type model (33) and (34) (open circles) and of its Fokker-Planck approximation (36) (stars) in the quasi-invariant regime at times (a) $t=1$, (b) $t=5$, (c) $t=10$, and (d) $t=25$. The following parameters have been used: $\beta =\delta =0.2, M=1$, and $\kappa ={10}^{-2}$. The value of the approximate collision invariant $\chi$ is estimated from the multiple-interaction Boltzmann-type model like in Fig. 7.

Figure 9

Test 3. Relative $L^1$ error (53) between the solutions to the Fokker-Planck equations (23) and (36). The numerical solution of both models is obtained by means of semi-implicit SP methods over the computational domain [0,10] in the $x$ variable, with $\mathrm{\Delta }t=\mathrm{\Delta }x=10/N_x$ and $N_x=401$ nodes.