Multi-Group Dynamics with Tolerant Switching in the Kolkata Paise Restaurant Problem with Dining Clubs

We study the Kolkata Paise Restaurant Problem (KPRP) with multiple dining clubs, extending work in [A. Harlalka, A. Belmonte and C. Griffin, \textit{Physica A}, 620:128767, 2023]. In classical KPRP, $N$ agents chose among $N$ restaurants at random. If multiple users choose the same restaurant, only one will eat. In a dining club, agents coordinate to avoid choosing the same restaurant, but may collide with users outside the club. We consider a dynamic in which agents switch among clubs or the unaffiliated (free agent) group based on their comparative probability of eating. Agents' affiliations are sticky in the sense that they are insensitive (tolerate) to differences in eating probability below a threshold $\tau$ without switching groups. We study the tendency of one group (dining club or free agent group) to become dominant as a function of tolerance by studying the mean-field dynamics of group proportion. We then show empirically that the mean-field group dynamic (assuming infinite populations) differs from the finite population group dynamic. We derive a mathematical approximation in the latter case, showing good agreement with the data. The paper concludes by studying the impact of (food) taxation, redistribution and freeloading in the finite population case. We show that a group that redistributes food tends to become dominant more often as a function of increasing tolerance to a point, at which point agents do not switch frequently enough to enable group dynamics to emerge. This is negatively affected by freeloaders from the non-redistributing group.