# Phase transition in the Kolkata Paise Restaurant problem

## TL;DR

This paper investigates a phase transition in the Kolkata Paise Restaurant problem, analyzing how agents learn to optimize restaurant choices and how system parameters influence steady-state utilization and convergence times.

## Contribution

It introduces and analyzes two crowd-avoiding strategies in the KPR problem, revealing critical behavior and scaling laws near phase transition points.

## Key findings

- Steady state wastage fraction scales as a power law near critical points.
- Convergence time diverges with a power law near phase transitions.
- Critical exponents depend on the spatial dimension of the system.

## Abstract

A novel phase transition behaviour is observed in the Kolkata Paise Restaurant (KPR) problem where large number ($N$) of agents or customers collectively (and iteratively) learn to choose among the $N$ restaurants where she would expect to be alone that evening and would get the only dish available there (or may get randomly picked up if more than one agent arrive there that evening). The players are expected to evolve their strategy such that the publicly available information about past crowd in different restaurants can be utilized and each of them is able to make the best minority choice. For equally ranked restaurants we follow two crowd-avoiding strategies: Strategy I, where each of the $n_i(t)$ number of agents arriving at the $i$-th restaurant on the $t$-th evening goes back to the same restaurant on the next evening with probability $[n_i(t)]^{-\alpha}$, while in Strategy II, with probability $p$, when $n_i(t) > 1$. We study the steady state ($t$-independent) utilization fraction $f:(1-f)$ giving the steady state (wastage) fraction of restaurants going without any customer in any particular evening. With both the strategies we find, near $\alpha_c=0_+$ (in strategy I) or $p=1_-$ (in strategy II), the steady state wastage fraction $(1-f)\propto(\alpha - \alpha_c)^{\beta}$ or $(p_c - p)^\beta$ with $\beta \simeq 0.8, 0.87, 1.0$ and the convergence time $\tau$ (for $f(t)$ becoming independent of $t$) varies as $\tau\propto{(\alpha-\alpha_c)}^{-\gamma}$ or ${(p_c-p)}^{-\gamma}$, with $\gamma \simeq 1.18, 1.11, 1.05$ in infinite-dimension (rest of the $N-1$ neighboring restaurants), three-dimension ($6$ neighbors) and two-dimension ($4$ neighbors) respectively.