Statistics of the Kolkata Paise Restaurant problem

Abstract

We study the dynamics of a few stochastic learning strategies for the "Kolkata Paise Restaurant" problem, where N agents choose among N equally priced but differently ranked restaurants every evening such that each agent tries get to dinner in the best restaurant (each serving only one customer and the rest arriving there going without dinner that evening). We consider the learning strategies to be similar for all the agents and assume that each follow the same probabilistic or stochastic strategy dependent on the information of the past successes in the game. We show that some "naive" strategies lead to much better utilization of the services than some relatively "smarter" strategies. We also show that the service utilization fraction as high as 0.80 can result for a stochastic strategy, where each agent sticks to his past choice (independent of success achieved or not; with probability decreasing inversely in the past crowd size). The numerical results for utilization fraction of the services in some limiting cases are analytically examined.

Econophysics review: II. Agent-based models

“…The KPR problem Ghosh and Chakrabarti (2009);Ghosh et al (2010a,b)) is a repeated game, played between a large number N of agents having no interaction amongst themselves. In KPR problem, prospective customers (agents) choose from N restaurants each evening simultaneously (in parallel decision mode); N is fixed.…”

“…The utilization fraction f k of the k-th ranked restaurants on every evening is studied and their average (over k) distributions D(f ) are studied numerically, as well as analytically, and one finds Ghosh and Chakrabarti (2009) ;Ghosh et al (2010a)) their distributions to be Gaussian with the most probable utilization fractionf ≃ 0.63, 0.58 and 0.46 for the cases with α = 0, T → ∞; α = 1, T → ∞; and α = 0, T → 0 respectively. For the stochastic crowd-avoiding strategy discussed in Ghosh et al (2010b), where p k (t+1) = 1 n k (t) for k = k 0 the restaurant visited by the agent last evening, and = 1/(N − 1) for all other restaurants (k = k 0 ), one gets the best utilization fractionf ≃ 0.8, and the analytical estimates forf in these limits agree very well with the numerical observations. Also, the time required to converge to the above value off is independent of N .…”

Zipf's law in city size from a resource utilization model

“…One of the toy models to study resource utilization [18] is the Kolkata Paise Restaurant (KPR) [19,20] problem, which is similar to various adaptive games (see [21]). In the simplest version, N agents (customers) simultaneously choose between equal number R (= N ) of restaurants, each of which serve only one meal every evening (generalization to any other number is trivial).…”

Econophysics of Agent-Based Models

“…0, respectively. For the stochastic crowd-avoiding strategy discussed by Ghosh et al (2010b), where p k (t þ 1) ¼ 1/n k (t) for k ¼ k 0 , the restaurant visited by the agent the last evening, and ¼1/(N À 1) for all other restaurants (k 6 ¼ k 0 ), one obtains the best utilization fraction " f ' 0:8, and the analytical estimates for " f in these limits agree very well with the numerical observations. Also, the time required to converge to the above value of " f is independent of N. The KPR problem has similarities to the Minority Game Problem (Arthur 1994, Challet et al 2004 as, in both the games, herding behavior is punished and diversity is encouraged.…”

“…Study of the KPR problem shows that while a dictated solution leads to one of the best possible solutions to the problem, with each agent obtaining his dinner at the best ranked restaurant in a period of N evenings, and with a best possible value of " f (¼1) starting from the first evening, the parallel decision strategies (employing evolving algorithms used by the agents and past information, for example n(t)), which are necessarily parallel among the agents and stochastic (as in a democracy), are less efficient ( " f ( 1; the best is discussed by Ghosh et al (2010b), giving " f ' 0:8 only). Note here that the time required is not dependent on N. We also note that most of the 'smarter' strategies lead to much lower efficiency.…”