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Kolkata Paise Restaurant Problem

From EverybodyWiki Bios & Wiki

The Kolkata Paise Restaurant Problem (KPR) is named after the old (early nineteen hundred, through seventies) very cheap fixed price (‘Paise’ used to be the lowest denomination of the Indian Rupee) and limited offer 'Paise Restaurants' in Kolkata (see e.g.,[1] for the few still surviving). It is an anti-coordination game model that describes how a large number of individuals (players) compete for limited resources without any mutual consultation or direct coordination. The problem becomes trivial (though gives full utilization of resources and that too from the first day; food for all the customers/players and maximum revenue for all the restaurants in zero learning or convergence time) when a non-playing coordinator or dictator asks every player to form a queue and to go to the restaurant corresponding to the position in the queue on the first day and shift by one restaurant position every successive day (maintaining the periodic boundary condition). The non-triviality of the problem comes when the individuals decide or choose independently (based on their own past experience of success or failure and information about the past crowd sizes in different restaurants) and tries to maximize their own pay-off (which also helps the restaurants to maximize their sales). The El Farol Bar problem corresponds to only two choices for each individual (to go to the bar or remain at home to avoid the over-crowded bar) for each individual. The KPR game[2][3][4][5][6][7] is therefore an extension of the El Farol Bar problem for many choices for each individual. For a review on the early developments, one can see.[8] If limited to two players only, KPR has got structural similarity with the Chicken (game) or Hawk-Dove games,[9] while the matching game-like algorithmic aspect of the full KPR problem can be compared with that of Gale-Shapley algorithm.[10] See also [11][12] for similar studies on and comparisons with the "Kolkata Game" or "Kolkata Algorithm".

Background

The theory of KPR is rooted in the history of the social sciences and in particular computational sociology and Econo-physics.

The theory of social science was first introduced by August Comte in 1853[13] [14]. Based on positivistic philosophy Comte’s main goal was to describe human behavior using scientific tools. He wanted to reproduce the success of the hard sciences of his time. Accordingly, he coined his theory ‘Social Physics’. Comte described a hierarchy of the sciences from mathematics and physics at the bottom through chemistry biology and Social Physics at the top.

In its beginning social physics research was mainly established on the ground of Probability theory and early notions of Statistics, developed at the 17 and 18 centuries.  The first notions of probability theory came as early as the 17th century by the work of Pascal and de Fermat, in their 1654 correspondence on the problem of points[15][16] . Later 1657 Huygens wrote what is known as the first book on probability theory [17][18]. The Normal distribution was introduced by de-Moivre in 1733 [19][20]. In 1763 Bayes [21] introduced his probability theory, the Bayes Theorem. Gauss (1809) [22] introduced the use of normal distribution in error estimation and the least square estimator, thereby introducing the foundations of statistical inference.  Laplace [23][24] (1812) generalized the work of de-Moivre introducing the Central Limit Theorem, he also re-discovered and developed the Bayesian inference systematically.  Legendre[25] (1905) re-discovered the method of least square.  In 1933 Kolmogorov [26] formulated the notion of probability theory as we use it today, with random variables, measurable functions, sampling, etc.

Statistics soon became not just a theory used to collect data but as a way of thinking[27] . As early as 1662 Graunt [28] investigated birth and death rates in London to draw conclusions on life expectations and urban mortality, Petty [29][30] coined the notion of ‘political arithmetic’.  Later in 1842 Quetele [31] introduced the notion of the ‘average man’ (l’homme moyen). F.Galton in 1869 used correlation and regression to study human traits and to define biometrics.

During the 20th century the major bulk of social scientists used what is now called quantitative methods, these were simple statistical methods including questionnaires, correlations, regressions, Bayesian decision theory etc.  In 1921 Wright [32] introduced the Path analysis, a graphical model where the nodes are social variables and the edges’ weights are the correlations between the variables. It was aimed to suggest possible causations between correlated variables.

Over the years the Social Sciences were moving apart from the hard-core sciences, the term ‘social Physics’ was hardly used. Toward the end of the 20th century new methods known as qualitative methods were introduced. These included interviews, ethnographic observations, narrative analysis, grounded theory and more [33]. 

The emergence of the computers brought new research tools such as simulation. In the midst of the 20th century Von-Neumann and Morgenstein [34] were investigating game theory and economic behavior. In 1967 Swanson [35] discussed social change, political instability, revolution and culture structure, his theory, although non-computational highly influenced the later computational sociology theory. Coleman [36] was investigating the micro-macro social relations. These were the first steps towards a simulation-based theory, later to become the Agent Based Modelling. In 1957 Orcutt [37] suggested simulating society as an ensemble of units having attributes as income and age, simulating their’ transitions and hence a socio-economic system. In 1971 Schelling [38] introduces his famous segregation model. Schelling’s work stressed the fact that simulations could ‘explain’ phenomena that the quantitative methods could not.

Agent Based Modeling introduces the use of ‘emergent properties’ into the social sciences, where local interactions between neighboring agents yield some global emergent property [39].  ‘More is different’ was the name Anderson [40] gave the phenomenon. Anderson also discussed the question whether the emergent phenomenon should be considered as a new physical law and when do we need it. Best examples of emergent properties are the behavior of swarm of social insects like bees and swaps, or flocks of birds and fish, but humans are also social animals. Theories like epidemiology can be easily simulated by ABM, likewise trends in economy, traffic in urban design, flow of information, dispersion of viruses [41] etc. The way social animals or social agents solve problem by emergent behavior is also known today as ‘swarm intelligence’ [42]. 

The theory of computational sociology is a step forward in the direction of Comte’s original idea. Computational sociology is a subfield of sociology that uses computational methods—such as simulations, algorithms, network analysis, and data mining—to model, understand, and analyze social phenomena[43] . Computational sociology is a step towards putting Social Sciences back on the ground of hard-core sciences. The theory was boosted mainly by simulation methods such as Agent-Base Modeling, Spin Lattice (Ising, Pots, Random Fields Ising) Models etc.

The theory of Econo-Physics takes computational sociology a step forward. It introduces notions from physics (mainly statistical mechanics, condensed matter physics) into the social theory, such as phase transition, condensation, etc. A crowd of people could be in different phases, it could have well organized form, it could be clustered or organized in trails, or it could be random, we could fine-tune the parameters that change its behavior from one phase into the other. We could also investigate the type (rate) of phase change, is it continuous or is it sharp, very similar to what we can do in thermodynamics and statistical mechanics. Similarly, the clustering of a group of people could be sometime described by a Bose-Einstein condensation, where a preferential choice is made when one decides to which subgroup to join[44].  Inserting such terms into the social context is like introducing a new language that could easily formulate our insights and perhaps predict and explain new phenomena.

Econo-physics was first introduced by Stanley in 1996 [45]. Stanley and Mantegna discussed fluctuation of financial indices, and showed that the shape of the distributing of returns showed a self-similarity property. Moreover, the distribution had a large tail and was similar to a truncated Levi distribution. Clearly this resembles the behavior of well-known physical systems near criticality.  Since then, a plethora of research Econophysics papers appeared, to mention some, Plerou [46] discussed cross correlations in financial time series, Bouchaud, Mézard and Potters [47] discussed stock order books, Chakraborti & Chakrabarti [48] discussed statistical mechanics of money, Cont and Bouchaud [49] discussed herd behavior.

The KPR game is a prototype example by which we can demonstrate Econophysics principles. We could play with the parameters of the game and introduce new emergent behaviors. We can force local relationships between the agents and investigate the change in the overall emergent property.  

Problem Definition

  • There are N players (prospective customers) and n restaurants; typically N = n. Both N and n can be arbitrarily large.
  • Each day, customers independently choose a restaurant, based on their past experience of success or failure.
  • At each restaurant only one of the players arriving there (prospective customers) is randomly chosen and served (payoff = 1). The rest leave without food for that day (payoff = 0; no time/money left for another search).
  • Players do not know each other's choices but have access to historical data of past selections.
  • The ideal outcome is perfect coordination, where each player chooses and picks a different restaurant in short convergence time. However this becomes difficult (in absence parallel communication exchanges) in KPR game. This leads to inefficiencies (some restaurants overcrowded, others empty) or partial utilization. The KPR objective is to evolve the collective ‘parallel learning’ algorithms for maximizing the utilization fraction in minimum (preferably less than lnN order) time.

Real-World Applications

The KPR model can describe various real-life resource allocation problems, such as:

  • Hospital Resource Allocation – Patients prefer top-rated hospitals, even if local hospitals have available beds. Overcrowding in top hospitals leaves some patients unattended, while other hospitals remain underutilized.[50]
  • Ride-Hailing Services – Passengers compete for taxis, sometimes overwhelming some areas while others remain unpopulated.[51][52]
  • Online Service Access – Users competing for limited online resources (e.g., booking slots, bandwidth allocation, computer job allocation problem in Internet of Things computer.[53]).
  • Dynamics of Dining Clubs in the KPR Problem.[54][9]
  • Designing benchmarking algorithms for AI.[10]

It has been pointed-out that the anti-coordination game has broad real-world applications,[55] and that the use of lotteries to resolve conflicts over specific resources would turn any anti-coordination game (or MAD Chairs game) into KPR.[56] For example, if contests for positions of authority were settled via lottery, instead of via election, then the game to win such positions would become KPR. However, lottery enforcement might not be sustainable in some real-world situations (thus revealing Coordination or MAD Chairs as the underlying model). For example, if contests for jobs were settled randomly, then the unemployed might retaliate with political pressure to regulate employment.[56]

This dynamic also appears in the board game "Eketorp". [57]

Strategies & Optimization

Strategies are evaluated based on their aggregate payoff and/or the proportion of attended restaurants (utilization ratio). A leading stochastic strategy, with utilization fraction ~0.79,[50] gives each customer a probability p of choosing the same restaurant as yesterday (p varying inversely with the number of players who chose that restaurant yesterday), while choosing among other restaurants with uniform probability. This is a better result than deterministic algorithms or simple random choice (noise trader), with utilization fraction 1 - 1/e ≈ 0.63.[2][58] Increased utilization fraction for customers, each having a fixed low budget allowance for local search using Traveling Salesman Problem type algorithm, have also been studied.[59]

Strategies without an external dictator (such as caste and turn-taking strategies) are available to achieve full utilization of resources if objective orderings of players and resources are available to all players.[56] In some cases, a social norm (e.g., alphabetical sort) could provide that ordering; in others, history of play would eventually provide it (e.g., sorting resources by popularity and sorting players by winnings, skill-estimate, or debt-ranking).

Extensions of KPR for on-call car hire problems (restaurants effectively having also the options for moving to their chosen places) have been explored in[51][52] and see[60] for the mean field solution of a generalized KPR problem in the same resource competition in spatial settings of the vehicle-for-hire market. For application of KPR to optimized job allocation problems in Internet-of-Things, see.[53] Stability of the KPR, induced by the introduction of dining clubs have also studied.[54] One can see[61] for a study on the impact of expert opinions (or even of faiths) on such evolving mutually competing search strategies for the resources. See also[62] for application of KPR model to anthropological and sociological analysis of the models of polytheism, and for an algorithmic application to cancer therapy, see.[63]

Extensions to quantum games for three player KPR have been studied, where, dimensionality permitted, each player can achieve much higher mean payoff.[64][65] For some recent studies in the context of three player KPR Game on the advantages of higher payoffs in quantum games when the initial state is maximally entangled, see.[66][67][68] See[69] for a general introduction to econophysics, sociophysics, classical KPR, quantum games and quantum KPR, and see[70] for a later review on classical and quantum KPR games.

One may see[71] for an early attempt to exploit KPR type strategies for developing Artificial Intelligence or AI models. For a study of KPR-like approaches related to graph partitioning problem, see.[72] Recently the KPR game has been extended (to Musical Chair type games) and that has been exploited[56] for creating a new benchmark for evaluating the AI algorithms.

References

  1. ↑ J. Kishan (2021). "Pice hotels: A lifeline for Kolkata's hungry worker". BBC Travel: Food & Hospitality (June 10 Issue).
  2. ↑ 2.0 2.1 A. S. Chakrabarti; B. K. Chakrabarti; A. Chatterjee; M. Mitra (2009). "The Kolkata Paise Restaurant problem and resource utilization". Physica A. 388 (12): 2420–2426. arXiv:0711.1639. Bibcode:2009PhyA..388.2420C. doi:10.1016/j.physa.2009.02.039. Unknown parameter |s2cid= ignored (help)
  3. ↑ Asim Ghosh, Bikas K. Chakrabarti (2011). "Kolkata Paise Restaurant (KPR) Problem". Wolfram Alpha.
  4. ↑ A. Ghosh; D. D. Martino; A. Chatterjee; M. Marsili; B. K. Chakrabarti (2012). "Phase transition in crowd dynamics of resource allocation". Physical Review E. 85 (2): 021116. arXiv:1109.2541. Bibcode:2012PhRvE..85b1116G. doi:10.1103/physreve.85.021116. PMID 22463162. Unknown parameter |s2cid= ignored (help)
  5. ↑ Frédéric Abergel; Bikas K. Chakrabarti; Anirban Chakraborti; Asim Ghosh (2013). Econophysics of Systemic Risk and Network Dynamics (PDF). New Economic Windows. Bibcode:2013esrn.book.....A. doi:10.1007/978-88-470-2553-0. ISBN 978-88-470-2552-3. Search this book on
  6. ↑ A. Chakraborti; D. Challet; A. Chatterjee; M. Marsili; Y.-C. Zhang; B. K. Chakrabartpi (2015). "Statistical Mechanics of Competitive Resource Allocation using Agent-Based Models". Physics Reports. 552: 1–25. arXiv:1305.2121. Bibcode:2015PhR...552....1C. doi:10.1016/j.physrep.2014.09.006. Unknown parameter |s2cid= ignored (help)
  7. ↑ Bikas K Chakrabarti; Arnab Chatterjee; Asim Ghosh; Sudip Mukherjee; Boaz Tamir (27 July 2017). Econophysics of the Kolkata Restaurant Problem and Related Games: Classical and Quantum Strategies for Multi-agent, Multi-choice Repetitive Games. Springer. ISBN 978-3-319-61351-2. Search this book on
  8. ↑ A. Ghosh; S. Mukherjee (2018). "Kolkata Paise Restaurant (KPR) Problem" (PDF). Science and Culture. 84: 5–25.
  9. ↑ 9.0 9.1 A. Harlalka; C. Griffin (2025). "Multi-group dynamics with tolerant switching in the Kolkata Paise Restaurant problem with dining clubs". J. Phys. Complex. 6 (2): 025002. Bibcode:2025JPCom...6b5002H. doi:10.1088/2632-072X/adc52e.
  10. ↑ 10.0 10.1 B. Picano; R. Fantacci (2024). "Efficient task offloading and resource allocation in an intelligent UAV-MEC system". IEEE J. Communication Networks.
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  23. ↑ Laplace, Pierre-Simon (1812). Théorie analytique des probabilités (in French). Paris: Courcier.CS1 maint: Unrecognized language (link) Search this book on
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  27. ↑ Porter, Theodore M. (2020). The Rise of Statistical Thinking, 1820–1900. Princeton University Press. ISBN 9780691609930 Check |isbn= value: checksum (help). Search this book on
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  33. ↑ Denzin, Norman K.; Lincoln, Yvonna S., eds. (2018). The SAGE Handbook of Qualitative Research (5th ed.). SAGE Publications. ISBN 978-1-5063-2615-8 Check |isbn= value: checksum (help). Search this book on
  34. ↑ von Neumann, John; Morgenstern, Oskar (1953). Theory of Games and Economic Behavior. Princeton University Press. ISBN 978-0-691-05761-1 Check |isbn= value: checksum (help). Search this book on
  35. ↑ Swanson, Guy E. (1967). Religion and Regime: A Sociological Account of the Reformation. University of Michigan Press. ISBN 978-0-472-06194-2 Check |isbn= value: checksum (help). Search this book on
  36. ↑ Coleman, James S. (1967). "The Concept of Equality of Educational Opportunity". Harvard Educational Review. 38 (1): 7–22.
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  42. ↑ Kennedy, James; Eberhart, Russell C. (2001). Swarm Intelligence. Morgan Kaufmann Publishers. ISBN 978-1-55860-595-4. Search this book on
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  55. ↑ Lewis, David (2008). Convention: A philosophical study. John Wiley & Sons. Search this book on
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  63. ↑ Zhang, Yuqin and He, Hanxing and Fu, Xin and Liu, Ganzhi and Wang, Huiying and Zhong, Wen and Xu, Xia and Chen, Bo and Mei, Lin (2025). "Glioblastoma-associated macrophages in glioblastoma: from their function and mechanism to therapeutic advances". Cancer Gene Therapy: 1–13. doi:10.1038/s41417-025-00905-9.CS1 maint: Multiple names: authors list (link)
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  65. ↑ M. Ramzan (2013). "Three-player quantum Kolkata restaurant problem under decoherence". Quantum Inform. Process. 12 (1): 577. arXiv:1111.3913. Bibcode:2013QuIP...12..577R. doi:10.1007/s11128-012-0405-8.
  66. ↑ S.-X. Jiang; J. Shi (2024). "Multi-party three-dimensional asymmetric cyclic controlled quantum teleportation in noisy environment". Quantum Inform. Process. 23: 261.
  67. ↑ K. Bolonek-Lason. "Quantum Cournot model based on general entanglement operator". Quantum Inform. Process. 23: 371.
  68. ↑ R.-G. Zhou; X.-X. Zhang; L.-T. Du (2025). "Quantum Teleportation in Noisy Environment (Book Chapter)". Design of Quantum Teleportation Schemes (Springer): 119-186.
  69. ↑ B. Tamir (2018). "Econophysics and the Kolkata Paise Restaurant Problem: More is different" (PDF). Science and Culture. 84: 37–47.
  70. ↑ B. K. Chakrabarti; A. Rajak; A. Sinha (2022). "Stochastic Learning in Kolkata Paise Restaurant Problem: Classical and Quantum Strategies". Front. Artif. Intell. 5: 874061. doi:10.3389/frai.2022.874061. PMC 9181993 Check |pmc= value (help). PMID 35692940 Check |pmid= value (help).
  71. ↑ N. Alon; R. Meir; M. Tennenholtz (2013). "The Value of Ignorance about the Number of Players" (PDF). Proc. Twenty-Seventh AAAI Conference on Artificial Intelligence.
  72. ↑ M. Anastos; O. Cooley; M. Kang; M. Kwan (2024). "Partitioning problems via random processes". Journal of the London Mathematical Society. 110 (6). doi:10.1112/jlms.70010.


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