*4.2. Summary*

KPR is a multi-agent multi-choice repeated game where players try to learn from their past successes or failures, utilizing publicly available information on the crowd sizes at different restaurant in the past to decide which restaurant to visit that day such that she would be alone there for being served the only prepared dish. Here, asymmetric case such that *λN* (*λ* < 1) agents are considered against *N* restaurants, for sufficiently large *N*. End of each day (iteration), we have measured social utilization for agents *f <sup>a</sup>*(*t*) = ∑*Ni*=<sup>1</sup>[*δ*(*ni*(*t*))/*λN*] where *ni*(*t*) denotes number of customers visiting *i*th restaurant on day *t*.

As shown in Figure 6 (for strategy I) and Figure 8 (for strategy II), the social wastage fraction (1 − *f a*) vanishes at the effective critical point *<sup>λ</sup>c*(*N*) with the critical exponent *β* value near unity. In addition, from Figure 7 (for strategy I) and Figure 9 (for strategy II), we see that the the convergence or relaxation time *t*, required for *f a* to stabilize, divergence near the same critical points *<sup>λ</sup>c*(*N*) for the respective strategies, with the exponent *γ* value about 1/2. Additionally, the finite size scaling analysis *<sup>λ</sup>c*(*N*) ∼ *λc* + *const*.*N*−1/(*dν*), where *λc* corresponds to *<sup>λ</sup>c*(*N*) for *N* going to infinity and *d* corresponds to the effective dimension, suggests the effective correlation length exponent *dν* value to be around 2 for both the strategies, as expected for such mean field (infinite range systems).

In Reference [81], we have studied the dynamics of the KPR game following the same two strategies for the case *λ* = 1. For *λ* = 1.0, where the critical points *λc* (for both the strategies) vanish, the universality class (values of the critical exponents *β* and *γ* were observed to be distinctly different, and this point needs further investigations. We may, however, note that, since at *λ* = 1, the number of both agents and restaurants are same (*N*), full social utilization (where *f a* = 1 = *f r*, occurring at *α* = 0+ for strategy I and at *p* = 1− for strategy II) induces an additional frustrating constraint in the collective choice dynamics involved here.

The KPR game models have been extended already and used to study real life problems, like resource allocation in Internet of Things [82], vehicle for hire [83], matching in mobility markets [84], etc. We hope the KPR game models will be utilized much more effectively in the context of much wider practical areas of collective learning dynamics and choices.

#### **5. Future of Econophysics: Some Perspective**

One often says that the main purpose of economic activity is to optimize the limited funds of labor and capital, natural and technical resources and capital resources, to satisfy our (practically) unlimited needs. "Economic science is therefore the science of efficiency, and as such, it is a quantitative science." [89] (also see Reference [90]). We have already argued [14] in Section 2 that epistemologically economics belongs to natural science (and not mathematics). It begins with observation which are to be analyzed using logic or mathematics and eventually should end in observation, as in all natural sciences. Since 1990s, most Universities of the world offer Science Graduation degrees (Bachelor of Science or Master of Science degrees) in economics (in addition to Bachelor of Arts or Master of Arts from Fine Arts or Humanities Departments).

Robert Solow [91] pointed out that, in the 1940s, economics had been basically a descriptive and institutional subject for a 'gentleman scholar'. The textbooks of those days were 'civilized' and discursive. ... "Formal analysis were minimal and it made economics the domain of intuitive economists". He concluded his summary of the state of economics near the end of the 20th century "with a paraphrase of Oscar Wilde's description of a fox hunt - 'the unspeakable in pursuit of the inedible'-saying that perhaps economics was an example of 'the over-educated in pursuit of the unknowable'." [91]. Despite the ongoing controversies today in the field of economics, the "New Millennium economists are far more comfortable with what they do after the changes in the structure and content of economics over the last half century" [92]. The root cause of these changes have been identified by Colander [92] to be due to the rise of Complexity Science since early 1980s. In fact, concepts from physics had continually been absorbed into the main stream economic formulation of ideas and models. As Venkat Venkatasubramanian noted in his recent book [93], "Concepts such as equilibrium, forces of supply and demand, and elasticity reveal influence of classical mechanics on economics. The analytical model of utility-based preferences can be traced back to Daniel Bernoulli, the grea<sup>t</sup> Swiss mathematical physicist from nineteenth century. One of the founders of neoclassical economics, Irving Fisher, was trained under the legendary Yale physicist, Jisiah Willard Gibbs, a co-founder of the discipline of statistical mechanics. Similarly, Jan Tinbergen, who shared the first Nobel Prize in Economics in 1969, was the doctoral student of the grea<sup>t</sup> physicist Paul Ehrenfest at Leiden University".

Indeed, more specifically as discussed in Section 2, we would like to correlate these changes to occur following the successful development in econophysics of the Simulated Annealing technique [22] in 1983 for Traveling Salesman type multi-variable optimization problems, and other successive developments in econophysics of analyzing correlations in stock prices (see, e.g., Reference [3,21]) or the kinetic exchange modelings of income and wealth distributions (see, e.g., References [28,63]). The statistical physics of TSP, as an example of successful developments in econophysics, had already been introduced in our 2010 econophysics textbook [32], which has been the only 'suggested textbook' (since inception in 2012) of the formal course on econophysics, offered (by Diego Garlaschelli) at the Physics Department of the Leiden University (see the course prospectus for 2012–2013 through that of 2020–2021 [94]), where one of the first Nobel-laureates in economics Jan Tinbergen came from.

Econophysics has come as an exceptional development in interdisciplinary sciences (see, e.g., Reference [95] for a popular exposition on this development). Historically, economics, more specifically social sciences, belonged to the Humanities departments and not of Science. For earlier interdisciplinary developments of Astrophysics, Biophysics, or Geophysics, the scenario and ambiance had been quite different. The mother departments had been parts of the same science schools and even the corresponding resources, like books, journals, and also the faculty, had strong overlaps and could be shared. The marriage negotiations for Econophysics have been difficult, though extremely desirable and natural; as the saying goes: "marriage between the King of natural sciences with the Queen of social sciences!"

Regular interactions and collaborations between the communities of natural scientists and social scientists are, however, rare, even today! Though, as mentioned already, interdisciplinary research papers on econophysics and sociophysics are regularly being published at a steady and healthy rate, and a number of universities (including Universities of Bern, Leiden, London, Paris, and Tufts University) are offering the interdisciplinary courses on econophysics and sociophysics, not many clearly designated professor positions, or other faculty positions for that matter, are available ye<sup>t</sup> (except for econophysics in Universities of Leiden and London). Neither are there designated institutions on these interdisciplinary fields, nor separate departments or centers of studies for instance. Of course, there have been several positive and inspiring attempts and approaches from both economics and finance side (see, e.g., References [96,97], along with a number of those [66,67,98–100] from physics, which have already been appreciated in the literature). Indeed, the thesis [101] in August 2018, Department of History and Philosophy of Science, University of Cambridge, by financial economist Christophe Schinckus (one of the co-editors of this special issue), says that "In order to reconstruct the subfield of econophysics, I started with the group of the most influential authors in econophysics and tracked their papers in the literature using the Web of Science database of Thomson-Reuters (The sample is composed of: Eugene Stanley, Rosario Mantegna, Joseph McCauley, Jean-Pierre Bouchaud, Mauro Gallegati, Benoît Mandelbrot, Didier Sornette, Thomas Lux, Bikas Chakrabarti and Doyne Farmer). These key authors are often presented as the fathers of econophysics simply because they contributed significantly to its early definition and development. Because of their influential and seminal works, these scholars are actually the most quoted authors in econophysics. Having the 10 highest quoted fathers of econophysics as a sample sounds an acceptable approach to define bibliometrically the core of econophysics". In addition, the entry on 'Social Ontology' in The Stanford Encyclopedia of Philosophy [15], as discussed in Section 2, confirms positive impact of such econophysics and sociophysics research studies on the overall modern philosophical outlook of social sciences.

We may note, however, a recently published highly acclaimed massive (580 page) book [96] on economics ('landmark volume', said E. Roy Weintraub, 'creative, elegant and brilliant work', said W. Brian Arthur and 'written by master economists', said D. Colander) by (Late) Martin Shubik (Ex-Seymour Knox Professor Emeritus of Mathematical Institutional Economics, Yale University and Santa Fe Institute) and Eric Smith (Santa Fe Institute) discussed extensively on econophysics approaches and in general on the potential of interdisciplinary research studies inspired by the developments in natural sciences. Getting somewhat excited, I wrote to Martin Shubik in late 2016 that their book can also serve as an outstanding 'white-paper' document in favor of a possible Proposal for an International Center for Interdisciplinary Studies on Complexity in Social Sciences. He immediately responded and gave his impression about the difficulties involved and indicated very briefly about the minimal financial and structural requirements (both my letter to him and his response is appended below (Figures 11 and 12).

**Figure 11.** The first part of the email conversation between (late) Martin Shubik and BKC. Second part (email from BKC; appended to this part) is continued in Figure 12. The precise suggestions made in this immediate response indicate Shubik's prior plan for such 'interdisciplinary institutes' in economics.


**Figure 12.** Email conversation in the end of 2016 between (late) Martin Shubik and BKC regarding interdisciplinary developments in economics and the possibility of setting up an International Center for Interdisciplinary Studies on Complexity in Social Sciences. This email from BKC was appended to the response email (Figure 11) from Shubik. The (Yale) date and time mark in the mail-header (and that for BKC's in Figure 11, on arrival in Kolkata) indicate hardly any time gap between the two and the readiness with the precise suggestions indicate Shubik's prior thinking in similar line.

This ready and specific comments by Shubik clearly suggests that he actually had thought about the need of such an International Center for fostering interdisciplinary research which needs to be more inclusive than, for example, the Santa Fe Institute. The model of the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste (funded by UNESCO and IAEA), was considered to provide helpful guidance for us here. It was contemplated, if an ICTP-type interdisciplinary research institute could be initiated for research studies on econophysics and sociophysics (see, e.g, Reference [102]). Though Shubik (who died in 2018 at the age of 92) agreed also to be one of its founding members, we could not make any progress yet. We may also note that Dirk Helbing and colleagues have been trying for an European Union funded 'Complex Techno-Socio- Economic Analysis Center' or 'Economic and Social Observatory' for the last decade or so (see Ref. [103] containing the White Papers arguing for their proposed project). We are also aware that Indian Statistical Institute had taken a decision to initiate a similar Center in India (see 'Concluding Remarks' in Reference [104]).

Hope, some such international visiting centers will come up soon and with them the spread of such interdisciplinary ideas will achieve more coherence and will lead to major success in such research studies.

**Author Contributions:** Data curation, A.S.; Writing—review & editing, B.K.C. and A.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We acknowledge all our colleagues (mentioned by name in Section 2) for the collaborations. BKC is grateful to J.C. Bose National Fellowship (DST, Govt. of India) gran<sup>t</sup> for support.

**Conflicts of Interest:** The authors declare no conflict of interest.
