September  2020, 15(3): 353-368. doi: 10.3934/nhm.2020022

Swarms dynamics approach to behavioral economy: Theoretical tools and price sequences

1. 

University of Granada, Departamento de Matemática Aplicada, 18071-Granada, Spain, Collegio Carlo Alberto, Torino, Italy, Politecnico Torino, Italy

2. 

Joint Research Centre, European Commission, Ispra, VA, Italy

3. 

Centro de Investigación y Estudios de Matemática (CONICET) and Famaf (UNC), Medina Allende s/n, 5000 Córdoba, Argentina

4. 

Credimi S.p.A., Milano, MI, Italy

5. 

University of Torino, Torino, Italy, Collegio Carlo Alberto, Torino, Italy

Received  December 2019 Revised  February 2020 Published  September 2020 Early access  September 2020

This paper presents a development of the mathematical theory of swarms towards a systems approach to behavioral dynamics of social and economical systems. The modeling approach accounts for the ability of social entities are to develop a specific strategy which is heterogeneously distributed by interactions which are nonlinearly additive. A detailed application to the modeling of the dynamics of prices in the interaction between buyers and sellers is developed to describe the predictive ability of the model.

Citation: Nicola Bellomo, Sarah De Nigris, Damián Knopoff, Matteo Morini, Pietro Terna. Swarms dynamics approach to behavioral economy: Theoretical tools and price sequences. Networks and Heterogeneous Media, 2020, 15 (3) : 353-368. doi: 10.3934/nhm.2020022
References:
[1]

D. AcemogluD. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, Journal of European Economic Association, 9 (2011), 177-208.  doi: 10.3386/w12748.

[2]

S.-M. AhnH.-O. BaeS.-Y. SeungY. Kim and H. Lim, Application of flocking mechanisms to the modeling of stochastic volatily, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[3]

G. Ajmone MarsanN. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinetic & Related Models, 1 (2008), 249-278.  doi: 10.3934/krm.2008.1.249.

[4]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.

[5]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[6]

G. AlbiL. PareschiG. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 49-98. 

[7]

H.-O. BaeS.-Y. ChoS.-H. LeeJ. Yoo and S.-B. Yun, A particle model for herding phenomena induced by dynamic market signals, Journal of Statistical Physics, 177 (2019), 365-398.  doi: 10.1007/s10955-019-02371-8.

[8]

H.-O. BaeS. -Y.ChoJ. Kim and S.-B. Yun, A kinetic description for the herding behavior in financial market, Journal of Statistical Physics, 176 (2019), 398-424.  doi: 10.1007/s10955-019-02305-4.

[9] K. D. Baley, Sociology and New Systems Theory - Toward a Theoretical Syntesis, Suny Press, 1994. 
[10]

P. Ball, Why Society is a Complex Matter, Springer-Verlag, 2012. doi: 10.1007/978-3-642-29000-8.

[11]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proceedings of the Natural Academy of Sciences USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.

[12]

N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2017. doi: 10.1007/978-3-319-57436-3.

[13]

N. BellomoF. ColasuonnoD. Knopoff and J. Soler, From a systems theory of sociology to modeling the onset and evolution of criminality, Netw. Heterog. Media, 10 (2015), 421-441.  doi: 10.3934/nhm.2015.10.421.

[14]

N. BellomoG. DosiD. A.Knopoff and M.E. Virgillito, From particles to firms: on the kinetic theory of climbing up evolutionary landscapes, Math. Models Methods Appl. Sci., 30 (2020), 1441-14060.  doi: 10.1142/S021820252050027X.

[15]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[16]

N. BellomoM. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479.  doi: 10.3934/krm.2013.6.459.

[17]

N. BellomoD. Knopoff and J. Soler, On the difficult interplay between life "complexity" and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913.  doi: 10.1142/S021820251350053X.

[18]

J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Tipping Points: Modelling Social Problems and Health, Wiley, London, 2015. doi: 10.1002/9781118992005.

[19]

R. Boero, M. Morini, M. Sonnessa and P. Terna, Agent-based Models of the Economy From Theories to Applications, Palgrave Macmillan, 2015.

[20]

P. Bonacich and P. Lu, Introduction to Mathematical Sociology Princeton University Press, Princeton, NJ, 2012.

[21]

S. BowlesA. Kirman and R. Sethi, Retrospectives: Friedrich hayek and the market algorithm, Journal of Economic Perspectives, 31 (2017), 215-230.  doi: 10.1257/jep.31.3.215.

[22]

C. Brugna and G. Toscani, Kinetic models of opinion formation in the presence of personal conviction, Physical Review E, 92 (2015), 052818. doi: 10.1103/PhysRevE.92.052818.

[23]

C. Brugna and G. Toscani, Boltzmann-type models for price formation in the presence of behavioral aspects, Netw. Heterog. Media, 10 (2015), 543-557.  doi: 10.3934/nhm.2015.10.543.

[24]

C. Brugna and G. Toscani, Kinetic models for goods exchange in a multi-agent market, Physica A, 499 (2018), 362-375.  doi: 10.1016/j.physa.2018.02.070.

[25]

D. BuriniS. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Physics of Life Reviews, 16 (2016), 123-139. 

[26]

D. BuriniL. Gibelli and N. Outada, A kinetic theory approach to the modeling of complex living systems, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 229-258. 

[27]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 853-862.  doi: 10.1109/TAC.2007.895842.

[28]

A. Deaton, Measuring and understanding behavior, welfare, and poverty, American Economic Review, 106 (2016), 1221-1243.  doi: 10.1257/aer.106.6.1221.

[29]

M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2361-2381.  doi: 10.1142/S0218202514500237.

[30]

M. Dolfin and M. Lachowicz, Modeling opinion dynamics: How the network enhances consensus, Netw. Heterog. Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.

[31]

M. DolfinL. Leonida and N. Outada, Modeling human behaviour in economics and social science, Physics of Life Reviews, 22 (2017), 1-21. 

[32]

M. DolfinD. KnopoffL. Leonida and D. Maimone Ansaldo Patti, Escaping the trap of 'blocking': A kinetic model linking economic development and political competition, Kinet. Relat. Models, 10 (2017), 423-443.  doi: 10.3934/krm.2017016.

[33]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.

[34]

S. Gächter and J. F. Schultz, Intrinsic honesty and the prevalence of rule violations across societies, Nature, 531(7595) (2017), 496-499. 

[35]

S. Galam, Sociophysics. A Physicist's Modeling of Psycho-Political Phenomena, Understanding Complex Systems, Springer, New York, 2012. doi: 10.1007/978-1-4614-2032-3.

[36]

F. Gino and L. Pierce, The abundance effect: Unethical behavior in the presence of wealth, Organizational Behavior and Human Decision Processes, 109 (2009), 142-155. 

[37]

H. Gintis, Game Theory Evolving, Second edition, Princeton University Press, Princeton NJ, 2009.

[38]

R. Hegselmann, Thomas C. Shelling and James M. Sakoda: The intellectual, technical and social history of a model, Journal of Artificial Societies and Social Simulation, 20 (2017). doi: 10.18564/jass.5311.

[39]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic and leaders, and other constant signals: A simple unifying model, Netw. Heterog. Media, 10 (2015), 477-509.  doi: 10.3934/nhm.2015.10.477.

[40]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.

[41]

A. Kirman, Complex Economics: Individual and Collective Rationality, Routledge, London, 2011. doi: 10.4324/9780203847497.

[42]

A. P. Kirman and J. B. Zimmermann, Economics with Heterogeneous Interacting Agents, Lecture Notes in Economics and Mathematical Systems, 503. Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56472-7.

[43]

D. Knopoff, On the modeling of migration phenomena on small networks, Math. Models Methods Appl. Sci., 23 (2013), 541-563.  doi: 10.1142/S0218202512500558.

[44]

D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426.  doi: 10.1142/S0218202513400137.

[45] M. MazzoliM. Morini and P. Terna, Rethinking Macroeconomics with Endogenous Market Structure, Cambridge University Press, 2019.  doi: 10.1017/9781108697019.
[46]

S. McQuadeB. Piccoli and N. Pouradier Duteil, Social dynamics models with time-varying influence, Math. Models Methods Appl. Sci., 29 (2019), 681-716.  doi: 10.1142/S0218202519400037.

[47]

M. Nieddu, Brownian and More Complex Agents to Explain Markets Behavior, Master's thesis, University of Torino, 2018, https://terna.to.it/tesi/nieddu.pdf.

[48]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.

[49]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013.

[50]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A Boltzmann approach, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130396, 15 pp. doi: 10.1098/rsta.2013.0396.

[51]

B. PiccoliN. Pouradier Duteil and E. Trelat, Sparse control of Hegselmann-Krause models: Black hole and declustering, SIAM Journal on Control and Optimization, 57 (2019), 2628-2659.  doi: 10.1137/18M1168911.

[52]

P. K. PiffD. M. StancatoS. CotéR. Mendoza-Denton and D. Keltner, Higher social class predicts increased unethical behavior, Proceedings of the Natural Academy of Sciences USA, 109 (2014), 4086-4091.  doi: 10.1073/pnas.1118373109.

[53]

S. Salvi, Corruption corrupts: Society-level rule violations affect individuals' intrinsic honesty, Nature, 53 (2016), 456-457. 

[54]

F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, Springer Series in Synergetics, Springer-Verlag, Berlin, 2003.

[55]

K. Sigmund, The Calculus of Selfishness, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2010. doi: 10.1515/9781400832255.

[56]

J. E. Stiglitz, Information and the change in the paradigm in economics, The American Economic Review, 92 (2009), 460-501.  doi: 10.1017/CBO9780511754357.004.

[57]

N. N. Taleb, The Black Swan: The Impact of the Highly Improbable, Random House, New York City, 2007.

[58]

R. H. Thaler, Misbehaving: The Making of Behavioral Economics, W. W. Norton & Company, New York, 2015.

[59]

R. H. Thaler, Behavioral economics: Past, present, and future, The American Economic Review, 106 (2016), 1577-1600. 

[60]

T. A. Weber, Price theory in economics, in Ö. Özer, and R. Phillips, The Oxford Handbook of Pricing Management, (2012), 281–340.

[61]

E.G. Weyl, Price theory, Journal of Economic Literature, 57 (2019), 329-384.  doi: 10.1257/jel.20171321.

show all references

References:
[1]

D. AcemogluD. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, Journal of European Economic Association, 9 (2011), 177-208.  doi: 10.3386/w12748.

[2]

S.-M. AhnH.-O. BaeS.-Y. SeungY. Kim and H. Lim, Application of flocking mechanisms to the modeling of stochastic volatily, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628.  doi: 10.1142/S0218202513500176.

[3]

G. Ajmone MarsanN. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinetic & Related Models, 1 (2008), 249-278.  doi: 10.3934/krm.2008.1.249.

[4]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.

[5]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[6]

G. AlbiL. PareschiG. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 49-98. 

[7]

H.-O. BaeS.-Y. ChoS.-H. LeeJ. Yoo and S.-B. Yun, A particle model for herding phenomena induced by dynamic market signals, Journal of Statistical Physics, 177 (2019), 365-398.  doi: 10.1007/s10955-019-02371-8.

[8]

H.-O. BaeS. -Y.ChoJ. Kim and S.-B. Yun, A kinetic description for the herding behavior in financial market, Journal of Statistical Physics, 176 (2019), 398-424.  doi: 10.1007/s10955-019-02305-4.

[9] K. D. Baley, Sociology and New Systems Theory - Toward a Theoretical Syntesis, Suny Press, 1994. 
[10]

P. Ball, Why Society is a Complex Matter, Springer-Verlag, 2012. doi: 10.1007/978-3-642-29000-8.

[11]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proceedings of the Natural Academy of Sciences USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.

[12]

N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2017. doi: 10.1007/978-3-319-57436-3.

[13]

N. BellomoF. ColasuonnoD. Knopoff and J. Soler, From a systems theory of sociology to modeling the onset and evolution of criminality, Netw. Heterog. Media, 10 (2015), 421-441.  doi: 10.3934/nhm.2015.10.421.

[14]

N. BellomoG. DosiD. A.Knopoff and M.E. Virgillito, From particles to firms: on the kinetic theory of climbing up evolutionary landscapes, Math. Models Methods Appl. Sci., 30 (2020), 1441-14060.  doi: 10.1142/S021820252050027X.

[15]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[16]

N. BellomoM. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479.  doi: 10.3934/krm.2013.6.459.

[17]

N. BellomoD. Knopoff and J. Soler, On the difficult interplay between life "complexity" and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913.  doi: 10.1142/S021820251350053X.

[18]

J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Tipping Points: Modelling Social Problems and Health, Wiley, London, 2015. doi: 10.1002/9781118992005.

[19]

R. Boero, M. Morini, M. Sonnessa and P. Terna, Agent-based Models of the Economy From Theories to Applications, Palgrave Macmillan, 2015.

[20]

P. Bonacich and P. Lu, Introduction to Mathematical Sociology Princeton University Press, Princeton, NJ, 2012.

[21]

S. BowlesA. Kirman and R. Sethi, Retrospectives: Friedrich hayek and the market algorithm, Journal of Economic Perspectives, 31 (2017), 215-230.  doi: 10.1257/jep.31.3.215.

[22]

C. Brugna and G. Toscani, Kinetic models of opinion formation in the presence of personal conviction, Physical Review E, 92 (2015), 052818. doi: 10.1103/PhysRevE.92.052818.

[23]

C. Brugna and G. Toscani, Boltzmann-type models for price formation in the presence of behavioral aspects, Netw. Heterog. Media, 10 (2015), 543-557.  doi: 10.3934/nhm.2015.10.543.

[24]

C. Brugna and G. Toscani, Kinetic models for goods exchange in a multi-agent market, Physica A, 499 (2018), 362-375.  doi: 10.1016/j.physa.2018.02.070.

[25]

D. BuriniS. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Physics of Life Reviews, 16 (2016), 123-139. 

[26]

D. BuriniL. Gibelli and N. Outada, A kinetic theory approach to the modeling of complex living systems, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 229-258. 

[27]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 853-862.  doi: 10.1109/TAC.2007.895842.

[28]

A. Deaton, Measuring and understanding behavior, welfare, and poverty, American Economic Review, 106 (2016), 1221-1243.  doi: 10.1257/aer.106.6.1221.

[29]

M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2361-2381.  doi: 10.1142/S0218202514500237.

[30]

M. Dolfin and M. Lachowicz, Modeling opinion dynamics: How the network enhances consensus, Netw. Heterog. Media, 10 (2015), 877-896.  doi: 10.3934/nhm.2015.10.877.

[31]

M. DolfinL. Leonida and N. Outada, Modeling human behaviour in economics and social science, Physics of Life Reviews, 22 (2017), 1-21. 

[32]

M. DolfinD. KnopoffL. Leonida and D. Maimone Ansaldo Patti, Escaping the trap of 'blocking': A kinetic model linking economic development and political competition, Kinet. Relat. Models, 10 (2017), 423-443.  doi: 10.3934/krm.2017016.

[33]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.

[34]

S. Gächter and J. F. Schultz, Intrinsic honesty and the prevalence of rule violations across societies, Nature, 531(7595) (2017), 496-499. 

[35]

S. Galam, Sociophysics. A Physicist's Modeling of Psycho-Political Phenomena, Understanding Complex Systems, Springer, New York, 2012. doi: 10.1007/978-1-4614-2032-3.

[36]

F. Gino and L. Pierce, The abundance effect: Unethical behavior in the presence of wealth, Organizational Behavior and Human Decision Processes, 109 (2009), 142-155. 

[37]

H. Gintis, Game Theory Evolving, Second edition, Princeton University Press, Princeton NJ, 2009.

[38]

R. Hegselmann, Thomas C. Shelling and James M. Sakoda: The intellectual, technical and social history of a model, Journal of Artificial Societies and Social Simulation, 20 (2017). doi: 10.18564/jass.5311.

[39]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic and leaders, and other constant signals: A simple unifying model, Netw. Heterog. Media, 10 (2015), 477-509.  doi: 10.3934/nhm.2015.10.477.

[40]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.

[41]

A. Kirman, Complex Economics: Individual and Collective Rationality, Routledge, London, 2011. doi: 10.4324/9780203847497.

[42]

A. P. Kirman and J. B. Zimmermann, Economics with Heterogeneous Interacting Agents, Lecture Notes in Economics and Mathematical Systems, 503. Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56472-7.

[43]

D. Knopoff, On the modeling of migration phenomena on small networks, Math. Models Methods Appl. Sci., 23 (2013), 541-563.  doi: 10.1142/S0218202512500558.

[44]

D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426.  doi: 10.1142/S0218202513400137.

[45] M. MazzoliM. Morini and P. Terna, Rethinking Macroeconomics with Endogenous Market Structure, Cambridge University Press, 2019.  doi: 10.1017/9781108697019.
[46]

S. McQuadeB. Piccoli and N. Pouradier Duteil, Social dynamics models with time-varying influence, Math. Models Methods Appl. Sci., 29 (2019), 681-716.  doi: 10.1142/S0218202519400037.

[47]

M. Nieddu, Brownian and More Complex Agents to Explain Markets Behavior, Master's thesis, University of Torino, 2018, https://terna.to.it/tesi/nieddu.pdf.

[48]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.

[49]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013.

[50]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A Boltzmann approach, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130396, 15 pp. doi: 10.1098/rsta.2013.0396.

[51]

B. PiccoliN. Pouradier Duteil and E. Trelat, Sparse control of Hegselmann-Krause models: Black hole and declustering, SIAM Journal on Control and Optimization, 57 (2019), 2628-2659.  doi: 10.1137/18M1168911.

[52]

P. K. PiffD. M. StancatoS. CotéR. Mendoza-Denton and D. Keltner, Higher social class predicts increased unethical behavior, Proceedings of the Natural Academy of Sciences USA, 109 (2014), 4086-4091.  doi: 10.1073/pnas.1118373109.

[53]

S. Salvi, Corruption corrupts: Society-level rule violations affect individuals' intrinsic honesty, Nature, 53 (2016), 456-457. 

[54]

F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, Springer Series in Synergetics, Springer-Verlag, Berlin, 2003.

[55]

K. Sigmund, The Calculus of Selfishness, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2010. doi: 10.1515/9781400832255.

[56]

J. E. Stiglitz, Information and the change in the paradigm in economics, The American Economic Review, 92 (2009), 460-501.  doi: 10.1017/CBO9780511754357.004.

[57]

N. N. Taleb, The Black Swan: The Impact of the Highly Improbable, Random House, New York City, 2007.

[58]

R. H. Thaler, Misbehaving: The Making of Behavioral Economics, W. W. Norton & Company, New York, 2015.

[59]

R. H. Thaler, Behavioral economics: Past, present, and future, The American Economic Review, 106 (2016), 1577-1600. 

[60]

T. A. Weber, Price theory in economics, in Ö. Özer, and R. Phillips, The Oxford Handbook of Pricing Management, (2012), 281–340.

[61]

E.G. Weyl, Price theory, Journal of Economic Literature, 57 (2019), 329-384.  doi: 10.1257/jel.20171321.

Figure 1.  1.0, 5.0 ratios: 10/50 buyers (red) and 10/10 sellers (blue), mean price sequences; blue line hides in large part the red one
Figure 2.  1.0, 5.0 ratio: 10/50 buyers (red) and 10/10 sellers (blue), zoom on individual price sequences. Y axes do not share the same scale
Figure 3.  1.0, 5.0 ratio: 10/50 buyers (red) and 10/10 sellers (blue), standard deviation of mean prices within buyers and within sellers over time
Figure 4.  Buyers
Figure 5.  Sellers
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