American Institute of Mathematical Sciences

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September  2015, 10(3): 527-542. doi: 10.3934/nhm.2015.10.527

A kinetic model for an agent based market simulation

Dieter Armbruster 1, , Christian Ringhofer 1, and  Andrea Thatcher 2,

1. 

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804
2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, United States

Received  December 2014 Revised  February 2015 Published  July 2015

A kinetic model for a specific agent based simulation to generate the sales curves of successive generations of high-end computer chips is developed. The resulting continuum market model consists of transport equations in two variables, representing the availability of money and the desire to buy a new chip. In lieu of typical collision terms in the kinetic equations that discontinuously change the attributes of an agent, discontinuous changes are initiated via boundary conditions between sets of partial differential equations. A scaling analysis of the transport equations determines the different time scales that constitute the market forces, characterizing different sales scenarios. It is argued that the resulting model can be adjusted to generic markets of multi-generational technology products where the innovation time scale is an important driver of the market.
Keywords: agent based simulations, Kinetic theory, multi-generational technology products., market models.
Mathematics Subject Classification: 60K35, 91B26, 93A3.
Citation: Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks & Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527
References:
[1]

T. Adriaansen, D. Armbruster, K. G. Kempf and H. Li, An agent model for the high-end gamers market, Advances in Complex Systems, 16 (2013), 1350028, 33pp. doi: 10.1142/S0219525913500288.  Google Scholar

[2]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math, 66 (2006), 896-920. doi: 10.1137/040604625.  Google Scholar

[3]

F. M. Bass, A new product growth model for consumer durables, Mathematical Models in Marketing, Lecture Notes in Economics and Mathematical Systems, 132 (1976), 351-253. doi: 10.1007/978-3-642-51565-1_107.  Google Scholar

[4]

L. Boltzmann, The second law of thermodynamics, Theoretical Physics and Philosophical Problems, Vienna Circle Collection, 5 (1974), 13-32. doi: 10.1007/978-94-010-2091-6_2.  Google Scholar

[5]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory Of Dilute Gases, Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[6]

P. Degond, J.-G. Liu and C. Ringhofer, Large-scale dynamics of mean-field games driven by local Nash equilibria, J. Nonlinear Sci., 24 (2014), 93-115. doi: 10.1007/s00332-013-9185-2.  Google Scholar

[7]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria, J. Stat. Phys., 154 (2014), 751-780. doi: 10.1007/s10955-013-0888-4.  Google Scholar

[8]

D. Helbing, A mathematical model for attitude formation by pair interactions, Behavioral sciences, 37 (1992), 190-214. Google Scholar

[9]

R. J. LeVeque, Finite Volume Methods For Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[10]

H. Li, D. Armbruster and K. G. Kempf, A population-growth model for multiple generations of technology products, Manufacturing & Service Operations Management, 15 (2013), 343-360. doi: 10.1287/msom.2013.0430.  Google Scholar

[11]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations And Monte Carlo Methods, Oxford University Press, 2014. Google Scholar

[12]

G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566. doi: 10.1007/s10955-012-0653-0.  Google Scholar

[13]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061. doi: 10.2307/2937956.  Google Scholar

show all references

References:
[1]

T. Adriaansen, D. Armbruster, K. G. Kempf and H. Li, An agent model for the high-end gamers market, Advances in Complex Systems, 16 (2013), 1350028, 33pp. doi: 10.1142/S0219525913500288.  Google Scholar

[2]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math, 66 (2006), 896-920. doi: 10.1137/040604625.  Google Scholar

[3]

F. M. Bass, A new product growth model for consumer durables, Mathematical Models in Marketing, Lecture Notes in Economics and Mathematical Systems, 132 (1976), 351-253. doi: 10.1007/978-3-642-51565-1_107.  Google Scholar

[4]

L. Boltzmann, The second law of thermodynamics, Theoretical Physics and Philosophical Problems, Vienna Circle Collection, 5 (1974), 13-32. doi: 10.1007/978-94-010-2091-6_2.  Google Scholar

[5]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory Of Dilute Gases, Springer-Verlag, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[6]

P. Degond, J.-G. Liu and C. Ringhofer, Large-scale dynamics of mean-field games driven by local Nash equilibria, J. Nonlinear Sci., 24 (2014), 93-115. doi: 10.1007/s00332-013-9185-2.  Google Scholar

[7]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria, J. Stat. Phys., 154 (2014), 751-780. doi: 10.1007/s10955-013-0888-4.  Google Scholar

[8]

D. Helbing, A mathematical model for attitude formation by pair interactions, Behavioral sciences, 37 (1992), 190-214. Google Scholar

[9]

R. J. LeVeque, Finite Volume Methods For Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[10]

H. Li, D. Armbruster and K. G. Kempf, A population-growth model for multiple generations of technology products, Manufacturing & Service Operations Management, 15 (2013), 343-360. doi: 10.1287/msom.2013.0430.  Google Scholar

[11]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations And Monte Carlo Methods, Oxford University Press, 2014. Google Scholar

[12]

G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566. doi: 10.1007/s10955-012-0653-0.  Google Scholar

[13]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061. doi: 10.2307/2937956.  Google Scholar

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