A kinetic model for an agent based market simulation

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	D. Helbing, A mathematical model for attitude formation by pair interactions, Behavioral sciences, 37 (1992), 190-214.
[9]	R. J. LeVeque, Finite Volume Methods For Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[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.
[11]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations And Monte Carlo Methods, Oxford University Press, 2014.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	D. Helbing, A mathematical model for attitude formation by pair interactions, Behavioral sciences, 37 (1992), 190-214.
[9]	R. J. LeVeque, Finite Volume Methods For Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[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.
[11]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations And Monte Carlo Methods, Oxford University Press, 2014.
[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.
[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.

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