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

A kinetic model for an agent based market simulation

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.
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). 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. doi: 10.1137/040604625.

[3]

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

[4]

L. Boltzmann, The second law of thermodynamics,, Theoretical Physics and Philosophical Problems, 5 (1974), 13. 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. 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. doi: 10.1007/s10955-013-0888-4.

[8]

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

[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. 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. 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. 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). 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. doi: 10.1137/040604625.

[3]

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

[4]

L. Boltzmann, The second law of thermodynamics,, Theoretical Physics and Philosophical Problems, 5 (1974), 13. 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. 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. doi: 10.1007/s10955-013-0888-4.

[8]

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

[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. 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. 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. doi: 10.2307/2937956.

[1]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253

[2]

Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103

[3]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012

[4]

Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431

[5]

Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2018060

[6]

Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463

[7]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429

[8]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621

[9]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623

[10]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77

[11]

Pierre Degond, Hailiang Liu. Kinetic models for polymers with inertial effects. Networks & Heterogeneous Media, 2009, 4 (4) : 625-647. doi: 10.3934/nhm.2009.4.625

[12]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929

[13]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489

[14]

Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022

[15]

Abderrahman Iggidr, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Multi-compartment models. Conference Publications, 2007, 2007 (Special) : 506-519. doi: 10.3934/proc.2007.2007.506

[16]

Gianluca D'Antonio, Paul Macklin, Luigi Preziosi. An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Mathematical Biosciences & Engineering, 2013, 10 (1) : 75-101. doi: 10.3934/mbe.2013.10.75

[17]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[18]

Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic & Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235

[19]

Pierre Monmarché. Hypocoercive relaxation to equilibrium for some kinetic models. Kinetic & Related Models, 2014, 7 (2) : 341-360. doi: 10.3934/krm.2014.7.341

[20]

Michael Herty, Christian Ringhofer. Averaged kinetic models for flows on unstructured networks. Kinetic & Related Models, 2011, 4 (4) : 1081-1096. doi: 10.3934/krm.2011.4.1081

2016 Impact Factor: 1.2

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (3)

[Back to Top]