
Previous Article
Stability of the replicator dynamics for games in metric spaces
 JDG Home
 This Issue

Next Article
A new perspective on the classical Cournot duopoly
Nonlinear dynamics from discrete time twoplayer statusseeking games
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom 
We study the dynamics of twoplayer statusseeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both nonpositional goods and positional goods (the latter entering into "status"). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite logCobbDouglas, and KingPlosserRebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the twoplayer game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or nonexistence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game.
References:
[1] 
K. J. Arrow, H. B. Chenery, B. S. Minhas, R. M. Solow, Capitallabor substitution and economic efficiency, The Review of Economics and Statistics, 43 (1961), 225250. doi: 10.2307/1927286. 
[2] 
M. Boldrin and L. Montrucchio, The Emergence of Dynamic Complexities in Models of Optimal Growth: The Role of Impatience, Rochester Center for Economic Research, Working Paper, 1985. 
[3] 
K. A. Brekke, R. B. Howarth, K. Nyborg, Statusseeking and material affluence: Evaluation the Hirsch hypothesis, Ecological Economics, 45 (2003), 2939. doi: 10.1016/S09218009(02)002628. 
[4] 
A. Chao, J. B. Schor, Empirical tests of status consumption: Evidence from women's cosmetics, Journal of Economic Psychology, 19 (1998), 107131. doi: 10.1016/S01674870(97)00038X. 
[5] 
R. D. Congleton, Efficient status seeking: externalities and the evolution of status games, Journal of Economic Behavior and Organization, 11 (1989), 175190. 
[6] 
R.A. Dana, L. Montrucchio, Dynamic complexity in duopoly games, Journal of Economic Theory, 40 (1986), 4056. doi: 10.1016/00220531(86)900062. 
[7] 
A. Glazer, K. A. Konrad, A signaling explanation for charity, The American Economic Review, 86 (1996), 10191028. 
[8] 
N. J. Ireland, Statusseeking, income taxation and efficiency, Journal of Public Economics, 70 (1998), 99113. 
[9] 
J. James, Positional goods, conspicuous consumption and the international demonstration effect reconsidered, World Development, 15 (1987), 449462. 
[10] 
R. G. King, C. I. Plosser, S. T. Rebelo, Production, growth and business cycles, Journal of Monetary Economics, 21 (1988), 195232. 
[11] 
R. G. King, C. I. Plosser, S. T. Rebelo, Production, growth and business cycles: Technical appendix, Real Business Cycles, (1998), 108145. doi: 10.4324/9780203070710.ch7. 
[12] 
C. S. Kumru, L. Vesterlund, The effect of status on charitable giving, Journal of Public Economic Theory, 12 (2010), 709735. doi: 10.1111/j.14679779.2010.01471.x. 
[13] 
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, SpringerVerlag New York, 1995. 
[14] 
A. Matsumoto, Controlling the Cournotnash chaos, Journal of Optimization Theory and Applications, 128 (2006), 379392. doi: 10.1007/s109570069021z. 
[15] 
L. Montrucchio, Optimal decisions over time and strange attractors: An analysis by the Bellman principle, Mathematical Modelling, 7 (1986), 341352. doi: 10.1016/02700255(86)900552. 
[16] 
M. Papageorgiou, Technical note: Chaos may be an optimal plan, Journal of Optimization Theory and Applications, 119 (2003), 387393. doi: 10.1023/B:JOTA.0000005452.22744.66. 
[17] 
T. Puu, Complex dynamics with three oligopolists, Chaos, Solitons and Fractals, 7 (1996), 20752081. doi: 10.1016/S09600779(96)000732. 
[18] 
T. Puu, The chaotic duopolists revisited, Journal of Economics Behavior and Organization, 33 (1998), 385394. doi: 10.1016/S01672681(97)000644. 
[19] 
T. Puu, On the stability of Cournot equilibrium when the number of competitors increases, Journal of Economics Behavior and Organization, 66 (2008), 445456. doi: 10.1016/j.jebo.2006.06.010. 
[20] 
D. Rand, Exotic phenomena in games and duopoly models, Journal of Mathematical Economics, 5 (1978), 173184. doi: 10.1016/03044068(78)900228. 
[21] 
M. Rauscher, Keeping up with the Joneses: Chaotic patterns in a status game, Economics Letters, 40 (1992), 287290. doi: 10.1016/01651765(92)90006K. 
[22] 
B. C. Snyder, R. A. Van Gorder, K. Vajravelu, Continuoustime dynamic games for the cournot adjustment process for competing oligopolists, Applied Mathematics and Computation, 219 (2013), 64006409. doi: 10.1016/j.amc.2012.12.078. 
[23] 
R. M. Solow, A contribution to the theory of economic growth, The Quarterly Journal of Economics, 70 (1956), 6594. doi: 10.2307/1884513. 
[24] 
H. V. Stackbelberg, Market Structure and Equilibrium, SpringerVerlag Berlin Heidelberg, 2011. 
[25] 
A. van Ackere, C. Haxholdt, Clubs as status symbol: Would you belong to a club that accepts you as a member?, SocioEconomic Planning Sciences, 36 (2002), 93107. doi: 10.1016/S00380121(01)000143. 
[26] 
R. A. Van Gorder, M. R. Caputo, Envelope theorems for locally differentiable openloop Stackelberg equilibria of finite horizon differential games, Journal of Economic Dynamics and Control, 34 (2010), 11231139. doi: 10.1016/j.jedc.2010.01.016. 
show all references
References:
[1] 
K. J. Arrow, H. B. Chenery, B. S. Minhas, R. M. Solow, Capitallabor substitution and economic efficiency, The Review of Economics and Statistics, 43 (1961), 225250. doi: 10.2307/1927286. 
[2] 
M. Boldrin and L. Montrucchio, The Emergence of Dynamic Complexities in Models of Optimal Growth: The Role of Impatience, Rochester Center for Economic Research, Working Paper, 1985. 
[3] 
K. A. Brekke, R. B. Howarth, K. Nyborg, Statusseeking and material affluence: Evaluation the Hirsch hypothesis, Ecological Economics, 45 (2003), 2939. doi: 10.1016/S09218009(02)002628. 
[4] 
A. Chao, J. B. Schor, Empirical tests of status consumption: Evidence from women's cosmetics, Journal of Economic Psychology, 19 (1998), 107131. doi: 10.1016/S01674870(97)00038X. 
[5] 
R. D. Congleton, Efficient status seeking: externalities and the evolution of status games, Journal of Economic Behavior and Organization, 11 (1989), 175190. 
[6] 
R.A. Dana, L. Montrucchio, Dynamic complexity in duopoly games, Journal of Economic Theory, 40 (1986), 4056. doi: 10.1016/00220531(86)900062. 
[7] 
A. Glazer, K. A. Konrad, A signaling explanation for charity, The American Economic Review, 86 (1996), 10191028. 
[8] 
N. J. Ireland, Statusseeking, income taxation and efficiency, Journal of Public Economics, 70 (1998), 99113. 
[9] 
J. James, Positional goods, conspicuous consumption and the international demonstration effect reconsidered, World Development, 15 (1987), 449462. 
[10] 
R. G. King, C. I. Plosser, S. T. Rebelo, Production, growth and business cycles, Journal of Monetary Economics, 21 (1988), 195232. 
[11] 
R. G. King, C. I. Plosser, S. T. Rebelo, Production, growth and business cycles: Technical appendix, Real Business Cycles, (1998), 108145. doi: 10.4324/9780203070710.ch7. 
[12] 
C. S. Kumru, L. Vesterlund, The effect of status on charitable giving, Journal of Public Economic Theory, 12 (2010), 709735. doi: 10.1111/j.14679779.2010.01471.x. 
[13] 
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, SpringerVerlag New York, 1995. 
[14] 
A. Matsumoto, Controlling the Cournotnash chaos, Journal of Optimization Theory and Applications, 128 (2006), 379392. doi: 10.1007/s109570069021z. 
[15] 
L. Montrucchio, Optimal decisions over time and strange attractors: An analysis by the Bellman principle, Mathematical Modelling, 7 (1986), 341352. doi: 10.1016/02700255(86)900552. 
[16] 
M. Papageorgiou, Technical note: Chaos may be an optimal plan, Journal of Optimization Theory and Applications, 119 (2003), 387393. doi: 10.1023/B:JOTA.0000005452.22744.66. 
[17] 
T. Puu, Complex dynamics with three oligopolists, Chaos, Solitons and Fractals, 7 (1996), 20752081. doi: 10.1016/S09600779(96)000732. 
[18] 
T. Puu, The chaotic duopolists revisited, Journal of Economics Behavior and Organization, 33 (1998), 385394. doi: 10.1016/S01672681(97)000644. 
[19] 
T. Puu, On the stability of Cournot equilibrium when the number of competitors increases, Journal of Economics Behavior and Organization, 66 (2008), 445456. doi: 10.1016/j.jebo.2006.06.010. 
[20] 
D. Rand, Exotic phenomena in games and duopoly models, Journal of Mathematical Economics, 5 (1978), 173184. doi: 10.1016/03044068(78)900228. 
[21] 
M. Rauscher, Keeping up with the Joneses: Chaotic patterns in a status game, Economics Letters, 40 (1992), 287290. doi: 10.1016/01651765(92)90006K. 
[22] 
B. C. Snyder, R. A. Van Gorder, K. Vajravelu, Continuoustime dynamic games for the cournot adjustment process for competing oligopolists, Applied Mathematics and Computation, 219 (2013), 64006409. doi: 10.1016/j.amc.2012.12.078. 
[23] 
R. M. Solow, A contribution to the theory of economic growth, The Quarterly Journal of Economics, 70 (1956), 6594. doi: 10.2307/1884513. 
[24] 
H. V. Stackbelberg, Market Structure and Equilibrium, SpringerVerlag Berlin Heidelberg, 2011. 
[25] 
A. van Ackere, C. Haxholdt, Clubs as status symbol: Would you belong to a club that accepts you as a member?, SocioEconomic Planning Sciences, 36 (2002), 93107. doi: 10.1016/S00380121(01)000143. 
[26] 
R. A. Van Gorder, M. R. Caputo, Envelope theorems for locally differentiable openloop Stackelberg equilibria of finite horizon differential games, Journal of Economic Dynamics and Control, 34 (2010), 11231139. doi: 10.1016/j.jedc.2010.01.016. 
[1] 
Getachew K. Befekadu, Panos J. Antsaklis. On noncooperative $n$player principal eigenvalue games. Journal of Dynamics & Games, 2015, 2 (1) : 5163. doi: 10.3934/jdg.2015.2.51 
[2] 
Martin Burger, Marco Di Francesco, Peter A. Markowich, MarieTherese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems  B, 2014, 19 (5) : 13111333. doi: 10.3934/dcdsb.2014.19.1311 
[3] 
Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics & Games, 2016, 3 (1) : 101120. doi: 10.3934/jdg.2016005 
[4] 
Saul MendozaPalacios, Onésimo HernándezLerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319333. doi: 10.3934/jdg.2017017 
[5] 
Josef Hofbauer, Sylvain Sorin. Best response dynamics for continuous zerosum games. Discrete & Continuous Dynamical Systems  B, 2006, 6 (1) : 215224. doi: 10.3934/dcdsb.2006.6.215 
[6] 
Jeremias Epperlein, Stefan Siegmund, Petr Stehlík, Vladimír Švígler. Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics. Discrete & Continuous Dynamical Systems  B, 2016, 21 (3) : 803813. doi: 10.3934/dcdsb.2016.21.803 
[7] 
Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems  B, 2015, 20 (10) : 34873505. doi: 10.3934/dcdsb.2015.20.3487 
[8] 
Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303318. doi: 10.3934/jdg.2016016 
[9] 
Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 3349. doi: 10.3934/jdg.2015.2.33 
[10] 
Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179187. doi: 10.3934/jimo.2012.8.179 
[11] 
Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117140. doi: 10.3934/jdg.2015.2.117 
[12] 
Libin Mou, Jiongmin Yong. Twoperson zerosum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95117. doi: 10.3934/jimo.2006.2.95 
[13] 
Oliver JuarezRomero, William OlveraLopez, Francisco SanchezSanchez. A simple family of solutions for forest games. Journal of Dynamics & Games, 2017, 4 (2) : 8796. doi: 10.3934/jdg.2017006 
[14] 
Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429460. doi: 10.3934/jmd.2013.7.429 
[15] 
Andrzej Swierniak, Michal Krzeslak. Application of evolutionary games to modeling carcinogenesis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 873911. doi: 10.3934/mbe.2013.10.873 
[16] 
John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335354. doi: 10.3934/jdg.2016018 
[17] 
Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93116. doi: 10.3934/krm.2017004 
[18] 
Pierre Cardaliaguet, JeanMichel Lasry, PierreLouis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279301. doi: 10.3934/nhm.2012.7.279 
[19] 
Juan J. Manfredi, Julio D. Rossi, Stephanie J. Somersille. An obstacle problem for TugofWar games. Communications on Pure & Applied Analysis, 2015, 14 (1) : 217228. doi: 10.3934/cpaa.2015.14.217 
[20] 
Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zerosum games. Discrete & Continuous Dynamical Systems  A, 2015, 35 (9) : 39013931. doi: 10.3934/dcds.2015.35.3901 
Impact Factor:
Tools
Article outline
Figures and Tables
[Back to Top]