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Stability of the replicator dynamics for games in metric spaces
Nonlinear dynamics from discrete time two-player status-seeking games
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom |
We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional 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 log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player 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 non-existence 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 and R. M. Solow,
Capital-labor substitution and economic efficiency, The Review of Economics and Statistics, 43 (1961), 225-250.
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 and K. Nyborg,
Status-seeking and material affluence: Evaluation the Hirsch hypothesis, Ecological Economics, 45 (2003), 29-39.
doi: 10.1016/S0921-8009(02)00262-8. |
[4] |
A. Chao and J. B. Schor,
Empirical tests of status consumption: Evidence from women's cosmetics, Journal of Economic Psychology, 19 (1998), 107-131.
doi: 10.1016/S0167-4870(97)00038-X. |
[5] |
R. D. Congleton,
Efficient status seeking: externalities and the evolution of status games, Journal of Economic Behavior and Organization, 11 (1989), 175-190.
|
[6] |
R.-A. Dana and L. Montrucchio,
Dynamic complexity in duopoly games, Journal of Economic Theory, 40 (1986), 40-56.
doi: 10.1016/0022-0531(86)90006-2. |
[7] |
A. Glazer and K. A. Konrad,
A signaling explanation for charity, The American Economic Review, 86 (1996), 1019-1028.
|
[8] |
N. J. Ireland,
Status-seeking, income taxation and efficiency, Journal of Public Economics, 70 (1998), 99-113.
|
[9] |
J. James,
Positional goods, conspicuous consumption and the international demonstration effect reconsidered, World Development, 15 (1987), 449-462.
|
[10] |
R. G. King, C. I. Plosser and S. T. Rebelo,
Production, growth and business cycles, Journal of Monetary Economics, 21 (1988), 195-232.
|
[11] |
R. G. King, C. I. Plosser and S. T. Rebelo,
Production, growth and business cycles: Technical appendix, Real Business Cycles, (1998), 108-145.
doi: 10.4324/9780203070710.ch7. |
[12] |
C. S. Kumru and L. Vesterlund,
The effect of status on charitable giving, Journal of Public Economic Theory, 12 (2010), 709-735.
doi: 10.1111/j.1467-9779.2010.01471.x. |
[13] |
Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory, Springer-Verlag New York, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[14] |
A. Matsumoto,
Controlling the Cournot-nash chaos, Journal of Optimization Theory and Applications, 128 (2006), 379-392.
doi: 10.1007/s10957-006-9021-z. |
[15] |
L. Montrucchio,
Optimal decisions over time and strange attractors: An analysis by the Bellman principle, Mathematical Modelling, 7 (1986), 341-352.
doi: 10.1016/0270-0255(86)90055-2. |
[16] |
M. Papageorgiou,
Technical note: Chaos may be an optimal plan, Journal of Optimization Theory and Applications, 119 (2003), 387-393.
doi: 10.1023/B:JOTA.0000005452.22744.66. |
[17] |
T. Puu,
Complex dynamics with three oligopolists, Chaos, Solitons and Fractals, 7 (1996), 2075-2081.
doi: 10.1016/S0960-0779(96)00073-2. |
[18] |
T. Puu,
The chaotic duopolists revisited, Journal of Economics Behavior and Organization, 33 (1998), 385-394.
doi: 10.1016/S0167-2681(97)00064-4. |
[19] |
T. Puu,
On the stability of Cournot equilibrium when the number of competitors increases, Journal of Economics Behavior and Organization, 66 (2008), 445-456.
doi: 10.1016/j.jebo.2006.06.010. |
[20] |
D. Rand,
Exotic phenomena in games and duopoly models, Journal of Mathematical Economics, 5 (1978), 173-184.
doi: 10.1016/0304-4068(78)90022-8. |
[21] |
M. Rauscher,
Keeping up with the Joneses: Chaotic patterns in a status game, Economics Letters, 40 (1992), 287-290.
doi: 10.1016/0165-1765(92)90006-K. |
[22] |
B. C. Snyder, R. A. Van Gorder and K. Vajravelu,
Continuous-time dynamic games for the cournot adjustment process for competing oligopolists, Applied Mathematics and Computation, 219 (2013), 6400-6409.
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), 65-94.
doi: 10.2307/1884513. |
[24] |
H. V. Stackbelberg,
Market Structure and Equilibrium, Springer-Verlag Berlin Heidelberg, 2011. |
[25] |
A. van Ackere and C. Haxholdt,
Clubs as status symbol: Would you belong to a club that accepts you as a member?, Socio-Economic Planning Sciences, 36 (2002), 93-107.
doi: 10.1016/S0038-0121(01)00014-3. |
[26] |
R. A. Van Gorder and M. R. Caputo,
Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games, Journal of Economic Dynamics and Control, 34 (2010), 1123-1139.
doi: 10.1016/j.jedc.2010.01.016. |
show all references
References:
[1] |
K. J. Arrow, H. B. Chenery, B. S. Minhas and R. M. Solow,
Capital-labor substitution and economic efficiency, The Review of Economics and Statistics, 43 (1961), 225-250.
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 and K. Nyborg,
Status-seeking and material affluence: Evaluation the Hirsch hypothesis, Ecological Economics, 45 (2003), 29-39.
doi: 10.1016/S0921-8009(02)00262-8. |
[4] |
A. Chao and J. B. Schor,
Empirical tests of status consumption: Evidence from women's cosmetics, Journal of Economic Psychology, 19 (1998), 107-131.
doi: 10.1016/S0167-4870(97)00038-X. |
[5] |
R. D. Congleton,
Efficient status seeking: externalities and the evolution of status games, Journal of Economic Behavior and Organization, 11 (1989), 175-190.
|
[6] |
R.-A. Dana and L. Montrucchio,
Dynamic complexity in duopoly games, Journal of Economic Theory, 40 (1986), 40-56.
doi: 10.1016/0022-0531(86)90006-2. |
[7] |
A. Glazer and K. A. Konrad,
A signaling explanation for charity, The American Economic Review, 86 (1996), 1019-1028.
|
[8] |
N. J. Ireland,
Status-seeking, income taxation and efficiency, Journal of Public Economics, 70 (1998), 99-113.
|
[9] |
J. James,
Positional goods, conspicuous consumption and the international demonstration effect reconsidered, World Development, 15 (1987), 449-462.
|
[10] |
R. G. King, C. I. Plosser and S. T. Rebelo,
Production, growth and business cycles, Journal of Monetary Economics, 21 (1988), 195-232.
|
[11] |
R. G. King, C. I. Plosser and S. T. Rebelo,
Production, growth and business cycles: Technical appendix, Real Business Cycles, (1998), 108-145.
doi: 10.4324/9780203070710.ch7. |
[12] |
C. S. Kumru and L. Vesterlund,
The effect of status on charitable giving, Journal of Public Economic Theory, 12 (2010), 709-735.
doi: 10.1111/j.1467-9779.2010.01471.x. |
[13] |
Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory, Springer-Verlag New York, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[14] |
A. Matsumoto,
Controlling the Cournot-nash chaos, Journal of Optimization Theory and Applications, 128 (2006), 379-392.
doi: 10.1007/s10957-006-9021-z. |
[15] |
L. Montrucchio,
Optimal decisions over time and strange attractors: An analysis by the Bellman principle, Mathematical Modelling, 7 (1986), 341-352.
doi: 10.1016/0270-0255(86)90055-2. |
[16] |
M. Papageorgiou,
Technical note: Chaos may be an optimal plan, Journal of Optimization Theory and Applications, 119 (2003), 387-393.
doi: 10.1023/B:JOTA.0000005452.22744.66. |
[17] |
T. Puu,
Complex dynamics with three oligopolists, Chaos, Solitons and Fractals, 7 (1996), 2075-2081.
doi: 10.1016/S0960-0779(96)00073-2. |
[18] |
T. Puu,
The chaotic duopolists revisited, Journal of Economics Behavior and Organization, 33 (1998), 385-394.
doi: 10.1016/S0167-2681(97)00064-4. |
[19] |
T. Puu,
On the stability of Cournot equilibrium when the number of competitors increases, Journal of Economics Behavior and Organization, 66 (2008), 445-456.
doi: 10.1016/j.jebo.2006.06.010. |
[20] |
D. Rand,
Exotic phenomena in games and duopoly models, Journal of Mathematical Economics, 5 (1978), 173-184.
doi: 10.1016/0304-4068(78)90022-8. |
[21] |
M. Rauscher,
Keeping up with the Joneses: Chaotic patterns in a status game, Economics Letters, 40 (1992), 287-290.
doi: 10.1016/0165-1765(92)90006-K. |
[22] |
B. C. Snyder, R. A. Van Gorder and K. Vajravelu,
Continuous-time dynamic games for the cournot adjustment process for competing oligopolists, Applied Mathematics and Computation, 219 (2013), 6400-6409.
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), 65-94.
doi: 10.2307/1884513. |
[24] |
H. V. Stackbelberg,
Market Structure and Equilibrium, Springer-Verlag Berlin Heidelberg, 2011. |
[25] |
A. van Ackere and C. Haxholdt,
Clubs as status symbol: Would you belong to a club that accepts you as a member?, Socio-Economic Planning Sciences, 36 (2002), 93-107.
doi: 10.1016/S0038-0121(01)00014-3. |
[26] |
R. A. Van Gorder and M. R. Caputo,
Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games, Journal of Economic Dynamics and Control, 34 (2010), 1123-1139.
doi: 10.1016/j.jedc.2010.01.016. |



















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