-
Previous Article
Social networks and global transactions
- JDG Home
- This Issue
-
Next Article
Free mobility of capital and Labor force in a two-country model: The dynamic game for growth
July
2019, 6(3): 195-209.
doi: 10.3934/jdg.2019014
Cooperative dynamic advertising via state-dependent payoff weights
Paderborn University, Department of Economics and SFB 901, Paderborn, Germany |
We consider an infinite horizon cooperative advertising differential game with nontransferable utility (NTU). The values of each firm are parametrized by a common discount rate and advertising costs. First we characterize the set of efficient solutions with a constant payoff weight. We show that there does not exist a constant weight that supports an agreeable cooperative solution. Then we consider a linear state-dependent payoff weight and derive an agreeable cooperative solution for a restricted parameter space.
Keywords:
Cooperative differential games,
nontransferable utility,
state-dependent payoff weights,
agreeability,
Pareto optimality.
Mathematics Subject Classification:
Primary: 91A12, 91A23; Secondary: 58E17, 91A05.
Citation:
Simon Hoof. Cooperative dynamic advertising via state-dependent payoff weights. Journal of Dynamics and Games,
2019, 6
(3)
: 195-209.
doi: 10.3934/jdg.2019014
![]()
References:
| [1] |
M. R. Caputo, Foundations of Dynamic Economic Analysis, Cambridge University Press, 2005.
doi: 10.1017/CBO9780511806827.
|
| [2] |
E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000.
doi: 10.1017/CBO9780511805127.
|
| [3] |
A. de-Paz, J. Marín-Solano and J. Navas,
Time-consistent equilibria in common access resource games with asymmetric players under partial cooperation, Environmental Modeling & Assessment, 18 (2013), 171-184.
doi: 10.1007/s10666-012-9339-x. |
| [4] |
S. Jørgensen and G. Zaccour, Time consistency in cooperative differential games, in Decision & Control in Management Science (ed. G. Zaccour), Springer, (2002), 349–366. |
| [5] |
S. Jørgensen, G. Martín-Herrán and G. Zaccour,
Agreeability and time consistency in linear-state differential games, Journal of Optimization Theory and Applications, 119 (2003), 49-63.
doi: 10.1023/B:JOTA.0000005040.78280.a6. |
| [6] |
S. Jørgensen, G. Martín-Herrán and G. Zaccour,
Sustainability of cooperation over time in linear-quadratic differential games, International Game Theory Review, 7 (2005), 395-406.
doi: 10.1142/S0219198905000600. |
| [7] |
V. Kaitala and M. Pohjola,
Economic development and agreeable redistribution in capitalism: Efficient game equilibria in a two-class neo-classical growth model, International Economic Review, 31 (1990), 421-438.
doi: 10.2307/2526848. |
| [8] |
J. Marín-Solano, Time-consistent equilibria in a differential game model with time inconsistent preferences and partial cooperation, in Dynamic Games in Economics (eds. J. Haunschmied, V. Veliov and S. Wrzaczek), Springer, 16 (2014), 219–238.
doi: 10.1007/978-3-642-54248-0_11. |
| [9] |
J. Marín-Solano,
Group inefficiency in a common property resource game with asymmetric players, Economics Letters, 136 (2015), 214-217.
doi: 10.1016/j.econlet.2015.10.002. |
| [10] |
L. A. Petrosjan,
Agreeable solutions in differential games, International Journal of Mathematics, Game Theory and Algebra, 7 (1998), 165-177.
|
| [11] |
L. A. Petrosyan and G. Zaccour, Cooperative differential games with transferable payoffs, in Handbook of Dynamic Game Theory (eds. T. Başar and G. Zaccour), Springer, (2018), 595–632.
doi: 10.1007/978-3-319-44374-4_12. |
| [12] |
A. Prasad and S. P. Sethi,
Competitive advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185.
doi: 10.1023/B:JOTA.0000043996.62867.20. |
| [13] |
S. P. Sethi,
Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184.
doi: 10.1002/oca.4660040207. |
| [14] |
G. Sorger,
Competitive dynamic advertising: A modification of the case game, Journal of Economic Dynamics and Control, 13 (1989), 55-80.
doi: 10.1016/0165-1889(89)90011-0. |
| [15] |
G. Sorger,
Recursive Nash bargaining over a productive asset, Journal of Economic Dynamics and Control, 30 (2006), 2637-2659.
doi: 10.1016/j.jedc.2005.08.005. |
| [16] |
D. W. K. Yeung and L. A. Petrosyan,
Subgame consistent solutions of a cooperative stochastic differential game with nontransferable payoffs, Journal of Optimization Theory and Applications, 124 (2005), 701-724.
doi: 10.1007/s10957-004-1181-0. |
| [17] |
D. W. K. Yeung, L. A. Petrosyan and P. M. Yeung, Subgame consistent solutions for a class of cooperative stochastic differential games with nontransferable payoffs, in Advances in Dynamic Game Theory. Annals of the International Society of Dynamic Games 9 (eds. S. Jørgensen, M. Quincampoix and T. L. Vincent), Birkhäuser, (2007), 153–170.
doi: 10.1007/978-0-8176-4553-3_8. |
| [18] |
D. W. K. Yeung and L. A. Petrosyan,
Subgame consistent cooperative solution for NTU dynamic games via variable weights, Automatica, 59 (2015), 84-89.
doi: 10.1016/j.automatica.2015.01.030. |
| [19] |
D. W. K. Yeung and L. A. Petrosyan, Nontransferable utility cooperative dynamic games, in Handbook of Dynamic Game Theory (eds. T. Başar and G. Zaccour), Springer, (2018), 633–670. |
| [20] |
G. Zaccour,
Time consistency in cooperative differential games: A tutorial, INFOR: Information Systems and Operational Research, 46 (2008), 81-92.
doi: 10.3138/infor.46.1.81. |
show all references
References:
| [1] |
M. R. Caputo, Foundations of Dynamic Economic Analysis, Cambridge University Press, 2005.
doi: 10.1017/CBO9780511806827.
|
| [2] |
E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000.
doi: 10.1017/CBO9780511805127.
|
| [3] |
A. de-Paz, J. Marín-Solano and J. Navas,
Time-consistent equilibria in common access resource games with asymmetric players under partial cooperation, Environmental Modeling & Assessment, 18 (2013), 171-184.
doi: 10.1007/s10666-012-9339-x. |
| [4] |
S. Jørgensen and G. Zaccour, Time consistency in cooperative differential games, in Decision & Control in Management Science (ed. G. Zaccour), Springer, (2002), 349–366. |
| [5] |
S. Jørgensen, G. Martín-Herrán and G. Zaccour,
Agreeability and time consistency in linear-state differential games, Journal of Optimization Theory and Applications, 119 (2003), 49-63.
doi: 10.1023/B:JOTA.0000005040.78280.a6. |
| [6] |
S. Jørgensen, G. Martín-Herrán and G. Zaccour,
Sustainability of cooperation over time in linear-quadratic differential games, International Game Theory Review, 7 (2005), 395-406.
doi: 10.1142/S0219198905000600. |
| [7] |
V. Kaitala and M. Pohjola,
Economic development and agreeable redistribution in capitalism: Efficient game equilibria in a two-class neo-classical growth model, International Economic Review, 31 (1990), 421-438.
doi: 10.2307/2526848. |
| [8] |
J. Marín-Solano, Time-consistent equilibria in a differential game model with time inconsistent preferences and partial cooperation, in Dynamic Games in Economics (eds. J. Haunschmied, V. Veliov and S. Wrzaczek), Springer, 16 (2014), 219–238.
doi: 10.1007/978-3-642-54248-0_11. |
| [9] |
J. Marín-Solano,
Group inefficiency in a common property resource game with asymmetric players, Economics Letters, 136 (2015), 214-217.
doi: 10.1016/j.econlet.2015.10.002. |
| [10] |
L. A. Petrosjan,
Agreeable solutions in differential games, International Journal of Mathematics, Game Theory and Algebra, 7 (1998), 165-177.
|
| [11] |
L. A. Petrosyan and G. Zaccour, Cooperative differential games with transferable payoffs, in Handbook of Dynamic Game Theory (eds. T. Başar and G. Zaccour), Springer, (2018), 595–632.
doi: 10.1007/978-3-319-44374-4_12. |
| [12] |
A. Prasad and S. P. Sethi,
Competitive advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185.
doi: 10.1023/B:JOTA.0000043996.62867.20. |
| [13] |
S. P. Sethi,
Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184.
doi: 10.1002/oca.4660040207. |
| [14] |
G. Sorger,
Competitive dynamic advertising: A modification of the case game, Journal of Economic Dynamics and Control, 13 (1989), 55-80.
doi: 10.1016/0165-1889(89)90011-0. |
| [15] |
G. Sorger,
Recursive Nash bargaining over a productive asset, Journal of Economic Dynamics and Control, 30 (2006), 2637-2659.
doi: 10.1016/j.jedc.2005.08.005. |
| [16] |
D. W. K. Yeung and L. A. Petrosyan,
Subgame consistent solutions of a cooperative stochastic differential game with nontransferable payoffs, Journal of Optimization Theory and Applications, 124 (2005), 701-724.
doi: 10.1007/s10957-004-1181-0. |
| [17] |
D. W. K. Yeung, L. A. Petrosyan and P. M. Yeung, Subgame consistent solutions for a class of cooperative stochastic differential games with nontransferable payoffs, in Advances in Dynamic Game Theory. Annals of the International Society of Dynamic Games 9 (eds. S. Jørgensen, M. Quincampoix and T. L. Vincent), Birkhäuser, (2007), 153–170.
doi: 10.1007/978-0-8176-4553-3_8. |
| [18] |
D. W. K. Yeung and L. A. Petrosyan,
Subgame consistent cooperative solution for NTU dynamic games via variable weights, Automatica, 59 (2015), 84-89.
doi: 10.1016/j.automatica.2015.01.030. |
| [19] |
D. W. K. Yeung and L. A. Petrosyan, Nontransferable utility cooperative dynamic games, in Handbook of Dynamic Game Theory (eds. T. Başar and G. Zaccour), Springer, (2018), 633–670. |
| [20] |
G. Zaccour,
Time consistency in cooperative differential games: A tutorial, INFOR: Information Systems and Operational Research, 46 (2008), 81-92.
doi: 10.3138/infor.46.1.81. |
Figure 2.
(Non) cooperative strategies and values
Note: For$x \in [0, 1]$ and $\mathit{\boldsymbol{\rho}} = (\frac{3}{4}, 1)$ the figure illustrates the noncooperative $\phi_i(x)$ and cooperative strategies $\sigma_i(x)$ (top panels) as well as noncooperative $D_i(x)$ and cooperative values $A_i(x)$ (bottom panels).
Note: For
| [1] |
Zeyang Wang, Ovanes Petrosian. On class of non-transferable utility cooperative differential games with continuous updating. Journal of Dynamics and Games, 2020, 7 (4) : 291-302. doi: 10.3934/jdg.2020020 |
| [2] |
Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521 |
| [3] |
Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 |
| [4] |
Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 |
| [5] |
Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : i-i. doi: 10.3934/dcdss.20201i |
| [6] |
Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445 |
| [7] |
Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 |
| [8] |
Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 |
| [9] |
A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 |
| [10] |
Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure and Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 |
| [11] |
Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416 |
| [12] |
Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611 |
| [13] |
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 |
| [14] |
Hans-Otto Walther. On solution manifolds of differential systems with discrete state-dependent delays. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022108 |
| [15] |
Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002 |
| [16] |
Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607 |
| [17] |
F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005 |
| [18] |
Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134 |
| [19] |
Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 |
| [20] |
Hans-Otto Walther. On Poisson's state-dependent delay. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]