# American Institute of Mathematical Sciences

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

Received  November 2018 Revised  April 2019 Published  May 2019

Fund Project: This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center "On-The-Fly Computing" (SFB 901) under the project number 160364472-SFB901

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.

Citation: Simon Hoof. Cooperative dynamic advertising via state-dependent payoff weights. Journal of Dynamics & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] L. A. Petrosjan, Agreeable solutions in differential games, International Journal of Mathematics, Game Theory and Algebra, 7 (1998), 165-177. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar

show all references

##### References:
 [1] M. R. Caputo, Foundations of Dynamic Economic Analysis, Cambridge University Press, 2005. doi: 10.1017/CBO9780511806827. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] L. A. Petrosjan, Agreeable solutions in differential games, International Journal of Mathematics, Game Theory and Algebra, 7 (1998), 165-177. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar
Plot of $h(\kappa) - 1 + \ln(2)$ for $\kappa \in (0, \overline \kappa]$
(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).
 [1] 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 & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521 [2] Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 [3] Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 [4] Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445 [5] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [6] Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 [7] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 [8] Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 [9] 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 [10] Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611 [11] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [12] Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607 [13] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 [14] Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 31-46. doi: 10.3934/dcdss.2020002 [15] F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 85-104. doi: 10.3934/dcdss.2020005 [16] Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134 [17] Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 [18] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [19] Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009 [20] Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics & Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012

Impact Factor:

## Tools

Article outline

Figures and Tables

[Back to Top]