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July  2019, 6(3): 195-209. doi: 10.3934/jdg.2019014

Cooperative dynamic advertising via state-dependent payoff weights

Simon Hoof

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.

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 & 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

Figure 1.  Plot of $h(\kappa) - 1 + \ln(2)$ for $\kappa \in (0, \overline \kappa]$
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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).
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