Cooperative dynamic advertising via state-dependent payoff weights

Simon Hoof

doi:10.3934/jdg.2019014

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2019, Volume 6, Issue 3: 195-209. Doi: 10.3934/jdg.2019014

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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: August 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 91A12, 91A23; Secondary: 58E17, 91A05.

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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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