Abstract
In this paper we apply non-linear incentive strategies to sustain over time an agreement. We illustrate the use of these strategies in a linear-quadratic transboundary pollution differential game. The incentive strategies are constructed in such a way that in the long run the pollution stock (the state variable) is close to the steady state of the pollution stock under the cooperative mode of play. The non-linear incentive functions depend on the emission rates (control variables) of both players and on the current value of the pollution stock. The credibility of the incentive equilibrium strategies is analyzed and the performance of open-loop and feedback incentive strategies is compared in their role of helping to sustain an agreement over time. We present numerical experiments to illustrate the results.
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Notes
- 1.
We are indebted to an anonymous reviewer for bringing this interpretation to our attention.
- 2.
To simplify the notation we will drop the explicit dependence on the time variable when no confusion can arise.
- 3.
The pollution stock and emission time-paths for the Nash and incentive equilibria as well as the payoffs when the players use feedback strategies presented later in this paper have been taken from De Frutos and Martín-Herrán (2015).
- 4.
We thank an anonymous reviewer for pointing out this question.
References
Breton, M., Sokri, A., & Zaccour, G. (2008). Incentive equilibrium in an overlapping-generations environmental game. European Journal of Operational Research, 185, 687–699.
Buratto, A., & Zaccour, G. (2009). Coordination of advertising strategies in a fashion licensing contract. Journal of Optimization Theory and Applications, 142, 31–53.
De Frutos, J., & Martín-Herrán, G. (2015). Does flexibility facilitate sustainability of cooperation over time? A case study from environmental economics. Journal of Optimization Theory and Applications, 165, 657–677.
De Giovanni, P. (2018). A joint maximization incentive in closed-loop supply chains with competing retailers: The case of spent-battery recycling. European Journal of Operational Research, 268, 128–147.
De Giovanni, P., Reddy, P. V., & Zaccour, G. (2016). Incentive strategies for an optimal recovery program in a closed-loop supply chain. European Journal of Operational Research, 249, 605–617.
Dockner, E. J., & Long, N. V. (1993). International pollution control: Cooperative versus noncooperative strategies. Journal of Environmental Economics and Management, 25, 13–29.
Ehtamo, H., & Hämäläinen, R. P. (1986). On affine incentives for dynamic decision problems. In T. Basar (Ed.), Dynamic games and applications in economics (pp. 47–63). Berlin: Springer.
Ehtamo, H., & Hämäläinen, R. P. (1989). Incentive strategies and equilibria for dynamic games with delayed information. Journal of Optimization Theory and Applications, 63, 355–369.
Ehtamo, H., & Hämäläinen, R. P. (1993). Cooperative incentive equilibrium for a resource management problem. Journal of Economic Dynamics and Control, 17, 659–678.
Ehtamo, H., & Hämäläinen, R. P. (1995). Credibility of linear equilibrium strategies in a discrete time fishery management game. Group Decision and Negotiation, 4, 27–37.
Haurie, A., Krawczyk, J. B., & Zaccour, G. (2012). Games and dynamic games. Singapore: World Scientific.
Haurie, A., & Pohjola, M. (1987). Efficient equilibria in a differential game of capitalism. Journal of Economic Dynamics and Control, 11, 65–78.
Jørgensen, S., Martín-Herrán, G., & Zaccour, G. (2003). Agreeability and time-consistency in linear-state differential games. Journal of Optimization Theory and Applications, 119, 49–63.
Jørgensen, S., Martín-Herrán, G., & Zaccour, G. (2005). Sustainability of cooperation overtime in linear-quadratic differential games. International Game Theory Review, 7, 395–406.
Jørgensen, S., Taboubi, S., & Zaccour, G. (2006). Incentives for retailer promotion in a marketing channel. In A. Haurie, S. Muto, L. A. Petrosjan, & T. E. S. Raghavan (Eds.), Annals of the international society of dynamic games (Vol. 8, pp. 365–378). Boston: Birkhäuser.
Jørgensen, S., & Zaccour, G. (2001a). Time consistent side payments in a dynamic game of downstream pollution. Journal of Economic Dynamics and Control, 25, 1973–1987.
Jørgensen, S., & Zaccour, G. (2001b). Incentive equilibrium strategies and welfare allocation in a dynamic game of pollution control. Automatica, 37, 29–36.
Jørgensen, S., & Zaccour, G. (2002). Time consistency in cooperative differential games. In G. Zaccour (Ed.), Decision and control in management science (in Honor of Alain Haurie) (pp. 349–366). Dordrecht: Kluwer Academic Publishers.
Jørgensen, S., & Zaccour, G. (2003). Channel coordination over time: Incentive equilibria and credibility. Journal of Economic Dynamics and Control, 27, 801–822.
Kaitala, V., & Pohjola, M. (1990). Economic development and agreeable redistribution in capitalism: Efficient game equilibria in a two-class neoclassical growth model. International Economic Review, 31, 421–437.
Kaitala, V., & Pohjola, M. (1995). Sustainable international agreements on greenhouse warming: A game theory study. In C. Carraro & J. A. Filar (Eds.), Annals of the international society of dynamic games (Vol. 2, pp. 67–87). Boston: Birkhäuser.
Kierzenka, J., & Shampine, L. F. (2001). A BVP solver based on residual control and the MATLAB PSE. ACM TOMS, 27, 299–316.
Martín-Herrán, G., & Taboubi, S. (2005). Incentive strategies for shelf-space allocation in duopolies. In A. Haurie & G. Zaccour (Eds.), Dynamic games: Theory and applications. GERAD 25th anniversary series (pp. 231–253). New York: Springer.
Martín-Herrán, G., & Zaccour, G. (2005). Credibility of incentive equilibrium strategies in linear-state differential games. Journal of Optimization Theory and Applications, 126, 1–23.
Martín-Herrán, G., & Zaccour, G. (2009). Credible linear incentive equilibrium strategies in linear-quadratic differential games. In P. Bernhard, V. Gaitsgory, & O. Pourtallier (Eds.), Annals of the international society of dynamic games (Vol. 10, pp. 261–291). Boston: Birkhäuser.
Petrosjan, L. A. (1997). Agreeable solutions in differential games. International Journal of Mathematics, Game Theory and Algebra, 7, 165–177.
Petrosjan, L. A., & Zaccour, G. (2003). Time-consistent Shapley value allocation of pollution cost control. Journal of Economic Dynamics and Control, 27, 381–398.
Petrosjan, L. A., & Zenkevich, N. A. (1996). Game theory. Singapore: World Scientific.
Taboubi, S. (2019). Incentive mechanisms for price and advertising coordination in dynamic marketing channels. International Transactions in Operational Research, 26, 2281–2304.
Tolwinski, B., Haurie, A., & Leitmann, G. (1986). Cooperative equilibria in differential games. Journal of Mathematical Analysis and Applications, 119, 192–202.
Van der Ploeg, F., & De Zeeuw, A. J. (1992). International aspects of pollution control. Environmental and Resource Economics, 2, 117–139.
Acknowledgements
We are grateful to two anonymous reviewers for valuable comments and suggestions in an earlier draft of this paper. This research is partially supported by MINECO under projects MTM2016-78995-P (AEI) (Javier de Frutos) and ECO2014-52343-P and ECO2017-82227-P (AEI) (Guiomar Martín-Herrán) and by Junta de Castilla y León VA024P17 and VA105G18 co-financed by FEDER funds (EU).
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de Frutos, J., Martín-Herrán, G. (2020). Non-linear Incentive Equilibrium Strategies for a Transboundary Pollution Differential Game. In: Pineau, PO., Sigué, S., Taboubi, S. (eds) Games in Management Science. International Series in Operations Research & Management Science, vol 280. Springer, Cham. https://doi.org/10.1007/978-3-030-19107-8_11
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