Abstract
Grim trigger strategies can support any set of control paths as a cooperative equilibrium, if they yield at least the value of the noncooperative Nash equilibrium. We introduce the recursive Nash bargaining solution as an equilibrium selection device and study its properties by means of an analytically tractable n-person differential game. The idea is that the agents bargain over a tuple of stationary Markovian strategies, before the game has started. It is shown that under symmetry the bargaining solution yields efficient controls.
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- 1.
See Yeung and Petrosyan [9] for a recent treatment on subgame consistent cooperation in differential games.
- 2.
Wirl [8] showed that within the set of nonlinear Markovian strategies the Nash equilibrium is nonunique and that the efficient steady state is potentially reachable.
- 3.
See also Dockner et al. [1, Ch. 9.5] for a textbook treatment.
- 4.
The latter assumption is not too prohibitive. If payoffs were not transferable the individual cooperative value is simply given by \(C_i(x_t) = J_i(\hat u(s), t)\) where \(\hat u(s)\) are the Pareto efficient controls. It turns out that in the symmetric setup \(C_i(x_t) = \frac {C(x_t)}{n}\) which would also be the result under an equal sharing rule with transferable payoffs.
- 5.
See e.g. Dockner et al. [1, Ch. 4] for the theory on noncooperative differential games.
- 6.
See also Dockner et al. [1, Ch. 6].
- 7.
Agreeability is a stronger notion than time consistency. In the former the agreement payoff dominates the noncooperative play for any state while in the latter only along the cooperative path. Time consistency was introduced by Petrosjan [5] (originally 1977) and agreeability by Kaitala and Pohjola [3]. See also Zaccour [10] for a tutorial on cooperative differential games.
References
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Acknowledgements
I thank Mark Schopf and participants of the Doktorandenworkshop der Fakultät für Wirtschaftswissenschaften for valuable comments. This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901).
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Hoof, S. (2018). Dynamic Voluntary Provision of Public Goods: The Recursive Nash Bargaining Solution. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-92988-0_2
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