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17.01.2022

Jump Equilibria in Public-Good Differential Games with a Single State Variable

verfasst von: Johannes M. Schumacher, Puduru Viswanadha Reddy, Jacob C. Engwerda

Erschienen in: Dynamic Games and Applications

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Abstract

A simple sufficient condition is proved for symmetric Markov subgame perfect Nash equilibria in public-good differential games with a single state variable. The condition admits equilibria in feedback strategies that have discontinuous dependence on the state variable. The application of the condition is demonstrated in the Dockner–Long model for international pollution control. The existence is shown of equilibria that are arbitrarily close to Pareto dominance for all initial conditions. In the limit as the discount rate tends to 0, the equilibrium strategies differ from the optimal strategies under full coordination, but nevertheless the agents’ payoffs do converge to those obtained from the coordinated (first-best) solution. For positive values of the discount rate, the supremal value function associated with the globally Pareto dominant equilibrium is a continuously differentiable function that is not a solution of the Hamilton–Jacobi–Bellman equation.

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Fußnoten
1
In the theory of discontinuous dynamical systems [10], points that are subject to pressure at a positive rate from both sides, as expressed in (2.9), are sometimes referred to as “chattering points”. The terminology is avoided here in the one-dimensional case, because the modeling in this paper does not presume that actual chattering takes place.
 
2
Generally speaking, for a continuous function x(t) defined on an interval [0, T] and a given point \({\hat{x}}\), it is possible that the set \(\{ t \in [0,T] \mid x(t) = {\hat{x}}\}\) has measure arbitrarily close to the length T of the interval [0, T], while there is no interval \([\tau ,\tau '] \subset [0,T]\) with \(\tau ' > \tau \) such that \(x(t) = {\hat{x}}\) for almost all \(t \in [\tau ,\tau ']\). An example can be constructed by making use of the “\(\varepsilon \)-Cantor set” [2, p. 140]. Episodicity of the solution of \({\dot{x}}=g(x,u)\) therefore implies in particular that the control function u(t) does not induce this type of singular behavior.
 
3
When used with a subscript indicating a region of the state space, the symbol \(\mathbbm {1}\) denotes the function that takes the value 1 in the indicated region, and the value 0 elsewhere.
 
4
In the terminology of physics, this means that pollution is modeled here as an extensive quantity (such as mass), rather than as an intensive quantity (such as temperature). Such modeling can be reasonable, for instance, in the case of chemical pollution.
 
5
The symbol p is used both to denote a function of s in (4.10) and to denote a function of x in (4.9). This is an abuse of notation.
 
6
The prime mark, when applied to vectors, denotes transposition.
 
7
An analogous observation is in [11]; see their comments following Def. 4.13 in the cited paper.
 
Literatur
1.
Zurück zum Zitat Akao KI, Uchida K, Wasa Y (2018) International environmental agreements as an equilibrium choice in a differential game. In: Proceedings of the 6th world congress of environmental and resource economists, Gothenburg, Sweden Akao KI, Uchida K, Wasa Y (2018) International environmental agreements as an equilibrium choice in a differential game. In: Proceedings of the 6th world congress of environmental and resource economists, Gothenburg, Sweden
2.
Zurück zum Zitat Aliprantis CD, Burkinshaw O (1998) Principles of real analysis, 3rd edn. Academic Press, San Diego MATH Aliprantis CD, Burkinshaw O (1998) Principles of real analysis, 3rd edn. Academic Press, San Diego MATH
3.
Zurück zum Zitat Barles G, Briani A, Chasseigne E (2013) A Bellman approach for two-domains optimal control problems in \({\mathbb{R}}^N\). ESAIM Control Optim Calc Var 19:710–739 MathSciNetCrossRef Barles G, Briani A, Chasseigne E (2013) A Bellman approach for two-domains optimal control problems in \({\mathbb{R}}^N\). ESAIM Control Optim Calc Var 19:710–739 MathSciNetCrossRef
4.
Zurück zum Zitat Barles G, Briani A, Chasseigne E (2014) A Bellman approach for regional optimal control problems in \({\mathbb{R}}^N\). SIAM J Control Optim 52:1712–1744 MathSciNetCrossRef Barles G, Briani A, Chasseigne E (2014) A Bellman approach for regional optimal control problems in \({\mathbb{R}}^N\). SIAM J Control Optim 52:1712–1744 MathSciNetCrossRef
5.
Zurück zum Zitat Basar T, Olsder GJ (1995) Dynamic noncooperative game theory, 2nd edn. Academic Press, London MATH Basar T, Olsder GJ (1995) Dynamic noncooperative game theory, 2nd edn. Academic Press, London MATH
6.
Zurück zum Zitat Dockner E, Wagener F (2014) Markov perfect Nash equilibria in models with a single capital stock. Econ Theory 56:585–625 MathSciNetCrossRef Dockner E, Wagener F (2014) Markov perfect Nash equilibria in models with a single capital stock. Econ Theory 56:585–625 MathSciNetCrossRef
7.
Zurück zum Zitat Dockner EJ, Van Long N (1993) International pollution control: cooperative versus noncooperative strategies. J Environ Econ Manag 25:13–29 CrossRef Dockner EJ, Van Long N (1993) International pollution control: cooperative versus noncooperative strategies. J Environ Econ Manag 25:13–29 CrossRef
8.
Zurück zum Zitat Dockner EJ, Sorger G (1996) Existence and properties of equilibria for a dynamic game on productive assets. J Econ Theory 71:209–227 MathSciNetCrossRef Dockner EJ, Sorger G (1996) Existence and properties of equilibria for a dynamic game on productive assets. J Econ Theory 71:209–227 MathSciNetCrossRef
9.
Zurück zum Zitat Engwerda JC (2016) Properties of feedback Nash equilibria in scalar LQ differential games. Automatica 69:364–374 MathSciNetCrossRef Engwerda JC (2016) Properties of feedback Nash equilibria in scalar LQ differential games. Automatica 69:364–374 MathSciNetCrossRef
10.
Zurück zum Zitat Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht CrossRef Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht CrossRef
11.
Zurück zum Zitat Jaakkola N, Wagener F (2020) All symmetric equilibria in differential games with public goods. Discussion Paper 2020-020/II. Tinbergen Institute Jaakkola N, Wagener F (2020) All symmetric equilibria in differential games with public goods. Discussion Paper 2020-020/II. Tinbergen Institute
12.
Zurück zum Zitat Rowat C (2001) Additive externality games. PhD thesis, University of Birmingham, Birmingham, UK Rowat C (2001) Additive externality games. PhD thesis, University of Birmingham, Birmingham, UK
13.
Zurück zum Zitat Rowat C (2007) Non-linear strategies in a linear quadratic differential game. J Econ Dyn Control 31:3179–3202 MathSciNetCrossRef Rowat C (2007) Non-linear strategies in a linear quadratic differential game. J Econ Dyn Control 31:3179–3202 MathSciNetCrossRef
14.
Zurück zum Zitat Rubio SJ, Casino B (2002) A note on cooperative versus non-cooperative strategies in international pollution control. Resour Energy Econ 24:251–261 CrossRef Rubio SJ, Casino B (2002) A note on cooperative versus non-cooperative strategies in international pollution control. Resour Energy Econ 24:251–261 CrossRef
15.
Zurück zum Zitat Tsutsui S, Mino K (1990) Nonlinear strategies in dynamic duopolistic competition with sticky prices. J Econ Theory 52:136–161 MathSciNetCrossRef Tsutsui S, Mino K (1990) Nonlinear strategies in dynamic duopolistic competition with sticky prices. J Econ Theory 52:136–161 MathSciNetCrossRef
16.
Zurück zum Zitat van Damme E (1987) Stability and perfection of nash equilibria. Springer, Berlin CrossRef van Damme E (1987) Stability and perfection of nash equilibria. Springer, Berlin CrossRef
18.
Zurück zum Zitat van der Ploeg F, de Zeeuw AJ (1991) A differential game of international pollution control. Syst Control Lett 17:409–414 MathSciNetCrossRef van der Ploeg F, de Zeeuw AJ (1991) A differential game of international pollution control. Syst Control Lett 17:409–414 MathSciNetCrossRef
19.
Zurück zum Zitat van der Ploeg F, de Zeeuw AJ (1992) International aspects of pollution control. Environ Resour Econ 2:117–139 CrossRef van der Ploeg F, de Zeeuw AJ (1992) International aspects of pollution control. Environ Resour Econ 2:117–139 CrossRef
Metadaten
Titel
Jump Equilibria in Public-Good Differential Games with a Single State Variable
verfasst von
Johannes M. Schumacher
Puduru Viswanadha Reddy
Jacob C. Engwerda
Publikationsdatum
17.01.2022
Verlag
Springer US
Erschienen in
Dynamic Games and Applications
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00415-x

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