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17-01-2022

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

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

Published in: Dynamic Games and Applications

## 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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Appendix
Available only for authorised users
Footnotes
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.

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Title
Jump Equilibria in Public-Good Differential Games with a Single State Variable
Authors
Johannes M. Schumacher