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Published in:

21-02-2020

# Symmetric Markovian Games of Commons with Potentially Sustainable Endogenous Growth

Authors: Zaruhi Hakobyan, Christos Koulovatianos

Published in: Dynamic Games and Applications | Issue 1/2021

## Abstract

Differential games of common resources that are governed by linear accumulation constraints have several applications. Examples include political rent-seeking groups expropriating public infrastructure, oligopolies expropriating common resources, industries using specific common infrastructure or equipment, capital flight problems, pollution, etc. Most of the theoretical literature employs specific parametric examples of utility functions. For symmetric differential games with linear constraints and a general time-separable utility function depending only on the player’s control variable, we provide an exact formula for interior symmetric Markovian strategies. This exact solution (a) serves as a guide for obtaining some new closed-form solutions and for characterizing multiple equilibria and (b) implies that if the utility function is an analytic function, then the Markovian strategies are analytic functions, too. This analyticity property facilitates the numerical computation of interior solutions of such games using polynomial projection methods and gives potential for computing modified game versions with corner solutions by employing a homotopy approach.

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Appendix
Available only for authorised users
Footnotes
1
An earlier survey paper in differential games is Clemhout and Wan [8]. A recent paper by Kunieda and Nishimura [22] extends the Tornell and Velasco [34] model by introducing uncertainty and financial constraints. This study examines how commons problems are affected by imperfect financial markets and how the possibility for sustainable growth is affected by these commons problems. Although our model is deterministic, it can contribute to extending such analyses by using more general utility functions.

2
An early application of Markovian differential games to pollution is Dockner and Long [10].

3
Typically, Markovian differential game models require metric space or other functional analysis methods in order to prove that solutions exist, that they are well behaved, or that they possess certain desirable functional properties. Such approaches are necessitated by the complexity of dynamic programming problems, especially if their constraints are nonlinear. Regarding the approximation theory difficulties posed by dynamic programming problems and an exposition of metric space methods, see, for example, Chow and Tsitsiklis [6]. Theoretical foundations of differential games are provided by Basar and Olsder [2] and Dockner et al. [12].

4
See, for example, Tsutsui and Mino [35] and Dockner and Long [10], who use a similar approach for characterizing multiple Markovian equilibria, but who are restricted to linear quadratic games.

5
Such solutions can provide insights into other extensions of dynamic games of commons with piecewise linear constraints such as Colombo and Labrecciosa [9] or partly linear/partly nonlinear constraints, such as Benchekroun [3].

6
Notice that we exclude $$A=0$$, which is games with non-renewable resources. We focus on games with potentially sustainable resource outcomes.

7
For example, unlike in many papers, such as in Dockner and Sorger [11, p. 213], an upper bound is imposed on the consumption level, c, and the resource reproduction function is also bounded in their study. Here, in some cases of sustainable growth, c can grow to infinity. In examples that we present in a later section, we identify the cases where an upper bound must be placed on c and cases in which such a bound does not apply.

8
See Tsutsui and Mino [35, p. 144] and Dockner and Long [10, p. 22]. We demonstrate this point in a later section of this paper, too.

9
See Clairaut [7].

10
See, for example, Lang [24, Theorem 4.2, p. 60], proving that the composition of continuous functions gives a continuous function.

11
This property of continuity of strategies differs from Dockner and Sorger [11, Theorem 1, p. 2015], where the strategies can be discontinuous functions.

12
The next section, where we present several closed-form solutions, gives “hands-on” examples of how the choice of parameters affects whether a Markov perfect Nash equilibrium solution is interior or not.

13
Most of our examples, except the slightly more generalized case with “Gorman preferences” and the case of constant absolute risk aversion preferences, which we present below, have been thoroughly studied by Gaudet and Lohoues [16], who go beyond the use of linear resource reproduction functions, specifying the types of resource reproduction functions that allow for linear strategies. We thank Hassan Benchekroun for pointing this paper to us.

14
In Tasneem, Engle-Warnick and Benchekroun [31], there is experimental evidence that players may choose both linear and nonlinear strategies. The theoretical model employed in Tasneem, Engle-Warnick and Benchekroun [31] allows for multiple equilibria, providing a clear distinction between linear and nonlinear equilibria. The evidence that nonlinear strategies may be chosen by players supports the usefulness of our new example. We are indebted to Hassan Benchekroun for making this point to us.

15
Apparently, combining (39), (38) and (37) is necessary in order to identify parametric restrictions guaranteeing that $$C\left( k\left( t\right) \right) >0$$ for all $$t\ge 0$$, consistently with an interior solution.

16
A study explaining that nonlinear strategies can also exist is Tsutsui and Mino [35]. Nevertheless, focusing on interior solutions is important on whether such nonlinear strategies can exist or not in linear quadratic games.

17
Specifically, in the case of Gorman preferences, the parametric restrictions on N, A, $$\rho$$, $$\chi$$, $$\theta$$, $$k\left( 0\right)$$, given by conditions (35), (39) and (42), are needed in order to guarantee that $$k\left( t\right) >-N\chi /A$$ for all $$t\ge 0$$.

18
Fish reproduction is the application in Sorger [29], who also uses a linear, constant reproduction rate, Ak. Alternative interpretations would include exogenously supplied infrastructure by governments to users, such as public roads, assuming that users have an upper capacity of usage, $${\bar{c}}$$.

19
For the derivation of $$\phi \left( \lambda \right)$$, see “Appendix.”

20
For a proof of this result, see “Appendix.”

21
See “Appendix” for details on this point.

22
To see why (75) implies $$A>\left( 5N-2\right) \rho /\left[ 2\left( 3N-1\right) \right]$$ for all $$N\ge 1$$, define $$H\left( N\right) =\left( 5N-2\right) \rho /\left[ 2\left( 3N-1\right) \right]$$. Notice that $$H\left( 1\right) =3\rho /4$$, with $$H^{\prime }\left( N\right) =\rho /\left[ 2\left( 3N-1\right) ^{2}\right] >0$$ and with $$\lim _{N\rightarrow \infty }H\left( N\right) =5\rho /6$$.

23
See “Appendix” for a proof of this statement.

24
Think, for example, of a railroad that is provided exogenously by a government, with railway companies utilizing this railroad infrastructure in a rivalrous and non-excludable manner, at no cost. This infrastructure, k , can depreciate with utilization, i.e., by the number of passengers of each company, $$q_{i}$$, according to an endogenous depreciation function that is linear in $$q_{i}$$, say, $$\delta \left( q_{i}\right) =\psi k_{i}$$, and parameter A in the law of motion of k is normalized so as to set $$\psi =1$$. A discrete-time version of this setup is given, for example, in Koulovatianos and Mirman [19, p. 203].

25
We thank an anonymous referee for pointing this interpretation to us.

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Title
Symmetric Markovian Games of Commons with Potentially Sustainable Endogenous Growth
Authors
Zaruhi Hakobyan
Christos Koulovatianos
Publication date
21-02-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 1/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00349-w

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