Stabilization of capital accumulation games

In this paper, the potential differential game concept introduced by Fonseca-Morales and Hernández-Lerma (2018) is used in analyzing stabilization problems for n-player noncooperative capital accumulation games (CAGs). By first identifying a CAG as a potential game, an associated optimal control problem (OCP) of the CAG is obtained, whose optimal solution is an open-loop Nash equilibrium for the CAG. Compared with a saddle-point stability condition obtained for undiscounted CAG in the literature, a sufficient and easily verifiable condition is obtained for both discounted and undiscounted CAGs. In addition, the concept allows the turnpike property obtained for OCPs in Trélat and Zuazua (2015) to be verified for CAGs. Lastly, an illustrative example is given to verify the latter stability result for some CAGs.

The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund.” The first author was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic for Ph.D. Program at KMUTT. This project is supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation (CLASSIC), Faculty of Science, KMUTT.

Moreover, Poom Kumam was supported by the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi under the TRF Research Scholar Grant No.RSA6080047.

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

The research of the third author was partially supported by the Fondo SEP-CINVESTAV grant FIDSC 2018/196, and Conacyt grant CF-2019 263963.

Authors and Affiliations

1. KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

   Jewaidu Rilwan

2. Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand

   Poom Kumam

3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan

   Poom Kumam

4. Mathematics Department, CINVESTAV-IPN, A. Postal 14-740, 07000, Mexico City, Mexico

   Onésimo Hernández-Lerma

Corresponding author

Correspondence to Poom Kumam.

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Published in the topical collection Optimal Control and Dynamic Games: Large Time Behavior and Geometry.

Appendix

Theorem A1

(Theorem 1, [13]) Let \(p : X \times U \rightarrow \mathbb {R}\) be a certain function in differential game (1)–(2). We assume that one of the following conditions holds for every \(i \in N:\)

1. (a)

   There exists a function \(c^i : U_{-i}\rightarrow \mathbb {R}\) such that

   $$\begin{aligned} L^i(x,u) = p(x,u)+c^i(u_{-i}). \end{aligned}$$
2. (b)

   There exist functions \(c^i : X \times U_{-i} \rightarrow \mathbb {R}\ {and} \ g_i : X\rightarrow X_i\) such that

   $$\begin{aligned} L^i(x,u) = p(x,u)+c^i(x,u_{-i}) \end{aligned}$$

   and \(f^i(x,u) = g^i(x).\)

3. (c)

   There exist functions \(c^i : X_{-i} \times U_{-i}\rightarrow \mathbb {R} \ {and}\ g_i : X_i \times U_i\rightarrow X_i\) such that

   $$\begin{aligned} L^i(x,u) = p(x,u)+ {c^i(x_{-i},u_{-i})} \end{aligned}$$

   and \(f^i(x,u) = g^i(x_i,u_i).\)

Then differential game (1)–(2) is an open-loop potential game with potential function p.

Assumption A1

For each \(i\in N,\) the sets \(U_i\) and \(X_i\) are open and convex.

Assumption A2

The functions \(L^1,\ldots ,L^n\) satisfy that

where \(R \subseteq N\) is a nonempty subset of indices k such that \(l_k > 0\).

Condition A1

(Sufficient conditions) The function \(P : [0,\infty ) \times X \times U \rightarrow \mathbb {R}\) is in \(C^2(X \times U)\), is concave in (x, u), and for every \(i \in N\) satisfies

Condition A2

(Sufficient conditions) Let P be a function as in Condition A1. We assume that for each Lagrange multiplier \(\lambda ^*\) as in Remark 4 in [13] the function

is concave in (x, u).

In (11)–(12), the function P is called p. Let P satisfies Conditions A1 and A2, then we have:

Assumption A3

The open-loop multistrategies \(u_\uptau \) of the \((OCP)_\uptau \) defined as in OCP (11)–(12) on the interval \([0,\uptau ],\ \uptau < \infty ,\) are in \(L^\infty ([0,\uptau ])\), the space of essentially bounded functions on \([0,\uptau ]\).

Assumption A4

For each finite \(\uptau >0,\ (OCP)_\uptau \) has at least one optimal solution \((x_\uptau ^*,u_\uptau ^*)\).

Assumption A5

The static optimization problem

subject to

has at least one optimal solution \((\bar{x},\bar{u})\).

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