Stabilization of capital accumulation games

Abstract

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.

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 $i \in N:$

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

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

• (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 $f^i(x,u) = g^i(x).$

• (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

$$\begin{aligned} \nabla _{x_k} L^1 =\cdots =\nabla _{x_k} L^n\ \ \forall k\in R, \end{aligned}$$

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

$$\begin{aligned} \nabla _{u_i} P =\nabla _{u_i} L^i,\ \ \nabla _{x_i} P =\nabla _{x_i} L^i \end{aligned}$$

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

$$\begin{aligned} (x,u)\mapsto H (t,x,u,\lambda ^*(t)) \end{aligned}$$

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

$$\begin{aligned} \max _{(x,u)\in X\times U}P(x,u) \end{aligned}$$

subject to

$$\begin{aligned} f(x,u)=0. \end{aligned}$$

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