Open-Loop Nash Equilibria for Dynamic Games Involving Volterra Integral Equations

Abstract

In this paper, we consider a class of finite horizon dynamic games in which the state equation is a Volterra integral equation with infinite delay. This may be viewed as a game with coupled state and control constraints and is viewed as a game with coupled constraints in the spirit of Rosen. Our existence result is obtained by a fixed point argument using normalized equilibria. The results rely on convexity and seminormality conditions popularized by L. Cesari in the 1960s and 1970s as well as many others. As an example to which our results are applicable, we consider a competitive economic model originally appearing in the works of C. F. Roos in 1925.

Appendices

Appendix A: Proof of Theorem 6

1.1 Preliminaries

In this section we give a proof of the lower closure result, Theorem 6. To facilitate our presentation we begin by recalling the following two definition.

Definition A.1

A set-valued map $R:A\rightrightarrows \mathbb R^{2+m}$ satisfies the upper semicontinuity property (Q) with respect to x if for almost every t ∈ [a, b] and all $x\in \mathbb R^n$ one has

$$\displaystyle \begin{aligned} R(t,x)=\bigcap_{\delta >0}\mathrm{cl}\left(\bigcup \{R(t,x')\,:\ |x-x'|<\delta\}\right). \end{aligned}$$

A related upper semicontinuity condition originally defined by L. Cesari is given in the following definition.

Definition A.2

A set-valued map $R:A\rightrightarrows \mathbb R^{2+m}$ satisfies the upper semicontinuity property (K) with respect to x if for almost every t ∈ [a, b] and all $x\in \mathbb R^n$ one has

$$\displaystyle \begin{aligned} R(t,x)=\bigcap_{\delta >0}\mathrm{cl}\left(\mathrm{co}\left(\bigcup \{R(t,x')\,:\ |x-x'|<\delta\}\right)\right),\end{aligned}$$

where, for a set W, the notation co(W) denotes the convex hull of W.

Remark A.1

Clearly if a set-valued mapping R(⋅, ⋅) satisfies property (Q) with respect to x, it also satisfies property (K) with respect to x. Indeed if R(⋅, ⋅) satisfies property (Q) with respect to x one has

$$\displaystyle \begin{aligned} \begin{array}{rcl}R(t,x)&\subset& \bigcap_{\delta >0}\mathrm{cl}\left(\bigcup \{R(t,x')\,:\ |x-x'|<\delta\}\right)\\ &\subset& \bigcap_{\delta >0}\mathrm{cl}\left(\mathrm{co}\left(\bigcup \{R(t,x')\,:\ |x-x'|<\delta\}\right)\right) = R(t,x)\end{array} \end{aligned} $$

for almost every t ∈ [a, b] and $x\in \mathbb R^{2+m}$ , since for any set co(W) is the smallest convex set containing W, which implies the inclusions become equalities.

To present the proof of Theorem 6, we need to introduce an additional set-valued mapping, $R^*:A\rightrightarrows \mathbb R^{3+m}$ by the formula

$$\displaystyle \begin{aligned} R^*(t,x)\doteq\big\{(\rho, y^1,y^2,z):\ \rho\ge h(|z|),\ y^1\ge \lambda^1(t)-1,\\ y^2\ge \lambda^2(t)-1,\ (y^1,y^2,z)\in R(t,x)\big\}, \end{aligned} $$

(18)

where h : [0, ∞) → [0, ∞) is monotone decreasing, continuous, and convex and such that h(ζ)/ζ →∞ as ζ →∞, and $\lambda ^1,\lambda ^2:[a,b]\to \mathbb R$ are Lebesgue integrable functions. This set-valued mapping is clearly related to the set-valued mapping R(t, x). However, from Cesari [6, 10.5.ii], it follows that the set-valued mapping R ^∗ satisfies the stronger upper semicontinuity condition property (Q) with respect to x as given in the above definition. As we will see, this fact will enable us to prove Theorem 6.

1.2 The Proof

In what follows, we will find the need to extract a number of sequences. To avoid extensive detailed notations, we will always assume the sequences are relabeled with the same index.

To begin our proof, let T ₀ ⊂ [a, b] be the set of Lebesgue measure zero where A(t) is not closed and, for i = 1, 2 and $k\in \mathbb N$ put

$$\displaystyle \begin{aligned} j_k^i=\int_a^b\eta_k^i(t)\, dt,\end{aligned}$$

and let $\tau ^i=\liminf _{k\to \infty } j^i_k$. Observe that for i = 1, 2, we have τ ⁱ > −∞. Since x _k  → x in measure as k →∞, $\lambda _k^i\to \lambda ^i$ weakly in $L^1([a,b];\mathbb R)$ as k →∞ and $\eta _k^i(t)\ge \lambda _k^i(t)$ for almost every t ∈ [a, b], we can extract subsequence so that x _k (t) → x(t) pointwise almost everywhere on [a, b] and

$$\displaystyle \begin{aligned}\lim_{k\to \infty} j^i_k=\lim_{k\to \infty}\int_a^b\eta_k^i(t)\, dt =\tau^i,\quad i=1,2.\end{aligned}$$

We further observe that under our hypotheses, we have τ ¹ ≤ E. Now, for $s\in \mathbb N$, let $\delta _s^i=\max \{|j_k^i-\tau ^i|:\ k\ge s+1\}$, and notice that $\delta _s^i\to 0$ as s →∞.

Now let $T_0^{\prime }\subset [a,b]$ be the set of Lebesgue measure zero for which x _k (t)↛x(t) as k →∞ and observe that we have

$$\displaystyle \begin{aligned} x(t)\in A(t)\quad\mbox{for all } t\in [a,b]\setminus (T_0\cup T_0^{\prime}).\end{aligned}$$

The sequences $\{\lambda _k^i\}$ (i = 1, 2) and {ξ _k } converge weakly in L ¹ to $\lambda _k^i$ and ξ, respectively. By applying the Dunford-Pettis theorem (see Cesari [6, 10.3.i]), there exists a function h : [0, +∞) → [0, +∞) that is monotone nondecreasing, convex, continuous, and satisfying h(ζ)/ζ →∞ as ζ →∞, such that the sequence of functions {ρ _k }, ρ _k (t)≐h(|ξ _k (t)|) converges weakly in $L^1([a,b];\mathbb R)$ to some nonnegative integrable function $\rho :[a,b]\to \mathbb R$.

For any $s\in \mathbb N$, we have that the sequences {ρ _s+k }, $\{\lambda _{s+k}^i\}$ (i = 1, 2), and {ξ _s+k } converge weakly in L ¹ as k →∞, respectively, to ρ, λ ⁱ (i = 1, 2) and ξ. Consequently by appealing to the Banach-Saks-Mazur theorem (see [6, 10.1.i]), there is a set of real numbers $c^{(s)}_{Nk}\ge 0$, k = 1, 2, …, N, $N\in \mathbb N$ satisfying $\sum {k=1}^N c^{(s)}_{Nk}=1$ such that the sequences of functions $\{\rho ^{(s)}_N\}$, $\{\lambda ^{i(s)}_N\}$ (i = 1, 2) and $\{\xi ^{(s)}_N\}$, defined, for t ∈ [a, b] and $N\in \mathbb N$ by the formulas

$$\displaystyle \begin{aligned} \rho_N^{(s)}(t)\doteq \sum_{k=1}^N c^{(s)}_N \rho_{s+k}(t),\quad \lambda^{i(s)}_N(t)=\sum_{k=1}^n c^{(s)}_N\lambda^i_{s+k}(t)\ (i=1,2),\end{aligned}$$

and

$$\displaystyle \begin{aligned}\xi^{(s)}_N(t)=\sum_{k=1}^Nc^{(s)}_N\xi_{s+k}(t),\end{aligned}$$

converges strongly in L ¹, respectively, to ρ, λ ⁱ (i = 1, 2) and ξ. Moreover, this is true for every $s\in \mathbb N$.

Now, for every $s\in \mathbb N$, there exists a set T _s  ⊂ [a, b] of Lebesgue measure zero and a sequence of positive integers $N^{(s)}_l\to \infty$ such that for every t ∈ [a, b] ∖ T _s , one has that ρ(t), λ ⁱ(t) (i = 1, 2) and ξ(t) are finite and the sequences $\{\rho ^{(s)}_{N^{(s)}_l}(t)\}$, $\{\lambda _{N^{(s)}_l}^{i(s)}(t)\}$ (i = 1, 2) and $\{\xi _{N^{(s)}_l}^{(s)}\}$ converge to ρ(t), λ ⁱ(t) (i = 1, 2) and ξ(t) as l →∞.

Let $T=T_0\cup T_0^{\prime }\cup \{T_s:\ s\in \mathbb N\}$ and observe T has Lebesgue measure zero. Define the functions $\eta _N^{i(s)}:[a,b]\to \mathbb R$, for i = 1, 2 and $s,N\in \mathbb N$, by the formulas

$$\displaystyle \begin{aligned}\eta_N^{i(s)}(t)=\sum_{k=1}^N c^{(s)}_{Nk}\eta^i_{s+k}(t),\end{aligned}$$

and observe that for almost all t ∈ [a, b], we have

$$\displaystyle \begin{aligned}\rho_k(t)=h(|\xi_k(t)\|),\quad \eta^i_k(t)\ge \lambda_k^i(t), \quad\mbox{and}\quad \lim_{k\to\infty}\int_a^b\eta_k^i(t)=\tau^i.\end{aligned}$$

Consequently, for all $s\in \mathbb N$ and all $N\in \mathbb N$ we have

$$\displaystyle \begin{aligned}\eta_N^{i(s)}(t)\ge \lambda_N^{i(s)}(t)\quad\mbox{and}\quad \tau^i-\delta^i_s\le \int_a^b\eta_N^{i(s)}(t)\, dt \le \tau^i+\delta_s^i.\end{aligned}$$

For $s\in \mathbb N$ and i = 1, 2 define $\eta ^{i(s)}:[a,b]\to \mathbb R$ by the formula

$$\displaystyle \begin{aligned} \eta^{i(s)}(t)=\liminf_{l\to \infty}\eta_{N^{(s)}_l}^{i(s)}(t),\quad t\in T,\end{aligned} $$

and zero elsewhere, and observe that η ^i(s)(t) ≥ λ ⁱ(t) (i = 1, 2 and $s\in \mathbb N$) for almost all t. By Fatou’s lemma, we now have

$$\displaystyle \begin{aligned} \int_a^b\lambda^i(t)\, dt \le \int_a^b\eta^{i(s)}(t)\, dt \le \liminf_{l\to \infty}\int_a^b\eta^{i(s)}_{N^{(s)}_l}(t)\, dt \le \tau^i+\delta^i_s,\end{aligned}$$

which implies that η ^i(s) is integrable and finite almost everywhere.

For $s\in \mathbb N$ and i = 1, 2 let $T_s^{i\prime }\subset [a,b]$ be the set of Lebesgue measure zero where η ^i(s)(t) is not finite and define the function $\eta ^i:[a,b]\to \mathbb R$ by the formula η ⁱ(t) =liminf _s→∞ η ^i(s)(t) for $t\in T_s^{i\prime }$ and zero elsewhere and observe that we have η ⁱ(t) ≥ λ ⁱ(t) for almost all t ∈ [a, b] and that

$$\displaystyle \begin{aligned}\int_a^b\eta^i(t)\, dt \le \tau^i.\end{aligned}$$

For $t\in [a,b]\setminus T^i_s$ we have that λ ⁱ(t) is finite so that for all $l\in \mathbb N$ sufficiently large, say l ≥ l ₀(t, s), we have $\eta ^{i(s)}_{N^{(s)}_l}(t)\ge \lambda (t)-1$. This means we can drop finitely many terms (depending on t and s) so that $\eta ^{i(s)}_{N^{(s)}_l}(t)\ge \lambda (t)-1$ for all $l\in \mathbb N$. Let $T_0^{i\prime \prime }\subset [a,b]$ be the set of measure zero where η ⁱ(t) is not finite. Finally let T ^∗⊂ [a, b] be the set of measure zero defined as the union of the sets T ₀, $T_0^{\prime }$, $T_0^{i\prime \prime }$, T _s , and $T_s^{i\prime }$ ($s\in \mathbb N$ and i = 1, 2) and fix t ₀ ∈ [a, b] ∖ T ^∗ and set x ₀ = x(t ₀). Then (t ₀, x _k (t ₀)) → (t ₀, x ₀) ∈ A as k →∞ and for any 𝜖 > 0 there exists $s_0\in \mathbb N$ such that if s ≥ s ₀ one has |x _s (t ₀) − x ₀| < 𝜖. Now, for any s ≥ s ₀ and $k\in \mathbb N$, we have $(\eta _{s+k}^1(t_0),\eta _{s+k}^2(t_0),\xi _{s+k}(t_0))\in R(t_0,x_{s+k}(t_0))$ and |x _s+k (t ₀) − x ₀| < 𝜖. Moreover, we also have for l ≥ l(t ₀, s) that

$$\displaystyle \begin{aligned}\eta^{i(s)}(t_0)\ge \lambda^i(t_0)-1,\end{aligned}$$

which means that

$$\displaystyle \begin{aligned} \bigg(\sum_{k=1}^{N_l} c^{(s)}_{N^{(s)}_lk}\rho_{s+k}(t_0), \sum_{k=1}^{N_l} c^{(s)}_{N^{(s)}_lk}\eta^{1}_{s+k}(t_0),\\ \sum_{k=1}^{N_l} c^{(s)}_{N^{(s)}_lk}\eta^{2}_{s+k}(t_0), \sum_{k=1}^{N_l} c^{(s)}_{N^{(s)}_lk}\xi_{s+k}(t_0)\bigg)\in \mathrm{co}R^*(t_0,x_0;\epsilon), \end{aligned} $$

(19)

where

$$\displaystyle \begin{aligned}R^*(t_0,x_0;\epsilon)=\bigcup\{R^*(t_0,x)\,:\ |x-x_0|<\epsilon\},\end{aligned}$$

in which R ^∗(t ₀, x) is given by (18) with t = t ₀. The right side of (19) is a sequence of points in $\mathbb R^{m+3}$ which has the vector (ρ(t ₀), η ^1(s)(t ₀), η ^2(s)(t ₀), ξ(t ₀)) as an accumulation point. This means

$$\displaystyle \begin{aligned}(\rho(t_0),\eta^{1(s)}(t_0),\eta^{2(s)}(t_0),\xi(t_0))\in \mathrm{cl}\,\mathrm{co}R^*(t_0,x_0;\epsilon)\\ =\mathrm{cl}\left(\mathrm{co}\left(\bigcup \{R^*(t,x)\,:\ |x-x_0|<\delta\}\right)\right), \end{aligned} $$

and since η ⁱ(t ₀) =liminf _s→∞ η ^i(s)(t ₀) (i = 1, 2) is finite, it follows that

$$\displaystyle \begin{aligned}(\rho(t_0),\eta^{1}(t_0),\eta^{2}(t_0),\xi(t_0))\in \mathrm{cl}\,\mathrm{co}R^*(t_0,x_0;\epsilon)\\ =\mathrm{cl}\left(\mathrm{co}\left(\bigcup \{R^*(t,x)\,:\ |x-x_0|<\delta\}\right)\right). \end{aligned} $$

Since this holds for all 𝜖 > 0, it follows that

$$\displaystyle \begin{aligned}(\rho(t_0),\eta^{1}(t_0),\eta^{2}(t_0),\xi(t_0))&\in \bigcap_{\epsilon >0}\mathrm{cl}\left(\mathrm{co}\left(\bigcup \{R^*(t,x)\,:\ |x-x_0|<\delta\}\right)\right)\\ &=R^*(t_0,x_0), \end{aligned} $$

since the set-valued map R ^∗ satisfies property(Q).

The desired conclusion now follows since t ₀ ∈ [a, b] ∖ T ^∗ is arbitrary, and the definition of R ^∗ implies that for almost all t ∈ [a, b] that x(t) ∈ A(t), (η ¹(t), η ²(t), ξ(t)) ∈ R ^∗(t, x(t)) and that

$$\displaystyle \begin{aligned} \int_a^b\eta^1(t)\, dt \le \tau^1\le E\quad \text{and}\quad \int_a^b\eta^2(t)\, dt\le \tau^2,\end{aligned}$$

as desired.$\Box$

Appendix B: Measurable Selections

In this section we present some well-known results concerning measurable set-valued maps. We begin with the following definitions.

Definition B.1

A set-valued map $\mathcal {M}:[a,b]\rightrightarrows \mathbb R^m$ is said to be Lebesgue measurable if and only if for every open subset $G\subset \mathbb R^m$ , the set $\{t\in [a,b]:\ \mathcal {M}(t)\cap G\ne \emptyset \}$ is Lebesgue measurable.

Definition B.2

Given a set-valued mapping $\mathcal {M}:[a,b]\rightrightarrows \mathbb R^m$ a Lebesgue measurable function $m:[a,b]\to \mathbb R^m$ is called a Lebesgue measurable selection of $\mathcal {M}$ if and only if $m(t)\in \mathcal {M}(t)$ for almost all t ∈ [a, b].

Regarding the existence of Lebesgue measurable selections, we have the following theorem.

Theorem B.1

A closed-valued, Lebesgue measurable mapping $\mathcal {M}:[a,b]\rightrightarrows \mathbb R^m$ always admits a Lebesgue measurable selection. That is, there exists a Lebesgue measurable function $m:[a,b]\to \mathbb R^m$ such that $m(t)\in \mathcal {M}(t)$ for almost every t ∈ [a, b]

Proof

See Rockafellar and Wets [8, Corollary 14.6] □

Regarding set-valued maps related to the set-valued maps $\mathcal {M}$ defined in the proofs of Theorems 7 and 8 we have the following theorem.

Theorem B.2

Let $\mathcal {V}:[a,b]\rightrightarrows \mathbb R^m$ be a Lebesgue measurable set-valued mapping such that $\mathcal {V}(t)\subset \mathbb R^m$ is closed for almost every t ∈ [a, b], let $g_1,g_2,\dots , g_l:[a,b]\times \mathbb R^m\to \mathbb R$ and $f_1,f_2,\dots , f_k:[a,b]\times \mathbb R^m\to \mathbb R$ be continuous mappings, and let $\alpha _i:[a,b]\to \mathbb R$ , i = 1, 2, …, l, and $\beta _j:[a,b]\to \mathbb R$ , j = 1, 2, …, k, be Lebesgue measurable functions. Then the set-valued map $\mathcal {M}:[a,b]\rightrightarrows \mathbb R^m$ defined by the formula

$$\displaystyle \begin{aligned}\mathcal{M}(t)\doteq \{v\in \mathcal{V}(t):\ g_i(t,u)\ge \alpha_i(t), i=1,2,\dots,l,\\ f_j(t,u)=\beta_j(t),\ j=1,2,\dots,k\}\end{aligned} $$

is a Lebesgue measurable set-valued mapping. Therefore, there exists a Lebesgue measurable function $u^*:[a,b]\to \mathbb R^m$ such that $u^*(t)\in \mathcal {M}(t)$ for almost every t ∈ [a, b].

Proof

See Rockafellar and Wets [8, Theorem 14.36]. □

Remark B.1

In Rockafellar and Wets [ 8 , Theorem 14.36], the theorem is stated for normal integrands. This notion includes continuous functions as a special case.