Better Reply Security and Existence of Equilibria in Differential Games

Abstract

We establish the existence of Nash equilibria in a large class of differential games. More specifically, we show that certain differential games are compact and better reply secure when the strategy space is equipped with the weak topology. Therefore, we can claim that such games must have Nash equilibria even when the payoffs of some of the players are discontinuous. On one hand, verifying better reply security with respect to a nonmetrizable weak topology is technically involved and requires a number of theoretical tools from functional analysis. On the other hand, our approach will allow us to generalize a number of current existence results regarding linear-quadratic differential games, games of evasion and pursuit, as well as some classic differential games in economics.

Appendix

Lemma A.1

Let α(t), v ⁿ(t) and v(t) be functions in L ²([0,T];ℝⁿ). Assume $v^{n}\mathop{\rightarrow}\limits^{w} v$, and let $s^{n}(t)=\int^{t}_{0} \alpha(\tau)v^{n}(\tau)\, d\tau$ and $s(t)= \int^{t}_{0} \alpha (\tau)v(\tau)\, d\tau$. Then s ⁿ pointwise converge to s on [0,T].

Proof

Given $v^{n}\mathop{\rightarrow}\limits^{w} v$ in L ²([0,T];ℝⁿ), $s^{n}(t)=\int^{t}_{0} \alpha(\tau)v^{n}(\tau)\, d\tau$ can expressed as

$$s^n(t)=\int^T_0 \alpha(\tau) \delta_{[0,t]}(\tau) v^n(\tau)\delta_{[0,t]}(\tau) \, d \tau, $$

where δ _[0,t] is the indicator function of the interval [0,t], i.e., δ _[0,t](τ)=1 when τ∈[0,t] and δ _[0,t](τ)=0, otherwise. Clearly, for a fixed t, the function α(τ)δ _[0,t](τ) is in L ²([0,T];ℝⁿ) and by the definition of weak convergence, we must have

$$\int^T_0 \alpha(\tau)\delta_{[0,t]}( \tau) v^n(\tau)\delta_{[0,t]}(\tau) \, d\tau\to \int ^T_0 \alpha(\tau)\delta_{[0,t]}(\tau) v( \tau)\delta_{[0,t]}(\tau) \, d\tau $$

and, therefore, s ⁿ(t)→s(t) for every t∈[0,T]. □