Asymptotic stability of strongly uninvadable sets

Abstract

In this article, we study in detail asymptotic stability of strongly uninvadable faces generated by finite Borel sets in a continuous strategy space of an evolutionary game. It is proved that such a face is an asymptotically stable set for the associated replicator dynamics. This result is illustrated using examples.

Appendix

We provide an abstract theorem regarding asymptotic stability which is used to prove the main result (Theorem 3) in Sect. 3. Consider an abstract differential equation

$$\begin{aligned} \phi '(t) = H(\phi (t)) \end{aligned}$$

(9)

on a Banach space $(X,\Vert \cdot \Vert _X)$. Assume that the differential equation (9) has a unique solution $\phi (t) = \phi (t;\phi _0)$ defined for every $t \ge 0$ for each initial condition $\phi _0$ in an invariant set $Y \subset X$, which is closed with non-empty interior.

The system (9) is analyzed below around a closed set ${\varPi }\subseteq Y$ of its rest points. To this end, we recall the definition of ${\mathcal {K}}_0^\infty$ functions:

$$\begin{aligned} {\mathcal {K}}_0^\infty&= \{\omega : [0,\infty ) \rightarrow [0,\infty ) ~\mid ~\omega \text{ is } \text{ strictly } \text{ increasing, } \text{ continuous, } \\&~~~~~~~~~~~~~\omega (0) = 0 \text{ and } \lim _{s \rightarrow \infty } \omega (s) = \infty \}. \end{aligned}$$

Also, for $\epsilon > 0$, let $B(\phi ,\epsilon )$ be the set of all $\varphi \in X$ such that $\Vert \phi - \varphi \Vert _X < \epsilon$. Moreover, let $d(\phi ,{\varPi }) = \inf _{\varphi \in {\varPi }} \Vert \phi - \varphi \Vert _X$ and let $B({\varPi },\epsilon )$ be the set of all $\phi \in X$ such that $d(\phi ,{\varPi }) < \epsilon$.

Theorem 4

Let ${\varOmega }$ be an open subset of Y containing a closed set ${\varPi }$ of rest points of the system (9). Assume that $V : {\varOmega }\rightarrow {\mathbb {R}}$ is uniformly continuous on ${\varOmega }$ and satisfies

• (i) $V(\phi ) \ge 0$ on ${\varOmega }$ and $V(\phi ) = 0$ for every $\phi \in {\varPi }$;

• (ii) there exists $\omega \in {\mathcal {K}}_0^\infty$ such that $\omega (d(\phi ,{\varPi })) \le V(\phi )$ for all $\phi \in {\varOmega }$;

• (iii) V is non increasing along trajectories of (9) that lie in ${\varOmega }\setminus {\varPi }$;

Then ${\varPi }$ is Lyapunov stable.

Theorem 5

Let ${\varOmega }$ be an open subset of Y containing a closed set ${\varPi }$ of rest points of the system (9). Assume that $V : {\varOmega }\rightarrow {\mathbb {R}}$ is uniformly continuous on ${\varOmega }$ and satisfies

• (i) $V(\phi ) \ge 0$ on ${\varOmega }$ and $V(\phi ) = 0$ for every $\phi \in {\varPi }$;

• (ii) there exists $\omega \in {\mathcal {K}}_0^\infty$ such that $\omega (d(\phi ,{\varPi })) \le V(\phi )$ for all $\phi \in {\varOmega }$;

• (iii) V is strictly decreasing along trajectories of (9) that lie in ${\varOmega }\setminus {\varPi }$;

• (iv) there exists $\delta _1 > 0$ such that for every trajectory $\phi (t)$ emanating from $B({\varPi },\delta _1)$, there exists a sequence $t_n \rightarrow \infty$ such that $V(\phi (t_n))$ converges to $V(\psi )$ for some $\psi \in {\varOmega }$ and

  $$\begin{aligned} \lim _{s \downarrow 0 ,~ n \uparrow \infty } |V(\phi (s;\psi )) - V(\phi (s;\phi (t_n)))| = 0. \end{aligned}$$

Then ${\varPi }$ is asymptotically stable.

The proofs of the above theorem are extensions of proofs of Theorem 8 and Theorem 9 provided in the Appendix in Hingu et al. (2016) and hence not repeated here.