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.
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Appendix
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
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:
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.
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Hingu, D. Asymptotic stability of strongly uninvadable sets. Ann Oper Res 287, 737–749 (2020). https://doi.org/10.1007/s10479-017-2695-9
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DOI: https://doi.org/10.1007/s10479-017-2695-9