Evolutionary Stability of Polymorphic Population States in Continuous Games

Abstract

Asymptotic stability of equilibrium in evolutionary games with continuous action spaces is an important question. Existing results in the literature require that the equilibrium state be monomorphic. In this article, we address this question when the equilibrium is polymorphic. We show that any uninvadable and finitely supported state is asymptotically stable equilibrium of replicator equation.

Appendix

Here, we establish two abstract stability theorems used to prove our main theorems in Sect. 3. To this end, we consider an abstract differential equation

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

(17)

on a Banach space $(X,\Vert \cdot \Vert _X)$. It is assumed that for each initial condition $\phi _0$ in an invariant set $Y \subset X$, the differential equation (17) has a unique solution $\phi (t) = \phi (t;\phi _0)$ defined for every $t \ge 0$. We want to analyse this system around a rest point $\phi ^* \in Y$. We recall the definition of $\mathscr {K}_0^\infty$ functions:

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

Theorem 8

Let $\Omega$ be an open subset of Y containing the rest point $\phi ^*$ of (17). Assume that $V : \Omega \rightarrow \mathbb {R}$ is continuous at $\phi ^*$ and satisfies

• (i) $V(\phi ) \ge 0$ on $\Omega$ and $V(\phi ^*) = 0$;

• (ii) there exists $\omega \in \mathscr {K}_0^\infty$ such that $\omega (\Vert \phi - \phi ^*\Vert _X) \le V(\phi )$ for all $\phi \in \Omega$;

• (iii) V is non-increasing along trajectories of (17) that lie in $\Omega$.

Then, $\phi ^*$ is Lyapunov stable.

Proof

Let $B(\phi ^*,\epsilon )$ be the open ball around $\phi ^*$ with radius $\epsilon$. Let $\epsilon > 0$ be small enough such that the closure of $B(\phi ^*,\epsilon )$ is contained in $\Omega$.

By continuity of V at $\phi ^*$ and condition (i), there exists $\delta > 0$ such that $V(\phi ) < \omega (\epsilon )$ whenever $\phi \in B(\phi ^*,\delta )$.

Clearly, by condition (iii), the set $U = \{ \phi \in \Omega ~|~ V(\phi ) < \omega (\epsilon ) \}$ is invariant. Without loss of generality we can assume that closure of U is a subset of $\Omega$. Therefore, for every $\phi _0 \in B(\phi ^*,\delta )$, the trajectory $\phi (t) = \phi (t;\phi _0)$ lies in U and hence,

$$\begin{aligned} \omega \left( \Vert \phi (t) - \phi ^*\Vert _X \right) \le V(\phi (t)) < \omega (\epsilon ). \end{aligned}$$

As $\omega \in \mathscr {K}_0^\infty$, it is invertible and from above it follows that

$$\begin{aligned} \Vert \phi (t) - \phi ^*\Vert _X < \epsilon . \end{aligned}$$

Hence, the trajectory $\phi (t)$ lies in $B(\phi ^*,\epsilon )$ whenever $\phi _0 \in B(\phi ^*,\delta )$. $\square$

Theorem 9

Let $\Omega$ be an open subset of Y containing the rest point $\phi ^*$ of (17). Assume that $V : \Omega \rightarrow \mathbb {R}$ is continuous on $\Omega$ and satisfies

• (i) $V(\phi ) \ge 0$ on $\Omega$ and $V(\phi ^*) = 0$;

• (ii) there exists $\omega \in \mathscr {K}_0^\infty$ such that $\omega (\Vert \phi - \phi ^*\Vert _X) \le V(\phi )$ for all $\phi \in \Omega$;

• (iii) V is strictly decreasing along trajectories of (17) that lie in $\Omega {\setminus } \{\phi ^*\}$;

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

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

Then, $\phi ^*$ is asymptotically stable.

Proof

As the Lyapunov stability follows from the above theorem, it remains to show that $\phi ^*$ is attracting.

Let $B(\phi ^*,\delta )$ and $B(\phi ^*,\epsilon )$ be as defined in the proof of the above theorem. Without loss of generality, we may assume that $\delta \le \epsilon$. Similarly, there exists $\delta _2 > 0$ such that all trajectories emanating from $B(\phi ^*,\delta _2)$ lie in $B(\phi ^*,\frac{\delta }{2})$.

Let $\bar{\delta } = \min \{\delta _1,\delta _2\}$ and $\phi (t) = \phi (t;\phi _0)$ be the trajectory of the differential Eq. (17) with the initial condition $\phi _0 \in B(\phi ^*,\bar{\delta })$. Then, by condition (iv), there exists a sequence $t_n \rightarrow \infty$ such that $V(\phi (t_n))$ converges to $V(\psi )$ for some $\psi \in \Omega$.

We need to show that $\psi = \phi ^*$. By condition (iii), $V(\phi (t)) > V(\psi )$ for every $t \ge 0$. If $\psi \not = \phi ^*$, let $\psi (t) = \phi (t;\psi )$. For any $t > 0$, $V(\psi (t)) < V(\psi )$. By condition (iv),

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

and hence

$$\begin{aligned} V(\phi (s,\phi (t_n))) < V(\psi ) \end{aligned}$$

for $s > 0$ small enough and n large enough which is a contradiction because $\phi (s,\phi (t_n)) = \phi (s+t_n;\phi _0)$. Hence, $\psi = \phi ^*$. $\square$