Evolutionary Stability of Polymorphic Profiles in Asymmetric Games

Abstract

Mendoza-Palacios and Hernández-Lerma (J Differ Equ 259(11):5709–5733, 2015) have introduced the concept of a strong uninvadable profile for asymmetric games with continuous pure strategy space and proved that such a profile is Lyapunov stable for the associated replicator dynamics when the profile is monomorphic. In the present paper, we establish that a polymorphic strong uninvadable profile is necessarily monomorphic. Further, it is shown that strong unbeatability is enough to guarantee Lyapunov stability of polymorphic profiles. A stability theorem for sets of polymorphic profiles is also presented and is illustrated using examples.

Appendix

We begin with the statement of an abstract stability theorem which is used to establish the main stability result, namely Theorem 3. (The proof of this abstract theorem follows from the proofs of Theorem 5 of [11] and Theorem 9 of [12].) To this end, consider an abstract differential equation

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

(29)

on a Banach Space $(X,\Vert \cdot \Vert _{X})$. Assume that the differential equation (29) 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 (29) is analyzed below around a closed set $\Pi \subseteq Y$ of its rest points. Also, we recall the definition of $\mathcal {K}_0^{\infty }$ functions:

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

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 $d(\phi ,\Pi )=\text {inf}_{\varphi \in \Pi } \Vert \phi -\varphi \Vert _X$ and let $B(\Pi ,\epsilon )$ be the set of all $\phi \in X$ such that $d(\phi ,\Pi ) < \epsilon$.

Theorem 4

([11]) Let G be an open set of Y containing a closed set $\Pi$ of rest points of the system (29). Assume that $V: G \rightarrow \mathbb {R}$ is uniformly continuous on G and satisfies

• (i) $V(\phi ) \ge 0$ on G and $V(\phi )=0$ for every $\phi \in \Pi$;

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

• (iii) V is strictly decreasing along the trajectories of (29) that lie in $G {\setminus } \Pi$;

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

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

  Then $\Pi$ is asymptotically stable.

The remainder of the appendix is devoted to prove conditions (d) and (e) in the proof of Theorem 3.

Proof of (d). Suppose $\mu (t)$ is the trajectory of the replicator dynamics emanating from $\mu \in \Omega (\bar{\mu },\theta (\bar{\mu })) {\setminus } \Pi ^*$. This implies $\mu _i \notin \Pi _i^*$ for some $i \in \{1,2\dots ,n\}$ which gives (by the definition of strong uninvadable set), $J_i(\mu _i(t),\mu _{-i}(t))-J_i(\bar{\mu }_i,\mu _{-i}(t)) < 0$ for all t. For $j \ne i$, $J_j(\mu _j(t),\mu _{-j}(t))-J_j(\bar{\mu }_j,\mu _{-j}(t)) \le 0$ for all t. Hence, from (c)(in the proof of Theorem 3), we get $\frac{d}{dt} V_{\bar{\mu }}(\mu (t)) < 0$ which proves (d).

Proof of (e). For $0< \epsilon _1 < \theta (\bar{\mu })$, by Theorem 1, there exists $\eta (\bar{\mu }) >0$ such that for every trajectory of the replicator dynamics with $\mu (0) \in \Omega (\bar{\mu },\eta (\bar{\mu }))$, we have $\, \mu (t) \in \Omega (\bar{\mu },\frac{\epsilon _{1}}{k})$ where $k= \text {max}\{k_1, k_2, \dots , k_n\}$.

Let us consider a fixed trajectory $\mu (t)= \mu (t;\mu _0)$ with ${\mu _0 (=\mu (0)) \in \Omega (\bar{\mu },\eta (\bar{\mu }))}$. It is clear that there exists a sequence $t_n \rightarrow \infty$ such that $\mu _i(t_n)(\{a_i^j\})$ is convergent to a limit, say ${\gamma _i^j}^*$; $1 \le j \le k_i , 1 \le i \le n$.

As $\mu (t_n) \in \Omega (\bar{\mu },\frac{\epsilon _{1}}{k})$, by the definition of the infinity norm [see (6)], we have for all i, $\Vert \mu _i(t_n)-\bar{\mu }_i\Vert < \frac{\epsilon _1}{k} \le \frac{\epsilon _1}{k_i}$. This together with (7) of [12] implies that $|\alpha _i^j - {\gamma _i^j}^*| \le \frac{1}{2} \left( \frac{\epsilon _1}{k_i}\right) ; \; 1 \le j \le k_i , 1 \le i \le n$. This gives

$$\begin{aligned} \sum _{j=1}^{k_i}|\alpha _i^j-{\gamma _i^j}^*| \le \frac{\epsilon _1}{2} < \frac{\theta (\bar{\mu })}{2}; \quad 1 \le j \le k_i , 1 \le i \le n. \end{aligned}$$

(30)

Also, we know that

$$\begin{aligned} \frac{\theta (\bar{\mu })}{2} \le \frac{\delta (\bar{\mu })}{2} < \frac{\epsilon (\bar{\mu })}{2} \le \alpha _i^j; \quad 1 \le j \le k_i , 1 \le i \le n. \end{aligned}$$

(31)

From (30) and (31), we get $|\alpha _i^j-{\gamma _i^j}^*| < \alpha _i^j ; \; 1 \le j \le k_i , 1 \le i \le n$, which in turn gives ${\gamma _i^j}^* > 0 ; \quad 1 \le j \le k_i, 1 \le i \le n$. Therefore,

$$\begin{aligned} V_{\bar{\mu }}(\mu (t_n)) = \sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{\mu _i(t_n)(\{a_i^j\})}\right) \end{aligned}$$

(32)

is converging to

$$\begin{aligned} V_{\bar{\mu }}(\mu ^*) = \sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{{\gamma _i^j}^*}\right) \end{aligned}$$

(33)

for some $\mu ^*$$\in \Sigma \subset \Omega (\bar{\mu },\theta (\bar{\mu }))$ where

Using (23), for $s>0$, we get

$$\begin{aligned} \mu _i(s;\mu _i^*)(\{a_i^j\})&= \mu _i(0;\mu _i^*)(\{a_i^j\}) \; \text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu ^*))\;\mathrm{d}t\Bigg ) \nonumber \\ ~&= {\gamma _i^j}^* \; \text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu ^*))\; \mathrm{d}t\Bigg ) \nonumber \\ ~&= {\gamma _i^j}^* \;\; T(s) \quad \mathrm{(say)}. \end{aligned}$$

(34)

We also have

$$\begin{aligned} \mu _i(s;\mu _i(t_n))(\{a_i^j\})&= \mu _i(0;\mu _i(t_n))(\{a_i^j\}) \; \text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu (t_n))) \; \mathrm{d}t\Bigg ) \nonumber \\ ~&= \mu _i(t_n)(\{a_i^j\}) \; \text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu (t_n)))\; \mathrm{d}t\Bigg ) \nonumber \\ ~&= \mu _i(t_n)(\{a_i^j\}) \;\; T_n(s) \quad \mathrm{(say)}. \end{aligned}$$

(35)

Hence, from (24),

$$\begin{aligned} |V_{\bar{\mu }}(\mu (s;\mu ^*)) - V_{\bar{\mu }}(\mu (s;\mu (t_n)))| \end{aligned}$$

$$\begin{aligned}&= \Bigg |\sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{\mu _i(s;\mu _i^*)(\{a_i^j\})}\right) - \sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{\mu _i(s;\mu _i(t_n))(\{a_i^j\})}\right) \Bigg | \nonumber \\ ~&= \Bigg |\sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{{\gamma _i^j}^* \, T(s)}\right) - \sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \, \, \mathrm {ln} \, \left( \frac{\alpha _i^j}{\mu _i(t_n)(\{a_i^j\}) \, T_n(s)}\right) \Bigg | \quad (\text {from} \, (34)\, \text {and} \, (35)) \nonumber \\ ~&= \sum _{i=1}^{n} \sum _{j=1}^{k_i} \alpha _i^j \; \Bigg | \, \mathrm {ln} \, \left( \frac{\mu _i(t_n)(\{a_i^j\}) \, T_n(s)}{{\gamma _i^j}^* \, T(s)}\right) \Bigg |. \end{aligned}$$

(36)

Consider

$$\begin{aligned} \frac{T_n(s)}{T(s)} = \frac{\text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu (t_n))) \; \mathrm{d}t\Bigg )}{\text {exp} \Bigg (\int _{0}^{s} \sigma _i(a_i^j\,|\, \mu (t;\mu ^*))\;\mathrm{d}t\Bigg )}. \end{aligned}$$

Since $U_i(\cdot )$ is bounded, it follows that $\sigma _i(\cdot |\mu )$ is bounded (see Proposition 4.4 and Theorem 4.3 of [16]) which in turn implies that $\frac{T_n(s)}{T(s)}$ converges to 1 uniformly in n as $s \downarrow 0$, and hence, from (36), we obtain

$$\begin{aligned} \lim _{s \downarrow 0 \,, \, n \uparrow \infty }|V_{\bar{\mu }}(\mu (s;\mu ^*)) - V_{\bar{\mu }}(\mu (s;\mu (t_n)))|=0. \end{aligned}$$

This establishes condition (e).