Asymmetric Replicator Dynamics on Polish Spaces: Invariance, Stability, and Convergence

Abstract

We study a class of asymmetric games with compact Polish strategy sets and provide sufficient conditions for the stability and convergence of profiles under the infinite-dimensional replicator dynamics on such games. We apply these results to analyze the dynamic behavior of the Cournot duopoly with different pricing mechanisms, the rope-pulling game, and a game with a Nash equilibrium profile consisting of uniform distributions. Further, we prove that the set of all Gaussian profiles remains invariant under the replicator dynamics on a large class of quadratic games. Moreover, we study the dynamics restricted to the set of Gaussian profiles, both analytically and numerically.

Appendix A Proof of Theorem 10: Bounding \(k_1\) by an Integrable Function

We now bound $k_1$ by a $P_t$-integrable function. Using Eq. (27), we obtain

$$\begin{aligned} l_1(t,x)=l_1(0,x)+\int \limits _{0}^{t}\sigma _1(x,P_s,Q_s)\textrm{d}s. \end{aligned}$$

(A1)

Substituting this in Eq. (26), we get

$$\begin{aligned} k_1(h,x)=\frac{1}{h}\left( e^{\int \limits _{t}^{t+h}\sigma _1(x,P_s,Q_s)\textrm{d}s}-1\right) -\sigma _1(x,P_t,Q_t). \end{aligned}$$

(A2)

We now define

$$\begin{aligned} i_1(h,x):=e^{\int \limits _{t}^{t+h}\sigma _1(x,P_s,Q_s)\textrm{d}s}. \end{aligned}$$

(A3)

Observe that

$$\begin{aligned} i_1(0,x)=1 \quad \textrm{and} \quad \frac{\partial }{\partial h}i_1(0,x)=\sigma _1(x,P_t,Q_t). \end{aligned}$$

(A4)

From Eqs. (A2) and (A4), we get

$$\begin{aligned} \nonumber \left| k_1(h,x)\right|&=\left| \frac{1}{h}\left( i_1(h,x)-1\right) -\sigma _1(x,P_t,Q_t)\right| \\ \nonumber&=\frac{1}{h}\bigg |i_1(h,x)-1-h\sigma _1(x,P_t,Q_t)\bigg |\\&=\frac{1}{h}\bigg |i_1(h,x)-i_1(0,x)-h \frac{\partial }{\partial h}i_1(0,x)\bigg |. \end{aligned}$$

(A5)

Note that, for any $C^2$-function $f:\mathbb {R\xrightarrow []{}\mathbb {R}}$ and $h>0$, we have

$$\begin{aligned} \nonumber \int \limits _{0}^{h}(h-u)f''(u)\textrm{d}u&=(h-u)f'(u)\big |_0^h-\int \limits _{0}^{h}(-1)f'(u)\textrm{d}u\\&=-hf'(0)+f(h)-f(0). \end{aligned}$$

(A6)

Since $i_1$ is infinitely differentiable in the variable h, from Eqs. (A5) and (A6), we get

$$\begin{aligned} \left| k_1(h,x)\right| =\frac{1}{h}\bigg |\int \limits _{0}^{h}(h-u)\frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)\textrm{d}u\bigg | = \bigg |\int \limits _{0}^{h}\left( \frac{h-u}{h}\right) \frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)\textrm{d}u\bigg |. \end{aligned}$$

Since $0\le u\le h$, we obtain

$$\begin{aligned} \left| k_1(h,x)\right|&\le \bigg |\int \limits _{0}^{h}\frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)\textrm{d}u\bigg |. \end{aligned}$$

(A7)

We now bound the integrand appearing in the inequality (A7). Note that

$$\begin{aligned} \nonumber \frac{\partial }{\partial {h}}i_1(u,x)&=i_1(u,x)\sigma _1(x,P_{t+u},Q_{t+u}),\\ \frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)&=\bigg ( \frac{\partial }{\partial {h}}\sigma _1(x,P_{t+u},Q_{t+u})+\sigma _1^2(x,P_{t+u},Q_{t+u}) \bigg )i_1(u,x). \end{aligned}$$

(A8)

Since $m_i$ and $v_i$, $i=1,2$ are $C^1$-functions, these functions and their derivatives are bounded on $[t,t+h]$. Hence, we have the following bounds:

$$\begin{aligned}&\bigg |\frac{\partial }{\partial {h}}\sigma _1(x,P_{t+u},Q_{t+u})\bigg | =\bigg |\frac{\partial }{\partial {h}}\big |_{h=u}\bigg (a_1x^2+b_1m_2(t+h)x\\&\quad -[a_1m_1(t+h)^2+b_1m_1(t+h)m_2(t+h)+a_1v_1(t+h)]\bigg )\bigg |\\&\quad \le A_1(1+|x|^4), \; \forall u\in [0,h], \end{aligned}$$

for some $A_1>0$. From Eq. (13), it follows that $\sigma _1(\cdot ,P_t,Q_t)$ is a quadratic function in the variable x and consequently $\sigma _1^2(\cdot ,P_{t+u},Q_{t+u})$ is a fourth-degree polynomial in the variable x. Using the fact that the trajectories are $C^1$, $\sigma _1^2$ can be bounded as follows:

$$\begin{aligned} \bigg |\sigma _1^2(x,P_{t+u},Q_{t+u})\bigg |&\le A_2(1+|x|^4),\; \forall u\in [0,h], \end{aligned}$$

for some $A_2>0$. From Eq. (A8), we get

$$\begin{aligned} \bigg |\frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)\bigg |&\le A(1+|x|^4)|i_1(u,x)|,\; \forall u\in [0,h], \end{aligned}$$

(A9)

where $A=A_1+A_2$. We next bound $i_1$ in the expression (A9). From Eqs. (13), (A3) and the fact that $a_1<0$, we obtain

$$\begin{aligned} i_1(u,x)&=e^{\int \limits _{t}^{t+u} (a_1x^2+b_1m_2(s)x-[a_1m_1(s)^2+b_1m_1(s)m_2(s)+a_1v_1(s)]) \textrm{d}s}\\&\le e^{\int \limits _{t}^{t+u}(b_1m_2(s)x- [a_1m_1(s)^2+b_1m_1(s)m_2(s)+a_1v_1(s)]) \textrm{d}s}. \end{aligned}$$

Using the fact that $m_i$ and $v_i$, $i=1,2$, being continuous, are bounded on the compact interval $[t,t+h]$ , we get

$$\begin{aligned} \left| i_1(u,x)\right| \le e^{B\left| x\right| +C},\; \forall u\in [0,h], \end{aligned}$$

(A10)

for some B,  $C>0$. Combining Eqs. (A9) and (A10), we obtain

$$\begin{aligned} \bigg |\frac{\partial ^{2}}{\partial {h}^{2}}i_1(u,x)\bigg |\le A(1+|x|^4)e^{B\left| x\right| +C},\; \forall u\in [0,h]. \end{aligned}$$

(A11)

From Eqs. (A7) and (A11), we get

$$\begin{aligned} \nonumber \left| k_1(h,x)\right|&\le Ah(1+|x|^4)e^{B\left| x\right| +C}\\&\le A(1+|x|^4)e^{B\left| x\right| +C}, \; \forall h\in [0,1]. \end{aligned}$$

(A12)

This yields the necessary $P_t$-integrable upper bound for $k_1$.