An Effective Replicator Equation for Games with a Continuous Strategy Set

Abstract

The replicator equation for a two-person symmetric game, which has an interval of the real line as strategy space, is extended with a mutation term. Assuming that the distribution of the strategies has a continuous density, a partial differential equation for this density is derived. The equation is analysed for two examples. A connection is made with the canonical equation from adaptive dynamics and the continuous stable strategy criterion.

Appendix

We have

$$\begin{aligned} \hat{\rho }(\xi ,\tau )&=n(\tau )e^{g_X X}e^{g_Y Y}e^{g_Z Z}\hat{\rho }_0(\xi ) \nonumber \\&=n(\tau )e^{g_X X}e^{g_Y Y}\sum _{n=0}^{\infty }e^{-2n\tau }a_{n}\phi _n(\xi ) \nonumber \\&=n(\tau )e^{g_X \xi }\sum _{n=0}^{\infty }e^{-2n\tau }a_{n}\phi _n(\xi +g_Y) \nonumber \\&=n(\tau )e^{g_X \xi }e^{-(\xi +g_Y)^2/2}\sum _{n=0}^{\infty }e^{-2n\tau }a_{n}c_n H_n(\xi +g_Y) \nonumber \\&=n(\tau )e^{(-g_X^2+2g_X g_Y)/2 }e^{-(\xi -g_X+g_Y)^2/2}\sum _{n=0}^{\infty }e^{-2n\tau }a_{n}c_n H_n(\xi +g_Y). \end{aligned}$$

(47)

Introducing the variable $y=\xi -g_X+g_Y$, we can expand $e^{-2n\tau } H_n(\xi +g_Y)=e^{-2n\tau } H_n(y+g_X)$, using $H_n(z)=h_{n,n}z^n+h_{n,n-1}z^{n-1}+\ldots +h_{n,0}$:

$$\begin{aligned} e^{-2n\tau } H_n(y+g_X)&=e^{-2n\tau }\big ( H_n(g_X)+y H_n'(g_X)+\ldots +\frac{1}{n!}y^n H_n^{(n)} \big ) \nonumber \\&= h_{n,n}(e^{-2\tau }g_X)^n+h_{n,n-1}e^{-2\tau }(e^{-2\tau }g_X)^{n-1}+\ldots +h_{n,0}e^{-2n\tau } \nonumber \\&\quad +e^{-2n\tau }\sum _{k=1}^n \frac{1}{k!}y^k H_n^{(k)}(g_X) \nonumber \\&=h_{n,n}(e^{-2\tau }g_X)^{n}\big (1+\frac{h_{n,n-1}}{h_{n,n}}(g_X)^{-1}+\ldots +\frac{h_{n,0}}{h_{n,n}}(g_X)^{-n}\big )\nonumber \\&\quad +e^{-2n\tau }\sum _{k=1}^n \frac{1}{k!}y^k H_n^{(k)}(g_X) \nonumber \\&=h_{n,n}(e^{-2\tau }g_X)^{n}\big (1+e^{-2\tau }r_1(\tau )\big )+ R_n(y,\tau ), \end{aligned}$$

(48)

with $|r_1(\tau )| \le C_1$, for some $C_1>0$ and all $\tau >0$. Here we have used that there exists an $M>0$, such that $e^{-2\tau }|g_X|\ge M$, for all $\tau >0$.

The $L^1$ norm of $e^{-y^2/2} R_n(y,\tau )$ can be estimated as

$$\begin{aligned} ||e^{-y^2/2} R_n(y,\tau )||_1&=\int _S|e^{-2n\tau }e^{-y^2/2}\sum _{k=1}^n \frac{1}{k!}y^k H_n^{(k)}(g_X)|\, \mathrm{d}y \nonumber \\&\quad \le e^{-2n\tau }\sum _{k=1}^n \frac{1}{k!}H_n^{(k)}(g_X)\int _S|y|^k e^{-y^2/2}\, \mathrm{d}y =e^{-2n\tau }\sum _{k=1}^n \frac{1}{k!}H_n^{(k)}(g_X)\alpha _k \nonumber \\&=h_{n,n}e^{-2n\tau } (g_X)^{(n-1)}\big (1+r_2(\tau )\big )=h_{n,n} e^{-2\tau }(e^{-2\tau }g_X)^{n-1}\big (1+r_2(\tau )\big ), \end{aligned}$$

(49)

where $|r_2(\tau )| \le C_2$, for some $C_2>0$ and all $\tau >0$.

Combining (48) and (49), we find

$$\begin{aligned} e^{-y^2/2}e^{-2n\tau } H_n(y+g_X)&=e^{-y^2/2}\big (h_{n,n}(e^{-2\tau }g_X)^{n}(1+e^{-2\tau }r_1(\tau ))+R(y,\tau )\big ) \nonumber \\&=e^{-y^2/2}\big (h_{n,n}(e^{-2\tau }g_X)^{n}(1+e^{-2\tau }r_1(\tau )) \nonumber \\&\quad +h_{n,n} e^{-2\tau }(e^{-2\tau }g_X)^{n-1}(1+r_2(\tau )\tilde{R_n}(y,\tau ))\big ) \nonumber \\&=e^{-y^2/2}h_{n,n}(e^{-2\tau }g_X)^{n}\big (1+e^{-2\tau }\tilde{R_n}(y,\tau )\big ). \end{aligned}$$

(50)

Here, with some abuse of notation, $\tilde{R_n}(y,\tau )$ has absorbed in each step all bounded terms, so that we have $||\tilde{R_n}(y,\tau )||_1<M_n$, for some $M_n>0$ and all $\tau >0$. Moreover, since $\tilde{R_n}(y,\tau )$ is a polynomial in $y$ whose time-dependent coefficients are either exponential functions in $\tau$, or are bounded from below by such functions, we have that $||\frac{\partial }{\partial \tau }\tilde{R_n}(y,\tau )||_1$ is also bounded. We will assume, without proof, that there exists an $M>0$ such that $M_n\le M$ for all $n$, so that there exists a function $R(y,\tau )$ with $||\tilde{R_n}(y,\tau )||_1\le ||R(y,\tau )||_1\le M$, such that

$$\begin{aligned} e^{-y^2/2}e^{-2n\tau } H_n(y+g_X) =e^{-y^2/2}h_{n,n}(e^{-2\tau }g_X)^{n}\big (1+e^{-2\tau }R(y,\tau )\big ). \end{aligned}$$

(51)

We then have

$$\begin{aligned} \hat{\rho }(\xi ,\tau )&=n(\tau )e^{(-g_X^2+2g_X g_Y)/2 }\left( \sum _{n=0}^{\infty }a_{n}c_n h_{n,n}\left( e^{-2\tau }g_X\right) ^{n}\right) e^{-\left( \xi -g_X+g_Y\right) ^2/2}\left( 1\!+\!e^{-2\tau }R(\xi ,\tau )\right) . \end{aligned}$$

(52)

Since $\hat{\rho }(\xi ,\tau )>0$, we must have that $\sum _{n=0}^{\infty }a_{n}c_n h_{n,n}(e^{-2\tau }g_X)^{n}>0$, for $\tau$ sufficiently large. The normalization term factor, for large $\tau$, now becomes

$$\begin{aligned} (n(\tau ))^{-1}&=e^{(-g_X^2+2g_X g_Y)/2 }\sum _{n=0}^{\infty }a_{n}c_n h_{n,n}\left( e^{-2\tau }g_X\right) ^{n} \int _S e^{-(\xi -g_X+g_Y)^2/2}(1+e^{-2\tau }R(\xi ,\tau )\big )\, \mathrm{d}\xi \nonumber \\&=\sqrt{2 \pi } e^{(-g_X^2+2g_X g_Y)/2 }\sum _{n=0}^{\infty }a_{n}c_n h_{n,n}(e^{-2\tau }g_X)^{n}(1+r_3(\tau )), \end{aligned}$$

(53)

with $|r_3(\tau )|\le C_3$, for some $C_3>0$ and all $\tau >0$.

This then yields, as $\tau$ grows large,

$$\begin{aligned} \hat{\rho }(\xi ,\tau )&=\frac{1}{\sqrt{2 \pi }}e^{-(\xi -g_X+g_Y)^2/2}(1+e^{-2\tau }R(\xi ,\tau )\big ), \end{aligned}$$

(54)

with $R(\xi ,\tau )$ uniformly bounded in the $L^1$ norm (and again with some abuse of notation with regards to $R(\xi ,\tau )$.)

It follows from the construction that $\int _S \xi R(\xi ,\tau )\, \mathrm{d}\xi$ is bounded, so

$$\begin{aligned} \overline{\xi }&=\int _S \xi \hat{\rho }(\xi ,\tau )\, \mathrm{d}\xi =\int _S \xi \frac{1}{\sqrt{2 \pi }}e^{-(\xi -g_X+g_Y)^2/2}+e^{-2\tau }r(\tau ) \nonumber \\&=g_X-g_Y+e^{-2\tau }r(\tau ), \end{aligned}$$

(55)

where both $r(\tau )$ and $r'(\tau )$ are uniformly bounded.