Stochastic Evolutionary Selection in Heterogeneous Populations for Asymmetric Games

Abstract

Although a large and growing literature has emerged to study evolutionary selection in finite populations recently, there is rare consideration on the effects of learning mechanisms’ heterogeneity. Based on a selection-mutation Moran process, this paper makes an attempt to construct and analyze a stochastic evolutionary game dynamic in two heterogeneous populations. Precisely speaking, one population is clever in the sense that its every individual mutates according to a pairwise comparison rule while the other is simple since each of its individuals mutates completely randomly. As a criterion of equilibrium selection, related stationary distributions of strategies are then analytically derived under weak selection. Besides, some novel findings can be obtained due to the explicit consideration of rational behavior. For example, it turns out that for coordination games the introduction of the rationality may inhibit the clever population from selecting the risk-dominant strategy, which is inconsistent with our common sense.

Appendix

Proof of Proposition 1

It can be observed that the selection-mutation Moran process reduces to the model by Ohtsuki (2010) when $w=0$. It thus follows from Ohtsuki (2010) that

$$\begin{aligned} \left\{ \begin{array}{l} \left\langle {x_1 [(A{{\varvec{y}}})_1 -{{{\varvec{x}}}}'A{{\varvec{y}}}]} \right\rangle _0 =\frac{(M-1)u}{8(1+Mu-u)}(a_{11} +a_{12} -a_{21} -a_{22} )\\ \\ \left\langle {y_1 [(B{{\varvec{x}}})_1 -{{{\varvec{y}}}}'B{{\varvec{x}}}]} \right\rangle _0 =\frac{(N-1)v}{8(1+Nv-v)}(b_{11} +b_{12} -b_{21} -b_{22} ) \\ \\ \left\langle {(A{{\varvec{y}}})_1 -(A{{\varvec{y}}})_2} \right\rangle _0 =\frac{1}{2}(a_{11} +a_{12} -a_{21} -a_{22} )\end{array}\right. \end{aligned}$$

By virtue of Eq. (12), it is not difficult to derive

$$\begin{aligned} \left\{ \begin{array}{l} \left\langle {x_1} \right\rangle _w =\frac{1}{2}+w(a_{11} +a_{12} -a_{21} -a_{22} )\left[ {\frac{(M-1)(1-u)}{8(1+Mu-u)}+\frac{1}{2}\delta } \right] +O(w^{2})\\ \\ \left\langle {y_1} \right\rangle _w =\frac{1}{2}+w\frac{(N-1)(1-v)}{8(1+Nv-v)}(b_{11} +b_{12} -b_{21} -b_{22} )+O(w^{2})\end{array} \right. \end{aligned}$$

(14)

Hence,

$$\begin{aligned} \!\!\left\{ \!\! \begin{array}{l} \left\langle {x_2} \right\rangle _w {=}1-\left\langle {x_1} \right\rangle _w {=}\frac{1}{2}-w(a_{11} +a_{12} -a_{21} -a_{22} )\left[ {\frac{(M-1)(1-u)}{8(1+Mu-u)}+\frac{1}{2}\delta } \right] {+}O(w^{2}) \\ \\ \left\langle {y_2} \right\rangle _w =1-\left\langle {y_1} \right\rangle _w =\frac{1}{2}-w\frac{(N-1)(1-v)}{8(1+Nv-v)}(b_{11} +b_{12} -b_{21} -b_{22} )+O(w^{2})\end{array} \right. \end{aligned}$$

(15)

Some simple calculations show that

$$\begin{aligned} \left\{ \begin{array}{l} {\left\langle {x_1 y_1 [(A{{\varvec{y}}})_1 -{{{\varvec{x}}}}'A{{\varvec{y}}}]} \right\rangle _0 =\frac{(M-1)[2(a_{11} -a_{21} )+v(N-1)(a_{11} +a_{12} -a_{21} -a_{22} )]u}{16(1+Mu-u)(1+Nv-v)}} \\ \\ {\left\langle {x_1 y_1 [(B{{\varvec{x}}})_1 -{{{\varvec{y}}}}'B{{\varvec{x}}}]} \right\rangle _0 =\frac{(N-1)[2(b_{11} -b_{21} )+u(M-1)(b_{11} +b_{12} -b_{21} -b_{22} )]v}{16(1+Mu-u)(1+Nv-v)}} \\ \\ \left\langle {y_1 [(A{{\varvec{y}}})_1 -(A{{\varvec{y}}})_2 ]} \right\rangle _0 =\frac{[2(a_{11} -a_{21} )+v(N-1)(a_{11} +a_{12} -a_{21} -a_{22} )]}{4(1+Nv-v)}\end{array}\right. . \end{aligned}$$

(16)

The proof can be completed by means of Eq. (14)–(16) and Eq. (11).