Evolutionary Games in Interacting Communities

Abstract

In this paper, we extend the evolutionary games framework by considering a population composed of communities with each having its set of strategies and payoff functions. Assuming that the interactions among the communities occur with different probabilities, we define new evolutionarily stable strategies (ESS) with different levels of stability against mutations. In particular, through the analysis of two-community two-strategy model, we derive the conditions of existence of ESSs under different levels of stability. We also study the evolutionary game dynamics both in its classic form and with delays. The delays may be strategic, i.e., associated with the strategies, spatial, i.e., associated with the communities, or spatial strategic. We apply our model to the Hawk–Dove game played in two communities with an asymmetric level of aggressiveness, and we characterize the regions of ESSs as function of the interaction probabilities and the parameters of the model. We also show through numerical examples how the delays and the game parameters affect the stability of the mixed ESS.

Appendix

1.1 Proof of Proposition 1

Let us first prove that Definition 1 implies Definition 2. Since the condition in Definition 1 holds for any sufficiently small $\epsilon$, as $\epsilon \rightarrow 0$, $\bar{U}_i(s_i,\mathbf{\epsilon }\mathbf{s}+(1-\epsilon ) \mathbf{s}^*,p) < \bar{U}_i(s_i^*,\epsilon \mathbf{s}+(1-\epsilon ) \mathbf{s}^*,p)$ for all $i=1,\ldots ,N$, implies $\bar{U}_i(s_i,\mathbf{s}^*,p) \le \bar{U}_i(s_i^*, \mathbf{s}^*,p)$ for all $i=1,\ldots ,N$. Therefore, the first condition in Definition 2 is established. Now we suppose there exists i such that $\bar{U}_i(s_i,\mathbf{s}^*,p)=\bar{U}_i(s_i^*,\mathbf{s}^*,p)$. Since the expected utility is linear in $\mathbf{s}$, the condition $\bar{U}_i(s_i,\mathbf{\epsilon }{} \mathbf{s}+(1-\epsilon ) \mathbf{s}^*,p) < \bar{U}_i(s_i^*,\epsilon \mathbf{s}+(1-\epsilon ) \mathbf{s}^*,p)$ can be written as $\mathbf{\epsilon }\bar{U}_i(s_i,\mathbf{s},p)+(1-\epsilon ) \bar{U}_i(s_i,\mathbf{s}^*,p) < \epsilon \bar{U}_i(s_i^*,\mathbf{s},p)+(1-\epsilon )\bar{U}_i(s_i^*,\mathbf{s}^*,p)$. Since $\bar{U}_i(s_i,\mathbf{s}^*,p)=\bar{U}_i(s_i^*,\mathbf{s}^*,p)$, the last inequality can be written $\mathbf{\epsilon }\bar{U}_i(s_i,\mathbf{s},p) < \epsilon \bar{U}_i(s_i^*,\mathbf{s},p)$; which yields $\bar{U}_i(s_i,\mathbf{s},p)<\bar{U}_i(s_i^*,\mathbf{s},p)$ since $\epsilon >0$. Therefore, the second condition in Definition 2 is established.

Let us now prove that the Definition 2 implies Definition 1. We have for all i and for any $\mathbf{s} \not = \mathbf{s}^*$, $\bar{U}_i(s_i, \mathbf{s}^*,p) \le \bar{U}_i(s_i^*,\mathbf{s}^*,p)$. If for some i, this inequality is strict, then the condition in Definition 1 is satisfied for $\epsilon =0$ and so for sufficiently small $\epsilon$. If for some i, $\bar{U}_i(s_i, \mathbf{s}^*,p)=\bar{U}_i(s_i^*,\mathbf{s}^*,p)$, then the second condition in Definition 2 implies $\bar{U}_i(s_i,\mathbf{s},p) < \bar{U}_i(s_i^*,\mathbf{s},p)$. If we multiply this relation by $\epsilon$ and add $(1-\epsilon )\bar{U}_i(s_i, \mathbf{s}^*,p)$ to the left-hand side, and $(1-\epsilon ) \bar{U}_i(s_i^*,\mathbf{s}^*,p)$ to the right-hand side, we get the condition in Definition 1.

1.2 Proof of Proposition 2

First, let us prove that the Definition 3 implies the Definition 4. If we take $\epsilon _i \rightarrow 0$ in Definition 3, we get: $\bar{U}_i(s_i,\mathbf{s}^*,p) \le \bar{U}_i(s_i^*,\mathbf{s}^*,p)$ for all i, and so condition (7). Now, to establish the condition (8) in Definition 4, we suppose there exists i such that $\bar{U}_i(s_i^*, \mathbf{s}^*,p) = \bar{U}_i(s_i, \mathbf{s}^*,p)$, we need to prove that $\bar{U}_i(s_i,(s_i,\mathbf{s}^*_{-i}),p)<\bar{U}_i(s_i^*,(s_i,\mathbf{s}^*_{-i}),p)$. We can write condition (6) as follows:

$$\begin{aligned}&\bar{U}_i\left( s_i,\left( \epsilon _is_1^*+(1-\epsilon _i)s_1^*,\ldots ,\epsilon _is_i+(1-\epsilon _i)s_i^*,\ldots ,\epsilon _is_N^*+(1-\epsilon _i)s_N^*\right) ,p\right) \nonumber \\&\quad <\bar{U}_i\left( s_i^*,\left( \epsilon _is_1^*+(1-\epsilon _i)s_1^*,\ldots ,\epsilon _is_i+(1-\epsilon _i)s_i^*,\ldots ,\epsilon _is_N^*+(1-\epsilon _i)s_N^*\right) ,p\right) .\nonumber \end{aligned}$$

By exploring the linearity of $\bar{U}_i$, we get:

$$\begin{aligned}&{} \epsilon _i\bar{U}_i(s_i,(s_1^*,\ldots ,s_i,\ldots ,s_N^*),p) + (1-\epsilon _i)\bar{U}_i(s_i,\mathbf{s}^*,p)\\&\quad <\epsilon _i\bar{U}_i(s_i^*,(s_1^*,\ldots ,s_i,\ldots ,s_N^*),p) + (1-\epsilon _i)\bar{U}_i(s_i^*,\mathbf{s}^*,p). \end{aligned}$$

Since we have $\epsilon _i>0$ and we suppose $\bar{U}_i(s_i^*, \mathbf{s}^*,p) = \bar{U}_i(s_i, \mathbf{s}^*,p)$, the above inequality yields:

$$\begin{aligned} \bar{U}_i(s_i,(s_i,\mathbf{s}_{-i}^*),p)<\bar{U}_i(s_i^*,(s_i,\mathbf{s}_{-i}^*),p), \end{aligned}$$

and so condition (8).

Now we prove that the Definition 4 implies the Definition 3. We have for all i and for all $\mathbf{s} \not =\mathbf{s}^*$

$$\begin{aligned} \bar{U}_i(s_i, \mathbf{s}^*,p) \le \bar{U}_i(s_i^*, \mathbf{s}^*,p). \end{aligned}$$

If this inequality is strict for all i, then condition (6) holds for $\epsilon _i=0$ and thus for sufficiently small $\epsilon _i$. If there exists i such that the comparison in (7) is an equality, then we obtain $\bar{U}_i(s_i,(s_i,\mathbf{s}^*_{-i}),p) < \bar{U}_i(s_i^*,(s_i,\mathbf{s}^*_{-i}),p)$ [condition (8)]. We multiply both sides by $\epsilon _i$, and by observing that $\bar{U}_i(s_i, \mathbf{s}^*,p)=\bar{U}_i(s_i^*, \mathbf{s}^*,p)$, we add $(1-\epsilon _i)\bar{U}_i(s_i,\mathbf{s}^*,p)$ to the left side and $(1-\epsilon _i)\bar{U}_i(s_i^*,\mathbf{s}^*,p)$ to the right side, we get condition (6).

1.3 Proof of Theorem  1

• There exists a mixed Nash equilibrium strategy $\mathbf{s}^*=(s_1^*,s_2^*)$, when users from any group are indifferent from playing strategy $G_i$ or $H_i$, i.e., all (pure) strategies are equally fit. At the equilibrium, we have the following system of equations:

  $$\begin{aligned} {} \left\{ \begin{array}{ccc} {} U_1(G_1,\mathbf{s}^*,p)&{}=&{} U_1(H_1,\mathbf{s}^*,p),\\ {} U_2(G_2,\mathbf{s}^*,p)&{}=&{} U_2(H_2,\mathbf{s}^*,p). \end{array} \right. \end{aligned}$$

  Thus, we obtain the following system:

  $$\begin{aligned} {} \left\{ \begin{array}{ccc} {} p_1s_1^*L_1+(1-p_1)s_2^*L_{12}+K_1&{}=0,&{} \text{(a) }\\ {} p_2s_2^*L_2+(1-p_2)s_1^*L_{21}+K_2&{}=0,&{} \text{(b) } \end{array} \right. \end{aligned}$$

  where $L_1=a_1-b_1-c_1+d_1$, $L_{12}=a_{12}-b_{12}-c_{12}+d_{12}$, $L_2=a_{2}-b_{2}-c_{2}+d_{2}$, $L_{21}=a_{21}-b_{21}-c_{21}+d_{21}$, $K_1=p_1(b_1-d_1)+(1-p_1)(b_{12}-d_{12})$, $K_2=p_2(b_{2}-d_{2})+(1-p_2)(b_{21}-d_{21})$. The solution of this system is given by $\mathbf{s}^*=(s_1^*,s_2^*)$, with $s_1^*=\frac{(1-p_1)L_{12}K_2-p_2L_2K_1}{{\varDelta }}$ and $s_2^*=\frac{(1-p_2)L_{21}K_1-p_1L_1K_2}{{\varDelta }}$; where ${\varDelta }=p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21}$. Clearly, $0<s_i^*<1$, $i=1,2$, if:

  • $\circ$   $0<{\varDelta }$, $0<(1-p_1)L_{12}K_2-p_2L_2K_1$, $(1-p_1)L_{12}K_2-p_2L_2K_1<{\varDelta }$, $0<(1-p_2)L_{21}K_1-p_1L_1K_2$, and $(1-p_2)L_{21}K_1-p_1L_1K_2<{\varDelta }$, or

  • $\circ$  ${\varDelta }<0$ , $(1-p_1)L_{12}K_2-p_2L_2K_1<0$, ${\varDelta }<(1-p_1)L_{12}K_2-p_2L_2K_1$, $0<(1-p_2)L_{21}K_1-p_1L_1K_2<0$, and ${\varDelta }<(1-p_2)L_{21}K_1-p_1L_1K_2$.

• Let us check for which conditions $\mathbf{s}^*=(s_1^*, s_2^*)$, if exists, is a strong ESS. Assume there is a small proportion of “mutants” that uses another strategy $\mathbf{s}=(s_1, s_2)$. Using the definition of the expected utility, we obtain:

  $$\begin{aligned} \bar{U}_1(s_1^*,\mathbf{s}^*,p)- \bar{U}_1(s_1, \mathbf{s}^*,p) = (s_1^*-s_1)(p_1 s_1^* L_1 +(1-p_1)s_2^* L_{12} + K_1)=0. \end{aligned}$$

  Following the same procedure for group 2, we obtain

  $$\begin{aligned} {} \bar{U}_2(s_2^*, \mathbf{s}^*,p)- \bar{U}_2(s_2, \mathbf{s}^*,p)= 0. \end{aligned}$$

  From (5), $\mathbf{s}^*$ is a strong ESS if $\bar{U}_i(s_i^*, \mathbf{s},p)- \bar{U}_i(s_i, \mathbf{s},p)>0$ for $i=1, 2$. But

  $$\begin{aligned} \begin{array}{cc} {} \bar{U}_1(s_1^*,\mathbf{s},p)- \bar{U}_1(s_1, \mathbf{s},p)=(s_1^*-s_1)\big ( p_1 s_1 L_1 +(1-p_1)s_2 L_{12}+K_1\big ),\\ {} \bar{U}_2(s_2^*,\mathbf{s},p)- \bar{U}_2(s_2, \mathbf{s},p)=(s_2^*-s_2)\big ( p_2 s_2 L_2 +(1-p_2)s_1L_{21}+K_2\big ). \end{array} \end{aligned}$$

  We define $f_i$, i=1,2 as follows:

  $$\begin{aligned} {} \left\{ \begin{array}{ccc} {} f_1(s_1,s_2)&{}=&{}(s_1^*-s_1)\big (p_1s_1L_1+(1-p_1)s_2L_{12}+K_1\big ),\\ {} f_2(s_1,s_2)&{}=&{}(s_2^*-s_2)\big (p_2s_2L_2+(1-p_2)s_1L_{21}+K_2\big ). \end{array} \right. \end{aligned}$$

  We have ${\nabla f_1}^T=\big [ 2p_1L_1(s_1^*-s_1)+(1-p_1)L_{12}(s_2^*-s_2) ,\;\; (1-p_1)L_{12}(s_1^*-s_1)\big ]$. Hence, $\displaystyle \frac{\partial ^2f_1}{{\partial s_1}^2}\displaystyle \frac{\partial ^2f_1}{{\partial s_2}^2}-\displaystyle \frac{\partial ^2f_1}{\partial s_1 \partial s_2}\displaystyle \frac{\partial ^2f_1}{\partial s_2\partial s_1}=-(1-p_1)^2L_{12}^2$ $<0$ at $\mathbf{{s}^*}$ (if $p_1 \not = 1$). Consequently, $\mathbf{{s}^*}$ is a saddle point. Since $f_1(\mathbf{{s}^*})=0$, $f_1$ changes the sign around $\mathbf{{s}^*}$. Therefore, the first community cannot resist invasions by mutants. Following the same procedure with $f_2$, we find that $(s_1^*,s_2^*)$ is a saddle point. Therefore, the condition of stability (5) does not hold and consequently $\mathbf{s}^*$ is not a strong ESS.

• Now, let us study for which condition $\mathbf{s}^*=(s_1^*,s_2^*)$ is a weak ESS. $\mathbf{s}^*=(s_1^*,s_2^*)$ is a weak ESS if $\bar{U}_1(s_1^*,(s_1,s_2^*),p) > \bar{U}_1(s_1,(s_1,s_2^*),p)$ and $\bar{U}_2(s_2^*,(s_1^*,s_2),p) > \bar{U}_2(s_2,(s_1^*,s_2),p)$. But

  $$\begin{aligned} {} \bar{U}_1(s_1^*, (s_1, s_2^*),p)- \bar{U}_1(s_1,( s_1, s_2^*),p)= -p_1L_1(s_1^*-s_1)^2. \end{aligned}$$

  which is strictly positive if $L_1 <0$. Following the same procedure with the second population, we get:

  $$\begin{aligned} {} \bar{U}_2(s_2^*, (s_1^*, s_2),p)- \bar{U}_2(s_2, (s_1^*, s_2),p) =-p_2L_2(s_2^*-s_2)^2. \end{aligned}$$

  which is strictly positive if $L_2 <0$. Therefore, if $L_1<0$ and $L_2<0$, $\mathbf{s}^*$ is a weak ESS.

• Finally, $\mathbf{s}^*$ is an intermediate ESS if $\bar{U}_1(s_1,\mathbf{s},p)+\bar{U}_2(s_2,\mathbf{s},p)<\bar{U}_1(s_1^*,\mathbf{s},p)+\bar{U}_2(s_2^*,\mathbf{s},p)$.

  Let $g(s_1,s_2)=\bar{U}_1(s_1^*,\mathbf{s},p)+\bar{U}_2(s_2^*,\mathbf{s},p)-\bar{U}_1(s_1,\mathbf{s},p)-\bar{U}_2(s_2,\mathbf{s},p)$, we have:

  $$\begin{aligned} g(s_1,s_2)= & {} (s_1^*-s_1)(p_1s_1L_1+(1-p_1)s_2L_{12}+K_1)+(s_2^*-s_2)(p_2s_2L_2\nonumber \\&+\,(1-p_2)s_1L_{21}+K_2). \end{aligned}$$

  The Hessian matrix of g is given by:

  $$\begin{aligned} {}\mathcal{{H}}(g) = \left( \begin{array}{cc} -2p_1L_1 &{} {-(1-p1)L_{12}-(1-p_2)L_{21}} \\ {-(1-p_1)L_{12}-(1-p_2)L_{21}} &{} -2p_2L_2 \end{array}\right) . \end{aligned}$$

  The determinant of $\mathcal{H}(g)$ is ${\varDelta }_1=4p_1p_2L_1L_2-\big ((1-p_1)L_{12}+(1-p_2)L_{21}\big )^2$. Hence, g is strictly positive for all $s_1 \not = s_1^*$, $s_2 \not = s_2^*$, if $L_1<0$, $L_2<0$ and ${\varDelta }_1>0$.

1.4 Completely Pure ESSs

Proposition 5

• $\mathbf{s}^*=(1,1)$ is a completely pure weak ESS if $p_1L_1+(1-p_1)L_{12}+K_1>0$ or if ($p_1L_1+(1-p_1)L_{12}+K_1=0$ and $L_1<0$) and also if $p_2L_2+(1-p_2)L_{21}+K_2>0$ or if ($p_2L_2+(1-p_2)L_{21}+K_2=0$ and $L_2<0$).

• $\mathbf{s}^*=(1,1)$ is an intermediate ESS if it is a weak ESS and if $p_1L_1+(1-p_1)L_{12}+K_1>0$, $p_2L_2+(1-p_2)L_{21}+K_2>0$, or if ($p_1L_1+(1-p_1)L_{12}+K_1=0$, $p_2L_2+(1-p_2)L_{21}+K_2=0$, and ${\varDelta }_1>0$).

• $\mathbf{s}^*=(1,1)$ is a strong ESS if it is an intermediate ESS and also if $p_1L_1+(1-p_1)L_{12}+K_1>0$ or $p_2L_2+(1-p_2)L_{21}+K_2>0$.

• $\mathbf{s}^*=(0,1)$ is a weak ESS if $(1-p_1)L_{12}+K_1<0$ or ($(1-p_1)L_{12}+K_1=0$ and $L_1<0$); and also if $p_2L_2+K_2>0$ or ($p_2L_2+K_2=0$ and $L_2<0$).

• $\mathbf{s}^*=(0,1)$ is an intermediate ESS if it is a weak ESS and if $(1-p_1)L_{12}+K_1<0$, $p_2L_2+K_2>0$, or ($(1-p_1)L_{12}+K_1=0$, $p_2L_2+K_2=0$ and ${\varDelta }_1>0$).

• $\mathbf{s}^*=(0,1)$ is a strong ESS if it is an intermediate ESS and if $(1-p_1)L_{12}+K_1<0$ or $p_2L_2+K_2>0$.

• $\mathbf{s}^*=(1,0)$ is a weak ESS if $p_1L_1+K_1>0$ or if ($p_1L_1+K_1=0$ and $L_1<0$) and also if $(1-p_2)L_{21}+K_2<0$ or ($(1-p_2)L_{21}+K_2=0$ and $L_2<0$).

• $\mathbf{s}^*=(1,0)$ is an intermediate ESS if it is a weak ESS and if $p_1L_1+K_1>0$, $(1-p_2)L_{21}+K_2<0$, or ($p_1L_1+K_1=0$, $(1-p_2)L_{21}+K_2=0$, and ${\varDelta }_1>0$).

• $\mathbf{s}^*=(1,0)$ is a strong ESS if it is an intermediate ESS and if $p_1L_1+K_1>0$ or $(1-p_2)L_{21}+K_2<0$.

• $\mathbf{s}^*=(0,0)$ is a weak ESS if $K_1<0$ or $(K_1=0$ and $L_1<0$) and also if $K_2<0$ or ($K_2=0$ and $L_2<0$).

• $\mathbf{s}^*=(0,0)$ is an intermediate ESS if it is a weak ESS and if $K_1<0$, $K_2<0$, or ($K_1=0$, $K_2=0$, and ${\varDelta }_1>0$).

• $\mathbf{s}^*=(0,0)$ is a strong ESS if it is an intermediate ESS and if $K_1<0$ or $K_2<0$.

Proof

• $\mathbf{s}^*=(1,1)$ is a weak ESS if

  $$\begin{aligned}&\bullet \,\, \bar{U}_1(s_1,(1,1),p)\le {U}_1(G_1,(1,1),p) , \text{ and } \\&\bullet \,\, \text{ if } \bar{U}_1(s_1,(1,1),p)= {U}_1(G_1,(1,1),p) \text{ then } \bar{U}_1(s_1,(s_1,1),p)<U_1(G_1,(s_1,1),p),\\&\bullet \,\, \bar{U}_2(s_2,(1,1),p)\le \bar{U}_2(G_2,(1,1),p), \text{ and } \\&\bullet \,\, \text{ if } \bar{U}_2(s_2,(1,1),p)= \bar{U}_2(G_2,(1,1),p) \text{ then } \bar{U}_2(s_2,(1,s_2),p)< \bar{U}_2(G_2,(1,s_2),p). \end{aligned}$$

  The conditions above yield, for the first community $p_1L_1+(1-p_1)L_{12}+K_1>0$ or ( $p_1L_1+(1-p_1)L_{12}+K_1=0$ and $L_1<0$ ); for the second community, we have $p_2L_2+(1-p_2)L_{21}+K_2>0$ or ($p_2L_2+(1-p_2)L_{21}+K_2=0$ and $L_2<0$).

  Furthermore, $\mathbf{s}^*$ is an intermediate ESS if $\bar{U}_1(s_1,\mathbf{s}^*,p)+\bar{U}_2(s_2,\mathbf{s}^*,p)\le U_1(G_1,\mathbf{s}^*,p)+U_2(G_2,\mathbf{s}^*,p)$, and if there exists an $\mathbf{s}$ for which this condition is an equality then $\bar{U}_1(s_1,\mathbf{s},p)+\bar{U}_2(s_2,\mathbf{s},p)<{U}_1(G_1,\mathbf{s},p)+{U}_2(G_2,\mathbf{s},p)$. These conditions yield: $p_1L_1+(1-p_1)L_{12}+K_1>0$ or ( $p_1L_1+(1-p_1)L_{12}+K_1=0$ and $L_1<0$) and $p_2L_2+(1-p_2)L_{21}+K_2>0$ or ($p_2L_2+(1-p_2)L_{21}+K_2=0$ and $L_2<0$) or ($p_1L_1+(1-p_1)L_{12}+K_1=0$, $p_2L_2+(1-p_2)L_{21}+K_2=0$, $L_1<0$, $L_2<0$, and ${\varDelta }_1>0$). Therefore, $\mathbf{s}^*$ is an intermediate ESS if it is a weak ESS and if either ($p_1L_1+(1-p_1)L_{12}+K_1>0$ or $p_2L_2+(1-p_2)L_{21}+K_2>0$), or ($p_1L_1+(1-p_1)L_{12}+K_1=0$ and $p_2L_2+(1-p_2)L_{21}+K_2=0$, and ${\varDelta }_1>0$).

  Finally, $\mathbf{s}^*$ is a strong ESS if for all $\mathbf{s}\not =\mathbf{s}^*$

  $$\begin{aligned}&\bullet \; \bar{U}_1(s_1,\mathbf{s^*},p)<U_1(G_1,\mathbf{s}^*,p), \text{ and } \\&\bullet \; \bar{U}_2(s_2,\mathbf{s}^*,p)<U_2(G_2,\mathbf{s}^*,p). \end{aligned}$$

  Or if

  $$\begin{aligned}&\bullet \; \bar{U}_1(s_1,\mathbf{s^*},p)=U_1(G_1,\mathbf{s}^*,p) \text{ and } \bar{U}_1(s_1,(s_1,1),p)<U_1(G_1,(s_1,1),p)\\&\bullet \; \bar{U}_2(s_2,\mathbf{s}^*,p)<U_2(G_2,\mathbf{s}^*,p). \end{aligned}$$

  Or if

  $$\begin{aligned}&\bullet \; U_1(s_1,\mathbf{s^*},p)<U_1(G_1,\mathbf{s}^*,p), \text{ and } \\&\bullet \; U_2(s_2,\mathbf{s}^*,p)=U_2(G_2,\mathbf{s}^*,p) \text{ and } \bar{U}_2(s_2,(1,s_2),p)<\bar{U}_2(G_2,(1,s_2),p). \end{aligned}$$

  Or if

  $$\begin{aligned}&\bullet \; U_1(s_1,\mathbf{s^*},p)=U_1(G_1,\mathbf{s}^*,p), \text{ and } U_2(s_2,\mathbf{s}^*,p)=U_2(G_2,\mathbf{s}^*,p) \text{ and } \\&\bullet \; \bar{U}_2(s_2,(s_1,s_2),p)<\bar{U}_2(G_2,(s_1,s_2),p) \text{ and } \bar{U}_2(s_2,(s_1,s_2),p)<\bar{U}_2(G_2,(s_1,s_2),p). \end{aligned}$$

  The conditions above yield $p_1L_1+(1-p_1)L_{12}+K_1>0$ or ( $p_1L_1+(1-p_1)L_{12}+K_1=0$ and $L_1<0$) and $p_2L_2+(1-p_2)L_{21}+K_2>0$ or ($p_2L_2+(1-p_2)L_{21}+K_2=0$ and $L_2<0$). Therefore, $\mathbf{s}^*$ is strong ESS if it is an intermediate ESS and if either $p_1L_1+(1-p_1)L_{12}+K_1>0$ or $p_2L_2+(1-p_2)L_{21}+K_2>0$.

We can follow the same procedure for determining the conditions of existence of all fully pure ESS. $\square$

1.5 Partially mixed ESSs

Proposition 6

• $\mathbf{{s}}^*=(1,s_2^*)$ where $s_2^*=-\frac{(1-p_2)L_{21}+K_2}{p_2L_2}$ is a weak ESS if $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$ and $L_2<0$; or if $p_1L_1+(1-p_1)L_{12}s_2^*+K_1=0$, $L_1<0$, and $L_2<0$.

• $\mathbf{{s}}^*=(1,s_2^*)$ is an intermediate ESS if it is weak and either $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$ or ${\varDelta }_1>0$.

• $\mathbf{{s}}^*=(1,s_2^*)$ is a strong ESS if it is an intermediate ESS and if $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$.

• $\mathbf{s}^*=(0,s_2^*)$ where $s_2^*=-\frac{K_2}{p_2L_2}$ is a weak ESS if $(1-p_1)s_2^*L_{12}+K_1<0$ and $L_2<0$, or if $(1-p_1)s_2^*L_{12}+K_1=0$, $L_1<0$, and $L_2<0$.

• $\mathbf{s}^*=(0,s_2^*)$ $\mathbf{s}^*$ is an intermediate ESS if it is a weak ESS and either $(1-p_1)s_2^*L_{12}+K_1<0$ or ${\varDelta }_1>0$.

• $\mathbf{s}^*=(0,s_2^*)$ is a strong ESS if it is an intermediate ESS and if $(1-p_1)s_2^*L_{12}+K_1<0$.

• $\mathbf{s}^*=(s_1^*,1)$ where $s_1^*=-\frac{(1-p_1)L_{12}+K_1}{p_1L_1}$ is a weak ESS if $p_2L_2+(1-p_2)L_{21}s_1^*+K_2>0$ and $L_1<0$; or if $p_2L_2+(1-p_2)L_{21}s_1^*+K_2=0$, $L_1<0$ and $L_2<0$.

• $\mathbf{s}^*=(s_1^*,1)$ is an intermediate ESS if it is a weak ESS and either $p_2L_2+(1-p_2)L_{21}s_1^*+K_2>0$ or ${\varDelta }_1>0$.

• $\mathbf{s}^*=(s_1^*,1)$ is a strong ESS if it is an intermediate ESS and $p_2L_2+(1-p_2)L_{21}s_1^*+K_2>0$.

• $\mathbf{s}^*=(s_1^*,0)$ with $s_1^*=-\frac{K_1}{p_1L_1}$ is a weak ESS if $(1-p_2)L_{21}s_1^*+K_2<0$ and $L_1<0$ or if $(1-p_2)L_{21}s_1^*+K_2=0$, $L_1<0$ and $L_2<0$.

• $\mathbf{s}^*=(s_1^*,0)$ is an intermediate ESS if it is a weak ESS and either $(1-p_2)L_{21}s_1^*+K_2<0$ or ${\varDelta }_1>0$.

• $\mathbf{s}^*=(s_1^*,0)$ is a strong ESS if it is an intermediate ESS and $(1-p_2)L_{21}s_1^*+K_2<0$.

Proof

• $\mathbf{s}^*=(1,s_2^*)$ where $s_2^*=-\frac{(1-p_2)L_{21}+K_2}{p_2L_2}$ is a weak ESS if either:

  $\bullet$ $\bar{U}_1(s_1,\mathbf{s}^*,p)<U_1(G_1,\mathbf{s}^*,p)$ ; $\bar{U}_2(s_2,\mathbf{s}^*,p)=\bar{U}_2(s_2^*,\mathbf{s}^*,p)$, and $\bar{U}_2(s_2,(1,s_2),p)<\bar{U}_2(s_2^*,(1,s_2),p)$ for all $\mathbf{s}\not =\mathbf{s}^*$ (since the equilibrium is mixed in the second community); or

  $\bullet$ if $\bar{U}_1(s_1,\mathbf{s}^*,p)=U_1(G_1,\mathbf{s}^*,p)$ and $\bar{U}_1(s_1,(s_1,s_2^*),p)<U_1(G_1,(s_1,s_2^*),p)$ and $\bar{U}_2(s_2,\mathbf{s}^*,p)=\bar{U}_2(s_2^*,\mathbf{s}^*,p)$, and $\bar{U}_2(s_2,(1,s_2),p)<\bar{U}_2(s_2^*,(1,s_2),p)$.

  The first set of conditions yields $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$ and $L_2<0$.

  The second set of conditions yields $p_1L_1+(1-p_1)L_{12}s_2^*+K_1=0$, $L_1<0$, and $L_2<0$.

• $\mathbf{s}^*=(1,s_2^*)$ where $s_2^*=-\frac{(1-p_2)L_{21}+K_2}{p_2L_2}$ is an intermediate ESS if:

  $\bullet$ $\bar{U}_1(s_1,(1,s_2^*),p)+\bar{U}_1(s_2,1,s_2^*),p)\le U_1(G_1,(1,s_2^*),p)+\bar{U}_2(s_2^*,(1,s_2^*),p)$ .

  $\bullet$ If there exists $\mathbf{s}$ for which the above condition is an equality, then $\bar{U}_1(s_1,\mathbf{s},p)+\bar{U}_2(s_2,\mathbf{s},p)<\bar{U}_1(s_1^*,\mathbf{s},p)+\bar{U}_2(s_2^*,\mathbf{s},p)$.

  We conclude that the conditions of existence of the intermediate ESS are either $L_2<0$ and $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$, or $p_1L_1+(1-p_1)L_{12}s_2^*+K_1=0$, $L_1<0$, $L_2<0$ and ${\varDelta }_1>0$. Therefore, $\mathbf{s}^*$ is an intermediate ESS if it is a weak ESS and either $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$ or ${\varDelta }_1>0$.

• $\mathbf{s}^*=(1,s_2^*)$ is a partially mixed strong ESS if

  $$\begin{aligned}&\bar{U}_1(s_1,(1,s_2^*),p) <\bar{U}_1(G_1,(1,s_2^*),p)\\&\quad \bar{U}_2(s_2, (1,s_2^*),p)=\bar{U}_2(s_2^*,(1,s_2^*),p) \text{ and } \bar{U}_2(s_2,(1,s_2),p)<\bar{U}_2(s_2^*,(1,s_2),p). \end{aligned}$$

  or if

  $$\begin{aligned}&\bar{U}_1(s_1,(1,s_2^*),p) =\bar{U}_1(G_1,(1,s2^*),p) \text{ and } \bar{U}_1(s_1,(s_1,s_2),p)<U_1(G_1,(s_1,s_2),p)\\&\quad \bar{U}_2(s_2,(s_1,s_2),p)<\bar{U}_2(s_2^*,(s_1,s_2),p) \end{aligned}$$

  The first set of conditions yield $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$ and $L_2<0$. The second set of conditions cannot be satisfied (saddle point). We conclude that $\mathbf{s}^*$ is a strong ESS if its an intermediate ESS and if $p_1L_1+(1-p_1)L_{12}s_2^*+K_1>0$.

We follow the same procedure for determining the conditions of existence of the other partially mixed ESS. $\square$

1.6 Proof of Theorem  2

In order to examine the stability of the interior stationary point, we make a linearization of the system (12) around $\mathbf{s}^*$ and observe how the linearized system behaves. We introduce a small perturbation around $\mathbf{s}^*$ defined by $x_1(t)=s_1(t)-s_1^*$ and $x_2(t)=s_2(t)-s_2^*$. The replicator dynamics then writes:

$$\begin{aligned} \begin{array}{cc} {}\dot{x}_1(t)=(x_1(t)+s_1^*)(1-x_1(t)-s_1^*)\big (p_1x_1(t)L_1+(1-p_1)x_2(t)L_{12}\big ),\\ {} \dot{x}_2(t)= (x_2(t)+s_2^*)(1-x_2(t)-s_2^*)\big (p_2x_2(t)L_2+(1-p_2)x_1(t)L_{21}\big ). \end{array} \end{aligned}$$

Keeping only linear terms in $x_1$ and $x_2$, we obtain a linearized system of the form $\dot{x}(t)=Ax(t)$ where $x^t=(x_1,x_2)$,

$$\begin{aligned} A = \left( \begin{array}{cc} \gamma _1p_1L_1&{} \gamma _1(1-p_1)L_{12}\\ \gamma _{2}(1-p_2)L_{21} &{} \gamma _{2}p_2L_2\end{array} \right) , \end{aligned}$$

$\gamma _1=s_1^*(1-s_1^*)$, and $\gamma _{2}=s_2^*(1-s_2^*)$. The linearized system is asymptotically stable if all the eigenvalues of A have negative real parts. The eigenvalues of A are the roots of the characteristic polynomial $\mathcal{{X}}_A=\lambda ^2-tr(A)\lambda +det(A)$, with $tr(A)=\gamma _1p_1L_1 +\gamma _{2}p_2L_2$ and $det(A)=\gamma _1\gamma _{2}(p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21})$. We check that if ${\varDelta }=p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21}>0$, $L_1<0$ and $L_2<0$, then the two eigenvalues of A have negative real parts and the stability follows.

1.7 Proof of Corollary 1

We aim to prove that a mixed intermediate ESS is asymptotically stable in the replicator dynamics. From Theorem 1, the interior equilibrium $s^*$ is an intermediate ESS if $L_1<0$, $L_2<0$ and ${\varDelta }_1>0$. In addition, from Theorem 2, $s^*$ is asymptotically stable if $L_1<0$, $L_2<0$, and ${\varDelta }>0$. We can then prove that if ${\varDelta }_1=4p_1p_2L_1L_2-((1-p_1)L_{12}+(1-p_2)L_{21})^2>0$, then ${\varDelta }=p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21}>0$ (or equivalently $4{\varDelta }>0$). We have:

$$\begin{aligned}&4p_1p_2L_1L_2-((1-p_1)L_{12}+(1-p_2)L_{21})^2- 4\big (p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21}\big )\\&\quad =-((1-p_1)L_{12}-(1-p_2)L_{21})^2 <0. \end{aligned}$$

The proof follows.

1.8 Proof of Theorem 3

Let us prove that the partially mixed ESS $\mathbf{s}^*=(1,s_2^*)$ with $s_2^*=- \frac{(1-p_2)L_{21}+K_2}{p_2L_2}$ is asymptotically stable. We make a linearization of the replicator dynamics around $\mathbf{s}^*$ and we study the Jacobian matrix. If all the eigenvalues of the Jacobian matrix have negative real parts then the asymptotic stability follows. The Jacobian matrix is given by:

$$\begin{aligned} A=\left( \begin{array}{ccc} -p_1L_1-(1-p_1)s_2^*L_{12}-K_1 &{}0\\ \gamma _2(1-p_2)L_{21}&{}\gamma _2p_2L_2 \end{array} \right) , \end{aligned}$$

with $\gamma _2=s_2^*(1-s_2^*)$. The eigenvalues of the Jacobian matrix are the solutions of the following characteristic polynomial: $\mathcal{{X}}_A=\lambda ^2-tr(A)\lambda +det(A).$ We check that $\mathbf{s}^*$ is asymptotically stable if $p_1L_1+(1-p_1)s_2^*L_{12}+K_1>0$ and $L_2<0$. Therefore, by virtue of Proposition 6, the strong ESS $\mathbf{s}^*$ is asymptotically stable.

Similarly, we can prove this result for all other partially mixed and completely pure strong ESSs.

1.9 Proof of Theorem  4

We showed in Appendix “Proof of Theorem 2” that the eigenvalues of A which are solutions of $\lambda ^2-tr(A)\lambda +det(A)=0$ have negative real parts when ${\varDelta }=p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21}>0$, $L_1<0$ and $L_2<0$; the mixed intermediate ESS is asymptotically stable when $\tau _{st}=0$ (Corollary 1). For the remaining of the proof that gives the bound on $\tau _{st}$ for which the stability is unaffected, the reader should refer to [16], pp.82, Theorem 3.4.

1.10 Proof of Theorem  5

The proof of this theorem is based on that given by Freedman and Kuang (Theorem 4.1, page 202), related to the location of roots of the characteristic equation (14), and stated as follows:

• If $\beta ^2 < \delta ^2$, $\Rightarrow$ if $\mathbf{s^*}$ is unstable for $\tau =0$, then it is unstable for any $\tau \ge 0$; if $\mathbf{s^*}$ is stable at $\tau =0$, then it remains stable for $\tau$ inferior than some $\tau _s \ge 0$. But, if $\mathbf{s}^*$ is stable at $\tau =0$, then ${\varDelta }=p_1p_2L_1L_2-(1-p_1)(1-p_2)L_{12}L_{21} > 0$ $\Rightarrow$ $\beta ^2 > \delta ^2$. Therefore, this case is excluded.

• If $\beta ^2 > \delta ^2$, $2\beta -\alpha ^2>0$, and $\big (2\beta -\alpha ^2\big )^2>4 (\beta ^2-\delta ^2)$, then the stability of the stationary point can change a finite number of times at most as $\tau$ is increased, and eventually it becomes unstable. But

  $$\begin{aligned} {} 2\beta -\alpha ^2= & {} 2\gamma _1\gamma _{2}p_1p_2L_1L_2-(p_1\gamma _1L_1+p_2\gamma _{2}L_2)^2\\= & {} -p_1^2\gamma _1^2L_1^2-p_2^2\gamma _{2}^2L_2^2. <0 \end{aligned}$$

  Therefore, this case is excluded in our model.

• Otherwise, (this is the only case when $\mathbf{s}^*$ is stable at $\tau =0$), the stability of the stationary point $\mathbf{s^*}$ does not change for any $\tau \ge 0$.