Conditions for Cooperation to be More Abundant than Defection in a Hierarchically Structured Population

Abstract

We study conditions for weak selection to favor tit-for-tat (TFT) over AllD in a repeated Prisoner’s Dilemma game played in a finite population subdivided into three subpopulations under the assumption of cyclic dominance in asymmetric interactions. Assuming parent-independent mutation and uniform migration, we show that TFT is more abundant that AllD in the stationary state if the defection cost incurred by individuals in interaction with dominant defecting individuals exceeds some threshold value. This threshold value decreases as the number of repetitions of the game, the population size, or the mutation rate increases, but increases as the migration rate increases. The same conclusions hold in the case of linear dominance.

Appendix: Model of Linear Dominance

In this section, we consider $m$ rounds of the Prisoner’s Dilemma played between individuals chosen at random in three subpopulations of the same finite size $N$ represented by $S_1,$ $S_2$, and $S_3$ under the assumption of linear hierarchy: an individual in $S_1$ dominates an individual in $S_2$ and in $S_3$, while an individual in $S_2$ dominates an individual in $S_3$. With respect to the strategies TFT and AllD, the payoff matrices $A_{i,j}$ for an individual in $S_i$ in interaction with an individual in $S_j$, for $i,j=1, 2, 3$, are given by

$$\begin{aligned} A_{1,1}=A_{2,2}=A_{3,3}=A_{1,2}=A_{1,3}=A_{2,3}=\begin{pmatrix} m(b-c) &{} -c \\ b &{} 0 \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} A_{2,1}=A_{3,1}=A_{3,2}=\begin{pmatrix} m(b-c) &{} -c-(m-1)\beta \\ b-(m-1)\beta &{} -m\beta \end{pmatrix}. \end{aligned}$$

The expected payoffs $w_1^{(i)}$ and $w_2^{(i)}$ to TFT and AllD, respectively, in subpopulation $S_i$, for $i=1,2,3$, are then given by

$$\begin{aligned} w_1^{(1)}&= \frac{3N}{3N-1}\Big [m x(b-c)-\frac{m(b-c)}{3N}-(1-x)c\Big ],\\ w_1^{(2)}&= \frac{3N}{3N-1}\Big [m x(b-c)-\frac{m(b-c)}{3N}-(1-x)c-\frac{1}{3}(1-x_1)(m-1)\beta \Big ],\\ w_1^{(3)}&= \frac{3N}{3N-1}\Big [m x(b-c)-\frac{m(b-c)}{3N}-(1-x)c-\frac{1}{3}(2-x_1-x_2)(m-1)\beta \Big ],\\ w_2^{(1)}&= \frac{3N}{3N-1}xb,\\ w_2^{(2)}&= \frac{3N}{3N-1}\Big [xb-\frac{1}{3}(m-x_1)\beta \Big ],\\ w_2^{(3)}&= \frac{3N}{3N-1}\Big [xb-\frac{1}{3}(m-x_1)\beta -\frac{1}{3}(m-x_2)\beta \Big ]. \end{aligned}$$

These expected payoffs can be expressed in the form:

$$\begin{aligned} w_1^{(i)}=\frac{3N}{3N-1}\Bigg [m x(b-c)-\frac{m(b-c)}{3N}-(1-x)c-\frac{1}{3}\sum _{l<i}(1-x_l)(m-1)\beta \Bigg ], \end{aligned}$$

(38)

and

$$\begin{aligned} w_2^{(i)}=\frac{3N}{3N-1}\Bigg [xb-\frac{1}{3}\sum _{l<i}(m-x_l)\beta \Bigg ], \end{aligned}$$

(39)

for $i=1,2,3$. This yields differences in the form:

$$\begin{aligned}&w_{1}^{(i)}-w_{2}^{(j)}=\frac{3N}{3N-1}\Bigg \{(m-1)x(b-c)-\Bigg [\frac{m}{3N}(b-c)+c\Bigg ]\\&\quad + \frac{1}{3}\sum _{l<j}(m-x_l)\beta -\frac{1}{3}\sum _{l<i}(1-x_l)(m-1)\beta \Bigg \},\nonumber \end{aligned}$$

for $i,j=1,2,3$.

Plugging these expressions into (13) for the conditional expected change in frequency of TFT in the whole population given an intensity of selection $\delta > 0$ and no mutation, which remains valid under linear hierarchy, leads to

$$\begin{aligned}&{\mathbb {E}}_{\delta }[\Delta X_{\mathrm {sel}}|\mathbf {X}=\mathbf {x}] =\frac{\delta (m-1)}{3(3N-1)}\Bigg \{(1-v)\sum _{i=1}^3 x_i(1-x_i)x+ \frac{v}{2}\sum _{i\ne j=1}^3x_i(1-x_j)x\Bigg \}(b-c)\\&\quad -\frac{\delta }{3(3N-1)}\Bigg \{(1-v)\sum _{i=1}^3 x_i(1-x_i)+\frac{v}{2}\sum _{i\ne j=1}^3x_i(1-x_j)\Bigg \}\Bigg [\frac{m}{3N}(b-c)+c\Bigg ]\\&\quad +\frac{\delta }{9(3N-1)}\Bigg \{(1-v)\sum _{1\le l<i\le 3}x_i(1-x_i)(m-x_l)\\&\quad +\frac{v}{2}\sum _{1\le l<j\le 3}\sum _{^{i=1}_{i\ne j}}^3x_i(1-x_j)(m-x_l)-(1-v)\sum _{1\le l<i\le 3}x_i(1-x_i)(m-1)(1-x_l)\\&\quad -\frac{v}{2}\sum _{1\le l<i\le 3}\sum _{^{j=1}_{j\ne i}}^3x_i(1-x_j)(m-1)(1-x_l)\Bigg \}\beta +o(\delta ). \end{aligned}$$

Taking the expected value in the stationary state gives

$$\begin{aligned} (3N-1){\mathbb {E}}_{\delta }[\Delta X_{\mathrm {sel}}]=\frac{\delta }{3}\Bigg \{\Bigg (M'_{1}-\frac{mM'_2}{3N}\Bigg )(b-c)-M'_2c+M'_3\beta \Bigg \}+o(\delta ), \end{aligned}$$

(40)

where

$$\begin{aligned} M'_{1}&= (m-1)\Bigg \{(1-v)\sum _{i=1}^3 {\mathbb {E}}_0\left[ x_i(1-x_i)x\right] +\frac{v}{2}\sum _{i\ne j=1}^3{\mathbb {E}}_0[x_i(1-x_j)x]\Bigg \},\\ M'_{2}&= (1-v)\sum _{i=1}^3 {\mathbb {E}}_0[x_i(1-x_i)]+\frac{v}{2}\sum _{i\ne j=1}^3{\mathbb {E}}_0[x_i(1-x_j)],\\ M'_{3}&= \frac{(1-v)}{3}\sum _{1\le l<i\le 3}{\mathbb {E}}_0[ x_i(1-x_i)(m-x_l)]+\frac{v}{6}\sum _{1\le l<j\le 3}\sum _{^{i=1}_{i\ne j}}^3{\mathbb {E}}_0[x_i(1-x_j)(m-x_l)]\\&\quad -\frac{(m-1)}{3}\\&\quad \times \Bigg \{(1-v)\sum _{1\le l<i\le 3} {\mathbb {E}}_0[x_i(1-x_i)(1-x_l)]+\frac{v}{2}\sum _{1\le l<i\le 3}\sum _{^{j=1}_{j\ne i}}^3{\mathbb {E}}_0[x_i(1-x_j)(1-x_l)]\Bigg \}. \end{aligned}$$

Comparing with (15), (16), and (17), we note that $M'_1=M_1$ and $M'_2=M_2$. Moreover, using the symmetry of the model in the neutral case, we obtain that

$$\begin{aligned} M'_3=M_3&= (1-v)E_0[ x_1(1-x_1)(m-x_2)]\\&+\frac{v}{2}\Bigg \{E_0[ x_1(1-x_2)(m-x_1)]+E_0[ x_1(1-x_2)(m-x_3)]\Bigg \} \\&-(1-v)(m-1)E_0[ x_1(1-x_1)(1-x_2)]-\frac{v}{2}\Bigg \{E_0[ x_1(1-x_2)^2]+E_0[ x_1(1-x_2)(1-x_3)]\Bigg \}. \end{aligned}$$

Therefore, the sufficient condition (21) for weak selection to favor TFT in the case of cyclic hierarchy is valid as well in the case of linear hierarchy.