Group Selection Under the Replicator Dynamic

Abstract

We consider a society where individuals get divided into different groups. All groups play the same normal form game, and each group is in the basin of attraction of a particular strict Nash equilibrium of the game. Group selection refers to the evolution of the mass of each group under the replicator dynamic. We provide microfoundations to the replicator dynamic using revision protocols, wherein agents migrate between groups based on average payoffs of the groups. Individual selection of strategies within each group happens in two possible ways. Either all agents instantaneously coordinate on a strict Nash equilibrium or all agents change their strategies under the logit dynamic. In either case, the group playing the Pareto efficient Nash equilibrium gets selected. Individual selection under the logit dynamic may slow down the process and introduce non-monotonicity. We then apply the model to the stag hunt game.

Appendix

Proof of Proposition 3.1

We provide the proof only for the revision protocol (11). Applying (11) to (10), we obtain

$$\begin{aligned} {\dot{m}}^i&=\sum _{j\in S}m^jm^i[{\bar{F}}^i(x^i,m^i)-{\bar{F}}^j(x^j,m^j)]_+-m^i\sum _{j\in S}m^j[{\bar{F}}^j(x^j,m^j)-{\bar{F}}^i(x^i,m^i)]_+\nonumber \\&=\sum _{{\bar{F}}^i>{\bar{F}}^j}m^jm^i[{\bar{F}}^i(x^i,m^i)-{\bar{F}}^j(x^j,m^j)]-m^i\sum _{{\bar{F}}^j>{\bar{F}}^i}m^j[{\bar{F}}^j(x^j,m^j)-{\bar{F}}^i(x^i,m^i)]\nonumber \\&=\sum _{{\bar{F}}^i>{\bar{F}}^j}m^jm^i[{\bar{F}}^i(x^i,m^i)-{\bar{F}}^j(x^j,m^j)]+m^i\sum _{{\bar{F}}^j>{\bar{F}}^i}m^j[{\bar{F}}^i(x^i,m^i)-{\bar{F}}^j(x^j,m^j)]\nonumber \\&=m^i\left[ {\bar{F}}^i(x^i,m^i)\sum _{j\in S}m^j-\sum _{j\in S}m^j{\bar{F}}^j(x^j,m^j)\right] \nonumber \\&=m^i\left( {\bar{F}}^i(x^i,m^i)-{\bar{F}}(x,m)\right) , \end{aligned}$$

(27)

where ${\bar{F}}(x,m)$ is the average social payoff (8). But (27) is the replicator dynamic (9). $\square$

Proof of Lemma 4.1

By our assumptions about the game (1), $e_i$ is a strict Nash equilibrium for every $i\in \{1,2,\ldots ,n\}$. Instantaneous coordination implies that all agents who are in group i at time $\tau =0$ will play $i\in S$ resulting in the Nash equilibrium $e_i$. At subsequent rounds of random matching, they continue playing the best response i. Revision protocols in Sect. 3 imply that at most one migrant will enter or leave group i at any given time $\tau$. When a single agent leaves the group that does not affect the best response of agents who remain. Hence, those agents continue to play $i\in S$. When a single agent arrives, that agent will best respond to the current strategy distribution $e_i$ in the group immediately upon arrival. The unique best response to $m^i(\tau )e_i$ is $i\in S$. Hence, at any given $\tau$, $x^i(\tau )=m^i(\tau )e_i$. $\square$

Proof of Proposition 4.2

At any time $\tau \ge 0$, the state of group i is $m^i(\tau )e_i$. We denote the vector of group masses as $m(\tau )=(m^1(\tau ),m^2(\tau ),\ldots ,m^n(\tau ))$ and the vector of group states as $m(\tau )e=(m^1(\tau )e_1,m^2(\tau )e_2,\ldots ,m^n(\tau )e_n)$.

From (6), we now obtain $F_i^i(m^i(\tau )e_i,m^i(\tau ))=v_{ii}$ and $F_s^i(m^i(\tau )e_i,m^i(\tau ))=v_{si}$ for all other strategies $s\ne i$. Therefore, (7) implies ${\bar{F}}^i(m^i(\tau )e_i,m^i(\tau ))=v_{ii}$ and (8) implies ${\bar{F}}(e,m(\tau )e)={\bar{v}}(\tau )$ as defined following (14). Hence, the replicator dynamic (9) takes the form

$$\begin{aligned} {\dot{m}}^i(\tau )&=m^i(\tau )\left[ {\bar{F}}^i(m^i(\tau )e_i,m^i(\tau ))-{\bar{F}}(e,m(\tau )e)\right] \\&=m^i(\tau )(v_{ii}-{\bar{v}}(\tau )). \end{aligned}$$

$\square$

Proof of Proposition 4.3

By Proposition 4.2, $m^i(t)$ evolves according to the replicator dynamic (14). We can interpret group i as “strategy i” in (14). The payoff of strategy i in (14) is the constant $v_{ii}$. But $v_{11}>v_{22}>\cdots >v_{nn}$. Hence, strategy 1 is strictly dominant. Under the replicator dynamic, the mass of strictly dominated strategies go to zero (Theorem 1, Samuelson and Zhang [23]). Therefore, $m^j(t)\rightarrow 0$ for all $j\ne 1$ and $m^1(t)\rightarrow 1$. $\square$

Proof of Proposition 6.1

As $v_{11}=V$ and $v_{22}=V-c$ in (23), the average payoff in (14) is²⁶

$$\begin{aligned} {\bar{v}}(\tau )=m^1(\tau )V+m^2(\tau )(V-c)=V-(1-m^1(\tau ))c. \end{aligned}$$

(28)

Using (28), we can write the rate of change in the mass of the stag group will under the replicator dynamic (14) as

$$\begin{aligned} {\dot{m}}^1(\tau )=m^1(\tau )\left( v_{11}-{\bar{v}}(\tau )\right) =m^1(\tau )(1-m^1(\tau ))c. \end{aligned}$$

$\square$