Homophilic replicator equations

Abstract

Tags are conspicuous attributes of organisms that affect the behaviour of other organisms toward the holder, and have previously been used to explore group formation and altruism. Homophilic imitation, a form of tag-based selection, occurs when organisms imitate those with similar tags. Here we further explore the use of tag-based selection by developing homophilic replicator equations to model homophilic imitation dynamics. We assume that replicators have both tags (sometimes called traits) and strategies. Fitnesses are determined by the strategy profile of the population, and imitation is based upon the strategy profile, fitness differences, and similarity in tag space. We show the characteristics of resulting fixed manifolds and conditions for stability. We discuss the phenomenon of coat-tailing (where tags associated with successful strategies increase in abundance, even though the tags are not inherently beneficial) and its implications for population diversity. We extend our model to incorporate recurrent mutations and invasions to explore their implications upon tag and strategy diversity. We find that homophilic imitation based upon tags significantly affects the diversity of the population, although not the ESS. We classify two different types of invasion scenarios by the strategy and tag compositions of the invaders and invaded. In one scenario, we find that novel tags introduced by invaders become more readily established with homophilic imitation than without it. In the other, diversity decreases. Lastly, we find a negative correlation between homophily and the rate of convergence.

Appendix: Games

The following background describes the Snowdrift, Prisoner’s Dilemma, and Stag Hunt games. For simplicity in the work above, we alter payoff matrices so that the games are in terms of one parameter, $\kappa = c/b$.

Imagine that two people, who will be the players in this game, are driving along a highway during a blizzard. They careen off of the road into a Snowdrift. Snow must be shovelled away from the vehicle so that it may continue its journey. Each player has the choice to step outside and shovel away the snow or remain in the vehicle. If at least one of the players cleans away the snow, then they both can reach home, thereby receiving the benefit, b. If no one does, they receive a payoff of 0. Snow shovelling is exhausting; therefore, there is a cost, c, to shovelling. This energy cost decreases the overall payoff. If both players shovel, then they split the cost; if only one shovels, then that player suffers the entire energy cost. There are three Nash equilibria: (Shovel, Don’t Shovel), (Don’t Shovel, Shovel), and one mixed strategy. However, the ESS for this game is the internal equilibrium. The following payoff matrix represents this game:

$$\begin{aligned} \varvec{\Pi }_{\mathrm {SD}} = \begin{pmatrix} b-c/2 &{}\quad b-c \\ b &{}\quad 0 \\ \end{pmatrix}, \end{aligned}$$

(15)

where $b>c>0$.

In the Prisoner’s Dilemma we have two players that choose from a strategy set of cooperate and defect. Let $b>c>0$. If both cooperate, they receive the socially optimal payoff, $b-c$. If one cooperates and the other defects, the cooperator earns $-c$, and the defector earns b, the temptation. If both defect, they each receive nothing. The payoff matrix for this game is:

$$\begin{aligned} \varvec{\Pi }_{\mathrm {PD}} = \begin{pmatrix} b-c &{}\quad -c \\ b &{}\quad 0 \\ \end{pmatrix}. \end{aligned}$$

(16)

The socially optimal outcome of cooperation is unstable. At the ESS, the population is entirely composed of defectors.

Consider two hunters, the players, who have the choice between hunting a stag or a hare. Cooperation is required to successfully harvest a stag and receiving the benefit, b; if only one player hunts stag, then that player will fail at catching anything. Hunting stag requires an investment of c regardless of the success. A player who hunts hare will receive a payoff of 0 regardless of the strategy decision of the other player. Again, we have that $b> c > 0$. The payoff matrix for this game is:

$$\begin{aligned} \varvec{\Pi }_{\mathrm {SH}} = \begin{pmatrix} b-c &{}\quad -c \\ 0 &{}\quad 0 \\ \end{pmatrix}. \end{aligned}$$

(17)

There are two ESSes for this game: all Stag Hunting or all hare hunting. The internal equilibrium is unstable.