Online networks, social interaction and segregation: an evolutionary approach

Abstract

There is growing evidence that face-to-face interaction is declining in many countries, exacerbating the phenomenon of social isolation. On the other hand, social interaction through online networking sites is steeply rising. To analyze these societal dynamics, we have built an evolutionary game model in which agents can choose between three strategies of social participation: 1) interaction via both online social networks and face-to-face encounters; 2) interaction by exclusive means of face-to-face encounters; 3) opting out from both forms of participation in pursuit of social isolation. We illustrate the dynamics of interaction among these three types of agent that the model predicts, in light of the empirical evidence provided by previous literature. We then assess their welfare implications. We show that when online interaction is less gratifying than offline encounters, the dynamics of agents’ rational choices of interaction will lead to the extinction of the sub-population of online networks users, thereby making Facebook and similar platforms disappear in the long run. Furthermore, we show that the higher the propensity for discrimination of those who interact via online social networks and via face-to-face encounters (i.e., their preference for the interaction with agents of their same type), the greater the probability will be that they all will end up choosing social isolation in the long run, making society fall into a “social poverty trap”.

Appendices

Mathematical Appendix A

Dynamics (2) is equivalent (see Hofbauer, 1981) to the Lotka-Volterra system:

$$ \dot{X}=X\left(a+ bX\right) $$

(11)

$$ \dot{Y}=Y\left(d+ eX+ fY\right) $$

(12)

via the coordinate change:

$$ {x}_1=\frac{1}{1+X+Y},\kern0.5em {x}_2=\frac{X}{1+X+Y},\kern0.5em {x}_3=\frac{Y}{1+X+Y} $$

(13)

From which X = x₂/x₁ and Y = x₃/x₁.

Please note that by the coordinate change (13), the edge e₁ − e₂ of the simplex S (see Fig. 1) corresponds to the positive semi-axis Y = 0 of the plane (X,Y), the edge e₁ − e₃ corresponds to the positive semi-axis X = 0 and the vertex e₁ corresponds to the point (X,Y) = (0,0) (see Fig. 9).

Fig. 9

Arrow diagram of the Lotka-Volterra system. The edge e₁ − e₂ of the simplex S corresponds to the positive semi-axis Y = 0 of the plane (X,Y), the edge e₁ − e₃ corresponds to the positive semi-axis X = 0, and the vertex e₁ corresponds to the point (X,Y) = (0,0). The set in which EP_SN > EP_NS holds coincides with the region on the left of the vertical straight line X =  − a/b

According to Eq. (11), $\dot{X}=0$ holds along the axis X = 0 and along the vertical straight line X =  − a/b>0; furthermore, $\dot{X}>0$ ($\dot{X}<0$) holds on the right (respectively, on the left) of X =  − a/b. According to Eq. (12), $\dot{Y}=0$ holds along the axis Y = 0 and along the straight line Y =  − d/f − (e/f)X. Furthermore,$\dot{\ Y}>0$ ($\dot{Y}<0$) holds above (respectively, below) Y =  − d/f − (e/f)X.

Remembering that a < 0, b > 0, and f > 0, we have that a unique stationary state with X > 0 and Y > 0, $\left(\overline{X},\overline{Y}\right)=\left(-a/b,-d/f+(ae)/(bf)\right)$, exists if and only if ae > bd (condition (10) of Proposition 3). The Jacobian matrix of system (11)–(12), evaluated at $\left(\overline{X},\overline{Y}\right)$, is a triangular matrix:

$$ J\left(\overline{X},\overline{Y}\right)=\left(\begin{array}{cc}b\overline{X}& 0\\ {}\overline{Y}& f\overline{Y}\end{array}\right) $$

With eigenvalues $b\overline{X}>0$ (in direction of X = − a/b) and $f\overline{Y}>0$. So $\left(\overline{X},\overline{Y}\right)$ is always a repulsive node (this completes the proof of point one of Proposition 3).

By following similar steps, it is easy to verify that:

• 1) The Lotka-Volterra system (11)–(12) always admits a unique stationary state (X, Y) = (−a/b, 0), with −a/b > 0, belonging to the positive semi-axis Y = 0 (corresponding to the edge e₁ − e₂ of the simplex S; see Fig. 1). Such a stationary state is a saddle point (with unstable manifold lying in Y = 0, and stable manifold lying in X =  − a/b) if the internal stationary state $\left(\overline{X},\overline{Y}\right)$ exists; otherwise it is a source (point two of Proposition 3).

• 2) The Lotka-Volterra system (11)–(12) admits a unique stationary state (X, Y) = (0, −d/f), with −d/f > 0, belonging to the positive semi-axis X = 0 (corresponding to the edge e₁ − e₃ of the simplex S) if d < 0. Such a stationary state is always a saddle point with unstable manifold lying in X = 0. If d ≥ 0, then no stationary state with Y > 0 exists in the positive semi-axis X = 0 (point three of Proposition 3).

• 3) The state (X, Y) = (0, 0) (corresponding to the vertex e₁ of the simplex S; see Fig. 1) is always a stationary state; it is a saddle point (with unstable manifold lying in X = 0, and stable manifold lying in Y = 0) if d ≥ 0 (i.e., if the stationary state in the semi-axis X = 0 does not exist, see point two above), otherwise it is a sink (point one of Proposition 1).

The stability properties of the stationary states e₂ and e₃ (points two to three of Proposition 1) and the existence and stability properties of the stationary state belonging to the edge e₂ − e₃ (point four of Proposition 3)⁶ can be easily analyzed by applying Propositions 1, 2 and 5 in Bomze (1983), who provided a complete classification of two-dimensional replicator equations.

Mathematical Appendix B

The condition:

$$ {EP}_{SN}\left({x}_1,{x}_2\right)>{EP}_{NS}\left({x}_1,{x}_2\right) $$

can be written as follows:

ax₁ + bx₂ < 0,

bx₂ <  − ax₁,

$X<-\frac{a}{b}$,

where X = x₂/x₁. Consequently, in the positive quadrant of the plane (X, Y), the set in which EP_SN > EP_NS holds coincides with the region on the left of the vertical straight line (see Fig. 9):

$$ X=-\frac{a}{b}>0 $$

(14)

Along the straight line (14),$\dot{\ X}=0$ holds, while the set in which EP_SN < EP_NS holds corresponds to the region on the right of (14). Since (14) cannot be crossed by trajectories (see Fig. 9), the two regions separated by (14) are invariant. Consequently, every trajectory starting from the region in which EP_SN < EP_NS cannot converge to the stationary state (X, Y) = (0, 0), which corresponds to the stationary state e₁ = (1, 0, 0). This completes the proof of Proposition 4.