The economics of utopia: a co-evolutionary model of ideas, citizenship and socio-political change

Abstract

We propose a new history-friendly approach to evolutionary socio-economic dynamics based around competition between five ‘utopias’ as central ideas about which to order society: capitalism, socialism, civil liberty, nature, and nationalism. In our model, citizens contribute economic resources to support their preferred utopia, and societal dynamics are explained as a co-evolutionary process between these competing utopias. We apply the model to analyze certain aspects of socio-economic and political change in the US from the 1960s–present. We carry out a history-friendly analysis inspired by such episodes as the outbreak of civil movements in the 1970s, the rise of neo-liberalism in the 1980s, and the channels through which America has engendered an ‘age of fracture’. Further applications for empirical and theoretical research are suggested.

Appendix

In this appendix, we present further insights on the intra-subsystemic dynamics of the model, and on the way these dynamics co-evolve giving rise to the overall dynamics of utopia competition. The exhaustive mathematical exploration of the model goes beyond the scope of this paper. Nevertheless, we want to highlight here some possible lines of progress in the formal exploration; likewise, we present certain results which clarify the mechanisms supporting our socio-economic insights in Section 4. We do not incorporate these results in Section 4 because, perhaps, they might interrupt the history-friendly style of discussion of the paper.

As we show in this Appendix, it is interesting to note that, although we have presented the model as a co-evolution framework that contributes to evolutionary political economy in line with population dynamics thinking, we can use machinery from evolutionary game theory to better understand the dynamics and the results. This is a typical way to proceed in population models (see Weibull 1995; Hofbauer and Sigmund 1998; Sandholm 2010). Thus, in this appendix, we show, firstly, how the intra-subsystem dynamics can be decomposed for future analysis in two extreme subgames and infinite mixes of these subgames. This procedure allows us to better understand the role of parameter φ in the model and in our results (persistence of various co-existing utopias, etc). Afterwards, we consider these insights to reflect on the overall replicator process (expression (3) in Section 3) which is interlinked (in a bi-directional way) in the model with the distinct intra-subsystem replicators (expressions (1) and (2) in Section 3, and the bidirectional links with expression (3)). We present new simulations as supporting material for the socio-economic interpretations in Section 4. The appendix also helps us to pose possible future developments (departing from the current model as a benchmark) as we explain in Section 5.

1.1 Insights on the dynamics of the model

1.1.1 Decomposition of the intra-subsystemic dynamics

Note that the payoff for each level of contribution (eq. (1)) can be written as follows:

$$ \left[\begin{array}{c}{\ u}_{1 t}^{\pi}\\ {}{\ u}_{2 t}^{\pi}\\ {}{\ u}_{3 t}^{\pi}\end{array}\right]=\left({\gamma}_t^{\pi}\left(1-\varphi \right)\left[\begin{array}{ccc}{x}_1& {x}_1& {x}_1\\ {}{x}_2& {x}_2& {x}_2\\ {}{x}_3& {x}_3& {x}_3\end{array}\right]+\varphi \left[\begin{array}{ccc}0& {x}_1& 0\\ {}-{x}_2& 0& {x}_2\\ {}0& -{x}_3& 0\end{array}\right]\right)\left[\begin{array}{c}{s}_{1 t}^{\pi}\\ {}{s}_{2 t}^{\pi}\\ {}{s}_{3 t}^{\pi}\end{array}\right] $$

Thus, at the intra-subsystem level, eq. (2) can be seen as the replicator dynamics of a population game where players are randomly paired to play a 2-player 3-strategy game where the payoff matrix is:

$$ \left({\gamma}_t^{\pi}\left(1-\varphi \right)\left[\begin{array}{ccc}{x}_1& {x}_1& {x}_1\\ {}{x}_2& {x}_2& {x}_2\\ {}{x}_3& {x}_3& {x}_3\end{array}\right]+\varphi \left[\begin{array}{ccc}0& {x}_1& 0\\ {}-{x}_2& 0& {x}_2\\ {}0& -{x}_3& 0\end{array}\right]\right) $$

Let us consider the extreme values of φ. For φ = 0, we have the following game (henceforth SG1, for subgame 1):

$$ {\gamma}_t^{\pi}\left[\begin{array}{ccc}{x}_1& {x}_1& {x}_1\\ {}{x}_2& {x}_2& {x}_2\\ {}{x}_3& {x}_3& {x}_3\end{array}\right] $$

Given that x ₁ < x ₂ < x ₃, strategy 3 is dominant, and evolutionarily stable. Thus, the point s ₃ = 1 is asymptotically stable and the system converges to it from any initial condition with s ₃ > 0.¹⁴ The speed of convergence will be faster the greater the value of ${\gamma}_t^{\pi}$. Figure 11 below shows the phase portrait of the dynamics of this game in the 2-dimensional simplex.

Fig. 11

Phase portrait of the game SG1, with x₁ = 0.1, x₂ = 0.2, x₃ = 0.3, and ${\upgamma}_{\mathrm{t}}^{\uppi}$= 0.5. Rest points are shown as red circles

For the other extreme value φ = 1, we have the following game (henceforth SG2, for subgame 2):

$$ \left[\begin{array}{ccc}0& {x}_1& 0\\ {}-{x}_2& 0& {x}_2\\ {}0& -{x}_3& 0\end{array}\right] $$

In SG2, strategy 3 is weakly dominated by strategy 1.¹⁵ It is not difficult to prove that the rest points of the replicator dynamics for SG2 are:

• 1. All points in the line s ₂ = 0 (and s ₃ = 1 − s ₁).

• 2. Point: s ₂ = 1. This point is unstable, as it is invadable by strategy 1.

Figure 12 shows the phase portrait of the dynamics of this game in the 2-dimensional simplex.

Fig. 12

Phase portrait of the game SG2, with x₁ = 0.1, x₂ = 0.2, x₃ = 0.3. Rest points are shown in red

Therefore, in terms of our model, when φ = 0, and citizens (within their subsystems) are purely partisans (in the sense that they just care about the rise to prevalence of their utopia, without paying attention to possible opportunistic behaviors by their peers in (1)) then, said subsystem tends (in isolated conditions) to a maximum average degree of citizen contribution. On the contrary, when permeability is absolute (as given by φ = 1 in (1)), then citizens perceive (or take advantage of) possible opportunistic behaviors and the subsystem tends to stabilize (in isolated conditions) in the lowest degree of citizen contribution. Of course, we have a continuum of possibilities between subgames 1 and 2, but we can infer that the lower the value of φ in a subsystem, we should tend to obtain higher average levels of commitment in said subsystem. Likewise, when φ is high, then low levels of commitment in the subsystem, or fluctuating paths driven by the ongoing revision of strategies, are expected. In any case, notice that when we couple the subsystems (considering (1), (2) and (3) together in Section 3), then the shares of the subsystems in society also evolve, and the effect of φ in the payoffs gets mediated by endogenously changing subsystem shares, and intra-subsystem behaviors. This much more complex situation is the one we see below.

1.1.2 Insights on the overall dynamics

Taking into consideration the decomposition shown in the previous section, and assuming φ > 0, note that the dynamics of subsystems with very low share ${\gamma}_t^{\pi}$ are driven by SG2, so in such vanishing subsystems eventually strategy 1 becomes dominant, strategy 3 may hold some minor share, and strategy 2 effectively disappears. In the general case, the dynamics of subsystems with a non-negligible share ${\gamma}_t^{\pi}$ will depend on the value of φ.

1.2 Low values of φ

As pointed out above, in subsystems with low share ${\gamma}_t^{\pi}$, SG2 drives the dynamics, so eventually strategy 1 becomes prevalent, strategy 3 may hold some minor share, and strategy 2 effectively disappears.

In subsystems with high share ${\gamma}_t^{\pi}$, SG1 drives the dynamics, so strategy 3 is clearly favored, and the greater the value of ${\gamma}_t^{\pi}$, the faster the convergence to strategy 3. A greater share s ₃ induces an increase in ${\gamma}_t^{\pi}$, thus creating a self-reinforcing dynamic.

Which particular subsystem(s) will end up with a significant share ${\gamma}_t^{\pi}$ will depend on initial conditions. A high value of ${\gamma}_{t=0}^{\pi}$ and, particularly, a high value of ${s}_{3, t=0}^{\pi}$ will be key. As a representative example, consider Fig. 13, where φ = 0.03.

Fig. 13

Representative example of a situation where two subsystems (Group and Nature) coexist. In these two subsystems s ₃ ≈ 1, whilst in the subsystems that vanish s ₂ ≈ 0 and s ₁ is high

1.3 High values of φ

As in the previous case, in subsystems with low share ${\gamma}_t^{\pi}$, SG2 drives the dynamics. The analysis of the subsystem(s) with significant share ${\gamma}_t^{\pi}$ is more complicated, as both SG1 and SG2 influence the dynamics. As an example, consider the case where φ = 0.8 and there is a subsystem with ${\gamma}_t^{\pi}$ ≈ 1. This game shows cyclic dynamics, as can be seen in Fig. 14 (where x₁ = 0.3, x₂ = 0.45, x₃ = 0.6). Figure 15 shows the overall dynamics of a simulation run where the Market subsystem prevails, and its intra-subsystemic dynamics are cyclic.

Fig. 14

Phase portrait of a game with (x₁, x₂ , x₃) = (0.3, 0.45, 0.6), φ = 0.8 and ${\gamma}_t^{\pi}$ = 1. Rest points are shown as red circles. Rest point (1, 0, 0) has associated eigenvalues −0.33 and 0.06; rest point (0, 1, 0) has associated eigenvalues −0.45 and 0.21; rest point (0, 0, 1) has associated eigenvalues −0.06 and 0.33; and finally, interior rest point (0.472, 0.0833, 0.444) has associated eigenvalues 4.58·10⁻³ ± 0.095i

Fig. 15

Representative example of a situation where only one subsystem survives (Market). The intra-subsystemic dynamics of this subsystem are cyclic

1.4 A final example

In intermediate situations where both SG1 and SG2 play a role in the intra-subsystemic dynamics of some subsystems, the overall dynamics can be very different from the extreme cases outlined above. As a final example, consider a model with x₁ = 0.3, x₂ = 0.45, x₃ = 0.6, φ = 0.16, and two subsystems with ${\gamma}_t^{\pi}$ = 0.5. In this setting, strategy 2 is dominant, and the associated intra-subsystemic dynamics can be seen in Fig. 16.

Fig. 16

Phase portrait of a game with (x₁, x₂ , x₃) = (0.3, 0.45, 0.6), φ = 0.16 and ${\gamma}_t^{\pi}$ = 0.5. Rest points are shown as red circles. Rest point (1, 0, 0) has associated eigenvalues 0.126 and 0.004; rest point (0, 1, 0), which is almost globally stable, has associated eigenvalues −0.054 and −0.036; and finally, rest point (0, 0, 1) has associated eigenvalues −0.126 and 0.038

Figure 17 shows the overall dynamics of a simulation run where the conditions outlined above are approximately met.

Fig. 17

Representative example of a situation where two subsystems (Group and Nature) coexist. In these two subsystems s ₂ ≈ 1, whilst in the subsystems that vanish s ₂ ≈ 0 and s ₁ is high