Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society

The evolution of human cooperation has been the subject of much research, especially within the framework of evolutionary public goods games, where several mechanisms have been proposed to account for persistent cooperation. Yet, in addressing this issue, little attention has been given to games of a more adversarial nature, in which defecting players, rather than simply free riding, actively seek to harm others. Here, we develop an adversarial evolutionary game using the specific example of criminal activity, recasting the familiar public goods strategies of punishers, cooperators, and defectors in this light. We then introduce a strategy—the informant—with no clear analog in public goods games and show that individuals employing this strategy are a key to the emergence of systems where cooperation dominates. We also find that a defection-dominated regime may be transitioned to one that is cooperation-dominated by converting an optimal number of players into informants. We discuss these findings, the role of informants, and possible intervention strategies in extreme adversarial societies, such as those marred by wars and insurgencies.

Figure 1

(Color online) Our model society is composed of four citizen types, defined by their propensities to both commit crimes and serve as witnesses in criminal investigations. Arrows between types indicate the number of possible game pairings and outcomes in which the strategy update step leads to a net change from one type to the other (e.g., there are two combinations that convert a villain into a paladin). Circular arrows within each type indicate those strategy updates that keep a player’s strategy unchanged within the given type.

Figure 2

(Color online) There are two qualitatively different equilibrium states available in our model, displayed here for both the [(a) and (b)] stochastic and [(c) and (d)] deterministic formulations. Paladins are shown as (red) circles, apathetics as (blue) squares, informants as (orange) triangles, and villains as (green) diamonds. All simulations use parameters $N=1000$, $M=5$, $\delta =0.3$, $\theta =0.6$, and $ϵ=0.2$. [(a) and (c)] An initial population distribution $P=A=0$, $I=40$, $V=960$ evolves toward utopia, where only noncriminals remain. [(b) and (d)] An initial population distribution $P=400$, $A=I=0$, $V=600$ ends in dystopia, with only villains and apathetics remaining.

Figure 3

(Color online) Even a small number of informants substantially increase the likelihood that a society will evolve toward utopia. Here, we plot the probability of the stochastic system ending in utopia given the initial numbers of paladins $\left(P_0\right)$ and informants $\left(I_0\right)$, assuming zero initial apathetics; higher $I_0$ values lead to greater probability of ending in utopia. Simulations used $N=100$, $M=5$, $\delta =0.3$, $\theta =0.6$, and $ϵ=0.2$, and were run 1000 times for each initial condition.

Figure 4

(Color online) All trajectories with $I_0>0$ end in utopia in the deterministic system. This example phase portrait shows unstable fixed points in light gray (red), unstable fixed lines in thick light gray (red), stable fixed lines in thick dark gray (blue), and trajectories beginning (or ending) along various eigenvectors as thick medium gray (green) arrows. Note that the dystopian fixed point $d$ and saddle point $s$ are unstable to increases in $I$, so that the only attracting final states for $I_0>0$ are those utopias with $P>P_c$. Here, $\delta =0.3$, $\theta =0.6$, and $ϵ=0.2$, but the qualitative results are independent of parameters.

Figure 5

(Color online) An optimal number of villain-to-informant conversions $I_0^{\ast }$ may be made to undermine a dystopian society while minimizing reformation costs. The solid line and points are the averaged simulated costs $C$ versus $I_0$ for 1000 runs of the stochastic system, while the dashed (red) line is the result for the deterministic system. Both systems use parameters $N=100$, $M=5$, $\delta =0.3$, $\theta =0.6$, $ϵ=0.2$, and $\alpha =20$.