Cooperation, Norms, and Revolutions: Evolutionary Dynamics of Populations with Conflicting Interactions

Movie 1:

Vector fields (small arrows) and phase diagrams (colored areas) for two interacting populations with incompatible preferences (conflicting interactions), when population 1 is more powerful than population 2 (f = 0.8), i.e. population 1 is assumed to be more powerful. The movie shows the situation for the two-population snow- drift game (first half of the movie) and the two-population stag hunt game (second half), when in-group interactions are considered, while interactions between populations are neglected (ba = b, ca = c, Ba = 0 = Ca). Therefore, the dynamics in each population is independent of the dynamics in the other population. The size of the parameters B and C is varied according to the relation C = -B3. This serves to demonstrate the parameter-dependence of the fix points and dynamics of both games. The small moving dots illustrate trajectories. One can clearly see the discontinuous transitions in the system behavior when one of the parameters B, C, or 1 -|B|∕|C| changes its sign.

In the snowdrift game, we find a stable fraction p0 = |B|∕(|B| + |C|) of cooperative individuals in each population, i.e. p = p0 = q. This stationary fix point corresponds to the large black circle moving along the diagonal line. In the stag hunt game, the fix point located on the diagonal line is unstable (see empty circle). Therefore, trajectories move away from it. If the fraction of cooperative individuals in a population is larger than p0, it will grow further, otherwise it will continuously shrink. That is, each population will either end up with 0% or 100% cooperative individuals, depending on the initial conditions. Therefore, 22 = 4 stable fix points are possible - one in each corner.

Further details: p is the fraction of individuals in population 1 showing their preferred, cooperative behavior 1, and q is the fraction of cooperative individuals in population 2 showing their preferred behavior 2. A fraction 1-q of individuals in population 2 shows the non-preferred behavior 1, and a fraction 1-p of individuals in population 1 shows behavior 2. The vector fields displays (dpdt, dqdt), i.e. the direction and size of the expected temporal change of the behavioral distribution, if the fractions of cooperative individuals in populations 1 and 2 are p(t) and q(t). Trajectories are representative flow lines (p(t), q(t)) as time t passes. The flow lines move away from unstable stationary points (empty circles) and are attracted towards stable stationary points (black circles). The colored areas represent the basins of attraction, i.e. all initial conditions (p(0), q(0)) leading to the same fix point [red = (0,0), yellow = (1,1), blue = (0,1), green = (1,0), salmon = (u, 0), mustard = (v, 1), other colors = (u, v), with 0 < u, v < 1]. Saddle points (crosses) are attractive in one direction, but repulsive in another.

Movie 2:

Same as Movie 1, but while interactions between both populations are considered, self-interactions are neglected (ba = 0 = ca, Ba = B, Ca = C). The contrast to Movie 1 is pronounced: In the snowdrift game (first half of the movie), everybody is either cooperative or non-cooperative in both populations now, corresponding to the stable fix points at (0,0) and (1,1) (see black circles). In contrast, in the stag hunt game (second half of the movie), the evolutionary equilibria are located at (p, q) = (1, 0) and (p, q) = (0, 1). p = 1 means that 100% of the individuals in population 1 show behavior 1, while q = 0 implies that 0% of the individuals in in population 2 show behavior 2 (i.e. all of them show behavior 1 as well). Therefore, we find the establishment of a commonly shared behavior (the formation of a behavioral norm).

Movie 3:

Same as Movie 1, but considering both, interactions within and between the two populations. Assuming no difference between in-group and out-group interactions, we have ba = Ba = B and ca = Ca = C. While the multi-population stag hunt game (first half of the movie) shows a tendency to establish a commonly shared behavior ("behavioral norm"), the snowdrift game (second half) rather delineates situations of conflict between both populations. It is known that conflicts between two populations may sometimes cause a "revolution". According to our interpretation, this corresponds to the discontinuous transition of the evolutionary equilibrium, when the background color turns from salmon to mustard. The abrupt change of the q-coordinate from 0 to 1 means that all individuals in the weaker population show the non-preferred behavior before the revolution, but their preferred behavior afterwards. The discontinuous transition occurs, when |B| and |C| in the multi-population snowdrift game become the same. (Note that there is no such revolutionary transition, when individuals have compatible preferences.)

The dynamics for two interacting populations without self-interactions is clearly less differentiated (see Movie 2). In particular, Movie S2 shows no revolutionary transition in the snowdrift game. It also lacks cases where the phase diagram of the stag hunt game displays three different basins of attraction at the same time, corresponding to a coexistence of three stable fix points. While two of them correspond to the establishment of a commonly shared behavior (a behavioral norm), the third point represents the formation of different behaviors (separate "subcultures") in each population.

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