Evolutionary dynamics of populations with conflicting interactions: Classification and analytical treatment considering asymmetry and power

Evolutionary game theory has been successfully used to investigate the dynamics of systems, in which many entities have competitive interactions. From a physics point of view, it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur, be it spins, particles, bacteria, animals, or humans. Here, we analyze the case, where the entities are heterogeneous, particularly the case of two populations with conflicting interactions and two possible states. For such systems, explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues, which determine their stability. In this way, four different types of system dynamics can be classified and the various kinds of phase transitions between them will be discussed. While these results are interesting from a physics point of view, they are also relevant for social, economic, and biological systems, as they allow one to understand conditions for (1) the breakdown of cooperation, (2) the coexistence of different behaviors (“subcultures”), (3) the evolution of commonly shared behaviors (“norms”), and (4) the occurrence of polarization or conflict. We point out that norms have a similar function in social systems that forces have in physics.

Figure 1

(Color online) Illustration of the outcomes of symmetrical $2\times 2$ games as a function of the payoff-dependent parameters $b_a=B$ and $c_a=C$ if $f=0.8$ (i.e., 80% of individuals belong to population 1) and if the entities interact within their own population, but different populations do not have any interactions between each other $\left(B_a=0=C_a\right)$ [4, 23]. $p=p_1^1$ is the fraction of entities of population 1 in state 1 and $q=p_2^2$ the fraction of entities of population 2 in state 2. The vector fields show $\left(dp/dt,dq/dt\right)$, i.e., the direction and size of the expected change of the distribution $\left(p,q\right)$ of states with time $t$. Sample trajectories illustrate some representative flow lines $\left[p\left(t\right),q\left(t\right)\right]$ as time $t$ passes. The flow lines move away from unstable stationary points (empty circles). Saddle points (crosses) are attractive in one direction, but repulsive in another. Stable stationary points (black circles) attract the flow lines from all directions. Each color (gray shade) represents one basin of attraction. It subsumes all initial conditions $\left[p\left(0\right),q\left(0\right)\right]$ leading to the same stationary point [$\text{yellow}=\left(1,1\right)$, $\text{green}=\left(1,0\right)$, $\text{blue}=\left(0,1\right)$, $\text{red}=\left(0,0\right)$, and $\text{turquoise}=\left(p_0,p_0\right)$ with $p_0=\left|B\right|/\left(\left|B\right|+\left|C\right|\right)$]. Solid red lines indicate the thresholds at which continuous (second-order) phase transitions take place, i.e., at which the system behavior changes qualitatively (characterized by the appearance or disappearance of stationary points), while the stable stationary points change continuously when the parameters are varied. Dashed lines indicate an abrupt change of a stable stationary point, i.e., a discontinuous (first-order) phase transition. For multi-population prisoner's dilemmas (MPD), we have $B<0$ and $C<0$, and the final outcome is $\left(p,q\right)=\left(0,0\right)$. For multi-population snowdrift games (MSD), we have $B>0$ and $C<0$, and the stable stationary solution corresponds to a coexistence of a fraction $p_0=\left|B\right|/\left(\left|B\right|+\left|C\right|\right)$ of entities in one state and a fraction $1-p_0$ of entities in the other. For multi-population harmony games (MHG), we have $B>0$ and $C>0$, and the eventually resulting outcome is (1,1). Finally, for multi-population stag hunt games (MSH), we have $B<0$ and $C>0$, and there is a bistable situation, i.e., it depends on the initial fraction of entities in a state, whether everybody ends up in this state or in the other one [23].

Figure 2

(Color online) Illustration of the outcomes as a function of the payoff-dependent parameters $B_a=B$ and $C_a=C$ if $f=0.8$ (i.e., 80% of the entities belong to population 1) and if the entities do not interact within their own population $\left(b_a=0=c_a\right)$, while entities belonging to different populations have interactions with each other [4]. Small arrows illustrate again the vector field $\left(dp/dt,dq/dt\right)$ as a function of $p=p_1^1$ and $q=p_2^2$. Black circles represent stable fix points, empty circles stand for unstable fix points, and crosses represent saddle points. The basins of attraction of different stable fix points are represented in different gray shades (colors) [$\text{yellow}=\left(1,1\right)$, $\text{green}=\left(1,0\right)$, $\text{blue}=\left(0,1\right)$, and $\text{red}=\left(0,0\right)$]. Solid red lines indicate the thresholds at which continuous phase transitions take place, dashed lines indicate discontinuous phase transitions. For MPDs, we have $B<0$ and $C<0$, for MSDs, we have $B>0$ and $C<0$, for MHGs, we have $B>0$ and $C>0$, and for MSH, we have $B<0$ and $C>0$.

Figure 3

(Color online) Illustration of the parameter-dependent types of outcomes as a function of the payoff-dependent parameters $B_a=b_a=B$ and $C_a=c_a=C$ if $f=0.8$ (i.e., 80% of the entities belong to population 1) and if the entities have interactions with other entities, independently of the population they belong to. This corresponds to the multipopulation case with interactions and self-interactions. Small arrows illustrate the vector field $\left(dp/dt,dq/dt\right)$ as a function of $p$ and $q$. Empty circles stand for unstable fix points (repelling neighboring trajectories), black circles represent stable fix points (attracting neighboring trajectories), and crosses represent saddle points (i.e., they are attractive in one direction and repulsive in the other). The basins of attraction of different stable fix points are represented in different shades of gray (colors) [$\text{red}=\left(0,0\right)$, $\text{green}=\left(1,0\right)$, $\text{blue}=\left(0,1\right)$, $\text{yellow}=\left(1,1\right)$, $\text{salmon}=\left(u,0\right)$, and $\text{mustard}=\left(v,1\right)$, where $0<u,v<1$]. Solid red lines indicate the thresholds at which continuous phase transitions take place and dashed lines indicate discontinuous phase transitions. For MPDs, we have $B<0$ and $C<0$, for MSDs, we have $B>0$ and $C<0$, for MHGs, we have $B>0$ and $C>0$, and for MSH, we have $B<0$ and $C>0$.

Figure 4

(Color online) Illustration of the parameter-dependent types of outcomes in the multipopulation stag-hunt game if $\left|C\right|/\left|B\right|$ and/or $f$ are varied and interaction between populations as well as self-interactions is considered. The representation and gray shades (colors) are the same as in Fig. 3. Solid red lines indicate the thresholds at which continuous phase transitions take place and dashed lines indicate discontinuous phase transitions.

Figure 5

(Color online) Illustration of the parameter-dependent types of outcomes in the multipopulation snowdrift game if $\left|C\right|/\left|B\right|$ and/or $f$ are varied and interaction between populations as well as self-interactions are considered. The representation and gray shades (colors) are the same as in Fig. 3.