Synchronization and extinction in cyclic games with mixed strategies

We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing $\beta$ balls of three and four types, respectively, where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of $\beta$, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.

Figure 1

Time-dependent densities for a system of two agents obtained through numerical integration of Eqs. (11) (solid black line, agent 1; green dashed line, agent 2) and compared with results from stochastic simulations with $\beta ={10}^7$ (red circles, agent 1; blue squares, agent 2). The same initial conditions $A_1\left(0\right)=0.8,B_1\left(0\right)=0.1,A_2\left(0\right)=0.1$, and $B_2\left(0\right)=0.8$ are used for both methods. In (d) the symbols represent the absolute values of the difference between strategy densities in the simulation data, while the lines are obtained from Eqs. (11). An exponentially fast synchronization is observed. The dot-dashed blue line indicates an exponential of slope $-2$.

Figure 2

Space-time plots for three-strategy games on a ring with different numbers of balls $\beta$ on each site: (a) $\beta =3$, (b) $\beta =4$, (c) $\beta =6$, and (d) $\beta =100$. Time progresses from top to bottom. For (a)–(c) 1000 time steps are shown for a system composed of 1000 sites, whereas for (d) we show 500 time steps for a system with 500 lattice points. To determine the color of a lattice site we use the RGB color model and map the percentage of $A,B$, and $C$ to the percentage of the colors red, green, and blue, respectively. In this scheme a site that contains the same number of balls for all three species is assigned a grayish color.

Figure 3

Average lattice domination time $T$ as a function of the total number of balls $N\beta$ in the system. When changing the number of balls $\beta$ per agent, a transition takes place between (a) an algebraic dependence on the total number of balls in the system for $\beta \le 3$, indicating a neutrally stable system, and (b) an exponential dependence for $\beta \ge 4$, characterizing a stable system. Each data point results from an average over 2000 runs with different realizations of the noise. Error bars are smaller than the symbol sizes.

Figure 4

Spatial covariance at time $t=1000$ for different numbers of balls $\beta$ [increasing from (a) 1 to (f) 6] and different system sizes. The data result from averaging over $500000$ independent runs.

Figure 5

Space-time plots from (a) numerical integration of the spatial mean-field equations (16) and (b) the numerical simulation of a system of 100 lattice sites over 100 time steps with $\beta ={10}^6$ balls at each site. In both cases the same initial condition (17) was used. Time increases in the downward direction. The system rapidly synchronizes and coherent waves are formed.

Figure 6

Space-time plots for four-strategy games on a ring with different numbers of balls $\beta$ on each site: (a) $\beta =1$, (b) $\beta =10$, (c) $\beta =20$, and (d) $\beta =50$. Time progresses from top to bottom. Shown are 1000 time steps for systems composed of 1000 sites. To determine the color of a lattice site we use the RGB color model and map the percentage of $A,B$, and $C$ to the percentage of the colors red, green, and blue, respectively. This is adequate as the percentage of $D$ is readily obtained from the fact that the sum over all four densities is 1.

Figure 7

Time-dependent lengths extracted from the space-time covariance for the four-strategy mixed model on a one-dimensional lattice for different values of $\beta$: (a) $\beta =1$, (b) $\beta =5$, (c) $\beta =15$, and (d) $\beta =30$. The length scale $L_i \left(L_n\right)$ is obtained from the individual (neutral) spatial covariance. In (a) the system size is $N=5000$, whereas in (b)–(d) data for a system of 2500 sites are shown. The data result from averaging over typically a few thousand independent runs. In (a) the dashed line indicates an exponent of 2/3, whereas in (c) and (d) the short red lines indicate an algebraic growth with an exponent of 1/2.

Figure 8

Average lattice domination time as a function of the total number of balls $N\beta$ in the system. Each data point results from an average over 2000 runs, the standard error being smaller than the sizes of the symbols.