Limit cycles and the benefits of a short memory in rock-paper-scissors games

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last $m$ interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.

Figure 1

Domains in which each of the three strategies are assessed to be the best.

Figure 2

Probability weights of rock (open circles), scissors (dots), and paper (squares) agents in a group of size $L=100$ with $\alpha =2,\beta =\gamma =1$. All agents have memory $m=100$ and update rate $\epsilon =2.5\times {10}^{-4}$. Dashed lines show solutions to delay Eqs. (16) and (17) using the same parameter values.

Figure 3

Probability weights of rock (open circles), scissors (dots), and paper (squares) agents in a group of size $L=100$ with $\alpha =2,\beta =\gamma =1$. All agents have memory $m=100$ and update rate $\epsilon ={10}^{-3}$. Dashed lines show solutions to delay Eqs. (16) and (17) using the same parameter values.

Figure 4

Dashed line shows the rock weight in a group of size $L=100$ with $\alpha =2,\beta =\gamma =1$. All agents have memory $m=100$ and update rate $\epsilon ={10}^{-3}$. Open circles show binomial logistic regression estimates of the rock fraction made using the memory of a single randomly selected agent at a sequence of regularly spaced intervals. Black dots show similar estimates made by simply computing the fraction of rock interactions in the agent's memory.

Figure 5

Given a sequence ${\left\{X_k\right\}}_{k=1}^m$ where $m=100$, of Bernoulli random variables with changing success probability $p\left(k\right)=p_0+(p_1-p_0)k/m$, where $p_0=0.2,p_1=0.25$. Black dots show the estimated probability density function of the logistic regression estimator ${\widehat{p}}_1$, having mean 0.256 and root mean squared error 0.07. Open circles show the estimated probability distribution of the number of successes in the sequence, serving as an alternative estimator for $p_1$, having mean 0.225 and root mean squared error 0.07.

Figure 6

Populations of $m=10$ (open circles) and $m=100$ (dots) agents in system of size $L=200$ with update rate $\epsilon =0.01$ initially composed of 199 short memory agents and a single long memory agent. Game parameters are $\alpha =2,\beta =\gamma =1$ and population dynamics parameters are $\rho =\kappa ={10}^{-3}$. Also shown in fraction of paper agents (black line), illustrating how the onset of oscillations coincides with stable equilibrium between memory lengths.

Figure 7

Bold line shows $r\left(t\right)$ from the numerical solutions to Eqs. (16) and (17) in the symmetric case when $m=100$ and $\epsilon =3\times {10}^{-4}$. Thin line shows corresponding solution when $\epsilon =5\times {10}^{-4}$. Open circles and black dots show corresponding simulation results in a system of size $L=1000$.

Figure 8

Dependence of the steady state standard deviation $\sigma$ of $r\left(t\right)$ on $\epsilon$ for $m=20$ (filled markers) and $m=80$ (open markers). Three different combinations of payoff values $(\alpha ,\beta ,\gamma )$ were used: $(1,1,1)$ [circles], $(1.5,2.5,1)$ [squares], and $(2,1,1)$ [triangles]. The vertical dotted line has horizontal coordinate $32\pi /27$, corresponding to the critical value of $\epsilon m^2$ in the symmetric game. $L=100$ in all cases.