Instability in Evolutionary Games

<div><h3>Background</h3><p>Phenomena of instability are widely observed in many dissimilar systems, with punctuated equilibrium in biological evolution and economic crises being noticeable examples. Recent studies suggested that such instabilities, quantified by the abrupt changes of the composition of individuals, could result within the framework of a collection of individuals interacting through the prisoner's dilemma and incorporating three mechanisms: (i) imitation and mutation, (ii) preferred selection on successful individuals, and (iii) networking effects.</p> <h3>Methodology/Principal Findings</h3><p>We study the importance of each mechanism using simplified models. The models are studied numerically and analytically via rate equations and mean-field approximation. It is shown that imitation and mutation alone can lead to the instability on the number of cooperators, and preferred selection modifies the instability in an asymmetric way. The co-evolution of network topology and game dynamics is not necessary to the occurrence of instability and the network topology is found to have almost no impact on instability if new links are added in a global manner. The results are valid in both the contexts of the snowdrift game and prisoner's dilemma.</p> <h3>Conclusions/Significance</h3><p>The imitation and mutation mechanism, which gives a heterogeneous rate of change in the system's composition, is the dominating reason of the instability on the number of cooperators. The effects of payoffs and network topology are relatively insignificant. Our work refines the understanding on the driving forces of system instability.</p> </div

The changes of (a) the average number of cooperators and (b) the average system payoff versus parameters and .

<p>Other parameters are fixed as and . The simulation lasts for time steps. The black squares and red circles represent the cases of and , respectively.</p

The distributions for different .

<p>With and , subgraphs (a) and (b) respectively show the time distributions for and . The simulations last for time steps. Distributions for different overlap each other, implying that the number of links in the network has no influence on the prosperity of cooperation and the system instability.</p

Transitions between extreme states consisting of all cooperators and all defectors.

<p>The system size is and the mutation rate is . The simulation was carried out for time steps. Each data point is an average over time steps.</p

Effects of payoff parameter on .

<p>Number of cooperators as a function of time for (a) and (b) . The data are obtained from simulations and the lines come from numerical solutions. The parameters are , and . Each data point represents an average over 1000 time steps. (c) Distributions for different values of . The data points are simulation results and the lines are analytic results.</p

Simulation and analytic results of the distribution of the number of cooperators for large values of .

<p>The system size is and the mutation rates are and , respectively. Simulation lasts for time steps while the analytical solution is presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0049663#pone.0049663.e197" target="_blank">Eq. (8)</a>.</p

Number of cooperators shown together with (a) the instantaneous average degree of the network and (b) the number of disjoint components in the network.

<p>The parameters are , , and . <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0049663#s2" target="_blank">Results</a> are shown for the early stage. Each data point represents an average over 100 time steps.</p

Illustration of how to quantify the strength of instability.

<p>We set and . In this specific example, , contributed by 5 transitions, 3 drops and 5 raises. Transitions, drops and raises are labelled by , and in the plot.</p

How the strength of instability changes with the threshold for different and .

<p>The parameter has almost no impact on the strength of instability and the plot (a) compares two examples and , meanwhile , and are fixed. Inset of the plot (a) displays the same curves in log-linear scale. The plot (b) shows the considerable effects of on the strength of instability. In fact, is a borderline: when the tails of curves will decay quickly for large , namely the change of the composition of individuals is less drastic, while if , the strength of instability decreases as the increase of , which is of the similar varying tendency to the system payoff. Other parameters are , and . All simulations lasts for time steps.</p

Number of cooperators as a function of time with preferential selection.

<p>The parameters are , , and . The simulation was carried out for time steps. Each data point is an average over time steps. The system spends most of the time in a state of all cooperators, interrupted by instabilities that last for a short duration when defectors suddenly appear. This is analogous to the coexistence of prosperity and instability as observed in the model of Cavaliere <i>et al. </i><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0049663#pone.0049663-Cavaliere1" target="_blank">[35]</a>.</p