An example for the learning-mutation process.

<p>An example for the learning-mutation process in a population of four players from generation to generation . In generation , three players adopt and one adopts a mutant strategy . At the beginning, they are divided into two mini ultimatum games, and , and update their strategies by the quantal response learning ( means proposers and means responders). In the first group, players do not change their original strategies, while in the second group, the proposer will change his strategy to since 0.4 dominates 0.1. Mutations happen after all the pairs reach Nash equilibria. The responder in the first pair mutates to (the red number). As a result of learning and mutation, strategies in generation are , , and .</p

Pairwise invasibility plot.

<p> and are dominant in white and gray regions, respectively. The red dash line denotes and the red point denotes . Every offer lower than 0.5 is dominated by some higher offers and equal or greater than 0.4 is also dominated by lower offers. In particular, dominates almost all lower offers, the only exception is (the red point).</p

Time evolution of the population mean offer.

<p>The population size is 100 and evolves under the learning-mutation process. Mutation rates are taken as (weak mutation rate) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0074540#pone-0074540-g003" target="_blank">Figures 3A and 3B</a> and (intermediate mutation rate) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0074540#pone-0074540-g003" target="_blank">Figure 3C</a>. Mutant strategies follow -distributions, where (local mutation) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0074540#pone-0074540-g003" target="_blank">Figures 3A and 3C</a>, and (global mutation) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0074540#pone-0074540-g003" target="_blank">Figure 3B</a>. Red dash lines denote . In all figures, the population mean offer increases if it is smaller than 0.4 but oscillates if it is between 0.4 to 0.5.</p

Effects of mutations and population sizes on the long-term mean offer.

<p>Mutation rates are taken as (weak mutation rate), (intermediate mutation rate) and (high mutation rate) in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0074540#pone-0074540-g003" target="_blank">Figures 3A, 3B and 3C</a>, respectively. The long-term mean offer depends significantly on the mutation rate and the mutation range, but is robust to different population sizes.</p

The vertices and degrees in a heterogeneous graph.

<p>The vertices are labeled vertex 1, vertex 2, , vertex . The degree of vertex is , and where is the number of -neighbors and the number of -neighbors.</p

Phase portrait of the adaptive dynamics Eqs (1) and (2).

<p>For each of the three graphs, there is a curve <i>p</i> = <i>p</i>^*(<i>x</i>) (the blue dash curve) separating the (<i>x</i>, <i>p</i>)-plane such that <i>dx</i>/<i>dt</i> > 0 for <i>p</i> > <i>p</i>^*(<i>x</i>) and <i>dx</i>/<i>dt</i> < 0 for <i>p</i> < <i>p</i>^*(<i>x</i>). Stable equilibria and unstable equilibria of the adaptive dynamics are marked by solid dots and empty dots, respectively. Trajectories with large initial <i>p</i> converge to <i>x</i> = 1, and with small initial <i>p</i> converge to <i>x</i> = 0. <b>(a)</b> Repeated PD game with </p><p></p><p></p><p></p><p><mi>n</mi><mo>¯</mo></p><mo>=</mo><mn>6</mn><p></p><p></p><p></p>, <i>R</i> = 4, <i>P</i> = 2, <i>S</i> = 0 and <i>T</i> = 5. <b>(b)</b> Repeated PD game with <p></p><p></p><p></p><p><mi>n</mi><mo>¯</mo></p><mo>=</mo><mn>6</mn><p></p><p></p><p></p>, <i>R</i> = 3, <i>P</i> = 1, <i>S</i> = 0 and <i>T</i> = 4. Because <i>R</i> + <i>P</i> = <i>S</i> + <i>T</i>, there exists a critical <i>p</i>^* = 0.1, where a trajectory of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137435#pone.0137435.e009" target="_blank">2</a>) starting from (<i>x</i>, <i>p</i>) converges to (1, <i>p</i>) if <i>p</i> > <i>p</i>^*, and converges to (0, <i>p</i>) if <i>p</i> < <i>p</i>^*. <b>(c)</b> Repeated PD game with <p></p><p></p><p></p><p><mi>n</mi><mo>¯</mo></p><mo>=</mo><mn>6</mn><p></p><p></p><p></p>, <i>R</i> = 3, <i>P</i> = 1, <i>S</i> = 0 and <i>T</i> = 5.<p></p

Evolution of Cooperation in a Heterogeneous Graph: Fixation Probabilities under Weak Selection

<div><p>It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.</p></div

Evolution of Conformity in Social Dilemmas

<div><p>People often deviate from their individual Nash equilibrium strategy in game experiments based on the prisoner’s dilemma (PD) game and the public goods game (PGG), whereas conditional cooperation, or conformity, is supported by the data from these experiments. In a complicated environment with no obvious “dominant” strategy, conformists who choose the average strategy of the other players in their group could be able to avoid risk by guaranteeing their income will be close to the group average. In this paper, we study the repeated PD game and the repeated <i>m</i>-person PGG, where individuals’ strategies are restricted to the set of conforming strategies. We define a conforming strategy by two parameters, initial action in the game and the influence of the other players’ choices in the previous round. We are particularly interested in the tit-for-tat (TFT) strategy, which is the well-known conforming strategy in theoretical and empirical studies. In both the PD game and the PGG, TFT can prevent the invasion of non-cooperative strategy if the expected number of rounds exceeds a critical value. The stability analysis of adaptive dynamics shows that conformity in general promotes the evolution of cooperation, and that a regime of cooperation can be established in an AllD population through TFT-like strategies. These results provide insight into the emergence of cooperation in social dilemma games.</p></div

Effect of graph heterogeneity on the fixation probability under weak selection and fixed cost per game.

<p>Simulation results for the fixation probability in random graphs and scale-free graphs with different heterogeneities are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A and 3B</a>, respectively (see the generation of these graphs in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560.s001" target="_blank">Supporting Information S1</a>). In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A</a>, and . The benefit-to-cost ratio is taken as 16, 20, 24, 28 and 32, respectively, and the selection intensity is . In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3B</a>, and . The benefit-to-cost ratio is taken as 5, 10, 15, 20 and 25, respectively, and the selection intensity is . In both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figures 3A and 3B</a>, the -axis denotes and the -axis the fixation probability of a single cooperator with neighbors. The fixation probability is measured using the fraction of runs where cooperators reached fixation out of runs. The dash line in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A</a> and the dash line in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3B</a> denote respectively the fixation probability under neutral selection (). Both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figures 3A and 3B</a> show the tendency that for all different values of , fixation probability approaches that of neutral selection as increases.</p

Evolutionary stabilities for TFT in the repeated PD game and the repeated PGG.

<p><b>(a)</b> PD game with payoffs <i>R</i> = 3, <i>P</i> = 1, <i>S</i> = 0 and 3 < <i>T</i> < 5. A TFT population can prevent the invasion of any non-cooperative strategy if </p><p></p><p></p><p></p><p><mi>n</mi><mo>¯</mo></p><mo>></mo>max<mo>{</mo><p></p><p><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><mi>P</mi><mo stretchy="false">)</mo></p><mo>/</mo><p><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><mi>P</mi><mo stretchy="false">)</mo></p><p></p><mo>,</mo><p></p><p><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><mi>S</mi><mo stretchy="false">)</mo></p><mo>/</mo><p><mo stretchy="false">(</mo><mn>2</mn><mi>R</mi><mo>−</mo><mi>S</mi><mo>−</mo><mi>T</mi><mo stretchy="false">)</mo></p><p></p><mo>}</mo><p></p><p></p><p></p> (the blue region). <b>(b)</b> 4-person PGG with 1 < <i>r</i> < 4. A TFT population can prevent the invasion of any non-cooperative strategy if <p></p><p></p><p><mi>n</mi><mo>¯</mo></p><p></p><p></p> is above the blue curve (the blue region).<p></p