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

The evolution of conformity in the repeated PGG.

<p>Repeated PGG with <i>n</i> = 10, <i>m</i> = 4 and <i>r</i> = 1.6. <b>(a)</b> Phase portrait of the adaptive dynamics Eq (4). Stable equilibria and unstable equilibria are marked by solid dots and empty dots, respectively.<i>p</i>^* = 1/3 (the blue dash line). A trajectory of Eq (4) starting from (<i>x</i>, <i>p</i>) converges to the stable cooperative equilibrium (1, <i>p</i>) if <i>p</i> > <i>p</i>^*, and converges to the unstable defective equilibrium (0, <i>p</i>) if <i>p</i> < <i>p</i>^*. <b>(b)</b> Monte-Carlo simulation result for a population of size 100. At the beginning of each time step, individuals are randomly divided into 25 groups and play the repeated PGG. In each time step, an average of 10 individuals are chosen to update, where they imitate actions that perform better with a probability proportional to the payoffs obtained in the repeated PGG. With probability 0.1, one of the 100 individuals is chosen to adopt a new strategy (i.e., the average individual mutation rate is 0.001) by adding a small random value (draw from Gaussian noise (0, 0.1)) on its former strategy. Monte-Carlo simulation also shows that conditional altruistic strategies (i.e., <i>x</i> = 1 with large <i>p</i>) and unconditional selfish strategies (i.e., <i>x</i> = 1 with small <i>p</i>) are bistable, and the population oscillates between <i>x</i> = 1 and <i>x</i> = 0.</p

Cross-Interface Emulsification for Generating Size-Tunable Droplets

We report cross-interface emulsification (XiE), a simple method for the generation of monodisperse droplets of controllable volumes from picoliter to nanoliter. A device is set up in which a fused-silica capillary is vibrating across the surface of the continuous phase (mineral oil) in a reservoir, and the flow of the dispersed phase (aqueous solution) in the capillary is segmented into monodisperse droplets at the air/oil interface. We find that the volume of droplets is mainly dominated by the flow rate and vibrating frequency and not significantly influenced by other factors, such as the viscosity of the continuous phase and dispersed phase, the inner diameter of the capillary (20–100 μm), or the shape of the tip (tapered or flat). These features reflect high robustness, flexibility, and precision of XiE for on-demand volume control of droplets. The droplets automatically assemble into planar monolayer droplet arrays (PMDA) in flat-bottomed microwells of 96-well plates, offering excellent convenience for imaging of droplets. As a representative application, we carry out digital loop-mediated isothermal amplification using PMDAs with multivolume droplets for the absolute quantification of nucleic acids. Our results demonstrate that XiE is simple and controllable for the production of monodisperse size-tunable droplets, and it offers opportunities for common laboratories, even without microfabrication facilities, to perform digital quantification, single cell analysis, and other biochemical assays with high throughput

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

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

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

Cross-Interface Emulsification for Generating Size-Tunable Droplets

We report cross-interface emulsification (XiE), a simple method for the generation of monodisperse droplets of controllable volumes from picoliter to nanoliter. A device is set up in which a fused-silica capillary is vibrating across the surface of the continuous phase (mineral oil) in a reservoir, and the flow of the dispersed phase (aqueous solution) in the capillary is segmented into monodisperse droplets at the air/oil interface. We find that the volume of droplets is mainly dominated by the flow rate and vibrating frequency and not significantly influenced by other factors, such as the viscosity of the continuous phase and dispersed phase, the inner diameter of the capillary (20–100 μm), or the shape of the tip (tapered or flat). These features reflect high robustness, flexibility, and precision of XiE for on-demand volume control of droplets. The droplets automatically assemble into planar monolayer droplet arrays (PMDA) in flat-bottomed microwells of 96-well plates, offering excellent convenience for imaging of droplets. As a representative application, we carry out digital loop-mediated isothermal amplification using PMDAs with multivolume droplets for the absolute quantification of nucleic acids. Our results demonstrate that XiE is simple and controllable for the production of monodisperse size-tunable droplets, and it offers opportunities for common laboratories, even without microfabrication facilities, to perform digital quantification, single cell analysis, and other biochemical assays with high throughput

Monte-Carlo simulations for the evolution of conformity in the repeated PD game.

<p>The graphs show two typical simulation runs for a population of size 100. At the beginning of each time step, individuals are randomly divided into 50 pairs and play the repeated PD game. In each time step an average of 10 individuals are chosen to update, where they imitate actions that perform better with a probability proportional to the payoffs obtained in the repeated game (i.e., this updating process can be approximately described by the replicator dynamics [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137435#pone.0137435.ref048" target="_blank">48</a>]). In addition, with probability 0.1, one of the 100 individuals is chosen to adopt a new strategy (i.e., the average individual mutation rate is 0.001) by adding a small random value (draw from Gaussian noise (0, 0.1)) on its former strategy. <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> = 3, <i>P</i> = 1, <i>S</i> = 0 and <i>T</i> = 5. The population oscillates between the cooperative boundary <i>x</i> = 1 and the defective boundary <i>x</i> = 0. <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> = 4, <i>P</i> = 2, <i>S</i> = 0 and <i>T</i> = 5. The population can be stabilized at the cooperative boundary.<p></p

Effect of the fixed cost per individual on the fixation probability under the weak selection.

<p>Four heterogeneous graphs, SW-I (the Small-World graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Watts2" target="_blank">[27]</a> with rewiring probability 0.1), SW-II (the Small-World graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Watts2" target="_blank">[27]</a> with rewiring probability 1), RD (random graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Nobari1" target="_blank">[62]</a>) and SF (scale-free graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Barabsi1" target="_blank">[28]</a>) are used to test the theoretical predictions. Simulation results for the fixation probability of a single cooperator with neighbors in SW-I, SW-II and RD are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figure 4A</a>. For , the harmonic means of the degree distribution are , , in SW-I, , , in SW-II, and , , in RD, respectively. The vertical dash line represents the harmonic mean of degree distribution, , for each of , where the red, green and blue vertical dash lines correspond to SW-I, SW-II and RD, respectively. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figure 4B</a> shows the simulation results for the fixation probability of a single cooperator with neighbors in SF. For , the harmonic means of the degree distribution are , , , respectively. The vertical dash line represents the harmonic mean of degree distribution, , where the blue, green and red vertical dash lines correspond to , respectively. In both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figures 4A and 4B</a>, the -axis denotes the benefit-to-cost ratio, , the -axis the fixation probability, and the horizontal dash-point line denotes the fixation probability of a single cooperator under neutral selection (i.e. ) which is . The fixation probability of a single cooperator is measured using the fraction of runs where cooperators reached fixation out of runs (based on graphs and runs per graph). Both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figures 4A and 4B</a> show that the theoretical predictions present a good approximation to the numerical results.</p

Cross-Interface Emulsification for Generating Size-Tunable Droplets

We report cross-interface emulsification (XiE), a simple method for the generation of monodisperse droplets of controllable volumes from picoliter to nanoliter. A device is set up in which a fused-silica capillary is vibrating across the surface of the continuous phase (mineral oil) in a reservoir, and the flow of the dispersed phase (aqueous solution) in the capillary is segmented into monodisperse droplets at the air/oil interface. We find that the volume of droplets is mainly dominated by the flow rate and vibrating frequency and not significantly influenced by other factors, such as the viscosity of the continuous phase and dispersed phase, the inner diameter of the capillary (20–100 μm), or the shape of the tip (tapered or flat). These features reflect high robustness, flexibility, and precision of XiE for on-demand volume control of droplets. The droplets automatically assemble into planar monolayer droplet arrays (PMDA) in flat-bottomed microwells of 96-well plates, offering excellent convenience for imaging of droplets. As a representative application, we carry out digital loop-mediated isothermal amplification using PMDAs with multivolume droplets for the absolute quantification of nucleic acids. Our results demonstrate that XiE is simple and controllable for the production of monodisperse size-tunable droplets, and it offers opportunities for common laboratories, even without microfabrication facilities, to perform digital quantification, single cell analysis, and other biochemical assays with high throughput