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Aspiration dynamics in structured population acts as if in a well-mixed one.

Du J, Wu B, Wang L - Sci Rep (2015)

Bottom Line: Studies via simulations show that, instead of comparison rules, self-evaluation driven updating rules may explain why spatial structure does not alter the evolutionary outcome.Though inspiring, there is a lack of theoretical result to show the existence of such evolutionary updating rule.Although how humans update their strategies is an open question to be studied, our results provide a theoretical foundation of the updating rules that may capture the real human updating rules.

View Article: PubMed Central - PubMed

Affiliation: Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China.

ABSTRACT
Understanding the evolution of human interactive behaviors is important. Recent experimental results suggest that human cooperation in spatial structured population is not enhanced as predicted in previous works, when payoff-dependent imitation updating rules are used. This constraint opens up an avenue to shed light on how humans update their strategies in real life. Studies via simulations show that, instead of comparison rules, self-evaluation driven updating rules may explain why spatial structure does not alter the evolutionary outcome. Though inspiring, there is a lack of theoretical result to show the existence of such evolutionary updating rule. Here we study the aspiration dynamics, and show that it does not alter the evolutionary outcome in various population structures. Under weak selection, by analytical approximation, we find that the favored strategy in regular graphs is invariant. Further, we show that this is because the criterion under which a strategy is favored is the same as that of a well-mixed population. By simulation, we show that this holds for random networks. Although how humans update their strategies is an open question to be studied, our results provide a theoretical foundation of the updating rules that may capture the real human updating rules.

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Simulations for aspiration dynamics confirm the criterion a + b > c+d.We study a payoff matrix of a = 1, −1 ≤ b ≤ 1, −0.5 ≤ c ≤ 1.5, and d = 0. The change trends of the abundance of strategy A for different structures are shown. According to the linear inequality σ a + b > c + σ d in [43], the equilibrium condition is σ = c − b, which is shown as the fitting (red dash dot) line in each panel. Below the line strategy A is favored. For the structures considered, the simulation results fit for the theoretical prediction σ = 1. (a) A well-mixed population with N = 8. (b) A cycle with N = 8. (c) A regular graph with k = 3 and N = 8. (d) A lattice with k = 4 and N = 9. (e) A star with N = 8. (f) A random graph with N = 8 and average degree . For all the simulations, we use selection intensity ω = 0.01. Each point is an average over 2 × 108 runs.
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f2: Simulations for aspiration dynamics confirm the criterion a + b > c+d.We study a payoff matrix of a = 1, −1 ≤ b ≤ 1, −0.5 ≤ c ≤ 1.5, and d = 0. The change trends of the abundance of strategy A for different structures are shown. According to the linear inequality σ a + b > c + σ d in [43], the equilibrium condition is σ = c − b, which is shown as the fitting (red dash dot) line in each panel. Below the line strategy A is favored. For the structures considered, the simulation results fit for the theoretical prediction σ = 1. (a) A well-mixed population with N = 8. (b) A cycle with N = 8. (c) A regular graph with k = 3 and N = 8. (d) A lattice with k = 4 and N = 9. (e) A star with N = 8. (f) A random graph with N = 8 and average degree . For all the simulations, we use selection intensity ω = 0.01. Each point is an average over 2 × 108 runs.

Mentions: We analytically derive a unified condition, a + b > c + d, for one strategy to be favored over the other in regular graphs, which is the same with that in a well-mixed population. The analytical derivation process is detailed in Methods and Supplementary Information. Further by simulation, we verify that under the limit of weak selection, the criteria of strategy dominance for aspiration dynamics with various population structures are the same (see Fig. 2). Particularly, this criterion is the well-known condition for risk dominance. Thus the risk-dominant strategy which has the bigger basin of attraction always dominates the population for any population structures. For random networks, we maintain that the criterion holds for different population sizes and average degrees via simulation. The results are depicted in Fig. 3.


Aspiration dynamics in structured population acts as if in a well-mixed one.

Du J, Wu B, Wang L - Sci Rep (2015)

Simulations for aspiration dynamics confirm the criterion a + b > c+d.We study a payoff matrix of a = 1, −1 ≤ b ≤ 1, −0.5 ≤ c ≤ 1.5, and d = 0. The change trends of the abundance of strategy A for different structures are shown. According to the linear inequality σ a + b > c + σ d in [43], the equilibrium condition is σ = c − b, which is shown as the fitting (red dash dot) line in each panel. Below the line strategy A is favored. For the structures considered, the simulation results fit for the theoretical prediction σ = 1. (a) A well-mixed population with N = 8. (b) A cycle with N = 8. (c) A regular graph with k = 3 and N = 8. (d) A lattice with k = 4 and N = 9. (e) A star with N = 8. (f) A random graph with N = 8 and average degree . For all the simulations, we use selection intensity ω = 0.01. Each point is an average over 2 × 108 runs.
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4306144&req=5

f2: Simulations for aspiration dynamics confirm the criterion a + b > c+d.We study a payoff matrix of a = 1, −1 ≤ b ≤ 1, −0.5 ≤ c ≤ 1.5, and d = 0. The change trends of the abundance of strategy A for different structures are shown. According to the linear inequality σ a + b > c + σ d in [43], the equilibrium condition is σ = c − b, which is shown as the fitting (red dash dot) line in each panel. Below the line strategy A is favored. For the structures considered, the simulation results fit for the theoretical prediction σ = 1. (a) A well-mixed population with N = 8. (b) A cycle with N = 8. (c) A regular graph with k = 3 and N = 8. (d) A lattice with k = 4 and N = 9. (e) A star with N = 8. (f) A random graph with N = 8 and average degree . For all the simulations, we use selection intensity ω = 0.01. Each point is an average over 2 × 108 runs.
Mentions: We analytically derive a unified condition, a + b > c + d, for one strategy to be favored over the other in regular graphs, which is the same with that in a well-mixed population. The analytical derivation process is detailed in Methods and Supplementary Information. Further by simulation, we verify that under the limit of weak selection, the criteria of strategy dominance for aspiration dynamics with various population structures are the same (see Fig. 2). Particularly, this criterion is the well-known condition for risk dominance. Thus the risk-dominant strategy which has the bigger basin of attraction always dominates the population for any population structures. For random networks, we maintain that the criterion holds for different population sizes and average degrees via simulation. The results are depicted in Fig. 3.

Bottom Line: Studies via simulations show that, instead of comparison rules, self-evaluation driven updating rules may explain why spatial structure does not alter the evolutionary outcome.Though inspiring, there is a lack of theoretical result to show the existence of such evolutionary updating rule.Although how humans update their strategies is an open question to be studied, our results provide a theoretical foundation of the updating rules that may capture the real human updating rules.

View Article: PubMed Central - PubMed

Affiliation: Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China.

ABSTRACT
Understanding the evolution of human interactive behaviors is important. Recent experimental results suggest that human cooperation in spatial structured population is not enhanced as predicted in previous works, when payoff-dependent imitation updating rules are used. This constraint opens up an avenue to shed light on how humans update their strategies in real life. Studies via simulations show that, instead of comparison rules, self-evaluation driven updating rules may explain why spatial structure does not alter the evolutionary outcome. Though inspiring, there is a lack of theoretical result to show the existence of such evolutionary updating rule. Here we study the aspiration dynamics, and show that it does not alter the evolutionary outcome in various population structures. Under weak selection, by analytical approximation, we find that the favored strategy in regular graphs is invariant. Further, we show that this is because the criterion under which a strategy is favored is the same as that of a well-mixed population. By simulation, we show that this holds for random networks. Although how humans update their strategies is an open question to be studied, our results provide a theoretical foundation of the updating rules that may capture the real human updating rules.

Show MeSH
Related in: MedlinePlus