Source-linked AI summary
Heterogeneous aspirations promote cooperation in the prisoner's dilemma game
Matjaz Perc, Zhen Wang
TL;DR
The paper asks how structured populations should select strategy sources when neighboring players differ in performance, and whether aspiration heterogeneity affects cooperation. It models aspiration strength and prevalence with u and v, tests robustness across uncertainty and networks, and finds that cooperation is strongest with appropriately tuned heterogeneity, including an intermediate v ≈0.5 for large u.
Problem
The study addresses how aspiration-driven selection among neighboring strategy sources influences cooperation in the evolutionary prisoner’s dilemma.
Method
The model varies aspiration parameter u and the fraction v of type A players, tests different uncertainty levels and interaction networks, and compares results with evolving individual aspirations.
Results
For u = 1 and v ≈0.5, the fraction of cooperators rises from 0.18 to 0.87 and the critical cost-to-benefit ratio rises from rc = 0.022 to 0.31.
Takeaways & Limitations
Heterogeneous aspiration to the fittest can promote cooperation across different interaction networks and levels of stochasticity, while an intermediate aspiration level can emerge through natural selection.
Takeaways & Limitations
The baseline prisoner’s dilemma simulations use a specific payoff parametrization and examine thresholds that depend on the interaction network, adoption rule, and simulation details.
Abstract
from arXiv · showhide
To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter $u$ that interpolates between these two extreme options. Setting $u$ equal to zero returns the random selection of the opponent, while positive $u$ favor the fitter players. In addition, we divide the population into two groups. Players from group $A$ select their opponents as dictated by the parameter $u$, while players from group $B$ do so randomly irrespective of $u$. We denote the fraction of players contained in groups $A$ and $B$ by $v$ and $1-v$, respectively. The two parameters $u$ and $v$ allow us to analyze in detail how aspirations in the context of the prisoner's dilemma game influence the evolution of cooperation. We find that for sufficiently positive values of $u$ there exist a robust intermediate $v \approx 0.5$ for which cooperation thrives best. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process $K$ and for different interaction networks. We also provide complete phase diagrams depicting the dependence of the impact of $u$ and $v$ for different values of $K$, and contrast the validity of our conclusions by means of an alternative model where individual aspiration levels are subject to evolution as well. Our study indicates that heterogeneity in aspirations may be key for the sustainability of cooperation in structured populations.
Introduction
The paper examines how aspiration-driven strategy copying and heterogeneous player types affect cooperation in structured prisoner’s dilemma populations. It proposes tuning both the strength and prevalence of aspirations to identify conditions under which cooperation thrives.
- Model design: The study introduces type A players who follow aspiration parameter u and type B players who choose potential role models randomly.The fractions of types A and B are v and 1−v, respectively.
- Motivation: The approach is motivated by prior evidence that heterogeneity in networks and evolving interaction structures can promote cooperative behavior.Examples include small-world, random regular, scale-free, adaptive, and growing networks.
- Research question: Parameter u controls whether players copy randomly selected neighbors or favor fitter neighbors, while v controls how widely aspiration-based selection is used.The two parameters jointly characterize the strength and prevalence of heterogeneous aspirations.
- Research question: Strong aspirations among most players can harm cooperation, whereas appropriately distributed heterogeneous aspirations may eliminate defectors.The paper tests these effects across interaction networks and strategy-adoption uncertainty levels.
- Model comparison: The study also contrasts its population-level aspiration design with a coevolutionary model in which individual aspiration levels evolve through natural selection.This comparison examines whether suitable aspiration levels can emerge rather than being imposed externally.
Results
Cooperation is most strongly promoted when aspiration-driven strategy sourcing is combined with random sourcing for an intermediate share of players. This effect remains under varied networks and adoption uncertainty, while coevolutionary aspirations can converge toward an intermediate level.
- Parameter effects: For large u, cooperation is favored when approximately half the players aspire to their fittest neighbors and the remainder choose role models randomly.For low u, both cooperation and the critical cost-to-benefit ratio increase monotonically with v.
- Parameter effects: At u = 1 and v ≈0.5, the fraction of cooperators rises from 0.18 to 0.87 and rc increases from 0.022 to 0.31.The critical cost-to-benefit ratio increases by a full order of magnitude under the optimal parameter combination.
- Network robustness: Positive u and v promote cooperation across interaction networks, although complex networks shift the optimal v toward 0.6 and reduce the effect relative to square lattices.The authors describe the overall promotive effect as largely universal and predictable.
- Uncertainty robustness: The promotive impact of positive u and v persists irrespective of adoption uncertainty K.For small u, v = 1 remains best; for large u, v ≈0.5 remains best across the phase diagrams.
- Mechanism: Positive u can initially favor defectors, but rapid depletion of cooperators to exploit produces negative feedback that halts and reverses defector dominance.Only 20–30% of cooperators survive during the initial downfall described in the time courses.
- Coevolutionary test: In the coevolutionary model, initially Gaussian-distributed aspiration levels sharpen rapidly around an intermediate value that becomes increasingly frequent.When a player copies another’s strategy, its aspiration level also becomes equal to the source player’s level.
Discussion
Heterogeneous aspiration to successful neighbors promotes cooperation across interaction networks and stochasticity levels. The strongest outcomes require tuning both the aspiration strength and the fraction of players using preferential selection.
- Heterogeneous aspiration to the fittest promotes cooperation across different interaction networks and levels of stochasticity.
- For low and moderate u, cooperation thrives best when the entire population aspires to the fittest.
- For large u, cooperation is optimal when approximately half the players copy their most successful neighbors and the rest select randomly.
- The optimal outcome requires jointly tuning the aspiration parameter and the density of players prone to aspiring to the fittest.
- When aspiration levels also evolve, an intermediate aspiration level emerges spontaneously through natural selection.
- The model extensions are presented as realistic because people generally follow successful individuals, while adverse circumstances may favor copying less successful partners.
Methods
The simulations use a rescaled evolutionary prisoner’s dilemma on several network topologies, with players differing in how they select strategy sources. A selection parameter biases type A players toward high-payoff neighbors, while type B players select randomly, and strategy adoption remains noisy.
- The prisoner’s dilemma uses payoffs T = 1 + r, R = 1, S = −r, and P = 0, where r = c/(b −c).
- Players occupy a square lattice, random regular graph, or small-world network, with type A assigned probability v and type B probability 1 −v.
- When u = 0, neighbor selection is uniformly random regardless of the fraction v of type A players.
- Type A players use aspiration parameter u to preferentially select high-payoff neighbors, whereas type B players select neighbors uniformly at random.
- After a neighbor is selected, the focal player adopts that neighbor’s strategy probabilistically, with K controlling noise or selection intensity.
- An alternative coevolutionary model assigns individual aspiration values from a Gaussian distribution and allows those values to evolve.
- Simulations use populations of 100×100 to 400×400 individuals, 10^5 iteration steps, discarded transients, and up to 40 independent runs.