Source-linked AI summary
Emergence of multilevel selection in the prisoner's dilemma game on coevolving random networks
Attila Szolnoki, Matjaz Perc
TL;DR
The paper asks how cooperation can evolve in the prisoner’s dilemma when individual incentives favor defection and network heterogeneity is not the decisive explanation. It introduces strategy-independent link deletion and random link addition on an evolving random network, finding that the resulting dynamics spontaneously generate multilevel selection and sustain cooperation across the full temptation range.
Problem
In well-mixed prisoner’s dilemma populations, individual defection leads to mutual defection even though mutual cooperation yields the highest shared payoff.
Method
The model coevolves strategies and a random interaction network by deleting links after strategy adoption or excessive degree and adding random links every τ Monte Carlo steps.
Results
The coevolutionary rule spontaneously generates multilevel selection, with cooperative groups favored macroscopically and complete cooperation maintained across the full temptation range when τ is sufficiently large.
Takeaways & Limitations
Cooperation can be promoted despite qualitatively preserved random-network heterogeneity, with the effect depending on the link-addition interval τ.
Abstract
from arXiv · showhide
We study the evolution of cooperation in the prisoner's dilemma game, whereby a coevolutionary rule is introduced that molds the random topology of the interaction network in two ways. First, existing links are deleted whenever a player adopts a new strategy or its degree exceeds a threshold value, and second, new links are added randomly after a given number of game iterations. These coevolutionary processes correspond to the generic formation of new and deletion of existing links that, especially in human societies, appear frequently as a consequence of ongoing socialization, change of lifestyle or death. Due to the counteraction of deletions and additions of links the initial heterogeneity of the interaction network is qualitatively preserved, and thus cannot be held responsible for the observed promotion of cooperation. Indeed, the coevolutionary rule evokes the spontaneous emergence of a powerful multilevel selection mechanism, which despite of the sustained random topology of the evolving network, maintains cooperation across the whole span of defection temptation values.
1. Introduction
The paper addresses why cooperation evolves in social dilemmas despite individual incentives to defect, focusing on whether network coevolution can promote cooperation without strong degree heterogeneity. It proposes a strategy-independent rule that preserves the random network’s qualitative degree distribution while generating multilevel selection.
- In well-mixed prisoner’s dilemma populations, defection leads to mutual defection and collective impoverishment despite mutual cooperation producing the highest shared payoff.
- Complex-network studies have linked cooperation promotion to heterogeneous degree distributions, but open questions remain about other network-mediated mechanisms.
- The paper aims to show that coevolutionary network rules can influence cooperation through macroscopic strategy-adoption dynamics rather than strong degree heterogeneity.
- The model deletes links after strategy adoption or excessive degree and adds random links every τ Monte Carlo steps, preserving the initial Poissonian degree distribution qualitatively.
- The resulting multilevel selection favors cooperative groups macroscopically while defectors retain microscopic advantages, producing complete cooperation across the temptation range when τ is sufficiently large.
2. Mathematical model
The model combines a parametrized prisoner’s dilemma with strategy-dependent link deletion and random link additions, while examining how these rules affect interaction-network structure and cooperation.
- Game definition: The prisoner’s dilemma uses T = b, R = 1, P = 0, S = 0, with 1 < b ≤2.Mutual cooperators receive R, mutual defectors receive P, and defectors gain T when facing cooperators.
- Coevolutionary rule: Random link additions can create dashed connections and occur every τ full game iterations.Figure 1 illustrates strategy transmission followed by link deletion, while noting that some dashed links may result from random additions.
- Coevolutionary rule: When player x adopts a strategy, it deletes every link except the one connecting it to the strategy donor, leaving kx = 1.The rule represents separation from former allies following changes in lifestyle, moral values, political orientation, or religious beliefs.
- Network structure: For τ = 70, degree distributions at b = 1.6 and b = 2.7 retain the initial network’s Poissonian profile despite the coevolutionary rule.The figure compares filled green squares and open blue circles with the initial distribution shown as open red squares.
- Network structure: Although network heterogeneity changes little, cooperation promotion depends crucially on τ.The text presents this dependence as an outcome visible in Figure 2 and notes that the deletion process may occasionally detach players.
3. Results
The coevolutionary network promotes cooperation most strongly at intermediate link-addition intervals, while larger intervals generate dormant phases and multilevel selection.
- Cooperation and fixation: The final state is always absorbing cooperation or defection, so ψC measures the probability of reaching ρC = 1 across independent runs.This probability is used instead of cooperator density near sharp transition points.
- Cooperation and fixation: Cooperation fully dominates up to b = 1.1 at τ = 1, b = 1.6 at τ = 10, and b = 2.2 at τ = 70.The last threshold exceeds the prisoner’s dilemma range considered here.
- Multilevel-selection dynamics: At b = 1.5, increasing τ produces longer constant-ρC intervals, revealing increasingly pronounced dormant phases and cascade-like dynamics.At τ = 500, dormant phases cumulatively exceed active phases.
- Multilevel-selection dynamics: Rare link additions let deletion isolate homogeneous groups, which later reconnect and trigger avalanches of strategy adoption.Dormant periods last roughly as long as τ because new links reconnect groups gradually.
- Optimal link-addition interval: At small τ, rapid additions prevent sufficient isolation and produce mean-field-like conditions that are harmful for cooperators.Thorough isolation is needed because mixed-strategy groups remain vulnerable to defector overrule.
- Optimal link-addition interval: The critical temptation to defect bc peaks at approximately τ = 70, then decreases slightly and saturates as dormant phases lengthen.Multilevel selection remains significant beyond the optimum, but prolonged dormancy gives defectors more time to overtake groups.
4. Summary
The paper shows that strategy-independent link turnover can promote cooperation without generating strong degree heterogeneity. Its effect is strongest near τ = 70, where multilevel selection emerges, and remains significant at larger τ.
- Summary: A simple coevolutionary rule preserves the initial random network’s degree heterogeneity while spontaneously evoking multilevel selection.The rule’s cooperation-promoting effect depends on τ, which controls how frequently new links form.
- Summary: Cooperation promotion reaches a local maximum at τ = 70 and remains intact for substantially larger τ across the full temptation range.Small τ prevents isolated homogeneous groups, whereas larger τ prolongs delays between active periods.