Source-linked AI summary
Making new connections towards cooperation in the prisoner's dilemma game
Attila Szolnoki, Matjaz Perc, Zsuzsa Danku
TL;DR
The paper examines whether evolving permanent connections can promote cooperation in the prisoner’s dilemma beyond what homogeneous interaction networks support. It finds that successful strategy transmission generates heterogeneous networks and that cooperation is best sustained under an intermediate maximal degree.
Problem
It remains important to determine whether cooperation-promoting network heterogeneity can emerge from initially homogeneous interactions through coevolutionary connection changes.
Method
The model coevolves prisoner’s dilemma strategies and permanent connections, rewarding successful strategy transmission with one new link from an initially degree-four square lattice.
Results
The rule improves cooperation in defection-prone environments, produces exponentially distributed heterogeneous degrees, and performs best with an intermediate maximal degree.
Takeaways & Limitations
Cooperation is promoted when successful players can become influential but their neighborhood growth and overall density remain bounded.
Takeaways & Limitations
Heterogeneous degree distributions alone do not guarantee substantial cooperation at high temptations to defect.
Abstract
from arXiv · showhide
Evolution of cooperation in the prisoner's dilemma game is studied where initially all players are linked via a regular graph, having four neighbors each. Simultaneously with the strategy evolution, players are allowed to make new connections and thus permanently extend their neighborhoods, provided they have been successful in passing their strategy to the opponents. We show that this simple coevolutionary rule shifts the survival barrier of cooperators towards high temptations to defect and results in highly heterogeneous interaction networks with an exponential fit best characterizing their degree distributions. In particular, there exist an optimal maximal degree for the promotion of cooperation, warranting the best exchange of information between influential players.
Introduction. –
The introduction situates cooperation research within heterogeneous network structures and prior coevolutionary models. It then proposes a prisoner’s dilemma model in which successful strategy transmission permanently expands players’ neighborhoods.
- Related work: Complex and scale-free networks have been identified as potent promoters of cooperation across major social dilemmas.The prisoner’s dilemma, snowdrift, and ultimatum games have been studied on diluted, hierarchical, random, small-world, empirical, and general graph networks.
- Related work: Prior work showed that coevolutionary rules can spontaneously generate heterogeneous teaching activity from an initially non-preferential setup, promoting cooperation.This result was reported for social dilemmas including the prisoner’s dilemma and snowdrift game.
- Model contribution: The proposed model coevolves prisoner’s dilemma strategies with players’ neighborhoods by allowing new permanent connections to previously unlinked neighbors.The rule operates alongside cooperative and defective strategy evolution.
- Model contribution: A player may create such a connection only after successfully passing its strategy to one of its current opponents.Each reproduction is treated as evidence of the donor player’s success at that time.
- Model contribution: The model embodies the premise that more successful individuals in real social systems typically have more associates than less successful individuals.Neighborhood expansion rewards donors whose strategies have been successfully reproduced.
Game definitions and setup. –
The study models an evolutionary prisoner’s dilemma with cooperation and defection, using payoff parameters that ensure the ranking T > R > P = S. Strategy success can increase a player’s degree through permanent new connections, and cooperation is evaluated in Monte Carlo simulations.
- Game definitions: The game uses cooperation and defection as competing strategies, with temptation T = b, reward R = 1, and punishment P and sucker payoff S both equal to 0.The condition 1 < b ≤2 ensures a proper payoff ranking.
- Game definitions: Each player is initially designated as either a cooperator or defector, with strategy evolution governed by payoff-based strategy enforcement.The supplied passage states that a player x tries to enforce its strategy on player y when px > py.
- Network evolution: Successful strategy transmission increases the successful player’s degree by an integer Δk, implemented here as Δk = 1.The added degree is realized by establishing a permanent connection with a randomly selected player not yet connected to the successful player.
- Simulation procedure: One Monte Carlo step consists of L2 elementary updates, so each individual is selected once on average during a full step.Simulations used populations from 100 × 100 to 400 × 400 individuals and measured the stationary cooperator fraction ρC after 105 to 106 MCS following discarded transients.
Results. –
Coevolutionary neighborhood growth sustains cooperation where the regular lattice fails, with performance maximized at an intermediate maximal degree. The resulting heterogeneous networks initially benefit defectors, but cooperation recovers as cooperators occupy influential positions, provided influential players can exchange information efficiently.
- Cooperation promotion: Coevolutionary neighborhood growth recovers and maintains cooperation at ρC = 0.66, whereas the regular square lattice fails to sustain cooperative behavior.The advantage concerns the final outcome; during the first 100 MCS, cooperation initially appears likely to decline.
- Cooperation promotion: The optimal maximal degree fluctuates between 50 and 70 across temptation values, producing the highest stationary cooperation.This dependence on kmax is non-monotonic and holds for all three tested b values.
- Cooperation promotion: At kmax = 50, cooperators survive almost halfway across the b range, while without coevolution they go extinct at b = 1.115.The no-coevolution case is kmax = 4, which leaves the initial topology unchanged.
- Time-scale separation: Faster network evolution can slightly reduce ρC because influential cooperators cannot exploit new neighbors quickly, allowing defectors a persistent advantage.This moderate reduction is virtually absent at very high b, where the final heterogeneous network dominates.
- Emergent networks: Highly heterogeneous degree distributions are not sufficient for strong cooperation: at higher kmax, cooperation remains moderate despite similar heterogeneity.Influential players near kmax can robustly source cooperation, but influential players with small neighborhoods cannot communicate efficiently.
Summary. –
A simple coevolutionary process improves cooperators’ survival in highly defection-prone environments and enhances their dominance at moderate temptations to defect. This promotion operates through spontaneously emerging heterogeneous networks and influential leaders.
- Summary. –: The coevolutionary process markedly improves cooperators’ survival under high temptation to defect and enhances their dominance at moderate temptations.The rule is introduced into a spatial prisoner’s dilemma game.
- Summary. –: Highly heterogeneous networks emerge spontaneously because successful strategy transmitters extend their neighborhoods through new connections to previously unlinked players.The setup is initially non-preferential, while the rule indirectly promotes players able to pass their strategy.
- Summary. –: Influential leaders are advantageous for cooperation, and the coevolutionary rule can generate them from an initially non-preferential state.The rule may create appropriate diversity among participating players when appropriately timed.