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Evolution of public cooperation on interdependent networks: The impact of biased utility functions
Zhen Wang, Attila Szolnoki, Matjaz Perc
TL;DR
Research on evolutionary cooperation has focused mainly on isolated networks, despite real networks influencing one another. This paper models public-goods cooperation across two interdependent networks linked by a biased utility function and finds that stronger bias raises aggregate cooperation, though unevenly across networks.
Problem
Research on evolutionary games has focused mainly on isolated networks, although real networks can influence one another and produce cross-network consequences.
Method
The paper models public-goods games on two interdependent lattices whose strategy updates use a utility function combining payoffs from corresponding players across networks.
Results
Stronger utility bias increases cooperation, but its benefits are unevenly distributed across networks while aggregate cooperation exceeds the isolated-network level.
Takeaways & Limitations
Biased utility functions can promote cooperation by suppressing feedback and separating evolutionary time scales, slowing defector invasion more than cooperative group formation.
Takeaways & Limitations
The model is too simple to be directly applicable to a concrete situation.
Abstract
from arXiv · showhide
We study the evolution of public cooperation on two interdependent networks that are connected by means of a utility function, which determines to what extent payoffs in one network influence the success of players in the other network. We find that the stronger the bias in the utility function, the higher the level of public cooperation. Yet the benefits of enhanced public cooperation on the two networks are just as biased as the utility functions themselves. While cooperation may thrive on one network, the other may still be plagued by defectors. Nevertheless, the aggregate level of cooperation on both networks is higher than the one attainable on an isolated network. This positive effect of biased utility functions is due to the suppressed feedback of individual success, which leads to a spontaneous separation of characteristic time scales of the evolutionary process on the two interdependent networks. As a result, cooperation is promoted because the aggressive invasion of defectors is more sensitive to the slowing down than the build-up of collective efforts in sizable groups.
Introduction. –
The introduction motivates studying cooperation on interdependent rather than isolated networks, where cross-network influence can produce unexpected consequences. It presents a model linking two networks through a biased utility function combining payoffs of corresponding players.
- Motivation: Research on evolutionary games has shown that network structure can help cooperation emerge and persist among selfish, unrelated individuals.
- Motivation: Complex-network research has focused predominantly on isolated networks, despite real networks simultaneously existing and influencing one another.
- Motivation: Cross-network dependence matters because a player’s success may depend on players in another network, and seemingly irrelevant changes can cause catastrophic consequences elsewhere.
- Contribution: This paper studies public cooperation on two interdependent networks linked by a utility function combining individual payoffs of corresponding players from different graphs.
Model definition. –
The model places public-goods players on two interdependent square lattices, coupling their evolutionary success through biased utility functions. Strategies spread locally via Fermi-rule invasions under random sequential Monte Carlo updating.
- Network and game structure: Players occupy two L × L square lattices with periodic boundaries, forming overlapping groups of size G = 5 and k = G−1 nearest neighbors.Each player belongs to g = G groups, and initial strategies are assigned as cooperators or defectors with equal probability.
- Biased utility functions: Interdependence is introduced through utilities Ux = αPx + (1 − α)Px′ and Ux′ = (1 − α)Px′ + αPx, although the networks are not physically connected.The parameter α controls the bias in how each corresponding player weighs payoffs from the two networks.
- Biased utility functions: At α = 0.5, both players consider the two-network payoffs equally, whereas α = 1 or α = 0 makes one network’s game completely guided by the other network’s payoffs.For α > 0.5, the roles of the two networks are exchanged and the treatment becomes fully symmetric.
- Evolutionary updating: Strategy invasions occur between nearest neighbors on each network, with adoption probabilities determined by the Fermi function using the corresponding players’ utilities.The same invasion procedure is applied on network B using its corresponding utilities.
- Evolutionary updating: Simulations use random sequential updating, giving each player on both networks one strategy-passing opportunity per Monte Carlo step on average.The linear system size varies from L = 200 to 800 to avoid finite-size effects, and K = 0.5 is used in the Fermi rule.
Results. –
Biased utility functions substantially increase cooperation on interdependent networks, but the benefit is unevenly distributed: cooperation can dominate one network while defectors remain prevalent on the other. This promotion arises from suppressed feedback and separated evolutionary time scales, which retard defector invasion more than cooperative organization.
- Results: At α = 0.01 and r/G = 0.76, defectors dominate network B while compact cooperator clusters persist, whereas cooperation is strongly promoted on network A.Network B evolves similarly to an isolated square lattice, where cooperators survive only above r/G = 0.745.
- Results: The difference in cooperator density between networks is largest near α = 0.01, shrinks at α = 0.4 and α = 0.49, and vanishes completely at α = 0.5.The interval where cooperators and defectors coexist is virtually independent of α, despite the differing densities between networks.
- Results: Near α = 0, the critical synergy factor r2c for full cooperator dominance on network A approaches the isolated-lattice threshold r1.The comparison uses r1 = 0.745, corresponding to r2 = 1.1, for full defector or cooperator dominance on an isolated square lattice.
- Results: The aggregate cooperation level across both interdependent networks exceeds the level attainable on an isolated network, showing that biased utility does not merely shift cooperation between networks.This aggregate improvement is examined specifically to determine whether interdependence truly promotes cooperative behavior despite possible negative effects on network B.
- Results: For α = 0.01 and r/G = 0.76, the mixed phase on network B forms after ∼10^3 MCS, while network A evolves about ten times longer and favors cooperation.Defector invasion is retarded on network A, and cooperation begins spreading after defectors exhaust nearby cooperators to exploit.
Summary. –
The study uses utility functions to interconnect two networks and examine why selfish individuals may engage in collaborative efforts in sizable groups. Its biased utility function captures a scenario where sacrificing some spatial-reciprocity benefits may prevent defectors from invading effectively, although the model is too simple for direct application.
- Utility functions interconnect two networks, providing insights into why selfish and unrelated individuals engage in collaborative efforts in sizable groups.
- The biased utility function captures a scenario in which sacrificing some spatial-reciprocity benefits can yield a net advance for cooperators by preventing effective defector invasion.
- Although too simple for direct application to a concrete situation, the model captures the essence of an everyday example involving reliance on governmental sources.