Source-linked AI summary
Rewarding evolutionary fitness with links between populations promotes cooperation
Zhen Wang, Attila Szolnoki, Matjaz Perc
TL;DR
The paper asks how cooperation evolves when initially independent structured populations can become interdependent through rewards for current fitness. It models threshold-triggered external links in the prisoner’s dilemma and public goods game, finding that intermediate thresholds promote cooperation, whereas high thresholds eliminate coupling when rewarded players cannot percolate.
Problem
The paper examines how interdependence between structured populations influences cooperation and whether fitness-based rewarding can promote cooperation without relying on prior success or strategy.
Method
The authors simulate evolutionary games on two initially disjoint structured populations, adding directed external links whenever a player’s current utility reaches a threshold and removing them when it falls below.
Results
Intermediate utility thresholds significantly promote cooperation, while thresholds at or above the critical value leave populations independent and make traditional network reciprocity the sole support for cooperators.
Takeaways & Limitations
Optimal cooperation emerges when rewarded players are rare but sufficiently frequent to percolate, creating dynamically self-organized interdependence across populations.
Takeaways & Limitations
External partners are selected randomly rather than according to their tags, leaving possible bilateral tag-based-linking effects outside the study’s scope.
Abstract
from arXiv · showhide
Evolution of cooperation in the prisoner's dilemma and the public goods game is studied, where initially players belong to two independent structured populations. Simultaneously with the strategy evolution, players whose current utility exceeds a threshold are rewarded by an external link to a player belonging to the other population. Yet as soon as the utility drops below the threshold, the external link is terminated. The rewarding of current evolutionary fitness thus introduces a time-varying interdependence between the two populations. We show that, regardless of the details of the evolutionary game and the interaction structure, the self-organization of fitness and reward gives rise to distinguished players that act as strong catalysts of cooperative behavior. However, there also exist critical utility thresholds beyond which distinguished players are no longer able to percolate. The interdependence between the two populations then vanishes, and cooperators are forced to rely on traditional network reciprocity alone. We thus demonstrate that a simple strategy-independent form of rewarding may significantly expand the scope of cooperation on structured populations. The formation of links outside the immediate community seems particularly applicable in human societies, where an individual is typically member in many different social networks.
1. Introduction
The paper extends evolutionary games on interdependent networks by using fitness-dependent rewards to create links between initially disconnected populations. This approach builds on research into rewarding and network interdependence as mechanisms affecting cooperation.
- Rewarding has been studied as a means to promote public cooperation and may increase earnings without reputational damage, retaliation, or antisocial punishment.
- Interdependent-network research examines how changes in one network can produce consequences in another and how interdependence influences cooperation.
- The paper introduces a utility threshold E that rewards sufficiently fit players with external links to corresponding players in the other network.
- External links increase a rewarded player’s utility using part of the other player’s utility, but are terminated when the focal player falls below the threshold.
- The linking rule depends only on current fitness and strategy-independent eligibility, rather than previous evolutionary success or strategy.
2. Evolutionary games
The study models strategy evolution on two initially disjoint structured populations, while utilities and external links coevolve according to threshold-based rewards. It combines evolutionary-game payoffs, strategy updating, and dynamic interpopulation coupling, while deliberately excluding tag-based partner selection.
- The games run on two disjoint square lattices or random regular graphs with periodic boundaries, with initially random cooperation or defection.
- The weak prisoner’s dilemma uses T = b, R = 1, P = 0, S = 0, and 1 < b ≤2, while the public goods game uses overlapping groups with cooperative contributions multiplied by r.
- Monte Carlo updating selects a player and a same-network neighbor, computes both payoffs, and lets the neighbor adopt the selected player’s strategy probabilistically.
- Players with external links use utility Ux = Πx + αΠx′, whereas unlinked players retain Ux = Πx; links are directed and do not create direct interaction.
- The simulations average final cooperation over up to 100 independent initial conditions, while external partners are selected randomly rather than by tag.
- The tag qx switches to one when current utility reaches E and otherwise remains zero, producing time-varying interdependence through link addition and removal.
3. Results
Across interaction networks and evolutionary games, cooperation is promoted most strongly at intermediate utility thresholds, where rewarded players remain sufficiently common to percolate. Above a critical threshold, external links disappear or lose connectivity, leaving cooperators dependent on traditional network reciprocity.
- Intermediate utility thresholds optimally promote cooperation, whereas cooperation drops suddenly at a critical threshold Ec.At E = 0, full interdependence supports cooperation; at or above Ec, the populations become effectively independent.
- On random regular graphs, Ec lies below the maximally attainable cooperator payoff and decreases as temptation to defect b increases.Cooperators inside domains may remain rewarded, but vulnerable boundary cooperators allow defectors to erode cooperative clusters and external links.
- Cooperation is optimally promoted when rewarded players are rare but still able to percolate, requiring sufficiently high E while maintaining E < Ec.Players with external links have a distinct cooperation level, and their influence supports surrounding cooperative domains.
- The same threshold-dependent promotion occurs in the public goods game, where Ec decreases as the multiplication factor r decreases.Smaller cooperative clusters under harsher dilemmas reduce the ability of cooperators to reach payoffs sufficient for reward eligibility.
- A small threshold reduction below Ec can make rewarded players percolate and substantially increase the stationary cooperator density.For b = 1.1 on a degree-eight random regular graph, E = 5.9 enables percolation whereas E = 6.0 does not.
4. Discussion
Rewarding players when their current utility reaches a threshold creates a time-varying interdependence between initially independent populations. Cooperation benefits most at intermediate thresholds, where rewarded players are sufficiently rare yet able to percolate, while excessive thresholds eliminate the coupling.
- External links reward players whose current utility reaches the threshold, increase their evolutionary potential, and disappear when utility falls below it.
- The coevolution of strategy-dependent fitness and external links produces a threshold-dependent interdependency that dynamically couples the two populations.
- Cooperation is promoted most strongly at intermediate utility thresholds, especially just below the critical threshold, when distinguished players can percolate.These players act as strong catalysts by providing a cooperation-supporting mechanism at a higher level.
- Distinguished players must be rare enough to remain optimal but frequent enough to percolate through the network.The relevant density is connected to percolation on isolated populations with pairwise social dilemmas and public goods games.
- The critical threshold generally lies below the maximal attainable cooperator payoff and decreases as the social dilemma becomes more severe.Harsher conditions produce smaller cooperative clusters, reducing the supply of cooperators that can reach near-optimal payoffs.
- The reported results appear independent of interaction-network structure and the studied social dilemma, suggesting a high degree of universality.