How does cooperation survive in a small world?
Cooperation is observed across multiple species and a range of life histories, from slime molds to apes, hence the interest in a general explanation for the emergence and persistence of this behaviour within social groups. Evolutionary game theory, using models like the Prisoner’s dilemma, has been employed to investigate these questions.
The main aim is to study importance of the prisoner’s dilemma in an evolutionary game theory framework regarding the evolution and maintenance of cooperation in the presence of cheaters, reflected in the final proportion of cooperators seen in each simulation using the Prisoners Dilemma in a Small World Network setting. The distribution of links among interacting players in games like the spatially iterated Prisoner’s dilemma provides an interesting avenue to study how social populations evolve under different interaction networks.
Small world network (SWN) connectivity allows regular (e.g., nearest neighbour) networks to gradually be altered to completely random interaction networks. We studied how SWNs affected the invasion of cheaters into a spatially structured population of cooperators, varying the relative pay-offs for cheaters and the proportion of randomised links among players in an otherwise regular interaction network. We studied how small world networks affect the invasion of cheaters into a spatially structured population of cooperators, varying the relative pay-offs for cheaters and the proportion of randomised links among players in an otherwise regular interaction network.
A system with four neighbours and self-play was used to analyse the relationship between spatial patterns and population dynamics with different sizes of players lattices. Individuals then play their strategies against their neighbours, either adjacent cells (Nowak & May, 1992) or further ones determined by the small world connectivity pathways. The highest scoring strategy will be adopted by the focal player of each neighbourhood.
The introduction of stochasticity in the system using small world network connectivity can complete change the behaviour of the system and how the scoring sensitivity in of the Prisoners Dilemma can reached to immobile systems when a deterministic mechanism is applied. While increasing the proportion of small world links among player consistently leads to more variation in transient time, spatial patterns and final mean proportion of cooperators in the population do not scale up in a predictable way. There are chaotic fluctuations in the proportion of cooperators, within a given range, allowing the coexistence of both strategies also when stability has not been reached. As the number of players increases, the frequency dependent dynamics of the system are highly dependent on total population density. The comparison between the temporal and spatial dynamics did not show a simple relationship with changing player numbers across different small world spatial structures
Deterministic Sytem (p = 0)
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Figure 2. Black squares represent defectors and green squares represent cooperators. Square lattice illustrating the evolution of spatial structures in a deterministic spatial prisoner’s dilemma game with four neighbours and self-interaction. Panel a shows a payoff (b) = 1.5, p = 0, lattice = 272 . The system (a) reached static stability at round two. Panel b shows a payoff (b) = 1.8, p = 0, lattice = 272. System b reached cyclical stability with a 2-point cycle at round 27 and the image showing 1000 rounds.
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Figure 4. Black squares represent defectors and green squares represent cooperators. Square lattice illustrating the evolution of spatial structures in a small world network system playing the spatial prisoner’s dilemma game with each player interacting with at least four neighbours and itself. Panel a shows a payoff (b) = 1.5, p = 0.5, lattice = 102, reaching cyclical stability with a 2 point cycle at round 3, image shows 1000 rounds. Panel b shows a payoff (b) = 1.8, p = 0.5, lattice = 102, reaching cyclical stability with a 2 point cycle at round 2258, rounds 1 to 100 and 2300 to 3000 are shown.
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Figure 5. Black squares represent defectors and green squares represent cooperators. Square lattice illustrating the evolution of spatial structures in a small world network system playing the spatial prisoner’s dilemma game with each player interacting with at least four neighbours and itself. Panel a shows a payoff (b) = 1.5, p = 0.5, lattice = 102, reaching static stability at round 4. Panel b shows a payoff (b) = 1.8, p = 0.5, lattice = 102, reaching static stability at round 692. Each panel image loops through 1000 rounds.
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Glossary
- Payoff (b): represent the individual’s fitness. Payoff is only varied for defectors, while cooperators payoff is always 1.
- Lattice: Size of the player’s arena, each cell is occupied by a defector or a cooperator at every point.
- Small World Network Value (p): shows the probability of being connected to a directly adjacent neighbour.
- Round: one round is defined as one iteration of the game when the players interact with each other and imitate the highest scoring strategy of their interaction neighbourhood.
- Transient time: rounds taken by the simulation to reach stability
- Cyclical Stability: period after transient time when the system fluctuates between specific proportions of cooperators for the rest of the game.
- Static stability: period after transient time when the proportion of cooperators in the system remains unchanged for the rest of the game.
- Chaotic bounded fluctuations: behaviour shown when the system does not reach stability, but the proportion of cooperators fluctuate within a given range.