Zhang et al (2011) Aspiration-induced reconnection in spatial public goods game

Zhang et al (2011) Aspiration-induced reconnection in spatial public goods game. Eur Phys Lett, 94, 18006. 

Mechanism: If player’s payoff obtained from the group centered on one of its neighbors does not exceed the aspiration level, the player will cut the link with the neighbor and rewire the link to one randomly selected player.

Method:

• Newman-Watts small-world network. A number of long-range links N_add are randomly added to the 2-D lattice with periodic boundaries. 
• Asynchronous updating: Players update strategies in random sequential order. Before updating strategies, there is potential re-wiring:
• A random player x evaluates payoff from neighborhood centered on random neighbor y. If this does not meet the aspiration level E, player x removes the link and creates a new link with a random non-neighbor node. 
• Local (von Neumann) neighbors cannot re-connect (for no good reason, really).
• Evolution: player x randomly select’s a neighbor z and adopts its strategy using a Fermi updating rule with noise parameter K. 

Results:

• For small values of production value r, there is an optimal aspiration level which maximizes the frequency of cooperators. For larger r, lower aspirations levels are always good. 
• Explanation: Negative feedback. 
• In the early stages of the evolution process, though cooperators were decimated by defectors, the peak of reconnection induced by the intermediate aspiration level will change the downfall of cooperation and facilitate the fast spreading of cooperation. 
• While for small or large aspiration levels, non- reconnection or excessive reconnection do not provide any possibility for the emergence of such a feedback effect.
• Degree distributions: The heterogeneous degree distribution induced by the intermediate aspiration level warrants a potent promotion of cooperation.

Take home: An intermediate aspiration level – only keep those connections that meet them, produce negative feedback and promote cooperation through negative feedback. This also gives rise to a power-law-like degree distribution.