Contents - Max-Planck-Institut für Physik komplexer Systeme

Cooperation is a central phenomenon in biology. The
functioning of biological cells depends crucially on the
cooperation between different genes. On higher levels,
cells cooperate to form tissues, organs, and organisms
and organisms cooperate to form social groups. Cooperation
is not restricted to humans, but can be observed
in almost every major family of species. Yet,
only humans have succeeded in scaling cooperation beyond
the level of relatives and direct acquaintances to
form companies, nations and supernational structures.
In modern society the stability of these higher forms of
cooperation is a major concern.
Understanding the evolution of cooperation between
selfish players is of importance not only because of
the central role of cooperation in biology but also to
address the emergence and failure of cooperation in
human societies. The central challenge in this field
is to understand how costly cooperative behavior can
emerge from a process of natural selection that acts on
the level of the individual.
Previous work has identified several key mechanisms
leading to the evolution and fixation of cooperation [1].
Among others, it was found that cooperation is promoted
if the interactions between agents are confined
to a certain topology such as a lattice or a complex
network. A recent development is the investigation of
games on adaptive networks [2,3], in which the agents’
behavior feeds back on the network topology. Cooperation
is promoted if cooperating players can secure
an advantageous topological position, directly due to
avoidance of defectors, or indirectly due to the continuous
arrival of new players or other ongoing changes
of the topology. Furthermore, cooperation in adaptive
networks can profit from the emergence of a selforganized
leadership structure and the formation of
strongly heterogeneous topologies, which are known
to promote cooperation.
In a recent publication [4], we studied a model of the
evolution of cooperation in an adaptive network. In
this model a state can be reached in which every agent
cooperates, although this state is dynamically unstable
against certain perturbations. The model thereby reveals
a powerful mechanism by which asymptotically
full cooperation can be achieved, but also points to a
possible mode of failure of this fragile cooperation.
We consider an undirected network of nodes, representing
agents, and links, representing interactions.
Each agent can either cooperate or defect, i.e. refuse
to cooperate. The outcome of interactions is modeled
by a snowdrift game, a paradigmatic model of cooperation.
If two agents cooperate then they share the benefit
2.22 Evolution of Fragile Cooperation
THILO GROSS, GERD ZSCHALER
and cost of the cooperation. If one of the agents defects
while the other cooperates, the defecting agent (the defector)
reaps the whole benefit without contributing to
the cost. However if both agents defect, neither cost
nor benefit is generated, which is considered the worst
outcome in this type of game.
The proposed model describes the fundamental processes
of social adaptation, i.e. adoption of social traits
from other agents, and social distancing, i.e., the avoidance
of certain other agents. Starting from a random
graph and randomly assigned equiprobable strategies,
we evolve the network as follows: In every time step,
one link is selected at random. With probability p,
this link is rewired. Otherwise, i.e., with probability
q = 1 − p, one of the linked players adopts the other
player’s strategy. For large p, players thus tend to
change their interaction partners, whereas for small p
they tend to revise their behavior.
Finally, we have to specify which agent copies the
other’s strategy in a strategy adoption event and which
agent keeps the link in a rewiring event. Concerning
strategy adoption, it is reasonable to assume that the
less successful agent is more likely to adopt the more
successful agents strategy than vice versa. Concerning
rewiring, we assume that the more successful agent
is more likely to keep the link. This implies that the
agents following the more successful strategy will on
average have a higher number of links than agents following
the less successful strategy.
But how do agents estimate the success of their neighbors?
Most previous models assume that the agents’
access to information is governed by the same network
as the underlying games, forcing the agents to base
their decisions on information from direct interactions.
The same network topology thus determines three different
aspects of the system: the interaction partners
of an agent, against whom the game is played, the potential
role models, whose strategies can be adopted,
and the agents from which information can be obtained.
However, for intelligent agents, and especially
humans, there is no reason to assume that these three
network roles are all fulfilled by coinciding topologies.
We therefore assume that information transfer in the
population is not governed exclusively by the interaction
network. As a first approximation, we consider
the simplest case in which the information transmission
network is replaced by an effective global coupling,
as information can be rapidly transmitted and in
the human population is also transported by the mass
media. In the main part of this work, we assume that
the agents rely on their perception of a strategy’s general
performance, i.e. the average payoff received by all
84 Selection of Research Results