How Cooperation Arises in Evolving Social Networks
By Ariana Strandburg-Peshkin
Abstract
The origins of cooperation and altruism have been a central question in fields such as
biology and economics. Here, I present a simple agent-based model of an evolving social
network, where agents play prisoner’s dilemma with one another and copy each other’s
strategies. The model shows that cooperation arises when agents are able to break ties
with agents that are not cooperating with them.
Background - What is a Prisoner’s Dilemma?
The prisoner’s dilemma is a famous problem in game theory.
The standard story goes like this:
Two suspects are being held separately and interrogated. Each prisoner has two options:
he can either keep quiet (cooperate) or rat out his buddy (defect). Each prisoner knows
that if he and his buddy both keep quiet, they will serve 1-year terms. If one rats and the
other doesn’t, the one who ratted will go free and the one who kept quiet will get 3 years
in prison. However, if both rat, they will serve 2-year terms. What do they do? Well, if
prisoner 1 thinks prisoner 2 will keep quiet (cooperate), he would do better to rat him out
(defect), in order to go free rather than serving a 1-year jail term. If prisoner 1 thinks
prisoner 2 will rat him out (defect), he would still do better to defect so as to serve a 1year term rather than a 3-year term. Prisoner 2 sees things the same way, so logically
both defect, when in fact both would have been better off had they both cooperated.
The dilemma can be written as a table of payoffs, where in this case the “payoff”
numbers are positive in value (unlike jail sentences).
Cooperate Defect
2,2
0,3
Cooperate
3,0
1,1*
Defect
Prisoner 1 Prisoner 2
The equilibrium is both players defect, because neither player can improve his payoff by
switching to a different strategy unilaterally.
The prisoner’s dilemma has been used as a model of many social dilemmas faced by
individuals in a community, in areas ranging from economics to biology. Although it has
classically been used to describe human interactions, the prisoner’s dilemma can apply to
other systems as well. For example, why do trees grow so tall? Wouldn’t they all be
better off if they were all shorter, so they wouldn’t have to expend so much energy
growing? They are trapped in a prisoner’s dilemma.
The Model
The agent-based model described here is intended to reflect human interactions in a
social network. Such a social network could be an economic market, a community of
individuals, or even a larger society. The model pertains to any network where humans
interact with one another and are faced with social dilemmas. The model is not
representative of “simpler” life forms because of various assumptions of the agents’
intelligence, as will be discussed in the conclusion.
The model consists of a network of agents that repeatedly play prisoner’s dilemma with
one another.
- Each agent knows its strategy: a real number p, between 0 and 1 – this number
represents the probability that the agent will cooperate in a given interaction.
- Each agent knows its links and their weights: each agent is connected to a certain
number of other agents, and each link is weighted based on how much payoff the
agent has received from that individual in the past.
- Each agent knows the strategies and payoffs of all of its neighbors (the agents to
which it is linked).
The network is initialized by creating L random (bidirectional) links between the N
agents. The following then takes place each iteration:
1. Agents play all of their neighbors in prisoner’s dilemma many times.
2. Agents tally up their payoffs, then find which of their neighbors has the highest
payoff and move towards that strategy.
3. Agents break ties with their lowest weighted neighbor, with a certain probability,
b. If an agent breaks a tie, it chooses another agent to link with at random. In
this way, the total number of links in the network remains constant.
To speed up the code, a probability function is used to calculate the payoffs for each
interaction (rather than having the agents interact manually). A small amount of local
noise is also added each iteration in order to prevent the agents from quickly converging
to one value simply because they are copying one another. This noise takes the form of
one agent each iteration adding or subtracting (at random) .001 from its strategy.
Results
When no ties are broken (b = 0), the network generally converges to an average strategy
of 0 (all agents defect). When any breaking of ties occurs, the network converges to an
average strategy of 1 (all agents cooperate). These results suggest that “punishment” by
breaking ties with defectors is sufficient to encourage cooperative behavior in social
networks.
Speed of Convergence
I investigated how the speed of convergence to an average strategy of 1 varies with the
probability of breaking ties, the size of the network, and the density of links. The results
can be summarized as follows.
- As the probability of breaking ties increases, the speed of convergence increases (See
Fig. 1). In other words, harsher punishment yields faster compliance. However, it is
notable that even very small probabilities of breaking ties (very gentle punishment) yield
cooperative behavior eventually.
- As the size of the network increases, the speed of convergence decreases (See Fig. 2). In
larger networks, cooperation spreads more slowly.
- As the density of links increases, the speed of convergence decreases (See Fig. 3). The
reason for this behavior is probably that in more highly-connected networks, links are not
as “valuable” to agents since they are so numerous. Therefore, the punishment of
breaking ties is not as effective, so the network is slower to converge to a cooperative
strategy.
Speed of Convergence vs. Probability of Breaking
Ties
180000
Iterations until Average Strategy = .99
160000
140000
120000
100000
80000
60000
40000
20000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Probability of Breaking Ties
Fig. 1 (a) Harsher punishment yields faster compliance.
0.9
1
Speed of Convergence vs. Probability of Breaking Ties
Iterations until Average Strategy = .99
1000000
100000
10000
y = 5271.2x
-
0 .7 0 2 1
1000
0.01
0.1
1
Probability of Breaking Ties
Fig. 1 (b) And here is my contribution to the log-log-plot-draw-a-line-and-call-it-apower-law approach…
Speed of Convergence vs. Size (Network Density = 2 Links / Agent)
Iterations to Average Strategy = .99
1000000
100000
10000
10
100
1000
Agents
Fig. 2 Larger networks take longer to converge to cooperation.
Convergence Speed vs. Number of Links
Convergence Speed (Iterations to .99)
25000
20000
15000
10000
5000
0
0
20
40
60
80
100
120
140
160
Number of Links (Agents = 25)
Fig 3 More densely-connected networks take longer to converge to cooperation.
Conclusion
This simple model of social interaction showed that social punishment (by agents
refusing to interact with a defector) can be effective in promoting cooperation. This
punishment tends to be most effective in small, loosely-connected networks, and harsher
punishment will yield cooperation faster. The model helps explain why individuals in a
society would conform to social norms, even against their individual interests.
As mentioned in the introduction, the model requires that agents be relatively intelligent.
They must have the capacity to keep track of who their “neighbors” are and how much
they value them. In addition, it requires that they have information about each of their
neighbors’ payoffs as well as strategies. For this reason, it is ideally suited for modeling
human interactions, but does not pertain to the interactions of “simpler” life forms.
In addition, while it requires complex social intelligence, the model fails to capture more
complex strategies. All agent strategies are probabilistic, thus neglecting more
complicated strategic possibilities, such as tit-for-tat.
Further research should explore the effects of adding more complex strategies. It would
also be interesting to look at games other than prisoner’s dilemma, and how the game
played on the network affects the network structure and vice versa.