Static Network also stimulates Social Cooperation

Prologue..

A fundamental aspect of all biological systems including human society is cooperation. It is well known that in unstructured dynamic populations where agents are allowed to make and break ties, natural selection favours defectors over cooperators. Defectors are those agents who do not wish to cooperate residing in social structure. On the other hand, cooperators are those agents who cooperate with their neighbours. These cooperative interactions, where individuals incur costs to benefit others, increase the greater good but are undercut by self-interest. It is very fascinating to study that how the selfish process of natural selection gave rise to cooperation and what is the role of social structure in promoting or stimulating cooperation. In this line, the recent studies on evolutionary game theory recognize the fact that who-meets-whom is not a random process, but determined by the spacial relationships that exist in structured social populations. A simple rule for the evolution of cooperation on social networks have been proposed by Ohtsuk et. al. [1] in the forum of Nature.

In a theoretical model, a cooperator pays a cost, c, for another individual to receive a benefit, b. A defector pays no cost and does not distribute any benefits. The evolutionary game of natural selection dictates the dynamics of cooperators and defectors in repeated game scenario. In the specified model, the players of an evolutionary game occupy the vertices of a graph. The edges define links between individuals in terms of  dynamic interaction with an assumption that the network is static for the duration of the evolutionary dynamics. In this settings, a population of N individuals consists of cooperators and defectors. A cooperator who helps all of its immediate neighbour. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi - ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighbouring cooperators. If a defector is connected to j cooperators, then its payoff is bj.

It has been theoretically established that cooperators have a fixation probability greater than 1/N (which is the case for neither cooperation or defection) if b/c > k. In short, the ratio of benefit (b) to cost (c) of the altruistic act has to exceed the average degree, k, which is given by the average number of neighbours per individual. The result beautifully aligns with Hamilton’s rule, which states that kin selection can favour cooperation provided b/c > 1/r, where r is the coefficient of genetic relatedness between individuals. In other way, the average degree of a graph is nothing but an inverse measure of social relatedness (or social viscosity). The fewer friends an agent have the more strongly the agent's fate is bound to theirs.

Static Network is not that bad..

Based on this theoretical establishment, Rand et. al. [2] tried explore a particular explanation with experimental support, i.e. the static network can ensure cooperation. The hypothesis is when the interaction is structed that is limited within your neighbours rather than the whole network, the emergence of clusters is facilitated. Though previous experiments were failed to established that static should encourage cooperation or clustering. Rand et. al. found the explanation behind this failure where they observed the criteria of having more cooperation is some specific structural property which involves combinations of payoff and network structure. The group studied the phenomena through two set of experiments.

Experiment 1:

 In laboratory setting, subjects were arranged on a ring conneted to k/2 neighbours on each side to initiate a game of Prisoner's Dilemma. In each round of game, each agent can defect (D) by doing nothing or cooperate (C) by paying an altruistic cost of c =10k units to contribute each of it's k neighbours a benfit of b units. All agents are constrained to choose same strategy for all of the neighbours. At every round, all of the subjects are concerned about the decisions of the neighbours as well as the as well as the total payoff for the round earned by themselves and by each neighbor.

In experiment 1, the b/c ratio has been fixed at 6 and k = 2 and 109 students from Yale University were asked to play 50 rounds of the game described (8.4 subjects per session on average). The subjects were assigned either to a “network” treatment i.e. fixed, in which their position on the ring is held constant every round, or a “well-mixed” treatment, in which their position is randomly shuffled every round (the information of shuffling is given to the subjects)

As the b/c > k condition is satisfied, it was expected that cooperation will rise in the network to sustain. The above graphs consolidates the fact. It has been also observed that a high stable level of cooperation exist when the subjects are embedded in the network (in turn, no significant relationship between cooperation and round number; P = 0.290). On the other hand, the well-mixed treatment, by
contrast, cooperation decreases over time (in turn, relationship between cooperation and round is significantly more negative in the well mixed treatment compared with the network treatment; P =
0.030). Hence, cooperation rates in the second half of the session are significantly higher in the network treatment than in the well-mixed treatment (due to negative dependency on round number). To conclude, it is observed that interaction structure does matter for stabilizing human cooperation and that static networks can facilitate cooperation under the right conditions (b/c > k).

Experiment 2:

The second experiment has been done in crowdfunding way where a huge number of recruits were made involved from online labor market Amazon Mechanical Turk (AMT). As k and b/c were varied systemically across the values of 2, 4, and 6 in  the network treatment generating nine main treatments: [k = 2, k = 4, k = 6] × [b/c = 2, b/c = 4, b/c = 6]. Also, well-mixed condition has been tested in different combination of b/c and k where the condition b/c>k was satisfied. The end-game effect which is a drawback in online experiment has been tactfully taken care of.

As observed, after an initial transient adjustment, cooperation in the b/c > k treatments stabilizes in the second half of the game (in turn, no significant relationship between cooperation and round number; P = 0.838), whereas cooperation continues to decline in the b/c ≤ k treatments (P = 0.028). The significant relation (P = 0.001) between round number and b/c > k indicator explained how cooperation unfolds over time.

On the other hand, in the final round, cooperation has been found to be significantly higher when b/c > k compared with b/c ≤ k. The more insightful fact is, there is no significant difference in final round cooperation in b/c < k compared to b/c = k and there is more final round cooperation in b/c > k comparison with b/c = k. The following figure tells the whole story:

To be or not to be?

These differences in the level of cooperation heavily depends on how players are distributed over the network. When b/c > k, clusters of cooperators emerge and are sustained, whereas no such clusters form when b/c ≤ k. According to evolutionary game theory, assortment is defined as a cooperator’s average fraction of cooperative neighbors minus a defector’s average fraction of cooperative neighbors. As observed, assortment rapidly emerges when b/c > k, but not when b/c ≤ k. parity of this, a level of assortment is significantly greater than zero for b/c > k , but not for b/c ≤ k. In summary, the b/c > k environment enables substantial clustering of cooperators, stabilizing cooperation. Despite similar initial levels of cooperation across the two networks with b/c  ≤  k and b/c > k, the distribution of cooperators rapidly change.

It's all about Money!

There lies a huge importance of this assortment in strategic implications. Defectors earn significantly higher payoffs than cooperators when b/c ≤ k. As clustering arises when b/c > k, it allows cooperators to interact preferentially with other cooperators. It is possible that the cost of cooperating is balanced out by increased access to the benefits created by other cooperators, improving the payoffs of cooperators compared to defectors. Cooperators earn significantly higher payoffs compared to defectors when b/c > k in compared to b/c ≤ k, so much so that when b/c > k, defectors no longer earn significantly more than cooperators.

 

The key factor determining outcomes is the b/c > k criterion, not the b/c ratio itself. A control for the b/c ratio shows a significant dependence between round number and a b/c > k indicator. Comparing b/c > k to b/c ≤ k, it has been observed that there is significantly more cooperation in the final round, significantly more assortment, and significantly higher payoffs of cooperators compared with defectors.

Oh! I don't know you!

On the other hand, shuffling the network results in a decay of cooperation even if the b/c > k condition is satisfied. In well-mixed b/c > k control conditions, there is significantly less assortment than in the networked treatments. As a result, we find that cooperation significantly declines in the second half of the game when the population is well mixed, even though b/c > k. Furthermore, the well-mixed controls show less cooperation in the final round and lower payoffs to cooperators compared with defectors. Hence it safe to conclude that it is not enough just to interact with k players in each round. Interactions must be embedded in static networks to achieve stable cooperation.

The whole article talks about the power of static interaction networks to promote human cooperation. With the right combination of payoffs and structure, networked interactions promote stable cooperation via the clustering of cooperators. This clustering offsets the altruistic costs of cooperating and makes it possible to sustain high levels of cooperation in sizable groups and to avoid the tragedy of the commons. The finding indicates that networks may play in the origins and maintenance of cooperation in human societies.

Food for thought..

Few interesting points along with with line of this article:

Experimentation was defined as switching to a strategy not currently played by any of one’s neighbors (a process similar to mutation in evolutionary models). Exploration/mutation disrupts the clustering of cooperators, because a player surrounded by cooperators might spontaneously switch to defection giving rise to different exploration dynamics. Thus, as the mutation rate increases, the b/c required to maintain cooperation rises above k.

The interesting fact is defectors with all defecting neighbors switch to cooperation 15.7% of the time when b/c ≤ k and 17.4% of the time when b/c > k, an insignificant difference. Thus, mutations from defection to cooperation, which do not hinder the clustering of cooperators, in both cases. However, spontaneous and sporadic changes from cooperation to defection are significantly less common when b/c > k compared with b/c ≤ k. Cooperators with all cooperating neighbors switch to defection 14.1% of the time when b/c ≤ k, but only 5.1% of the time when b/c > k. Importantly, this 5.1% mutation rate is low enough to ensure success of cooperation in b/c > k network condition. Exploring the evolutionary dynamics of strategies that can modify their mutation rates across settings is an important direction for future work.

There may be rationales for the b/c > k criterion that come from behavioral models or myopic learning models in addition to the evolutionary model. For example, when b/c > k is satisfied, cooperators need only one cooperative neighbor to break even (i.e., to earn more than the zero payoff they would earn if they had not played the game or if they had played in a group where all players defected). Thus, the b/c > k condition may be relevant for agents who, rather than maximizing their payoff through imitation as in most evolutionary models, engage in a variant of conditional cooperation, where they cooperate as long as doing so does not make them worse off than the baseline reference point. For similar reasons, the b/c > k condition may also be relevant for learning models that seek a satisficing payoff, rather than a maximal payoff.

Epilogue..

Thus, the whole idea suggests that regularity in network can stimulate cooperation and that might help to explain why structure exists and sustains. Also, in social world to facilitate cooperation a social institutions can be formed and maintained. The static nature of the network of these institutions can organize the spatial placement of different entities and can control the randomness in the who-connects-to-whom phenomena to ensure social altruistic cooperation.

References: 

[1] Hisashi Ohtsuki, Christoph Hauert, Erez Lieberman & Martin A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks, Nature (2006) 441(7092):502-505.

[2] David G. Randa, Martin A. Nowake, James H. Fowlerh, and Nicholas A. Christakisd, Static network structure can stabilize human cooperation, PNAS (2014), vol. 111, no. 48, 17093–17098