Emergence of Cooperation from Global Rules

I was gushing when I saw Dr. Helbing in Italy. I was a participant at the Lake Como School of Advances in Complex Systems and he was one of the speakers there. I mean why shouldn't I? His work back in 2009 was the main reference of the first half of my undergraduate thesis. The question remains, why write this now? Apparently none about the idea of global governance has been published yet. I'll delve into this in the succeeding paragraphs. 

For context, he together with Dr. Yu, published a paper on PNAS that tackles the phenomenon of the sudden outbreak of predominant cooperation in a noisy world controlled by selfishness and defection where individuals imitate superior strategies and show success-driven migration. 

Edit: The paper was published 3 years after this post at Physica A: Statistical Mechanics and its Applications.

Introduction

Cooperation is known to exist at all levels of biological hierarchies. Cooperative acts come in many forms, but they all share a crucial problem, that is, cooperation is much vulnerable from exploitations through selfishness [1]. Thomas Hobbes, in his masterpiece “Leviathan (1651)” argued that man is naturally a selfish hedonist. Hence, living in close proximity with each other may imply “war of every man against man”. Conflict is perceived to be a naturally occurring phenomenon while cooperation, on the other hand, is a field not yet extensively explored [2]. In any act of cooperation, an individual may decide to “pay a price” to another individual so as to provoke a reaction which happens to be beneficial to that former individual. Leimar and Connor cited an example where a bird drops its feces unto a bush aiding soil fertilization and nourishment. The bush grew and provided shelter for the bird’s nest. Given these circumstances, it is important to point out that the bush’s growth did not evolve as an act of “gratitude” [2] and is probably closer to the concept of “Invisible” hand by Adam Smith where our selfish desires improve the life of organisms or people around us.

Biologists and economists believe that individuals (in times of great need) tend to take full advantage of their own welfare at the expense of others, attempt to resolve cooperation with selfishness. Ideas in game theory were initially applied on evolutionary biology by Lewontin (1961). The game, he says, consists of players (species) who are continually in conflict with nature. Hence, every strategy they come up with is solely for the purpose of reducing their possible extinction [3]. We often refer this as Darwin’s theory of natural selection. Game theory provides the necessary platform to understand the nature of cooperation [4]. It provides techniques necessary to analyse situations involving two or more individuals where strategies taken by one individual not only depends on his own action, but also on other individuals’ actions as well [5]. For example, two publishers in a city set prices for their newspapers and are aware that both are dependent on each other in terms of profit. Then, they are said to be players in a game [6]. 

Cooperation modelled through game theory, especially prisoner’s dilemma (PD), is a well researched field [3,9-12]. However, it is surprising that Prisoner’s dilemma favors defection more than cooperation. To start with, PD is a two-person game (originally two prisoners accused of a joint crime) wherein players are made to choose either to cooperate (deny the crime) or defect (confess participation with the crime) [7]. Prisoner’s Dilemma payoff matrix in Table 3.1 hints cooperation as the most favorable strategy for both prisoners 1 and 2 since both receive far more favorable pair of payoff values (1,1) compared to the rest of the pairs in the matrix listed: (1.3,0), (0,1.3) and (0.1,0.1). This observation, however, does not hold in Prisoner’s Dilemma. If we further analyze the matrix, we see that if prisoner 1 chooses to cooperate and prisoner 2 chooses to cooperate as well, the latter individual receives payoff value equal to 1. If prisoner 2 changes his strategy and opt for defection, he receives a greater, hence, favorable payoff of 1.3. The same sets of conditions also apply in the event that Prisoner 2 chooses to cooperate first. Furthermore, if prisoner 1 chooses to defect, prisoner 2 chooses a more favorable strategy, which is defection (payoff 0.1) over cooperation (payoff 0). Summing all these consequences namely: 1) if prisoner 1 cooperates, prisoner 2 defects and 2) if prisoner 1 defects, prisoner 2 defects, and vice versa, implies defection as sole strategy most beneficial to both individuals. Whether prisoner 1 chooses to defect or to cooperate, prisoner 2, will and at all cost, defect since

his payoff, for whatever strategy prisoner 1 chooses, is larger thus, more favorable compared to that of cooperation. We refer to this phenomenon as Nash Equilibrium. Prisoner’s Dilemma follows the rules T>R>P>S and R>(S+T)/2 where T corresponds to the temptation to unilaterally defect, R (reward) to mutual cooperation, P (punishment) to defection for both parties and S (sucker’s payoff) to the cooperating individual [8; 7].

Our work seeks to analyze the evolution of cooperation among selfish individuals acted upon by local, global and combined conditions through implementation of spatial Prisoner’s Dilemma. Local interactions, pertaining to payoff comparisons among neighboring individuals, are implemented in the community and subjected to noiseless and noisy imitation, migration and combined rules. We show that cooperation survives in chosen cases. These results, however, only pertain to individuals who behave based on local rules, an interesting idea lies on a new concept which focuses on global interactions. This means that instead of comparing neighbour’s payoff values, individuals now regard the total payoff of the entire community before taking appropriate strategies.

We also measure segregation and clustering properties among individuals in the community by computing the index of dissimilarity (ID) and entropy as adapted from the work of J.F. Valenzuela and C. Monterola [9]. Demographers have developed the concept of index of dissimilarity as means to calculate degree of segregation in residence patterns among ethnic groups. For example in a country with individuals either belonging to one of two types of nationalities, an index of dissimilarity equal to zero means that nationality 1 is settled in the country the same way as nationality 2 is also settled [10]. For a geographic area partitioned by M number of boxes, the index of dissimilarity is calculated through the following formula:

$ID=\frac{1}{2}$

where p1i and p2i are the fractions of individual 1 and 2 respectively in box i. Entropy (S), on the other hand, is used to measure aggregation or clustering of individuals in a geographic area. Lower levels of entropy imply greater levels of clustering. It is represented by the following formula:

where pi is the fraction of individual  (either cooperating or defecting individual) in the ith box [9].

Methodology

Implementation of Global and Local Interactions in a virtual community

We adapted the method used in reference [8] where individuals behave based on Local Interactions (LI) in a Moore’s neighbourhood. Without loss of generality, we generated a 49x49 empty lattice and filled it randomly with 25 % cooperating individuals (deny in PD) and 25% defecting individuals (confess in PD). In order to analyze local community behaviour, three methods were implemented on a randomly-filled lattice (RL) under noiseless and noisy conditions namely: Imitation (I), Migration (M) and Imitation with Migration (IM). Rules on spatial PD were used for all three conditions with payoff values T=1.3, R=1, P=0.1, and S=0 in Table 3.1 [8]. In a noiseless Imitation scheme (figure a), a randomly selected individual was made to interact with its four immediate neighbours. After which, the selected individual compares payoffs and copies the strategy of the neighbour which exhibits the highest payoff. When noise is implemented on a Random Lattice,

the selected individual copies the strategy of its best performing neighbour with probability (1-r). Otherwise, it resets. If reset becomes an option, individual either cooperates with probability q or defects with probability 1-q. In a noiseless Migration scheme (figure 3.2b), a randomly selected individual with mobility m was made to relocate to the nearest empty site within its Moore neighbourhood of size (2m+1)x(2m+1) given that payoff is higher in the new location. Otherwise, it stays put. Noise implementation in the Migration scheme literally resets a neighbor at random with probability (1-q) of becoming a defector and cooperator otherwise. Imitation with Migration scheme (figure 3.2c) is preceded by a noiseless Migration followed by a noiseless or noisy Imitation depending on the type of IM implemented. All three methods were repeated for all individuals in the lattice in a random sequential manner for 200 iterations. Figure 3.2 presents schematic diagrams summarizing the three methods in both noisy and noiseless conditions for Local Interactions.

Global Interactions (GI) is very similar to Local Interactions only that this time, individuals do not base their strategies on their neighbor’s payoffs, and instead they take into consideration the total payoff of the entire community every time imitation or migration is performed. Global Interactions total payoff values are calculated based on the following formula:

where N is the total number of individuals in the community, n is the total number of individual i’s neighbors and i;j is the payoff value (based on PD payoff matrix) of individual iwith respect to neighbor j. All calculations were performed over periodic boundary conditions. Combining Global with Local rules allow individuals to execute their strategies initially, when maximum global payoff is achieved and secondly, if optimum local payoff is detected among neighboring entities.

Entropy and Index of Dissimilarity as tools for measuring segregation and clustering

As adapted from the work of Valenzuela and Monterola [9], index of dissimilarity and entropy were computed by first partitioning the entire 49x49 Random Lattice into 49 boxes of equal areas. After which, we count the number of  individuals in the box and take its fraction over total number of  individuals in the entire lattice. Calculations were based on equations 3.1 and 3.2. Index of dissimilarity and entropy values are then plotted for each iteration step.

Results and Discussion

Implementation of Global and Global with Local Interactions into a virtual community

Random Lattice site governed by local rules is shown to promote the evolution

of cooperation through implementation of combined imitation with migration scheme. We introduce the concept of Global Interactions (GI) permitting individuals to consider total payoff values of the entire community, much like how a worker ant collects food for the benefit of the whole colony. Or for human

systems, this corresponds to imposing taxation in building community structures

that allows production and delivery of services more efficient and cheaper.

Imitation

Imitation and Migration

Imitation and Migration

Figure. Comparisons among Local, Local-Global and Global Rules in noisy and noiseless cases. The Number of cooperators monitored per iteration step is plotted for Global (red), Global-Local (green) and Local(blue) conditions. Global conditions boost cooperation among individuals in a community regardless of the presence of noise for both Imitation and Imitation with Migration conditons. Cooperators are rendered blue while defectors are red.

The figure above keeps track of the number of cooperators for global, local and combined schemes as it changes through time in both noiseless and noisy conditions. The number of cooperators boost to a maximum number in the Global Interactions scheme even when subjected to noise in both Imitation and Imitation with Migration cases.

Dissimilarity Index vs Time Step

Imitation

Imitation and Migration

Migration

Entropy (Cooperators) vs Time Step

Imitation

Imitation and Migration

Migration

[1] P. Kappeler and C. van Schaik, Cooperation in primates and humans: mechanisms and evolution. Springer, 2005.

[2] P.Hammerstein, Genetic and Cultural Evolution of Cooperation. MIT Press, 2003.

[3] J. M. Smith, Evolution and the Theory of Games. Cambridge University Press, 2004.
[4] R. Ellickson, Order Without Law: How Neighbors Settle Disputes. Harvard University Press, 1991.
[5] F. Carmichael, A Guide to Game Theory. Dorset Press, 2005.
[6] E. Rasmusen, Games and Information: An Introduction to Game Theory (4th edition). Blackwell Publishing, 2007.
[7] B. Brembs, “Chaos, cheating and cooperation: Potential solutions to the prisoner’s dilemma,” OIKOS, 1996.
[8] D. Helbing and W. Yu, “Outbreak of cooperation among success-driven individuals under noisy condition,” PNAS, 2009.
[9] J. Valenzuela and C. Monterola, “Convective flow-induced short timescale segregation in a dilute bidisperse particle suspension,” International Journal of Modern Physics C, 2008.
[10] A. Fogleman, Hopeful Journeys: German Immigration, Settlement, and Political Culture in Colonial America. University of Pennsylvania Press, 1996.