Ethnocentrism, religion, and austerity: a science poster for the humanities

Artem Kaznatcheev and I presented a poster on May 4th at the University of British Columbia to a highly interdisciplinary conference on religion. The conference acronym is CERC, which translates as Cultural Evolution of Religion Research Consortium. Most of the 60-some attendees are religion scholars and social scientists from North American and European universities. Many are also participants in a large partnership grant from the Social Sciences and Humanities Research Council of Canada (SSHRC), spearheaded by Ted Slingerland, an East Asian scholar at UBC. Some preliminary conversations with attendees indicated considerable apprehension about how researchers from the humanities and sciences would get on. Many of us are familiar with collaborative difficulties even in our own narrow domains. Skepticism was fairly common.

As far as I know, our poster was the only computer simulation presented at the meeting. We titled it Agent-based modeling of the evolution of “religion”, with scare quotes around religion because of the superficial and off-hand way we treated it. Because we know from experience that simulations can be a tough sell even at a scientific psychology conference, we were curious about whether and how this poster would fly in this broader meeting.

We opted for a poster composed almost entirely of graphics. This makes for convenient talking points and does not demand much reading, which is in any case anathema to social interaction and difficult to do in the noisy environment of a poster session.

The major theme of this SSHRC grant is that humans might have evolved religion because religion enforces cooperation. Although I was late joiner of the SSHRC grant, our poster addressed two of the five major sub-questions posed in the grant application, from the standpoint of simulations of biological evolution: whether cooperation is inherently parochial, and whether cooperation is enhanced by existential insecurity. The left column of the A1-sized poster dealt with the parochial question, while the right column dealt with the insecurity question.

Cooperation is parochial

The table on the left shows the net outcomes for player 1 in a well-known non-zero-sum game which highlights the essential dilemma of cooperation. Each of two players can either cooperate with the other or not. The payoff for the row player is computed as the benefit of receiving cooperation minus the cost of giving it. If player 1 cooperates with a cooperator, player 1 gets the benefit of receiving cooperation and loses the cost of giving it. If player 1 cooperates against a defector, player 1 incurs the cost of giving cooperation but enjoys no benefit (the so-called sucker’s payoff). If player 1 defects against a cooperator, player 1 enjoys the benefit of cooperation without incurring any cost (the so-called temptation to defect). When defection is mutual, the outcome is 0, as no cooperation occurs.

Without knowing what the other player will do, mutual defection is the rational solution (Nash equilibrium). However, as long as benefit exceeds cost, which it often does in specialized populations, the best overall outcome for both parties is mutual cooperation (Pareto optimum). Because of the competitive nature of natural selection, cooperation between biological agents has been a mystery since Darwin, and yet cooperation is extremely common across biology. One explanation is that species have somehow evolved to cooperate.

In every evolutionary cycle of our computer simulations, each agent interacts with each of its immediate
von Neumann neighbors by playing a one-shot prisoner’s dilemma game, with the game’s outcomes added to the agent’s potential to reproduce. An agent can reproduce if a randomly-drawn proportion is less than this modified potential, and if there is an empty adjacent space for the offspring.

The world is a 50-by-50 toroidal (to ensure that each agent has the same number of potential neighbors) lattice, as illustrated in this sample figure after 50 evolutionary cycles. Each agent has three simple genes. Two genes specify cooperation or defection against in-group or out-group neighbors, and the third specifies a group tag so that an agent can identify whether their neighbor is from the same group. The world starts empty, and each evolutionary cycle a new, randomly-constructed agent immigrates into a randomly-chosen cell. Cloned offspring are just like their parent, except for a tiny mutation rate.

At 50 cycles, the world is still relatively empty, but beginning correlations between strategy, tag, and location can already be detected. Group tag is indicated by color, and strategy by letter. Selfish agents don’t cooperate with anyone, traitors cooperate outside but not inside their group, ethnocentrics cooperate inside but not outside their group, and humanitarians cooperate with everyone. These sample plots are for a smaller 25×25 lattice so that the world’s state is easier to view.

After 500 evolutionary cycles, this same world is much fuller, and exhibits ethnocentric dominance as seen on the left. Below right, a plot of strategy means and standard errors shows results for 50 worlds across 1000 cycles. When the population saturates, around 200 cycles, ethnocentrics start to pull away from their close competitor, humanitarians. The reason for this separation is that humanitarians cooperate across group frontiers, incurring the sucker’s payoff, while ethnocentrics do not, a benefit of succumbing to the temptation to defect. Ethnocentric dominance is robust across wide variation in parameter settings, as long as the temptation to defect exceeds the cost of cooperation. Selfish and traitorous strategies never get much traction because they don’t take care of their own kind. Due to mutations, these free-rider strategies do not disappear (Hartshorn, Kaznatcheev, & Shultz, in press).

It is also interesting to note that ethnocentric cooperation is common across a wide range of species, from bacteria and plants to people (witness voting patterns and ethnic cleansing). So, yes, cooperation does tend to be parochial, at least with viscous populations and group tags, and when group detection is not too costly (Kaznatcheev, 2010a, 2010b; Kaznatcheev & Shultz, 2011). Evolution favors ethnocentrism because it keeps cooperation high (here around 75% of interactions), sometimes even higher than humanitarianism does.

Austerity promotes cooperation.

We implemented existential insecurity by varying the austerity of the environment. In simulations, an austere environment can be created by either increasing death rate or decreasing birthrate. We varied birthrate, while holding death rate constant. This preliminary experiment was done similarly to the parochial simulation except for the absence of group tags, to keep things simple. The only two strategies here are cooperation or defection.

This figure on left shows standard errors of mean proportions of cooperation. There are more cooperative behaviors in austere than in easy worlds.

Below left, standard error values around the mean number of strategies in an austere environment reveal more cooperative strategies than defection strategies. Below right, the opposite is true for an easy environment.

So, in this very abstract study, austerity favors the evolution of cooperation. We do still need to explore the parametric space of this simulation and run it with group tags.

How well was all this received by our new multi-disciplinary colleagues? They seemed to get it, even if
they were unfamiliar with the methodology, they readily recognized the implications for religion, and they found it very interesting. Some even liked our deliberately evocative names for cooperation strategies. Most of the time, I inadvertently (or sometimes intentionally) neglected to mention that different religions could be represented by group tags. To my initial surprise, our poster goers did not need to hear me say that – they automatically inferred it. None objected to our superficial characterization of religion. Perhaps this connection is already evident from the many religion-fueled conflicts around the world, currently and throughout human history.

Of course, our simulations are considerably more general than religious tags. Unlike other species with relatively fixed group tags, humans have proven to be quite flexible in the tags they use to discriminate in- from out-group members. Humans use skin color, language, language accent (Cohen, 2012), location, and many other initially arbitrary tags to determine what sort of agent they are dealing with. In many urban centers around the world, it is important not to wear the wrong-color shirt in a particular neighborhood. Such tags are cultural products, but the important point is that humans are genetically prepared to use group tags in support of their ethnocentric patterns of cooperation.

Many excellent questions were asked about the poster, some of which could be answered with reference to
other simulations from our lab and this blog. To cite a single example, Professor Lyle Eslinger, a particularly astute religious-studies colleague from U. of Calgary noted that our austerity simulation reminded him of the pioneering (1902) observations of Prince Kropotkin in the austere regions of Siberia. Cooperation was more evident to Kropotkin than the competition noted by Darwin in the easier Galápagos Islands. Not many computational modelers would have spontaneously come up with that reference.

Post-conference, I received a batch of papers from Ian MacDonald, a student of David Sloan Wilson at U. of
Binghamton, who gave a conference talk on their study of church-goers in that city. The gist of these correlational studies is that religious activity is positively related to austere environmental conditions (Delamontagne, 2010; Gill & Lundsgaarde, 2004; Immerzeel & van Tubergen, 2013; Ruiter & van Tubergen, 2009). In a complementary fashion, our abstract simulation shows that increased austerity causes an increase in cooperation and cooperative strategies; perhaps additional simulations will provide further insights and qualifications.

We are pleased at this reception and at having an excellent opportunity to meet and interact with our new
colleagues.

It turns out that we humans and these abstract computer agents are not alone in exhibiting a positive relation between environmental austerity and cooperation. This figure shows that yeast populations with pure cooperators are better able to survive a sudden and harsh environmental change than are mixed populations of cooperators and defectors. All six pure-cooperator populations and only one of six mixed populations were able to withstand a sudden drop in the food supply on day 3 (Sanchez & Gore, 2013).

Notwithstanding all this harmony, I do anticipate some downstream controversy around the notion of group selection as an evolutionary mechanism. It is relevant to note that the effects in our simulations are products of simulated individual selection. Our groups are emergent properties of natural selection acting on the reproductive potentials of individual agents. Notions of group selection are not necessary to explain our simulation results.

References

Cohen, E. (2012). The evolution of tag-based cooperation in humans: the case for accent. Current Anthropology, 53, 588-616.

Delamontagne, R. G. (2010). High religiosity and societal dysfunction in the United States during the first decade of the twenty-first century. Evolutionary Psychology, 8(4), 617-657.

Gill, A., & Lundsgaarde, E. (2004). State welfare spending and religiosity: a cross-national analysis. and Society, 16(4), 399-436.

Hartshorn, M., Kaznatcheev, A., & Shultz, T. R. (in press). The evolutionary dominance of ethnocentric cooperation. Journal of Artificial Societies and Social Simulation.

Immerzeel, T., & van Tubergen, F. (2013). Religion as reassurance? Testing the insecurity theory in 26 European countries. European Sociological Review, 29(2), 359-372.

Kaznatcheev, A. (2010a). The cognitive cost of ethnocentrism. In S. Ohlsson & R. Catrambone (Eds.),Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 967-971). Austin, TX: Cognitive Science Society. [pdf]

Kaznatcheev, A. (2010b). Robustness of ethnocentrism to changes in inter-personal interactions. In Complex Adaptive Systemes – AAAI Fall Symposium. Arlington, VA. [pdf]

Kaznatcheev, Artem, & Shultz, Thomas R. (2011). Ethnocentrism maintains cooperation, but keeping one’s children close fuels it. Proceedings of the 33rd Annual Conference of the Cognitive Science Society, 3174-3179 [pdf]

Ruiter, S., & van Tubergen, F. (2009). Religious attendance in cross-national perspective: a multilevel analysis of 60 countries. American Journal of Sociology,
115(3), 863-895.

Sanchez, A., & Gore, J. (2013). Feedback between population and evolutionary dynamics determines the fate of social microbial populations. PLoS Biology, 11(4), e1001547.

Extra, Special Need for Social Connections

There is now evidence that the payoffs designed by researchers are not the only source of variation in human strategies in two-player symmetric games. In many cases, discrepancies from behavior predicted by variation in payoffs might be explained by social factors such as avoidance of inequality, desire for social harmony, or likability of the opponent.

An interesting example of a study showing the extra, special need for social connections had a woman, in an fMRI scanner playing iterative prisoner’s dilemma against either another woman or a computer (Rilling et al., 2002). Dollar payoffs for the row player are shown in Table 1 for each round. The authors compared BOLD signals across cells, conditions, and brain areas of interest. They found no main effects of the actions of either the scanned or non-scanned player. However, the interaction of these two actions was statistically significant. The comparison indicating the importance of social connections contrasted the sum of the blue and pink cells against the two white cells in Table 1. The Nash equilibrium is in the red cell at mutual defection, as the financial payoff is always better for a defector, whether the other (column) player cooperates or defects. The Pareto optimum, or best overall outcome, is in the blue cell, characterized by mutual cooperation. The BOLD signal was stronger, suggesting more neuronal activity, in these two colored cells with matched actions than in the two white cells with unmatched actions (one player cooperates and the other defects) in several brain areas that have been linked with reward processing: nucleus accumbens, caudate nucleus, ventromedial frontal/orbitofrontal cortex, and rostral anterior cingulate cortex. Because the sums of rewards for this comparison are equal (at $3), this pattern in the BOLD signal cannot be explained by game payoffs per se.

Table 1: Dollar Payoffs in Prisoner’s Dilemma Game

$	Cooperate	Defect
Cooperate
Defect

I computed the proportions of decision combinations, against a human partner, and averaged them across four different conditions. Table 2 shows that the two most common decision combinations are CC and DD, and CC (with the highest payoff) being the most common. This preference for identical decisions is mindful of the notion of Hofstadter’s (1985) super-rationality. He assumed that responses to a symmetric game will be the same for all super-rational players. They could find this strategy by maximizing the payoff to each player, on the assumption of sameness (i.e. only playing on the diagonal). The only two cells with identical responses are the colored ones – CC and DD. Because the payoff is higher for CC than for DD, they would both cooperate. If not everyone is super-rational, that could account for variation in the results.

Table 2: Average Frequency of Decision Combinations

Frequency	Cooperate	Defect
Cooperate
Defect

Much was also made of higher BOLD signals in the blue cell (mutual cooperation) than in the other three cells, perhaps because that seemed compatible with the authors’ hypothesis that social cooperation was especially rewarding. However, that particular comparison result could alternatively be explained by reward size alone, as the average reward is $2 in the blue cell and $1.33 in the other three cells, or by Hofstadter’s super-rationality. Nonetheless, the former contrast between the colored and white cells does suggest that something is going on which is not compatible with mere dollar payoffs. This conclusion was buttressed by post-game interviews with scanned participants, who often said that mutual cooperation was particularly satisfying, especially when compared with defect-cooperate which was uncomfortable because it provoked guilt over having profited at another’s expense or realization that the outcome would provoke subsequent defection by the exploited other player. Unfortunately, the authors do not present quantitative data on participant satisfaction with each of the four outcomes.

A more general, and very skillful, review of the literature on the neural basis of decision making in symmetric two-player games also supports the idea that designed payoffs may not capture all of the variance (Lee, 2008). Some decisions may be socially motivated. Players may decide what to do on the basis of wanting to enhance or reduce the well-being of their opponent. Such decisions are related to activity in brain circuits implicated in evaluation of rewards.

Among the highlights in Lee’s review:

1. Table 3 shows the row player’s utility function for prisoner’s dilemma adjusted to accommodate aversion to inequality (Fehr & Schmidt, 1999). Parameters and reflect sensitivity to disadvantageous and advantageous inequality, respectively. When , mutual cooperation and mutual defection both become Nash equilibria (Fehr & Camerer, 2007). More generally, the row player’s utility function can be defined as , where and are inequalities disadvantageous and advantageous to player 1, respectively. It is assumed that and . For a given payoff to player 1, is maximal when , expressing an equality preference. Presumably, some instantiation of this equation could account for the preference for identical decisions in the Rilling et al. study.
2. Aversion to inequality is also evident in other games (e.g., dictator, ultimatum, trust). For example, in the dictator game, a dictator receives an amount of money and can donate some of it to a recipient. Because any donation reduces the payoff to the dictator, the donation amount indexes altruism. Dictators tend to donate about 25% of their money (Camerer, 2003).
3. As predicted by temporal-difference learning algorithms, future rewards are often exponentially discounted during games, assuring that immediate rewards carry more weight than future rewards (Camerer, 2003; Erev & Roth, 1998; Lee, Conroy, McGreevy, & Barraclough, 2004; Lee, McGreevy, & Barraclough, 2005; Sandholm & Crites, 1996). This could be important in games such as iterative prisoner’s dilemma, in deciding whether to defect to gain an immediate reward (if the opponent cooperates) vs. cooperate in order to encourage an opponent to cooperate in future cycles.
4. In some such algorithms, fictive reward signals can be generated by inferences based on a person’s knowledge, and these have been shown to influence financial decision making (Lohrenz, McCabe, Camerer, & Montague, 2007).
5. Game payoffs can be estimated by a player mentally simulating hypothetical interactions (Camerer, 2003).
6. In games, reputation and moral character of an opponent influence a player’s tendency to cooperate with that opponent (Delgado, Frank, & Phelps, 2005; Nowak & Sigmund, 1998; Wedekind & Milinski, 2000).
7. In primates, midbrain dopamine neurons encode reward prediction errors (Schultz, 2006; Schultz, Dayan, & Montague, 1997) and decrease their activity when expected reward does not occur (Bayer & Glimcher, 2005; Roesch, Calu, & Schoenbaum, 2007; Schultz, et al., 1997). The following brain areas modulate their activity according to reward variation: amygdala, basal ganglia, posterior parietal cortex, lateral prefrontal cortex, medial frontal cortex, orbitofrontal cortex, striatum, insula. Imaging studies show that many of these brain areas are also active during social decision making (Knutson & Cooper, 2005; O’Doherty, 2004).
8. Some of the variability in social decisions may be due to genetic differences. For example, the minimum acceptable offer in ultimatum games is more similar between MZ than DZ twins (Wallace, Cesarini, Lichtenstein, & Johannesson, 2007). The ultimatum game is similar to the dictator game because a proposer offers some of the money to a recipient, who then can accept or reject the offer. If the offer is rejected, then neither player receives money. The mean offer in ultimatum games is about 40%, suggesting that proposers are motivated to avoid rejection.
9. Hormones can influence decisions in social games. For example, oxytocin increases the amount of money invested during trust games (Kosfeld, Heinrichs, Zak, Fischbacher, & Fehr, 2005). In the trust game, an investor invests a proportion of her own money. This money then is multiplied, and then transferred to a trustee. The trustee then decides how much of this transferred money is returned to the investor. The amount invested by the investor measures the trust of the investor in the trustee, and the amount of repayment reflects the trustee’s trustworthiness.

Table 3: Payoffs for Prisoner’s Dilemma, Adjusted for Inequality Aversion (adapted from Fehr & Camerer, 2007)

$	Cooperate	Defect
Cooperate
Defect

In conclusion, evidence is accumulating that objective payoffs designed for two-person, symmetric games may not explain all of the observed psychological phenomena. The door is now open for more serious investigation of social factors. Particularly in humans, and to a lesser extent in other primates, a player’s perceived payoffs may differ from the designed payoffs. Such payoff discrepancies could merit further exploration.

In terms of methodology, two ideas from these two papers stand out. One is the revised utility equation that supports inequality aversion (point 1 in the list of highlights from Lee’s review). The other is the simple technique of asking participants to evaluate the various decision combinations after Rilling et al.’s prisoner’s dilemma games. The utility equation could be helpful in modeling, and participant evaluation would be a cheap way to assess people’s evaluation of game outcomes.

Taken together, the material reviewed here could be very relevant to our planned research on objective and subjective rationality. Artem’s blog post on the possible discrepancy between objective and subjective rationality postulated that the objective game payoffs used by evolution may differ from human participants’ perceived reward values. Also relevant is Marcel’s post on habitual selfish agents and rationality, in which he points out that apparent irrationality is in game playing could result from rational processes being applied to subjectively perceived rewards that differ from reality. The evidence reviewed here (from Rilling et al., 2002 and Lee, 2008) identifies particular ways in which humans’ perceived reward values differ from experimenter-designed game payoffs. Inequality avoidance could be a key strategy with which to start exploring these phenomena, particularly in view of the payoff tweaks discussed in point 1 and Table 3.

References

Bayer, H. M., & Glimcher, P. W. (2005). Midbrain dopamine neurons encode a quantitative reward prediction error signal. Neuron, 47, 129-141. [NIH html]

Camerer, C. F. (2003). Behavioral game theory: Experiments in strategic interaction. Princeton, NJ: Princeton University Press.

Delgado, M. R., Frank, R. H., & Phelps, E. A. (2005). Perceptions of moral character modulate the neural systems of reward during the trust game. Nature Neuroscience, 8, 1611-1618. [pdf]

Erev, I., & Roth, A. E. (1998). Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria. The American Economic Review 88, 848-881. [pdf]

Fehr, E., & Camerer, C. F. (2007). Social neuroeconomics: the neural circuitry of social preferences. Trends in Cognitive Sciences, 11, 419-427. [pdf]

Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics, 114, 817-868. [pdf]

Hofstadter, D. R. (1985). Metamagical themas. New York: Basic Books.

Knutson, B., & Cooper, J. C. (2005). Functional magnetic resonance imaging of reward prediction. Current Opinion in Neurology, 18, 411-417. [pdf]

Kosfeld, M., Heinrichs, M., Zak, P., Fischbacher, U., & Fehr, E. (2005). Oxytocin increases trust in humans. Nature, 435, 673-676. [pdf]

Lee, D. (2008). Game theory and neural basis of social decision making Nature Neuroscience, 11 (4), 404-409 DOI: 10.1038/nn2065 [NIH html]

Lee, D., Conroy, M. L., McGreevy, B. P., & Barraclough, D. J. (2004). Reinforcement learning and decision making in monkeys during a competitive game. Brain Research: Cognitive Brain Research, 22, 45-58. [link]

Lee, D., McGreevy, B. P., & Barraclough, D. J. (2005). Learning and decision making in monkeys during a rock-paper-scissors game. Brain Research: Cognitive Brain Research, 25, 416-430. [link]

Lohrenz, T., McCabe, K., Camerer, C. F., & Montague, P. R. (2007). Neural signature of fictive learning signals in a sequential investment task. Proceedings National Academy of Sciences U.S.A., 104, 9493-9498. [link]

Nowak, M. A., & Sigmund, K. (1998). Evolution of indirect reciprocity by image scoring. Nature, 393, 573-577. [pdf]

O’Doherty, J. P. (2004). Reward representation and reward-related learning in the human brain: insights from neuroimaging. Current Opinion in Neurobiology, 14, 769-776. [pdf]

Rilling, J. K., Gutman, D. A., Zeh, T. R., Pagnoni, G., Berns, G., S., & Kilts, C. D. (2002). A neural basis for social cooperation. Neuron, 35, 395-405. [pdf]

Roesch, M. R., Calu, D. J., & Schoenbaum, G. (2007). Dopamine neurons encode the better option in rats deciding between differently delayed or sized rewards. Nature Neuroscience, 10, 1615-1624. [NIH html]

Sandholm, T. W., & Crites, R. H. (1996). Multiagent reinforcement learning in the iterated prisoner’s dilemma. Biosystems, 37, 147-166. [pdf]

Schultz, W. (2006). Behavioral theories and the neurophysiology of reward. Annual Review of Psychology, 57, 87-115. [pdf]

Schultz, W., Dayan, P., & Montague, P. R. (1997). A neural substrate of prediction and reward. Science 275, 1593-1599. [pdf]

Wallace, B., Cesarini, D., Lichtenstein, P., & Johannesson, M. (2007). Heritability of ultimatum game responder behavior. Proceedings of the National Academy of Sciences USA, 104, 15631-15634. [link]

Wedekind, C., & Milinski, M. (2000). Cooperation through image scoring in humans. Science, 228, 850-852. [link]

Fewer Friends, More Cooperation

Cooperation is fundamental to all social and biological systems. If cells did not cooperate, multi-cellular organisms would never have evolved [1]. If people did not cooperate, there would be no nation states [2]. But this wide-scale cooperation is somewhat of a mystery from the perspective of Darwinian evolution, which would seem to favor competition for scarce resources and reproductive success over cooperation. Cooperation incurs a cost to provide a benefit elsewhere. Indeed, a basic finding in agent-based computer simulations with unstructured populations is that evolution favors defection over cooperation [3]. As a result, there has been intensive research into spatially structured populations, attempting to explain the pervasive cooperation seen in nature. Under certain realistic spatial conditions, an agent is more likely to encounter members of its own gene pool than would be expected by chance; and this allows cooperation to evolve [4].

A particularly important study in this tradition is that of Ohtsuki and colleagues [5] at Harvard’s productive Program for Evolutionary Dynamics. Their complex mathematical derivations and computer simulations conveniently conform to a rather simple rule: that evolution favors cooperation if the benefit of receiving cooperation b divided by the cost c of giving it exceeds the average number of neighbors k in the population. Or, b/c > k, exactly parallel to Hamilton’s famous rule where the b/c ratio had to exceed relatedness r in order for cooperation to thrive [6].

Ohtsuki et al.’s simulations find that this rule holds in a wide variety of graph structures: lattices, cycles, random regular graphs, random graphs, and scale-free networks. Square lattices involve either von Neumann (k = 4) or Moore (k = 8) neighbors. Cycles involve a circular arrangement of agents where k = 2. In random regular graphs, the links between agents are random except that every agent has an equal number of links (k). Random graphs are similar except that agents have an average of k links, rather than exactly k. Scale-free networks are hub-like graphs generated according to the method of preferential attachment: an agent links with others in proportion to the other’s connectivity [7].

Similar results obtain for two different reproductive schemes: death-birth and imitation. For death-birth updating, in each time step a random individual is chosen to die, and its neighbors compete for the empty site in proportion to their fitness. At each cycle of imitation updating, a random agent keeps its own strategy or imitates a neighbor’s strategy proportional to their fitness. In all cases, fitness is determined by the outcome of each agent’s interactions with its neighbors.

Death-birth updating: Image from Ohtsuki et al. 2006

In this graph [5], a blue co-operator competes with a red defector for newly-freed location via death-birth updating. The co-operating candidate’s fitness is 2b-4c because it receives cooperation from two neighboring co-operators and gives cooperation to four neighbors. The defecting candidate’s fitness is b because it receives cooperation from one co-operator and gives no cooperation.

For imitation updating, cooperation evolves as long as b/c > k + 2, the plus 2 because each agent is effectively its own neighbor.

The last sentence of the Ohtsuki paper [5] nicely summarizes the results: “The fewer friends I have, the more strongly my fate is bound to theirs.” This derives from the effect of k, which varies from 2-10. As k decreases, it is exceeded by smaller values of the b/c ratio.

In a very crowded literature, this paper is particularly notable for including both simulations and mathematical analysis; in effect, the simulation results provide empirical confirmation of the mathematics. Typical studies use only one of these methods, leaving readers to wonder whether the results would generalize to parameter settings other than those used in simulations, or whether the mathematical analysis was done properly or can predict empirical simulation results. The paper is also notable for including a fairly wide variety of graph structures. More typically, one sees results for only one particular graph, most often a square lattice. In all of these ways, the Ohtsuki et al. paper serves as inspiration for future theoretical work on evolution.

References

1.           Axelrod, R. and W.D. Hamilton, The evolution of cooperation. Science, 1981. 211: p. 1390-1396.

2.           Wedekind, C. and M. Milinski, Cooperation through image scoring in humans. Science, 2000. 228: p. 850-852.

3.           Nowak, M.A., Evolutionary dynamics2006, Cambridge, MA: Harvard University Press.

4.           Lieberman, E., C. Hauert, and M.A. Nowak, Evolutionary dynamics on graphs. Nature, 2005. 433: p. 312-316.

5.           Ohtsuki, H., Hauert, C., Lieberman, E., & Nowak, M. (2006). A simple rule for the evolution of cooperation on graphs and social networks Nature, 441 (7092), 502-505 DOI: 10.1038/nature04605

6.           Hamilton, W.D., The genetical evolution of social behaviour, I. Journal of Theoretical Biology, 1964. 7: p. 1-16.

7.           Santos, F.C. and J.M. Pacheco, Scale-free networks provide a unifying framework for the emergence of cooperation. Physical Review Letters, 2005. 95.