Rikard Roitto, Associate Professor, Stockholm School of Theology
Conversion and Repentance Through Altruism in Two Modified
Versions of Hammond and Axelrod’s Simulation of Ethnocentric
Cooperation
Abstract
This paper explores the possibility of converting other agents to one’s own kind and one’s own
cooperation strategy through altruistic generosity, a feature that has never been built into a
simulation of cooperation strategies. This is done by modifying Hammond and Axelrod’s (2006)
simulation, which shows that ethnocentric behavior is the most successful strategy in a spatialized
game of cooperation where color is the only attribute that agents can act on. Their simulation is
altered in two different ways. First, the simulation is changed so that each agent can “convert”
neighboring agents to its own color with a low probability C through outgroup altruism toward noncooperative outgroup members. There is a probability R that the converted agent also “repents”,
that is, changes cooperation style to the cooperative agent’s style, after conversion. The result is that
the strategy of altruism can successfully compete with ethnocentric strategies if it leads to
conversion of the other player’s color around 1–2% of time, and change of cooperation style to
altruism around 0.4-0.5% of the time. Second, the simulation is altered so that only green agents can
convert the other colors through altruism. The most obvious result of this simulation is that green
agents become increasingly dominant with an increasing probability of conversion. More interesting
it that the overall proportion of altruists decreases and ethnocentrism becomes more dominant as a
cooperation strategy compared to the first modified simulation, although not as dominant as in the
original simulation by Hammond and Axelrod. That is, green can dominate the board through
conversion of other colors even though only some of the green agents are “evangelizing” altruists.
The simulations have bearing on the historical problem of why apparently self-destructive behaviors
of cooperation and non-self-defense towards outgroup individuals, conducted by some Christians
during the first centuries, seems to have contributed to the expansion of the Christian movement.
Introduction
In Hammond and Axelrod’s (2006) study, agents in a two-dimensional grid decide to cooperate or
defect with neighbors in four directions in games of prisoner’s dilemma. Agents have no memory of
what happened last round, so the decision whether to cooperate or defect is solely based on the
neighbor’s color. The “ethnocentric” strategy (cooperate with your own color, otherwise defect) is by
far more successful than both “egoistic” (cooperate with no one), generally “altruistic” (cooperate
with agents of all colors), and “cosmopolitan” (cooperate only with agents of a different color)
strategies. The foremost strength of the simulation is how stable the results are in favor of the
ethnocentric strategy, no matter how the parameters are tweaked. The dominance of the
ethnocentrists has been further established by the thorough examination of the simulation by Max
Hartshorn et al. (2013). Another strength is that it isolates the problem of ingroup-favoritism, since
color is the only variable that agents can act on. Therefore, I have chosen to their simulation as a
starting point in order to explore the effects of the possibility of conversion (change of color) and
repentance (change of cooperation style). The possibility of converting other agents has never before
been built into a simulation of cooperation strategies.
Simulations of cooperative behavior invariably show that unconditional cooperation with anyone is a
losing strategy. Rather, limited cooperation, restrained forgiveness, and cliquish cooperation, are the
winning strategies, which in the long run maximize resources. As a historian specialized in the earliest
centuries of Christianity, I can see that the historical data I work with does not entirely fit the logic of
these simulations. In the first centuries of Christianity, a certain degree unwarranted altruism
towards non-Christians and self-destructive abstinence from self-protective measures seem to have
been key factors that lead to the successful distribution of Christian religion in the population of the
Mediterranean world and beyond.
Already the pioneering work of Robert Axelrod (1984) shows that strategies which “forgive” – that is,
continue to cooperate with defectors – to a limited extent can be more efficient than tit-for-tatstrategies in games of iterated prisoner’s dilemma under conditions where the players make
mistakes with some frequency. However, ever-forgiving strategies are not very successful. Patrick
Grim (1995, 1996) demonstrates that the “spatialized” iterated prisoner’s dilemma, where one can
only cooperate with one’s neighbors in a grid and where one can copy the cooperation strategy of
one’s most successful neighbor, can lead to success for very cooperative strategies, forgiving
defectors about 2/3 of the time. Other simulations demonstrate that when defectors can be
punished and when non-punishers can be punished, this stimulates cooperative behavior (e.g.
Axelrod 1986; Boyd and Richerson 1992; Kendal et al. 2006). Several simulations establish the
effectiveness of cliquish cooperation. Some simulations demonstrate the effectiveness of
“ethnocentrism”, that is, favoring those similar to oneself (e.g. Hammond and Axelrod 2006;
Hartshort et al. 2013; Kim 2010; Riolo et al. 2001; 2002). Other simulations show that under
conditions where agents can establish cooperation partners, it pays to be very forgiving toward
partners but defect in cooperation with everyone else (e.g. Hales 2000; Hruschka and Henrich 2006).
Nevertheless, in all these simulations unconditional cooperation with defectors and outgroup
members is a losing strategy.
The just mentioned simulations do not take into account how altruistic behavior sometimes
influences the receiver of altruism. Some of our sources from early Christianity suggest that care of
outgroup members was one of the movement’s most important missionary strategies. In the
Gospels, Jesus teaches an ethics of unconditional forgiving, giving without distinction to anyone, nonretaliation, etc. (e.g. Matthew 5:38–48, 18:21–22), which according to the simulations above would
be a no-starter. It would be unrealistic to assume that all Christians during the first centuries actually
practiced this radical ethics. Nevertheless, during the first three expansive centuries of Christianity,
this seemingly self-destructive ethics seems to have been practiced at least to some extent. Several
sources even assume that the ethics of benefaction towards outsiders, non-retaliation, non-selfdefense and willingness to suffer, are vital to evangelization and conversion (e.g. Matthew 5:10–16,
38-48; Luke 10:25-37; 2 Corinthians 4; 1 Peter 3:13–4:18). Texts arguing for limiting altruism toward
outsiders are surprisingly rare. Several historians of early Christianity, particularly those who use
social-scientific perspectives, have concluded that the determination to cooperate beyond
established social categories was one of the most important factors behind the spread of Christianity
in the first centuries (e.g. Czachesz 2011; 2012; Runciman 2006; Stark 1996).
In the social world of the first Christians, the Roman empire of the first centuries A.D., authorities
were often corrupt and could not always trusted to resolve conflicts. Instead, society was organized
around various social networks that cooperated in a cliquish manner to ensure reciprocation. Family,
ethnicity, patron-client networks, etc., guaranteed reliable cooperation partners (Neufeld and
DeMaris 2011). That is, the general tendency of the social world of the first Christians fits into the
results of the above mentioned simulations of cliquish and ethnocentric behavior. Does this mean
that a movement that did not stick strictly to the strategy of ingroup-favoritism would have had a
significant disadvantage? As it seems, it was the other way around. Christianity spread precisely
because its cultural innovation to break with this logic.
If we are to translate what the historical data seem to indicate into a simulation, the sources claim
that there is a trade-off between cooperation with defectors outside your own group and
conversion. It is this effect that this paper simulates. How will the game change if there is a chance
that unwarranted altruistic and forgiving behavior causes conversion and sometimes even moral
reform? In this study, I will modify simulation of cooperation styles by Hammond and Axelrod, so
that it incorporates the possibility of conversion and repentance.
My theoretical justification for this modification is that humans are biologically evolved to learn how
to behave socially by imitation (Heyes and Galef 1996; Laland 2001). We behave like and affiliate
with those who we interact with. Most attempts to explain altruism tries to do so by proving that it is
really rational under certain circumstances. Altruism towards kin is easily explained by the fact that it
increases the fitness of the shared gene pool (Dawkins 1976). Altruism in established groups can be
explained by group level selection processes (Wilson 2002). Most work with simulations also work
from the assumption that a good simulation makes cooperation emerge because it is more rational
than other strategies. These kinds of explanations are relevant on a time scale of biological evolution,
but are not always applicable to cultural evolution within shorter historical time scales. Some human
behavioral patterns become frequent in a population even though they do not increase fitness.
(Think of how millions of people suddenly craved to play World of Warcraft.) They are just cultural
patterns that spread well in a population. Susan Blackmore (1999) has suggested that some forms of
altruism spread simply because altruists have more friends and therefore become “meme fountains”
who have influence over other people. (Meme theory is a controversial theory, but Blackmore’s
suggestion on altruism would be valid within the framework of almost any theory of cultural
influence.) Since we humans learn how to behave socially by imitation, altruism may spread in a
population even in situations where it does not increase fitness. It is just a matter of how contagious
the behavior is.
The First Modification of the Model: Conversion and Repentance
[The model can be found here: http://www.openabm.org/model/3880/version/1/]
The simulation of cooperation strategies by Hammond and Axelrod (2006) is a spatialized version of
the prisoner’s dilemma, where each agent decides whether to cooperate with its neighbors in four
directions. The space has wraparound borders so that all agents have four neighbors. Each agent has
a color and one of four cooperation strategies: cooperate with everyone (altruism), cooperate with
no one (egoism), cooperate only with neighbors of your own color (ethnocentrism), or cooperate
only with neighbor with a different color from your own (cosmopolitanism). The more successful an
agent is, the greater is its chance to reproduce in an adjacent empty site. Agents have no memory
and therefore no possibility to adapt depending on what happened last round. The only cognitive
capacity agents have is to see the color of its neighbors.
The main characteristic of the simulation is that the ethnocentric agents robustly come out as
winners, under a variety of conditions. Ethnocentrism is far more successful than both altruism,
egoism, and cosmopolitanism. Most importantly, since ethnocentrists can create cooperative clusters
yet isolate egoists of different colors, their strategy solves the problem of free-riders (at least
outgroup free-riders). As Hartshorn et al. (2013) point out, the ethnocentrists also have an
advantage in relation to altruists of other colors, since they can cooperate with their own cluster
while at the same time taking advantage of neighboring altruists.
Hammond and Axelrod (2006: 4) summarize the simulation as follows:
Each time period consists of four stages: immigration, interaction, reproduction, and
death.
1. An immigrant with random traits enters at a random empty site.
2. Each agent has its potential to reproduce (PTR) set to 12 percent. Each pair of
neighbors then interacts in a one-move prisoner’s dilemma in which each chooses
(independently) whether to help the other. Giving help has a cost—namely, a decrease
in the agent’s PTR by 1 percent. Receiving help has a benefit—namely, an increase in
the agent’s PTR by 3 percent.
3. Each agent is chosen in a random order and given a chance to reproduce with
probability equal to its PTR. Reproduction consists of creating an offspring in an
adjacent empty site, if there is one. An offspring receives the traits of its parent, with a
mutation rate of 0.5 percent per trait.
4. Each agent has a 10 percent chance of dying, making room for future offspring.
I have used Uri Wilensky’s (2003) implementation of this simulation in NetLogo and modified it so
that in a game between agenti and agentj there is a certain probability that whenever agenti gives to
a agentj although agentj would not give back, there is a probability C that agentj will “convert”, that
is, change color to the same color as agenti. If agenti successfully converts agentj, then there is a
probability R that agentj also “repents”, that is, changes its pattern of cooperation to the same
pattern as agenti. Expressed as pseudo-code, I have added the following to the original simulation:
If agenti gives to agentj AND color of agenti <> color of agentj then {
If agentj is an egoist OR agentj is an ethnocentrist {
If random < C then {
color of agentj = color of agenti
If random < R then {
cooperation-pattern of agentj = cooperation-pattern of agenti
}
}
}
}
IMG 1: Screen dump of the simulation in NetLogo, which is a modification of Wilensky’s (2003)
implementation.
Whenever an agent cooperates in a costly manner, there is a chance that the neighbor will convert
and become a more profitable future cooperation partner. First, there is a probability C that the
neighbor changes color, “converts”. If the neighbor converts, there is a probability R that the
neighbor changes cooperation style, “repents”, too. The probability that a non-cooperative neighbor
of a different color both converts and repents is thus CR. That is, altruism is contagious with the
probability CR. (It should be noted that the algorithm is constructed so that egoistic neighbors of the
agent’s own kind cannot “repent”.)
In the simulations, the environmental values were set to the standard values suggested by Hammond
and Axelrod (2006), with the addition of above mentioned modifications. Just like Hammond and
Axelrod, I measured the average of the final 100 periods of ten 2000-period runs. Different values of
C (0.00 to 0.10 with a 0.01 interval) and R (0.0 to 0.6 with a 0.1 interval) were tested to see how the
proportions of egoists, altruists, ethnocentrists and cosmopolitans would change.
Results
The egoists and the cosmopolitans are not particularly successful under any circumstances. However,
as R and C increases, the altruists become increasingly successful at the expense of the
ethnocentrists.
IMG 2: Ratio of altruists/ethnocentrists with increasing conversion propability (C), given repentance
rate (R) 0.0.
IMG 3: Ratio of altruists/ethnocentrists with increasing conversion propability (C), given repentance
rate (R) 0.5.
IMG 4: Ratio of altruists/ethnocentrists with increasing conversion probability (C), given repentance
rate (R) 0.0 to 0.6.
IMG 5: Ratio of cooperators/ethnocentrists with increasing repentance rate (R), given conversion
propability (C) 0.01.
IMG 6: Ratio of cooperators/ethnocentrists with increasing repentance rate (R), conversion
propability (C) 0.03.
IMG 7: Function field of the ratio of cooperators/ethnocentrists, given varying values of R and C. Each
line represents the ratio 1, 2, 3, etc. The leftmost line represents the threshold value 1 when
cooperators and ethnocentrists are equally successful.
With high values of C and R, altruists dominate the board completely, since they can convert anyone
who does not reciprocate. The simulation could be said to give an unfair advantage to the altruists
(and the cosmopolitans for that matter). However, the interesting question is not whether it is
possible to give the altruists this advantage, but how big the advantage of conversion has to be in
order to be able to compete with otherwise more successful strategies. The answer is: not that big.
The altruists break even with the ethnocentrists for instance if C is 0.01 and R is 0.5; or if C is 0.02 and
R is 0.2. Expressed differently, altruism is contagious with the probability CR, and altruists are as
successful as the ethnocentrists if CR is approximately 0.005 when C is 0.01; or if CR is approximately
0.004 when C is 0.02. In other words, if agents can afford to take the cost associated with
unwarranted altruism, it will of course decrease the resources of its members, but altruism does only
have to be a moderately successful conversion strategy in order to spread altruism across the board.
One possible interpretation of the result is that the altruistic strategy spreads well, even though it is
not beneficial for those exercising the behavior. Another possible interpretation is that given a high
enough value of C and R, the altruistic behavior will lead to more resources and therefore a higher
ratio of reproduction. Therefore, I measured the average number of reproductions before death for
ethnocentrists and altruists.
IMG 8: Function field of the ratio of the average reproductivity before death of cooperators and
ethnocentrists, given varying values of R and C.
The result is that the former interpretation is correct. Even though the prosperity of the altruists
improves with increasing C and R, it never quite pars with that of the ethnocentrists. The
reproduction success of the altruists never becomes more than about 60% of that of the
ethnocentrists. That is, conversion through altruism is dangerous but contagious. Nevertheless, it
should also be noted that altruism to some extent becomes less dangerous as C and R increases.
The Second Modification of the Model: Only One Group Can Convert Others
In the simulation above, all colors have the opportunity to convert others. However, in the Roman
Empire, most groups (religions, ethnic groups, collegia etc.) did not actively try to convert others to
their own group. Only a few groups (some branches of Judaism, the Mithras-cult, Christianity)
systematically recruited people from other groups. Therefore, when Christianity decided to
evangelize systematically, they had a significant advantage over groups that did not aspire to convert
people.
To build this into our simulation, we allow only green agents to convert others through altruism. The
simulation was run for different values of C and R in the same way as the first simulation.
Result
As we may have expected, green dominates the field s even at quite low values for C.
IMG 9: Function field of the proportion of green agents in the simulation, given varying C and R. The
leftmost line marks where the ratio of green agents/other agents is 1, the second leftmost line the
ratio 2, etc. (The erratic pattern in the rightmost half of the function field is just random variations,
due to mutation and immigration preventing green from reaching 100%.)
The graph shows that the success of green increases rapidly with an increasing C and stabilizes
around C=0.04 where the green agents outnumber other agents by about 9 to 1.
A more interesting result, however, is that R does not affect how successful the conversion efforts of
green is at all. This is perhaps a surprising result, given the result of the first simulation, where R had
a decisive impact on the success of altruism. The success of a strategy and the success of a group are
however two different phenomena. If we use this result to interpret the success of early Christianity,
we should at least consider the possibility that the efforts to convert people to the Christian faith
were far more important to the diffusion of the movement than the efforts to make new converts
altruistic towards outsiders. While this is not something that can be concluded solely on the basis of
a simulation, it seems reasonable.
Giving this advantage to one color dramatically decreases the amount of altruists on the board
compared to the first simulation. In the first simulation, altruists and ethnocentrists were equally
successful with C = 0.012 and R = 0.5 (see graphs above). However, when only green agents are
allowed to convert the others, the amount of altruists plunge:
IMG 10: The first graph shows how dominant green becomes if C = 0.012 and R = 0.5. The second
graph shows that the ratio of altruists is not higher in the green group than in other groups. The third
graph shows that the altruists of all colors (color green in the graph, labeled CC) are significantly less
successful than the ethnocentrists (color red in the graph, labeled CD) under these conditions.
Here, we have a situation where the evangelizing group (green) does not seem to need more altruists
than other groups in order to dominate the field through conversion, since they have the decisive
advantage of being the only group with evangelizing ambitions. If we examine the proportion of
altruists and ethnocentrists for green and other colors respectively, we see the following:
IMG 11: The left function field shows the proportion of altruists as compared to ethnocentrists for
green agents. The right function field shows the proportion of altruists as compared to ethnocentrists
for agents of other colors.
Our first observation is that altruism never becomes the dominant cooperation strategy for green
agents, even though it increases somewhat with increasing C and R. Among green agents,
ethnocentrists always outnumber altruists, as opposed to the situation in the first simulation when
agents of all colors were allowed to convert other agents. However, the ethnocentrists do not
dominate the board to the same extent as they do in Hammond and Axelrod’s original simulation.
The significance of this observation is that only some agents of an evangelizing group need to be
altruistic for conversion to be effective, if only one group evangelizes but the others do not. If we
translate this to reasonable assumptions about the spread of early Christianity, we can assume that it
was only necessary for some Christians to be altruistic toward outsiders in order to attract new
members. There was probably plenty of room for ingroup favoritism in Early Christianity.
Our second observation is that in the other (non-evangelizing) colors the altruists increasingly
outnumber the ethnocentrists. This might look weird, until one realizes that what is actually
happening is that the ethnocentrists become exceedingly rare when green altruists can convert all
non-green ethnocentrists with increasing probability. That is, with increasing C and R ethnocentrism
is a sure way to be converted to green. (This result cannot be used meaningfully for historical
interpretation, as far as I can see, but is just a side-effect of high values of C and R.)
Discussion
Hammond and Axelrod’s (2006) study, which this study modifies, indicates that ethnocentric
behavior very robustly out-competes both egoistic and altruistic behavior. The first simulation of his
study simulates the effect of a probability C of conversion (change of color) and R of repentance
(change of cooperation strategy) after conversion, whenever an altruist cooperates with an
ethnocentrist or egoist of another color. A rather small CR changes the balance of success between
generally altruistic agents and ethnocentric agents who only cooperate with their own kind. For
instance, altruists become as numerous as the ethnocentrists when C = 0.01; R = 0.5; CR =0.005. Or
when C=0.02; R=0.2; CR=0.004.However, in spite of this numeral success, individual altruistic agents
are not as successful as the ethnocentric agents in the sense that they do not reproduce as much as
the ethnocentrists before they die, not even with rather high values of C and R.
These simulations show that given the rather reasonable assumptions that we humans sometimes
affiliate with those who are good to us and imitate those who we interact with, altruism toward
outgroup individuals can spread within a population far beyond what could be considered rational,
even if recipients of altruism are only occasionally influenced by the altruist.
The second simulation, where only the color green is allowed to convert others, presents some
results that differ quite radically from the first simulation. When only one group is allowed to convert
the others, the altruists never outnumber the ethnocentrists, even though the altruists become
somewhat more successful with increasing C and R. Green only needed some of its agents to be
altruistic toward outsiders in order to convert other colors and rule the board. Moreover, the
dominance of the converting color green is only increasing with C but not with R.
Generosity and abstinence from self-defense as a strategy for evangelization in Early Christianity is
the historical problem that prompted me as a historian to conduct these simulations. But can we use
the results of the simulations to understand the historical expansion of Christianity in the first
centuries? The main problem here is whether it is historically realistic to assume that altruistic
behavior in early Christianity had an effect on conversion and repentance with a rate analogous to
the numbers C and R in the simulations. I believe this question is more or less impossible to answer,
since we do not have quantifiable historical data from this period, and if we had such data, what time
span of altruistic cooperation in real life would be equal to one tick in a simulation? We can only
conclude that the simulation confirms that even a rather small success rate of conversion and
repentance can make evangelization through altruism successful. Especially under conditions where
only one group evangelizes but the others do not (the second simulation), as was the case in Early
Christianity, the risk of outgroup altruism is balanced even by a rather small success-rate of
conversion. The results of this study’s modifications of Hammond and Axelrod’s simulation confirms
that an evangelizing movement operating in an ethnocentric environment can be successful if it
exercises altruistic behavior towards outsiders, given the not too far-fetched assumption that
altruism occasionally affected the loyalties and behavior of the receiver of altruism. Another
important finding from the simulations for anyone interested in the expansion of Early Christianity is
that only some agents in the evangelizing population have to be altruistic. That is, one does not have
to make the unrealistic assumption that each and every early Christian was a saint, in order to
understand how the altruism of some individuals contributed to the diffusion of the movement.
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