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Risk aversion as an evolutionary adaptation
Arend Hintze1,3,∗, Randal S. Olson2,3, Christoph Adami1,3, Ralph Hertwig4
1Department of Microbiology and Molecular Genetics
2Department of Computer Science and Engineering
3BEACON Center for the Study of Evolution in Action
Michigan State University, East Lansing, MI, USA
4Center for Adaptive Rationality
Max Planck Institute for Human Development, Berlin, Germany
∗E-mail: hintze@msu.edu
Abstract
Risk aversion is a common behavior universal to humans and animals alike. Economists have
traditionally defined risk preferences by the curvature of the utility function. Psychologists
and behavioral economists also make use of concepts such as loss aversion and probability
weighting to model risk aversion. Neurophysiological evidence suggests that loss aversion has
its origins in relatively ancient neural circuitries (e.g., ventral striatum). Could there thus be
an evolutionary origin to risk avoidance? We study this question by evolving strategies that
adapt to play the equivalent mean payoff gamble. We hypothesize that risk aversion in the
equivalent mean payoff gamble is beneficial as an adaptation to living in small groups, and
find that a preference for risk averse strategies only evolves in small populations of less than
1,000 individuals, while agents exhibit no such strategy preference in larger populations.
Further, we discover that risk aversion can also evolve in larger populations, but only when
the population is segmented into small groups of around 150 individuals. Finally, we observe
that risk aversion only evolves when the gamble is a rare event that has a large impact
on the individual’s fitness. These findings align with earlier reports that humans lived in
small groups for a large portion of their evolutionary history. As such, we suggest that rare,
high-risk, high-payoff events such as mating and mate competition could have driven the
evolution of risk averse behavior in humans living in small groups.
1
arXiv:1310.6338v1 [q-bio.PE] 23 Oct 2013
Risk aversion as an evolutionary adaptation 2
Introduction
When people are faced with dicey decisions, a well-documented trend holds (Bernoulli, 1954;
Pratt, 1964; Joseph Arrow, 1965): If the stakes are sufficiently high, people are risk averse.
Risk averseness is usually described as a resistance to accept a deal with risky payoff as
opposed to one that is less risky or even safe, even when the expected value of the safer
bargain is lower. This tendency can be explained in two ways. First, the classical economists’
account of risk aversion is in terms of the shape of the utility function in expected utility
theory. Specifically, the curvature of the utility function is interpreted to measure the agent’s
risk attitude, and the more concave the utility function, the more risk averse the agent will be.
A concave utility function corresponds to the notion of diminishing marginal utility of wealth
according to which “the additional benefit which a person derives from a given increase of his
stock of a thing, diminishes with every increase in the stock that he already has” (Marshall,
1920, p. 79). Second, cumulative prospect theory, perhaps the most influential descriptive
account of decision making under risk choice among psychologists and behavioral economists,
models risk aversion in terms of three different but related concepts: diminishing marginal
utility, loss aversion (i.e., the pain of losses is felt stronger than the joy of equivalent gains),
and probability weighting (i.e., the elevation of the weighting function and thus the degree
of over/underweighting of small probabilities of gains and losses, respectively) (Kahneman
and Tversky, 1979; Tversky and Kahneman, 1992).
Risk aversion is also observed outside of economic decisions. For example, foraging ani-
mals actively avoid foraging in an area if they cannot reliably find food there (Smallwood,
1996; Kachelnik and Bateson, 1996). This work suggests that the risk attitude of both hu-
mans and animals may be shaped by a common fundamental principle (Marsh and Kacelnik,
2002). Moreover, there is considerable evidence from cognitive neuroscience that loss aver-
sion has a neural basis (Trepel et al., 2005). Neurophysiological measurements suggests that
different regions of the brain process value and risk assessment (Christopoulos et al., 2009;
Symmonds et al., 2010). This work also implies that the neural circuitry that encodes risk
aversion (or its building blocks such as loss aversion) is phylogenetically ancient. However,
the origin of this neurally encoded risk aversion is rarely discussed (but see Okasha (2007);
Schulz (2008); Stern (2010)), even though there is strong evidence that risk-taking behavior
has significant genetic components (Cesarini et al., 2009; Bell, 2009). Here, we suggest that
selective pressures acting during the evolution of human populations in the past can explain
risk averse behavior.
To test whether people are risk averse or risk prone, subjects are usually presented with
a monetary gamble where there is a safe choice and a risky choice. For example, the safe
choice rewards the subject with a fixed payoff Pwith 100% certainty, whereas the risky
choice rewards the subject with a higher payoff Q(Q > P ) only half of the time. In this
equivalent mean payoff gamble (Silberberg et al., 1988), the rewards are designed such that
the mean payoff of the safe and the risky choice is the same. In other words, as long as the
same choice is repeated during the lifetime or over evolutionary time, both risk prone and
risk averse choices have the same payoff in the long run. Thus, no choice should be preferred.
Risk aversion as an evolutionary adaptation 3
In contrast to the equivalent mean payoff gamble studied here, it has been shown (Clark
and Yoshimura, 1993; Yoshimura and Clark, 1991; Robson, 1996) that if the risky choice
comes with a higher payoff than the certain one, evolution will favor risk prone behav-
ior. Similarly, the certain choice is favored if it has a higher payoff than the risky one.
While geometric mean fitness maximization or bet hedging can influence the payoffs re-
ceived (Donaldson-Matasci, 2008), these strategies do not affect the payoff in the equivalent
mean payoff gamble. Geometric mean fitness maximization and bet hedging requires multiple
games, which is not allowed in the gamble studied here. Utility theory also does not explain
risk aversion because it makes the same prediction for the certain and risky choice because
their payoffs are identical. While prospect theory correctly predicts the human choice, it
does not provide a reason for why these choices are beneficial or adaptive, which is the focus
of this study.
Of course, the gamble described above is a crude simplification of human choice under
risk, which can be shaped by many other factors. External factors such as framing (Tversky
and Kahneman, 1981), how the odds are presented (Hoffrage et al., 2000), or if the decision
has to be made from experience or from description (Hertwig and Erev, 2009) play a role in
human decision making under risk. The relative value of the payoff to the subject as well as
whether the gamble is real or hypothetical can have an effect on the subject’s preference (Holt
and Laury, 2002). Internal factors such as age (Harbaugh et al., 2002; Levin and Hart,
2003), cognitive ability (Boyle et al., 2011), and habits or personal circumstance (Campbell
and Cochrane, 1999; Constantinides, 1990) influence the subject’s decision, as well as how
the subject weighs the potential value of losses and gains (Kahneman and Tversky, 1979).
Without downplaying the importance of these factors, the question remains: Where did this
risk averse behavior originate from?
Intuitively, one would argue that risk-seeking behavior is not favored by evolution. In
fact, it has been proposed that animals actively avoid risk due to the increased mortality
risky decisions often entail (Stern, 2010). When foraging, animals only take risks when the
risk of the decision is outweighed by other factors (Bateson, 2007; Bednekoff, 1996; Houston,
1991; Poethke and Liebig, 2008). Additionally, foraging animals avoid risk when resources
are plentiful, but adopt riskier strategies when resources are scarce (Stephen and Krebs,
1986). Further, organisms ranging from bacteria (Beaumont et al., 2009) to birds (Bulmer,
1984) are suggested to mitigate risk in their reproductive success via bet hedging (Gillespie,
1974; Slatkin, 1974; Philippi and Seger, 1989; Lehmann and Balloux, 2007). However, in
these natural situations risky behavior often does not compensate for the potential cost of
taking the risk. Thus, while these circumstances could explain the evolutionary utility of
risk averse behavior in many natural settings, it does not explain why humans would be
prone to risk aversion in equivalent mean payoff gambles.
Previous studies have reported that in small populations of evolving organisms, the fit-
ness of riskier behaviors is significantly affected by the variance in the payoff of the be-
havior (Gillespie, 1974, 1977). This observation suggests that strategies that minimize the
variance in the payoff of a gamble should have a selective advantage only in small popula-
tions. Consequently, evolution in small populations could potentially explain the origin of
Risk aversion as an evolutionary adaptation 4
risk aversion by humans in equivalent mean payoff gambles. Throughout evolutionary his-
tory, humans have experienced at least two population bottlenecks that reduced the human
population to as few as 1,000 individuals (Cann et al., 1987; Vigilant et al., 1991). However,
a population size of 1,000 individuals is unlikely to be small enough to evolve risk averse
behavior as a dominant strategy in the population (Gillespie, 1974, 1977). Instead, a more
likely explanation is that humans have lived in groups of about 150 individuals throughout
their evolutionary history (Aiello and Dunbar, 1993; Dunbar, 1993), which plausibly could
have been a small enough effective population size for risk aversion to have evolved.
To test whether evolution can explain the emergence of risk averse strategies, we gener-
alize the equivalent mean payoff gamble (with a safe and a risky choice) so that there are an
infinite number of possible choices, parameterized by the probability χto obtain the high
payoff. We choose this payoff to be 1/χ, so that the mean payoff of any choice will be 1. We
will call any of the possible gambles a strategy, and denote each strategy by the probability
χ. The choice χ= 1 then implies that the agent chooses the safest gamble. In this game,
there is no limit on how risky the gamble is, except we do not allow strategies with χ= 0,
as they are not normalizable.
In order to study the evolution of the strategy, we simulate a population of agents whose
choice of strategy is determined genetically, and inherited by the agent’s offspring. The
payoff that the agent receives is taken as the agent’s fitness. A small probability of mutation
introduces variation, so that alternative strategies from the ancestral one can be explored.
Each agent makes only one decision during its lifetime that determines its fitness, which
means that the agents are potentially making a life (positive payoff 1/χ) or death (zero
payoff) decision. Such a life or death decision is akin to a rare lifetime event that has a large
impact on an individual’s fitness, such as mating and mate competition (Buss and Schmitt,
1993). We use this agent-based simulation to explore how small a single population has to
be in order to have a significant impact on the evolution of risk aversion. Additionally, we
implement an island-based model to test whether larger populations that were segmented
into small groups (with the possibility of migration between groups) could still select for
the evolution of risk aversion. Although this model cannot take into account the complexity
of human evolution nor the exact circumstances thereof, it can address the plausibility of
risk aversion as an evolutionary adaptation to equivalent mean payoff gambles due to small
population sizes.
Results
Evolution in a single population
Each agent in a population is represented by a single probability χ(the agent’s inherited
gambling strategy), where χdetermines the fitness of the agent. Every agent only plays the
Risk aversion as an evolutionary adaptation 5
Figure 1. Strategy evolution with a fixed population size of 100 individuals and a
mutation rate of 1%. A) Mean strategy ¯χon the line of descent at generation 950 over
1,000 replicate runs. Measurements were taken after selection happened, hence the value
for generation 1 is not 0.5. The agents on the line of decent show a preference for risk
averse strategy. The dotted line indicates the expected value 0.5 for unbiased evolution,
i.e., no strategy preference. B) The probability distribution of χat generation 950 of the
dominant strategy across 1,000 replicate runs. This is identical to the distribution of
strategies within the population at generation 950 (Figure S2), showing that there is no
considerable difference between line of descent and population averages. The agents evolve
a significant preference for risk averse strategies by generation 950 (Wilcoxon rank sum test
P = 7.7954e−22 between this distribution and a uniform random distribution).
gamble once in their lifetime, so their fitness is determined by polling a random variable X
X=1
χp=χ
0p= 1 −χ.(1)
exactly once, where pis the probability to receive the corresponding payoff and χis
the agent’s strategy. An agent equipped with a strategy χ > 0.5 is considered risk averse,
whereas an agent with a strategy χ < 0.5 is considered risk-prone. All else being equal, we
expect that evolution should not prefer any strategy over another, because the mean payoff
of a species (individuals with the same χ) should be the same regardless of χ.
If population size does not have a significant effect on the evolution of risk aversion, we
would expect the strategy preference of any individual to drift neutrally, so that at the end
of the evolutionary run (generation 950) the expected mean population strategy is ¯χ= 0.5
(the mean of a uniformly distributed random variable constrained between zero and one).
Instead, we observe in Figure 1A that the mean χconverges to 0.6941 ±0.0139 (mean ±
two standard errors).
Similarly, we would expect that if the strategy drifts neutrally, we should observe χto
be distributed in a uniform manner at the end of the evolutionary runs. Instead, for a
population size of N= 100, we observe in Figure 1B a distribution that departs significantly
Risk aversion as an evolutionary adaptation 6
Figure 2. Mean strategy ¯χat the end of 1,000 evolutionary runs as a function of
population size. Agents in smaller populations (e.g., 50 and 100) demonstrate a clear
preference for risk averse strategies. In contrast, agents in larger populations (e.g., size
5,000 and 10,000) display only weak risk aversion or no preference. Error bars are two
standard errors over 1,000 replicates. The dotted line indicates the expected mean value
for unbiased evolution, i.e., no strategy preference.
from uniformity (Wilcoxon rank sum test P = 7.7954e−22 between this distribution and a
uniform random distribution). This result suggests that population size plays a critical role
in shaping what strategies evolve in the agent population.
To further explore the effect of population size on evolved strategy preference, we ran the
evolutionary simulation with different fixed population sizes. Figure 2 demonstrates that
the final evolved strategy depends strongly on the population size. These results highlight
that agents in smaller populations prefer risk averse strategies that receive a lower payoff but
with higher reliability. In contrast, agents in larger populations do not show a preference for
risk averse nor risk prone strategies and converge on ¯χ= 0.5 because all strategies perform
roughly the same in large populations, that is, the χof individual strategies drifts neutrally.
Theory of selection for variance
The tendency for natural selection to select against variance in offspring number has been
discussed before. Indeed, Gillespie has argued that large variance in offspring number could
be selected against because adverse outcomes (few or zero offspring) cannot easily be bal-
anced by favorable outcomes (large broods) for individuals of the same species, because the
individuals without offspring may not get to try the “offspring lottery” again (Gillespie,
1974, 1977). Further, Gillespie proposed an approximate mean actual fitness that takes the
variance in the offspring number distribution into account (see Methods, Equation [6]).
Gillespie’s fitness estimate wact strongly depends on the strategy choice χ(because it
determines the variance in the offspring number distribution) as well as population size
Risk aversion as an evolutionary adaptation 7
A B
Figure 3. A) Fitness (wact ) as a function of strategy choice χand population size
according to Gillespie’s model Equation (6). Fitness differences between strategies with
different χare far more pronounced in smaller populations. Larger population sizes
effectively buffer risky strategies against immediate extinction when the risky strategy does
not pay off. See legend for population sizes. We note that the x-axis is log scale. B)
Fixation probability (Π) of a perfectly risk-averse strategy (χ= 1) within a uniform
background of strategies with choice χ, as a function of χ(solid line). Fixation
probabilities were estimated from 100,000 repeated runs, each seeded with a single invading
strategy with χ= 1 in a background population of N= 50 resident strategies with strategy
χ. The dashed line is Kimura’s fixation probability Π(s) = (1 −e−2s)/(1 −e−2N s) (see,
e.g., (Gillespie, 2004)), where sis the fitness advantage of the invading strategy calculated
using Equation (6). Error bars are two standard errors.
(Figure 3A). Thus, his theory explains why agents that evolved in small populations show
a preference for risk averse strategies, whereas agents that evolved in larger populations
showed no such strategy preference. We can test the theory directly by measuring the
probability Π (the fixation probability) that a perfectly risk-averse strategy (χ= 1) can
invade (and replace) a homogenous population consisting of strategies with choice χ≤1.
We find that the observed probability of fixation (shown in Figure 3B for a population size of
N= 50) agrees qualitatively with the fixation probability calculated using Gillespie’s fitness
in Kimura’s formula (dashed line in Figure 3B), but not quantitatively. Indeed, an effective
fitness of about half of Gillespie’s estimate reproduces the simulations almost exactly, which
corroborates earlier findings (Shpak, 2005).
Evolution in groups
In the previous section, we demonstrated that agents in small populations evolve a preference
for strategies with low variance in their payoff distribution, i.e., risk aversion. The group
size for humans throughout evolutionary history has been proposed to be around 150 indi-
Risk aversion as an evolutionary adaptation 8
Figure 4. Mean strategy ¯χon the line of descent at generation 950 as a function of the
migration rate in an island model genetic algorithm with 128 groups with 128 members in
each group. Regardless of the migration rate, it is the group size and not the total
population size that determines if the agents evolve risk averse strategies. Error bars
indicate two standard errors over 1,000 replicates. The dotted line indicates the expected
value for unbiased evolution, i.e., no strategy preference. A migration rate of 0.5 implies
that half of the agents in each group migrate every generation.
viduals (Aiello and Dunbar, 1993; Dunbar, 1993), which suggests that evolving in such small
groups could have been the reason behind the evolution of human risk aversion. However, a
small group size and a small population size are two different things. While humans might
have lived in small groups of 150 individuals, the total population size of humans has been
much larger, and were only at times as low as 1,000 individuals (Cann et al., 1987; Vigilant
et al., 1991). Even though selection may occur within groups of about 150, individuals likely
migrated between groups. Migration could have caused selection to effectively act on much
larger groups (or even the entire human population) negating the selection for variance effect.
We can simulate such an environment using an island-based evolutionary model (see
Methods), in which individuals live in groups (the “islands”) that randomly exchange indi-
viduals with each other via migration. For example, we can run 1,000 replicate evolutionary
experiments with 128 groups of 128 individuals each, with varying migration rates. In this
configuration the total population size is 16,384 individuals, which according to Figure 2
should result in agents evolving no strategy preference. Figure 4 shows that regardless of the
migration rate, the group size and not the total population size determines whether agents
evolve risk averse strategies. This result suggests that even with migration between groups,
the effective population size that selection acts on is determined by the group size and not
the total population size.
When we change the size of the groups but fix the total population size (i.e., increase the
group size and reduce the number of groups) while keeping the migration rate at a constant
0.1, we again observe that the group size critically determines the preferred evolved agent
Risk aversion as an evolutionary adaptation 9
Figure 5. Mean strategy ¯χon the line of descent at generation 950 as a function of the
ratio between group size and number of groups. Group size critically determines the
preferred evolved agent strategies, where risk averse strategies are preferred in smaller
groups and no strategy is preferred in larger groups. The x-axis tick labels are formatted
as group size
number of groups . Error bars indicate two standard errors over 1,000 replicates. The dotted
line indicates the expected value for unbiased evolution, i.e., no strategy preference.
strategies (Figure 5). Risk averse strategies are preferred in smaller groups and no strategy
is preferred in larger groups. This result recapitulates the results from Figure 2, which shows
that the preference for strategies with low payoff variation (i.e., risk aversion) depends on
the effective population size.
Relative value of the gamble
Another way to alter risk aversion in humans is by changing the relative value of the pay-
off (Holt and Laury, 2002). When the gamble is about small amounts of money (i.e.,
“peanuts” gambles or hypothetical money), humans tend to be less risk averse, whereas
raising the relative value of the gamble increases risk aversion. In our evolutionary simula-
tion, agents play the gamble a single time and the payoff they receive is their only source of
fitness. This constraint effectively turns the gamble into a life or death situation, similar to
a game with extraordinarily high stakes.
To simulate lower-stakes gambles, we add a baseline payoff (β) to the payoff so that the
fitness of the agent becomes
X=β+1
χp=χ
β p = 1 −χ(2)
where pis the probability to receive the corresponding payoff and χis the agent’s strategy.
Typical gambles humans partake in fall either in the loss or in the gain domain. In biological
systems, on the other hand, organisms accumulate resources in order to ultimately produce
offspring. The “gambles” these organisms undertake will influence the number of offspring,
Risk aversion as an evolutionary adaptation 10
Figure 6. Mean strategy ¯χon the line of descent at generation 950 depending on the
additional payoff (β). The larger the additional payoff βbecomes, the more often strategies
return to an unbiased choice. Error bars indicate two standard errors over 1,000 replicates.
The dotted line indicates the expected value for unbiased evolution, i.e., no strategy
preference.
which will be positive or zero. Thus, we can not differentiate between losses or gains like hu-
mans would think about gambles for money. Therefore, gains and losses must be considered
relative to fitness.
When we run the evolutionary simulation with a population size N= 100 for various
values of β, we observe that the larger the baseline βbecomes, the more often strategies
return to an unbiased choice (Figure 6). This result is expected because fitness differences
only matter if their relative impact is larger than 1
N(Kimura, 1962; Gillespie, 2004). Thus,
risk averse strategies will only be selected for when the outcome of the gamble represents a
significant portion of the individual’s fitness when taking the population size into account.
Repetition of the gamble
Thus far, we have only investigated one-time gambles. What happens when the agents engage
in the same gamble multiple times during their lifetime? Intuitively, repeating the gamble
should reduce the variance in the overall payoff the agents receive, and if gambles are played
infinitely, the payoffs will converge on the same mean. In this experiment, we do not consider
situations where agents can change their behavior based on previous experiences (Lopes,
1981), but rather focus on unconditional responses. Shown in Figure 7, we observe that the
agents do not evolve a preference for risk aversion if the gamble is repeated several times in
a lifetime. At the same time, this repetition effect depends strongly on the population size,
such that smaller populations still evolve a preference for risk averse strategies with as many
as 10 repetitions of the gamble. Therefore, a preference for risk aversion will only evolve for
gambles that are encountered only a few times during an individual’s lifetime.
Risk aversion as an evolutionary adaptation 11
Figure 7. Mean strategy ¯χon the line of descent at generation 950 for three different
population sizes (50, 500, and 5,000) depending on how many times the gambles are
repeated. The more often the gamble is repeated during an individual’s lifetime, the less
likely risk aversion will evolve as a preferred strategy. Error bars indicate two standard
errors over 1,000 replicates. The dotted line indicates the expected value 0.5 for unbiased
evolution, i.e., no strategy preference.
Discussion
We hypothesized that risk aversion in humans could have been an evolutionary adaptation
to living in small groups. We tested this hypothesis by evolving digital agents whose fitness
is determined by a single choice during their lifetime in groups of varying size, and where
that choice is encoded genetically and thus heritable. We observed that a preference for
risk averse strategies does indeed evolve, but only when the group size is sufficiently small.
These findings align with reports from earlier work that humans lived in groups of about 150
individuals for a large portion of their evolutionary history (Aiello and Dunbar, 1993; Dunbar,
1993), providing a plausible evolutionary explanation for the risk averse behavior commonly
observed in humans. In other words, these findings provide a quantitative foundation to the
idea that evolution can explain risk aversion (Okasha, 2007).
Additionally, we find that risk aversion is the preferred evolutionary adaptation to life in
small groups when these groups are embedded within much larger groups, even with a large
amount of migration between groups. However, it is important that the risky decisions occur
only rarely during an individual’s lifetime, and where the outcome of the risk represents a
significant effect on the individual’s fitness. If the gamble has a negligible impact on fitness
(e.g., only small gains are at stake) or if the risk is encountered regularly in the individual’s
lifetime, then the selective advantage of risk aversion will be lost. Examples of such rare,
high-risk, high-payoff gambles include mating and mate competition (Buss and Schmitt,
1993).
Our work does not imply that no risk-prone strategies can possibly evolve. What we
show is that risk aversion evolves on average, but the distribution of strategies within a
Risk aversion as an evolutionary adaptation 12
population is quite broad (Figure 1). Thus, while on average agents are risk-averse if they
evolve in a small population, there will always be some agents that are extremely risk-prone.
Such agents can do extraordinarily well by chance and persist, but their genes are ultimately
doomed for extinction.
While our model is only haploid and uses a single locus, we do not expect a diploid model
using multiple loci to have qualitatively different results from the results presented in this
paper. Regardless, gene flow in diploid organisms in an island model and its impact on the
evolution of risk aversion is likely an interesting extension of this experiment to pursue in
future work.
When exploring whether risk aversion in human decisions today can be explained by
evolution in the genes contributing to (risky) mate choice and mate competition during the
species’ evolutionary history, we must assume that genes involved in decisions that have
a huge impact on fitness like mate choice are also involved in general decision making.
This is supported by the evidence that the genes that subserve evaluation and reward are
ancient (Symmonds et al., 2010), suggesting that ancient risk averse behavior can still in-
fluence general decision mechanisms today. However, it is also clear that nature cannot be
the only force that has shaped our risk averse behavior, because there is ample evidence
that experience is a contributing factor. Thus, while this work has studied the impact that
evolutionary history can have on strategy choice, it should be seen only as an element in
understanding why humans shy away from risk.
Methods
Single population evolutionary model
We use a genetic algorithm applied on a population of agents to simulate evolution of the
population (Michalewicz, 1996). Each agent in this population is defined by a probability χ
(the “choice”), which encodes the agent’s strategy. We seed the initial agent population by
assigning every agent a random χdrawn from a uniform distribution (0,1] with a variance
of 1
12 . Varying the initial starting condition has no significant effect on the outcome of the
experiments. Every agent in the population only plays the gamble once in its lifetime to
determine its fitness, where χis the probability to receive a fitness of 1
χor receive 0.0 fitness
with a chance 1 −χ. The strategy of each agent can only change due to evolution, i.e.,
strategies cannot change during the agent’s lifetime.
Once all of the agent fitnesses are evaluated for a given generation, the agents produce
offspring into the next generation proportional to their fitness, i.e., we use fitness propor-
tional roulette wheel selection to determine the next generation of individuals Back (1996),
implementing the Wright-Fisher process. Offspring inherit the strategy χfrom their parent
(no sexual recombination), except that 1% of all offspring are subjected to mutation. If an
offspring is subject to mutation, its new strategy is drawn randomly from a uniform distri-
bution (0,1]. We repeat this evolutionary process every generation with a fixed population
size for 1,000 generations.
Risk aversion as an evolutionary adaptation 13
Theory of selection for variance in offspring number
Gillespie suggested that in finite populations where the fitness of individuals carries a stochas-
tic component (modeled by a mean µand a variance σ2), the actual realized fitness wact is
given by (Gillespie, 1974, 1977):
wact =µ−1
Nσ2,(3)
where Nis the population size. Because in the equivalent mean payoff gamble agents receive
a payoff
X=1
χp=χ
0p= 1 −χ(4)
the variance becomes
σ2(X) = 1
χ−1 (5)
and the actual fitness of a strategy χis
wact(χ)=1−1
N(1
χ−1) ,(6)
as the mean of Xin Equation (4) (in an infinite population) equals 1. The fitness advantage
sof a strategy with χ= 1 versus a strategy χis then
s=wact(1) −wact (χ)
wact(χ)=1/χ −1
N−(1/χ −1) .(7)
We use Equation (6) to compute the actual fitness of a strategy using a given χwhile taking
the size of the population Ninto account (Figure 3A), and we use the fitness advantage (7)
in the calculation of the fixation probability using Kimura’s formula in Figure 3B.
Island-based evolutionary model
In our second set of experiments, we use an island-based evolutionary model to simulate
an environment in which thousands of individuals are evolving in several small groups.
For an overview of island models and the effect of population size, see (Whitley et al.,
1998; Cant´u-Paz and Goldberg, 2003). Island models have three parameters: The size of a
single group, the number of groups, and a migration rate defining how many individuals per
group are moved randomly to new groups during each generation. If an agent is selected to
migrate, we randomly select a new group and a random agent within that group, and switch
agents. Thus, our island-based evolutionary model implements several single population
evolutionary models with a fixed fraction of individuals migrating between the populations
Risk aversion as an evolutionary adaptation 14
every generation. The migration rate is the probability that an agent will be picked for
migration per generation. For example, a migration rate of 0.1 implies that 10% of the
agents in the entire population are picked to switch (affecting up to 20% of the population,
as each switch affects two agents).
Typically, island models are used to speed up evolution in rugged fitness landscapes and
increase genetic diversity within the population. In this experiment, we are not concerned
with ruggedness nor diversity. Instead, we use an island model because it best approximates
the scenario of individuals evolving in multiple small groups with some level of inter-group
migration.
Retracing the line of descent
At the end of each evolutionary run, we reconstruct the line of descent (LOD) by picking a
random agent in the population and tracing back to the first generation using only direct
ancestors (Lenski et al., 2003). This procedure rapidly converges on the last most recent
common ancestor (LMRCA) that swept the population. In our experiments, we determined
that the agents on the LOD at generation 950 most often represented the LMRCA, thus
we used those agents as the final representative agent for their respective evolutionary run.
The LOD between the first agent and the LMRCA of a population contains all mutations
that fixed during evolution, while all other mutants were outcompeted. Thus, analyzing an
evolutionary run’s LOD enables us to retrace the evolutionary history of the population.
Acknowledgments
We thank Georg N¨oldeke for discussions and insights on risk aversion in economics, and
Robert Heckendorn for discussions of island models. We gratefully acknowledge the support
of the Michigan State University High Performance Computing Center and the Institute for
Cyber Enabled Research (iCER).
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