# Conflicts, Assortative Matching, and the Evolution of Signaling
Norms††thanks: We thank Wallice Ao, Jonathan Newton and Van Kolpin for their
comments.

Ethan Holdahl  Jiabin Wu Email<EMAIL_ADDRESS>Address: 515 PLC, 1285
University of Oregon, Eugene, Oregon, USA 97403.

###### Abstract

This paper proposes a model to explain the potential role of inter-group
conflicts in determining the rise and fall of signaling norms. In one
population, assortative matching according to types is sustained by signaling.
In the other population, individuals do not signal and they are randomly
matched. Types evolve within each population. At the same time, the two
populations may engage in conflicts. Due to assortative matching, high types
grow faster in the population with signaling, yet they bear the cost of
signaling, which lowers their population’s fitness in the long run. We show
that the survival of the signaling population depends crucially on the timing
and the intensity of inter-group conflicts.

Keywords: Signaling Norms, Conflicts, Assortative Matching, Evolutionary Game
Theory

##  1 Introduction

Tattooing has been an important part of the Polynesian cultures for thousands
of years (Pesapane et al., 2014; Bennett, 2017; Clark and Langley, 2020). It
is quite common for ancient Polynesian warriors to cover their whole bodies in
tattoos as tattoos were considered as a sign of masculinity and bravery. The
pain of Polynesian tattooing, however, was extreme and the risk of death by
infection was high. The healing process sometimes took over a year. Those who
shied away from tattooing, or failed to endure the pain and abandoned their
tattooing were considered as cowards and lived in shame for the rest of their
lives. Why did the Polynesians adopt such a practice? The classic theory of
signaling by Zahavi (1975) in biology and Spence (1973) in economics provides
an answer:111The theory has been widely applied to explain different phenomena
ranging from life sciences to social sciences. See further discussion of the
subject in Grafen (1990), Maynard Smith and Harper (1995), Johnstone (1997),
Zahavi and Zahavi (1997), Maynard Smith and Harper (2003), Searcy and Nowicki
(2005), Getty (2006), Grose (2011) and Számadó (2012), among many others.
Tattooing served as a costly signaling device to differentiate the strong from
the weak. Harmful and wasteful practices such as tattooing that serve as
markers to differentiate different types of individuals are called the
signaling norms in a recent paper by Przepiorka and Diekmann (2021).222Other
examples of signaling norms are elaborated body piercings, lethal initiation
rituals, excessive feasts, and wearing obstructive dressing codes.

A more subtle yet important question is, suppose no tattooing is the ancestral
condition, what determined the emergence of tattooing as a signaling norm in
the ancient Polynesian societies? On the one hand, using tattoos to assort its
members allowed a tribe to organize its social hierarchy, mating and
reproduction in a more efficient way, which might help boosting the tribe’s
population growth rate. On the other hand, the heath cost of tattooing might
slow the rate down. Larger tribes often had a higher chance to survive in
ancient tribal warfare and violent conflicts are argued to have played a
critical role in human’s ancestral past (Keeley, 1996; R.W. Wrangham, 1996;
Buss and Shackelford, 1997; S. LeBlanc, 2003; Guilaine and Zammit, 2004; Gat,
2006; M. Potts, 2008; Ferguson, 2012). Would the inter-tribe conflicts in
Polynesia matter for the selection of the tattooing tribes over the non-
tattooing tribes? If so, how? In this paper, we attempt to formally
investigate the role of conflicts in influencing the evolution of signaling
norms in an evolutionary game theoretical model.

In the model we propose, there are two populations. In each population, there
are two types of individuals: high and low. The types are genetically
determined and the high type has some hidden fitness advantages over the low
type. The individuals need to match in pairs to obtain payoffs (forming a
mating or foraging pair for example), which in turn determine their
reproduction rates. However, the individuals cannot directly observe others’
types. Assume that one population is equipped with signaling technology, while
the other is not. In the population without signaling technology, the
individuals are randomly matched in pairs because they cannot assort according
to types. In the population with signaling technology, the high type
individuals may adopt the signal to differentiate themselves from the low type
individuals so that they can identify one another and avoid being matched with
the low type individuals. As long as doing so is beneficial to the high type
individuals; and at the same time, the cost of signaling is sufficiently high
to deter the low type individuals to mimic the high type individuals, a
separating equilibrium can be sustained in the population. Note that we treat
signaling as a behavioral trait instead of a genetic trait, so the individuals
can choose whether to adopt it in the behavioral timescale.

The assortative benefit to high types in the signaling population allows them
to evolve faster than they would without signaling. However, even absent
signaling, high types would eventually prevail given their fitness advantage.
When both populations are homogeneously high type, the assortative advantage
the signaling population had over the non-signaling population disappears
leaving the signaling population with a cost but no comparative benefit. This
means the signaling population’s fitness is lower in the long run. Therefore,
there is a trade-off in utilizing signaling and it becomes crucial if the two
populations engage in conflicts. We show that the timing and the intensity of
inter-group conflicts play an essential role in determining the survival of
the signaling population. In particular, the signaling population has a higher
chance to survive in the early stage when it still has a fitness advantage if
the period of isolation before conflicts is shorter and the conflicts are
fiercer.

The theory of signaling has been used in different evolutionary models to
explain human behavior. Gintis et al. (2001) formalize the idea by Zahavi
(1975) and Miller (2000) that high-quality individuals may engage in pro-
social activities (cooperate in a public good provision game) to signal their
desirability to gain better mating opportunities. This is called the
“competitive altruism" in Roberts (1998). While quality can be signaled by
doing good, it can well be signaled by any costly activity. Hopkins (2014)
instead considers a model in which the individuals do not differ in qualities,
but in their degrees of altruism. In this model, it is natural for the
altruists to use pro-social activities to signal their altruism and the author
shows that altruists who can mentalize have a greater advantage over
nonaltruists. Przepiorka and Diekmann (2021) consider a trust game in which
there are two types of trustees: short-term (impatient) and long-term
(patient). They identify the condition under which a separated equilibrium
with only the long term trustees send a costly signal to the trusters and they
show that only when the probability of meeting a long-term trustee is
sufficiently low for the trusters, would the separated equilibrium be
collectively more beneficial than the case without any signaling opportunity.
Note that these papers all consider what is called the “dilemma" situations.
Without signaling opportunity, “ bad" type or behavior such as nonaltruist and
defection prevail. On the contrary, we consider a non-dilemma situation and
high types would eventually dominate the entire population with or without
signaling opportunity. In addition, these papers do not explicitly consider
the role of conflicts, while we investigate how different properties of
conflicts can affect the evolution of signaling norms.

The paper is organized as follows. Section 2 introduces the model, analyze how
types evolve within a population and how conflicts work between the two
populations. Section 3 conducts comparative statics on the timing and the
intensity of inter-group conflicts. Section 4 discusses alternative modelling
choices and provides some concluding remarks.

##  2 The Model and Analysis

In this section we outline how populations evolve in our model. We are
interested in the effect that signaling has on the evolution of a population
so we consider two populations: one with signaling technology and the other
without signaling technology. Both populations are endowed with high type
individuals who are better suited to their environment and thus have a higher
fitness, and low type individuals who are not as adapted to the current
environment and thus have a lower fitness. The types are genetically
determined, meaning that the individuals cannot change their own types and the
distribution of types in a population evolves through reproduction. We use a
discrete generational model where in each generation (period) individuals
match with another individual in their population. This matching can be
interpreted as a mating relationship, a foraging relationship, or any other
social relationship. The type of each individual is not observable to the
other members of the population, so where there is no signaling, individuals
match randomly with another individual in their population. However, when
signaling is present individuals are able to discriminate. As a result,
signalers randomly match with another signaling individual in their population
leaving the non-signalers in that population to randomly match with each
other. After matching, they reproduce according to the payoffs in the table
below minus the cost of signaling if applicable. Table 1 reports the payoffs.
An individual of type $i$, who is matched with an individual of type $j$,
obtains a payoff of $V(i,j)$, for $i,j\in\\{H,L\\}$. Note that since these
payoffs determine the individuals’ life long reproductive rates, the length of
each discrete period can be thought of as the amount of time needed for a
given individual to reach adulthood.

Table 1: Payoffs: Reproduction Rates |  | H | L
---|---|---|---
| H | $V(H,H)$ | $V(H,L)$
L | $V(L,H)$ | $V(L,L)$

We impose the following relation on the payoffs:

$V(H,H)>V(H,L)>V(L,H)>V(L,L).$ (1)

Viewing the matching as a mating relationship, we have a model where there are
no hybrids, only high or low type offspring can be produced. High types
produce $V(H,*)$ high type offspring which depend on who they match with.
Likewise, low types produce $V(L,*)$ low type offspring which depend on who
they match with. Then two high types will together produce $2V(H,H)$ high
types, a low and high type pairing will produce $V(H,L)$ high types and
$V(L,H)$ low types, and a 2 low type pairing will produce $2V(L,L)$ low types.

Alternatively, we can think of $V(*,*)$ as the amount of offspring per parent
that makes it to reproduction. Since high types are more fit than low types,
high types will be more successful than low types in making it to adulthood
thus explaining $V(H,L)>V(L,H)$, even if we expect high and low types to be
born in equal proportion in a mixed parent situation. Additionally, we can
think that the parents provide for their offspring in some capacity before
they reach adulthood. It is likely that high type parents are better providers
than low type parents which explains $V(H,H)>V(L,H)$, $V(H,H)>V(H,L)$,
$V(H,L)>V(L,L)$, and $V(L,H)>V(L,L)$. In other words, one’s reproductive rate
is increasing both in one’s type and in its matched partner’s type.

An analogous story can be told in a foraging relationship. Matched individuals
share some, but not all of their foraged goods together. Here, high types are
more successful than low types in procuring food. Since the amount of food is
directly related to reproductive rate, Inequality (1) follows. Here, high
types produce only high type offspring and low types produce only low type
offspring either asexually or otherwise.

In addition, we want to ensure that populations will not die out. Hence, we
require $V(H,H)>1$.

###  2.1 A Population without Signaling

In a population without signaling technology, the individuals cannot observe
others’ types. As such they are randomly matched with another individual in
their population. We define the non-signaling population at time $t$ as
$N_{t}$ where $N_{t}=N^{H}_{t}+N^{L}_{t}$. Here, $N^{H}_{t}$ and $N^{L}_{t}$
are the amount of high types and low types, respectively, in the non-signaling
population at time $t$.

Assuming a large $N_{t}$, the law of large numbers implies that $N^{H}_{t}$
and $N^{L}_{t}$ evolve according to their expected payoffs from Table 1. So we
have:

$\displaystyle
N^{H}_{t+1}=[\frac{N^{H}_{t}}{N_{t}}*V(H,H)+\frac{N^{L}_{t}}{N_{t}}*V(H,L)]*N^{H}_{t},$
(2) $\displaystyle
N^{L}_{t+1}=[\frac{N^{H}_{t}}{N_{t}}*V(L,H)+\frac{N^{L}_{t}}{N_{t}}*V(L,L)]*N^{L}_{t}.$
(3)

We simulate the dynamics described by Equations (2) and (3) in Figure 1. The
first graph shows the evolution of the average reproductive rate for each type
as well as for the population as a whole and the second shows the evolution of
the sizes of high and low type sub-populations as well as the population as a
whole. Note that the starting population has been normalized to 1 with high
types initially making up 20% of the population. The final graph shows the raw
growth ($N_{t}-N_{t-1}$) of the population as well as the raw growth of each
sub-population within the population.

Inequality (1) implies $V(H,H)>V(L,H)$ and $V(H,L)>V(L,L)$. Hence, in the
absence of signaling, high types are able to evolve within the population
because they realize higher payoffs than low types i.e. survival of the
fittest. As the proportion of high types increases in the population, the
average reproductive rate of the population will converge to the reproductive
rate of the high types. Note that there is a secondary effect here. Inequality
(1) also implies $V(H,H)>V(H,L)$ and $V(L,H)>V(L,L)$. Hence, as high types
make up a greater proportion of the population, individuals become more likely
to match with high types, thus the reproductive rates of both low types and
high types increase as high types come to make up a greater proportion of the
population, which can be seen in the first graph of Figure (1).

Figure 1: Population Dynamics Without Signaling

Parameters:

$N_{0}^{H}=.2$

$N_{0}^{L}=.8$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

###  2.2 A Population with Signaling

In this section we consider a population that has signaling technology. We
should emphasize that here we examine behavioral signaling that individuals
can choose whether to opt into. Hence, the individuals make their decisions on
signaling within each generation, while the genetically determined types
evolve across generations. Let $K$ be the cost of signaling. We assume that
the cost of signaling, $K$, is sufficiently costly such that that only high
types can afford it. Formally, we require:

$V(L,L)>V(L,H)-K.$ (4)

We also only want to consider signals that are potentially incentive
compatible for high types. That is, a high type receives a higher payoff by
being matched with another high type even after paying the cost of signaling
than it does by being matched with a low type:

$V(H,H)-K>V(H,L).$ (5)

Combining the two inequalities, we know that for a viable signal of cost $K$
to exist it must be the case that:

$V(H,H)+V(L,L)>V(L,H)+V(H,L).$ (6)

This is known as the single crossing property (Zahavi, 1975; Spence, 1973). We
enforce these conditions on the reproductive rates $V(.,.)$ and the cost of
signaling $K$.

Since we consider behavioral signaling, we assume that the individuals reach
an equilibrium on their decisions on adopting the signal within each
generation before reproduction. In an equilibrium, each individual has no
incentive to deviate from its current choice of whether to signal. As shown in
Spence (1973), given the incentive compatibility conditions (4) and (5), there
exists a separating equilibrium in which only high types signal. Moreover, all
high types are matched with one another, and low types are matched with one
another, resulting in perfectly assortative matching in types. The high types
earn a payoff of $V(H,H)-K$ and the low types earn a payoff of $V(L,L)$ in
equilibrium. Note that the separating equilibrium is independent of the
distribution of types in the population. Even when the group of low types is
vanishingly small, the high types would still pay the cost of signaling to
segregate themselves from the low types.

We define the population with signaling at time $t$ as $S_{t}$ where
$S_{t}=S^{H}_{t}+S^{L}_{t}$. Here, $S^{H}_{t}$ and $S^{L}_{t}$ are the amount
of high types and low types, respectively, in the signaling population at time
$t$. Because of signaling, the individuals are only matched with their own
types. Hence, $S^{H}_{t}$ and $S^{L}_{t}$ experience a constant growth rate
over time. Their evolutionary dynamics are as follows:

$\displaystyle S^{H}_{t+1}=[V(H,H)-K]*S^{H}_{t},$ (7) $\displaystyle
S^{L}_{t+1}=V(L,L)*S^{L}_{t}.$ (8)

Figure 2: Population Dynamics Where High Types Signal

Parameters:

$S_{0}^{H}=.2$

$S_{0}^{L}=.8$

$K=.1$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

We simulate the dynamics described by Equations (7) and (8) in Figure 2. We
have three similar graphs as in Figure (1) and use the same parameters. Note
that in the top graph, the reproductive rate of the high types and low types
are not influenced by the relative size of each sub-population since they
always match with their own types. This can be seen in Equations (7) and (8),
where the coefficients on $S^{H}_{t}$ and $S^{L}_{t}$, do not depend on the
population composition. As a result, the growth rate of the population
increases only because high types evolve to make up a greater proportion of
the population. It can not be seen in the graphs here, but it should be clear
that as $K$, the cost of signaling, decreases the speed of evolution
increases.

###  2.3 Comparing Signaling and Non-Signaling Populations

Figure 3 shows the overlap of Figure 1 and Figure 2, both of which have the
same parameters. Looking at the first graph of Figure 3, we can see that while
$N^{H}_{t}/N_{t}$ is sufficiently small, the reproductive rate for the
signaling high types is greater than the reproductive rate for the high types
in the non-signaling population. This is because $V(H,H)-K>V(H,L)$. As a
result, high types evolve faster within the signaling population compared to
the high types in the non-signaling population. In the first graph of Figure
3, we can see that it takes until period 10 before the high types in the non-
signaling population reach the same reproductive rate of their counterparts in
the signaling population. Because of this and because low types in the
signaling population have a lower reproductive rate than low types in the non-
signaling population, high types make up a larger proportion of the signaling
population compared to the non-signaling population. Therefore, it takes a few
more periods until the average reproductive rate across the non-signaling
population surpasses that of the signaling population. Looking at the second
graph in Figure 3, we can see that although the reproductive rate of the non-
signaling population surpasses that of the signaling population around period
13, the signaling population retains a population advantage until around
period 21. This is because the signaling population builds a significant
population advantage in the initial periods. However, once sufficient time has
passed and both populations are essentially homogeneous with only high types
left, the non-signaling population realizes a reproductive rate that is $K$
greater than that of the signaling population.

Figure 3: Comparison of Population Dynamics Between Signaling and Non-
Signaling Populations

Parameters:

$N_{0}^{H}=S_{0}^{H}=.2$

$N_{0}^{L}=S_{0}^{L}=.8$

$K=.1$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

###  2.4 Conflicts between the Two Populations

Assume that in period $T$ the non-signaling population and the signaling
population engage in conflicts. In periods $t\in[0,T)$ the populations evolve
according to the dynamics in the above sections, but starting at period $T$
they start to eliminate the other population according to the Lanchester’s
square law (Lanchester, 1916). The Lanchester’s square law has been applied to
a variety of human and non-human conflicts. See for example Wilson et al.
(2002), Johnson and MacKay (2015) and Clifton (2020). In each period, before
reproduction, each member of a population kills $\beta$ members of the other
population. Hence, the dynamics for $N^{H},N^{L},S^{H},$ and $S^{L}$ are now
given as:

$\displaystyle
N^{H}_{t+1}=[\frac{N^{H}_{t}}{N_{t}}*V(H,H)+\frac{N^{L}_{t}}{N_{t}}*V(H,L)]*max\\{[N^{H}_{t}-\beta\frac{N^{H}_{t}}{N_{t}}I_{t\geq
T}S_{t}],0\\},$ (9) $\displaystyle
N^{L}_{t+1}=[\frac{N^{H}_{t}}{N_{t}}*V(L,H)+\frac{N^{L}_{t}}{N_{t}}*V(L,L)]*max\\{[N^{L}_{t}-\beta\frac{N^{L}_{t}}{N_{t}}I_{t\geq
T}S_{t}],0\\},$ (10) $\displaystyle
S^{H}_{t+1}=[V(H,H)-K]*max\\{[S^{H}_{t}-\beta\frac{S^{H}_{t}}{S_{t}}I_{t\geq
T}N_{t}],0\\},$ (11) $\displaystyle
S^{L}_{t+1}=V(L,L)*max\\{[S^{L}_{t}-\beta\frac{S^{L}_{t}}{S_{t}}I_{t\geq
T}N_{t}],0\\}.$ (12)

In the equations above, $I_{t\geq T}$ is an indicator that the groups are
engaging in inter-group conflicts and the max argument simply ensures that the
populations would not reach a negative number. Note that we specify that
conflicts take place before reproduction in each period in this model. It is
clear that when applying this model, the time between each discrete period
would simply be the amount of time it takes for a generation to reproduce.
While conflicts may run at a different time scale, we can force conflicts into
the reproduction time scale by adjusting the value of $\beta$.

Figure 4: Conflicts Between Signaling and Non-Signaling Populations After a
Period of Peace

Parameters:

$N_{0}^{H}=S_{0}^{H}=.2$

$N_{0}^{L}=S_{0}^{L}=.8$

$K=.1$

$\beta=.2$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

The vertical lines indicate the last period before conflicts begin $(T-1)$.

Figure 4 provides three examples of outcomes when conflicts occur in our
model. In the top graph, the non-signaling population enters the periods of
conflicts with a population advantage. Because they do not need to pay the
cost of signaling, the fact that they have a population advantage indicates
that they also have a superior reproductive rate. As a result, the non-
signaling population always wins. In the second graph, the signaling
population enters the conflicts with a population advantage. At this point the
signaling population doesn’t necessarily have a higher reproductive rate,
however, if the conflicts are fierce enough ($\beta$ is sufficiently large),
the signaling population will be able to eliminate the non-signaling
population before the non-signaling population can catch up with their
superior long run reproductive rate. The third graph is a special case.
Although the signaling population enters the conflicts with a slightly larger
population, because $\beta$ is sufficiently small, the non-signaling
population is able to overcome the signaling population during the conflicts.

##  3 Comparative Statics

Figure 5: Conflicts Between Signaling and Non-Signaling Populations After a
Period of Peace: Comparative Statics

Parameters:

$N_{0}^{H}=S_{0}^{H}=.2$

$N_{0}^{L}=S_{0}^{L}=.8$

$K=.1$

$\beta=.2$

$T=20$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

The white intercept lines indicate the current parameters values.

We have shown that costly signaling in a population may be competitively
advantageous in inter-group conflicts in our model. The viability of signaling
depends largely on three different factors: the period that conflicts start
($T$), the cost of signaling ($K$), and the intensity of conflicts ($\beta$).
Here, we argue that the signaling population benefits from a shorter period of
isolation before conflicts, a smaller cost of signaling, and a higher
intensity of conflicts. To clearly see the effects of these three factors, we
run the model thousands of different times, varying the key parameters: $T,K$
and, $\beta$ and indicate the outcomes of the conflicts. The results are
reported in Figure 5. In areas colored orange, the non-signaling population
wins the conflicts and in the blue area the signaling population wins. The
white intercept lines indicate where the model is evaluated in the last graph
of Figure 4.

The first graph in Figure 5 shows the result of the conflicts between the
signaling population and the non-signaling population as we vary the period
that conflicts start and the intensity of the conflicts. As we can see, beyond
a certain period the non-signaling population will always win no matter what
the value of $\beta$ is. This is because the non-signaling population is both
greater and growing faster beyond that period. One can verify this by looking
at Figure 3. It is also necessary for $\beta$ to be sufficiently large for the
signaling population to possibly win the conflicts. As we can see, when
$\beta$ is sufficiently small, the non-signaling population always wins.
Looking at the second graph of Figure 5, we can see that a smaller signaling
cost benefits the signaling population. The third graph of Figure 5 shows the
interaction between the cost of signaling and the intensity of conflicts at
$T=20$. The non-signaling population always wins when $K>.1$ because the non-
signaling would have a larger population than the signaling population at
$T=20$. Before that point, higher intensity of conflicts can compensate for a
higher cost of signaling for the signaling population.

Please note that thus far we have used $N_{0}^{H}=.2$ and $N_{0}^{L}=.8$ and
$S_{0}^{H}=.2$ and $S_{0}^{L}=.8$. Recall that high types are the individuals
that are better suited to their current environment and thus have a higher
fitness than low types. With this in mind, it is useful to think about the
initial values of $N_{0}^{H},N_{0}^{L},S_{0}^{H},$ and $S_{0}^{L}$ as being
related to the speed at which the environment is changing around the people.
The faster the environment changes, the less time passes before a new genetic
mutation is considered the high type which would essentially restart our model
with a small proportion of high types. So, a faster changing environment would
translate to a smaller initial proportion of high types. With that in mind,
Figure 6 shows the results of the conflicts when we vary the initial
proportion of high types in each population, $N_{0}^{H}$ and $S_{0}^{H}$,
while keeping them consistent across populations, $N_{0}^{H}=S_{0}^{H}$, and
maintaining $N_{0}=S_{0}=1$ so that $N_{0}^{H}$ and $S_{0}^{H}$ is simply the
initial proportion of high types in each population. The results are clear:
The smaller the initial proportion of high types, the more likely it is that
the signaling population will win the conflicts. This is because the
assortative advantage is largest when the proportion of high types is smallest
which further exaggerates the speed at which the makeup of the population
changes from low type to high type. Hence, giving the populations more room to
evolve naturally favors the signaling population. Abstracting from the model,
this implies that signaling is more advantageous the quicker one’s environment
changes. In a rapidly changing environment, the amount of high types at any
given time is expected to be lower than in a slow changing environment since
we define high types as those best adapted to the current environment. As can
be seen in the graphs, the smaller the initial proportion of high types, the
wider the range of $T,K$ and $\beta$ values that result in conflicts that
favors the signaling population.

Figure 6: Conflicts Between Signaling and Non-Signaling Populations After a
Period of Peace: Initial Conditions Comparative Statics

Parameters:

$N_{0}^{H}=S_{0}^{H}=.2$

$N_{0}^{L}=S_{0}^{L}=.8$

$K=.1$

$\beta=.2$

$T=20$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

The white intercept lines indicate the current parameters values.

##  4 Discussions

###  4.1 Alternative Modeling Choices

In our model we assume that in the signaling population only high types
transmit a signal and as a result, the signal enforces perfect assortative
matching within the signaling population. One may question what if the signal
is ubiquitous across the signaling population such that low types as well as
high types transmit the signal. In this case, the signal would not be a signal
at all but just a costly trait since no one would be able to distinguish one
type from another. As a result, the once difference between the evolutionary
dynamics of the signaling and non-signaling population would be that the
signaling population experiences a level decrease in their reproductive rate
with no functional assortativity advantage. The only scenarios where the
signaling population is able the beat the non-signaling population in
conflicts in this setup are with extreme parameter values where the signaling
cost and the expression $V(H,H)-V(H,L)$ are so large that the low types in the
signaling population are effectively eliminated by the signaling cost
immediately, thus effectively giving the high types assortative matching
which, if the initial proportion of high types in each population is low
enough, may allow for a very narrow window under which the signaling
population will realize a population advantage.

Another question one may have is what if there are both signalers and non-
signalers in the same population. In this case, so long as the signaling high
types are able to gain an assortative advantage over the non-signaling high
types in the population, then that may be enough to offset the signal cost and
realize a higher reproductive rate for them in the short run even in the
absence of inter-group conflicts? We demonstrate that this is not the case in
a one-population model with four types (signaling high type, non-signaling
high type, signaling low type, non-signaling low type) as shown in Figure 7.
The rationale is as follows. In our settings, high types evolve to make up an
increasing proportion of the population with certainty as time increases until
there are only high types remaining. As a result, at some point the relative
difference in assortativity between the signaling high types and the non-
signaling high types will not be big enough to offset the cost of signaling.
Therefore, absent conflicts between the signalers and non-signalers, non-
signaling high types will dominate the entire population with certainty in the
long run.

The above discussions on alternative modelling choices suggest that signalling
being behavioral (only high types choose to adopt it) and conflicts between
the signaling population and the non-signaling population are prerequisites
for signaling to survive.

Figure 7: One Population Where Some High Types Signal and Some Don’t

Parameters:

$N_{0}^{H}=S_{0}^{H}=.1$

$N_{0}^{L}=.8$

$S_{0}^{L}=0$

$K=.1$

|  | H | L
---|---|---|---
| H | $1.15$ | $0.90$
L | $0.85$ | $0.80$

###  4.2 Concluding Remarks

It has been well understood that costly signaling can provide a competitive
advantage at the intra-group level, however, to our knowledge the effects of
signaling has not been examined at the inter-group level of conflicts. We
consider a model in which the same individuals (high types) that benefit from
signaling would evolve even in the absence of signaling. Thus, when there are
two populations, one with and one without signaling, it follows that they
would both eventually evolve to be homogeneously high type. At that point, if
the two populations were to engage in conflicts, the population without
signaling would have a competitive advantage because they do not need to pay
the cost of signaling. However, we argue that there can be an advantage to
signaling in inter-group conflicts: that the group with signaling evolves
faster than the group without signaling. As a result, if conflicts occur, the
prevailing population is largely a question of how long the populations evolve
in isolation before they engage in conflicts. Our model predicts that
societies that have shorter periods of isolation before conflicts, and
conflicts that are more intense may favor the rise of signaling norms such as
tattooing.

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