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METHODOLOGY IN BIOLOGICAL GAME THEORY
SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Abstract. Game theory has a prominent role in evolutionary biology, in par-
ticular in the ecological study of various phenomena ranging from conflict
behavior to altruism to signaling and beyond. The two central methodologi-
cal tools in biological game theory are the concepts of Nash equilibrium and
Evolutionarily Stable Strategy (ESS). While both were inspired by a dynamic
conception of evolution, these concepts are essentially static – they only show
that a population is uninvadable, but not that a population is likely to evolve.
In this paper we argue that a static methodology can lead to misleading views
about a dynamic evolutionary processes. Instead, we advocate a more plu-
ralistic methodology, which includes both static and dynamic game theoretic
tools. Such an approach provides a more complete picture of the evolution of
strategic behavior.
1. Introduction
When an ecologist or an evolutionary biologist is confronted with an apparently
maladaptive phenotype, she must answer two questions. Firstly, why is this phe-
notype stable? Since it is apparently maladaptive, why hasn’t this phenotype been
eliminated in favor of a more adaptive alternative? Secondly, what led to the evolu-
tion of that behavior in the first place? This second question is especially pressing
if it seems likely that an ancestral population did not possess the apparently mal-
adaptive phenotype.
These two questions seem quite similar. One might be inclined to think an
answer to the first will provide an answer to the second. We here suggest that in
the context of game theory these two questions are often conflated, and that this
conflation leads to incorrect judgments about evolutionary processes. There is one
case in particular – the case of signaling behavior – where the proffered answer to
the first question has been regarded as satisfactory despite the fact that scholars
have (unknowingly) introduced a yet more difficult to solve mystery in the form of
the second question.
In biological game theory there is a prevailing methodology which we will call the
equilibrium methodology. This methodology involves developing a model of evolu-
tion and considering potential endpoints of evolution utilizing so called equilibrium
concepts. Most common among these are the concept of a (strict) Nash equilibrium
1
2 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
and the concept of an Evolutionarily Stable Strategy (ESS). Since it looks primarily
at end points of evolutionary processes, this methodology is most clearly aimed at
answering the first question: why is a particular state stable? But it is also aimed
at providing a partial answer to the second question. Biologists often claim that the
equilibria they find are (at least) potential endpoints for an evolutionary process.
So, the claim that a state is an equilibrium entails that the state is stable and also
that it is reachable by evolution.
What is left out of the equilibrium methodology is any model of the dynamics
of evolutionary processes. This is in contrast to what we will call the dynamic
methodology, which models explicitly (to various degrees) the process of evolution.
We will argue that recent results from the game theoretic study of signaling in biol-
ogy demonstrates that the equilibrium methodology alone is inadequate to answer
the second major question, and that it has in fact been misleading. While the the-
oretical possibility of such problems has been known for some time, those pitfalls
have been regarded as either obvious or unrealistic. That there are biologically sig-
nificant examples where the methodology has failed suggests that the limitations of
the methodology may be endemic – a conclusion for which we shall indeed argue.
We do not argue that the static methodology should be abandoned – equilibrium
analysis is an important part of the process of understanding evolutionary games.
Instead, we suggest that any full analysis of strategic interaction must proceed by
utilizing both methodologies in tandem.
We will begin in Section 2 by describing the equilibrium methodology and dis-
cussing the already well known limitations of this method. In Sections 3 and 4
we describe two cases from the signaling games literature where we believe this
methodology has misled investigators in their search for explanations of behavior.
In contrast, we suggest that the dynamic methodology provides significant insight.
Finally, in Section 5, we conclude.
2. The equilibrium methodology
Game theory was initially developed in economics as a model for human strategic
interaction. A game in strategic form is a mathematical object which includes a
list of players, a set of strategies for each player, and a specification of a payoff for
every combination of strategies by each player. Game theory was later introduced to
biology by Maynard Smith and Price (1973) (although similar ways of approaching
problems go back at least as far as Fisher (1915)). In a biological setting the
strategies are interpreted as alternative phenotypes and the payoffs are interpreted
as fitnesses.
In analyzing games in both economics and biology, it has become common to
develop conditions required for a set of strategies to be in equilibrium. Most well
METHODOLOGY IN BIOLOGICAL GAME THEORY 3
known in game theory is the concept of Nash equilibrium, which merely requires
that no player could improve her situation by unilaterally switching. Nash equilibria
can be of two types. Pure strategy Nash equilibria represent situations where
an entire population is monomorphic with respect to the equilibrium phenotype.
Mixed strategy equilibria involve random distributions of strategies and require a
more nuanced interpretation. In economics, these equilibrium strategies represent
intentional randomization by individual players. In a biological setting, a mixed
strategy might either represent a single organism whose phenotype is determined
by a random process, or it might represent a population that is polymorphic – one
that has several different phenotypes represented.
The Nash equilibrium criterion picks out a set of strategies as deserving special
attention. Once there, no player has a positive incentive to leave, and so one might
expect that this set of strategies (in biology, phenotypes) would be stable. More
specifically, it’s stable in a weak sense. A player might do equally well by switching.
Consider, for example, the game in Figure 1. Here the strategy profile (A, A)
is a Nash equilibrium, because no one does strictly better by switching to B. But
considered from an evolutionary perspective, this equilibrium seems suspect. Sup-
pose one begins with a population of A-types. If a mutant B-type were introduced,
she would not be eliminated by natural selection since she does as well as any
other. Should another B-type arise and should they interact, their fitness will be
enhanced and natural selection should favor the B-types. The reason for this is
that Ais weakly dominated by B. This means that Balways gets at least as high
a payoff as Aand a higher payoff in at least one instance. As a result, the concept
of Nash equilibrium is too general from an evolutionary point of view. It includes
population-states which one would not expect to be stable.
A B
A0,0 0,0
B0,0 1,1
Figure 1. The strategy profile (A, A) is a Nash equilibrium, but
Ais weakly dominated by B.
One possibility is to restrict attention to strict Nash equilibria. To be a strict
Nash equilibrium it must be the case that every individual will do strictly worse
by switching. It is usually regarded that such a restriction is too strong. It seems
clear that strict Nash equilibria should count as stable for any evolutionary process
(at least in finite games). However, there are certain mixed Nash equilibria which
should also be considered stable from an evolutionary point view. Mixed strategy
4 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Nash equilibria cannot be strict. As a result, a concept that lies in between Nash
equilibrium and strict Nash equilibrium must be considered.1
Maynard Smith and Price (1973) and later Maynard Smith (1982) suggested a
notion of evolutionary stability that would coincide with the biological notion of
uninvadability. Those phenotypes which cannot be invaded by small mutations are
called Evolutionarily Stable Strategies (ESS). Formally, the definition of an ESS is,
Definition 1. A strategy (i.e. phenotype) s∗is an ESS if and only if the following
two conditions are met
(1) u(s∗, s∗)≥u(s, s∗)for all alternative strategies sand
(2) If u(s∗, s∗) = u(s, s∗), then u(s∗, s)> u(s, s).
u(x, y) represents the fitness (payoff) of strategy xagainst y. The first condition
states that s∗is in Nash equilibrium with itself, i.e., there is no other strategy
earning a higher payoff against s∗. The second condition guarantees stability in
case of a mutant strategy sthat earns the same payoff against s∗by requiring that
s∗is doing better against sthan the mutant strategy against itself.
The two conditions for the evolutionary stability of a strategy seem to be natural
for a first approximation where we assume strategies to be distributed in a large,
randomly interacting population, and where an incumbent strategy is confronted
only with one mutant at a time. There is an alternative characterization of ESS
that is particularly revealing in this context. It is easy to show that s∗is an ESS if
and only if
u(s∗, εs + (1 −ε)s∗)> u(s, εs + (1 −ε)s∗).
for all εthat are less than some sufficiently small ¯ε. That is to say, in a population
with a share of ε s strategies and 1 −ε s∗strategies, s∗gets a higher expected
payoff than s.
The strategy s∗under consideration might be a pure strategy or a mixed strat-
egy (a probability distribution over different pure strategies of the game). Like
interpreting mixed strategy Nash equilibrium, interpreting mixed strategy ESS is
a delicate matter (Bergstrom and Godfrey-Smith, 1998).
It should be emphasized that both Nash equilibria and ESS are static concepts.
In a biological context, one considers a population at a given state and asks if this
population would remain at that state. Therefore they cannot, necessarily, explain
how a population arrived at that state. Maynard Smith did, in part, recognize
this problem. He suggested that in two pathological cases, ecologists would have to
consider “change as well as constancy” (1982, 8). His particular focus was on the
issue of ESS, and we will limit our attention to that case here.
1The well-known evolutionary game Hawk-Dove is used as an illustration of this situation.
METHODOLOGY IN BIOLOGICAL GAME THEORY 5
The first case is one where no strategy is an ESS. The simplest example of
such a case is illustrated by the children’s game Rock-Paper-Scissors, illustrated in
Figure 2. In this game there is a single Nash equilibrium where each player plays
each strategy with equal probability. However, this strategy is not evolutionarily
stable. Because all strategies do equally well against this mixed strategy, we must
consider the second condition of the ESS definition. This requires that the uniform
mixture do better against any alternative than this alternative does against itself.
However, the uniform mixture secures an expected payoff of 0 against all alternative
strategies, which is precisely what an alternative strategy secures against itself. As
a result, there is no ESS in this game.
Rock P aper Scissor s
Rock 0,0−1,1 1,−1
P aper 1,−1 0,0−1,1
Scissors −1,1 1,−1 0,0
Figure 2. Rock-Paper-Scissors
Maynard Smith thought this represented no significant problem for his method-
ology for two reasons. First, the paucity of the ESS-methodology in this case was
clear; there was no ESS. A theoretical ecologist confronting such a game would
immediately see that the methodology was unhelpful. Second, Maynard Smith
suggested he was aware of no biologically realistic situation where there is no ESS.
However, it has since been discovered that the mating behavior of male side-blotched
lizards (Sinervo and Lively, 1996) and toxin production in some bacteria (Kerr
et al., 2002) both follow a Rock-Paper-Scissors structure, suggesting that this game
is biologically plausible. One could, just possibly, still maintain that strategic in-
teractions without an ESS are very rare.
Perhaps more importantly, Maynard Smith’s second situation where one must
consider “change as well as constancy” occurred when there was more than one
ESS. A trivial version of such a game is illustrated in Figure 3. In this game both
Aand Bare ESSs. Thus, the ESS theory cannot predict which should be expected
to evolve.
Again, however, Maynard Smith regarded such situations as obvious. The pres-
ence of two ESSs will alert one to the presence of a troubling case. In both these
cases, no ESS and more than one ESS, Maynard Smith cautions that one should not
A B
A1, 1 0, 0
B0, 0 1, 1
Figure 3. A coordination game
6 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
suppose that an ESS strategy is likely to evolve. Notice, though, that it is much
harder to downplay the case of having more than one ESS, since in a strategic
context such situations are presumably abundant.
After Maynard Smith, another potentially troubling case was suggested by Nowak
(1990). Nowak presents a game with nonlinear payoffs where an ESS may not be
the result of evolution. Again, however, these examples depend on fitness values
which are not linear in the population proportions of the other strategies – another
tipoff that the ESS analysis may be misleading.
Given the various limitations of the equilibrium methodology, why should May-
nard Smith have championed it? In the first place, he seemed to think that the
class of strategic interactions where it is applicable comprises nonetheless a large
part of real-world interactions. Secondly, the equilibrium methodology appears very
general. By focusing clearly on the stability question, it is hoped that the result of
equilibrium analysis will apply to many different types of underlying evolutionary
dynamics. One does not, for instance, need to make any assumptions about the
role of mutation, drift, population size or structure, environmental heterogeneity,
etc., in order to derive important conclusions from the model. We believe that it is
this “dynamics agnosticism” which motivates many biologists to utilize the equilib-
rium methodology. A motivation for this might be the desire to derive conclusions
that will apply to as many potential situations as possible and thus would require
few idealized assumptions. If one is too specific, then one runs the risk of being
inapplicable to many biologically realistic situations.
In the following two sections we will present two cases where the equilibrium
analysis has been employed, but is nonetheless misleading. The first one (Section 3)
is, in a sense, of Maynard Smith’s second type. Here we illustrate a subtle slide in
reasoning that has allowed scholars to ignore that the game is one of these cases, and
thus has led to incorrect assumptions about the evolvability of certain strategies.
The second case is more troubling. The Sir Philip Sidney game is a game invented
by Maynard Smith and analyzed using the equilibrium methodology. This game
does not conform to any of the above known pathological cases. In Section 4 we
show how, even here, the equilibrium analysis is misleading in the answer it gives
to the evolvability question.
3. Common interest signaling
Signaling has become a canonical example of the application of game theory to
biological phenomena. One might wonder how two organisms evolved to use some
METHODOLOGY IN BIOLOGICAL GAME THEORY 7
arbitrary mechanism for the exchange of information.2Of special interest to biol-
ogists is how such a system of information transfer might come about in situations
where the parties to the transfer might have diverging interests about what should
be done with that information. We will return to these cases of signaling in the
face of partial conflict of interest in Section 4. But for the time being we will focus
on the case where all parties have common interests.
3.1. Lewis’ signaling game. Perhaps the simplest example of signaling was first
discussed by Lewis (1969). Lewis described a game where there are two parties:
a sender and a receiver. The sender observes some feature of the world that is
relevant to both parties. She can send one of a set of messages to the receiver.
The receiver can observe the message, but not the feature of the world, and take
some action. Depending on the state of the world, different actions benefit both
the sender and the receiver equally.
We can consider a class of finite versions of these games known as N×N×N
signaling games. In these games there are Nstates of the world and Nactions,
where there is exactly one action which is appropriate in each state and it differs
from state to state. The sender has access to Nmessages which she can send to
the receiver. We assume that both the sender and the receiver benefit when the
receiver takes the appropriate action. This makes the Lewis signaling game a game
of common interest.
3.2. Static analysis. In the N×N×Nsignaling game, there are many Nash
equilibria. There are N! combinations of strategies which are dubbed by Lewis as
“signaling systems”.3In these equilibria the sender chooses a different message for
each state and the receiver chooses the appropriate act given the message. In these
equilibria both the sender and the receiver do as well as possible. It is also easy to
show that such states are only ESSs of this game (W¨arneryd, 1993).4
The fact that there is no conflict of interest between the sender and the receiver
in a Lewis signaling game has led some to conclude that common interest signaling
2The details of what should count as “signaling” are discussed in some detail in (Maynard Smith
and Harper, 2003) and (Searcy and Nowicki, 2005). While an interested theoretical problem, we
will leave it aside for the time being.
3The number of signaling systems corresponds to the number of all one-to-one functions from a
set with Nelements into a set with Nelements.
4It should be noted that ESS as a concept is only applicable to symmetric games – those where
all players have the same strategy set and the payoff of playing strategy xagainst ydoes not
depend on the identity of the player playing strategy x. Lewis signaling games as defined are
not symmetric, because the sender and receiver have different strategies. W¨arneryd considers the
symmetrized version of the game, where each player can be in the role of the sender and of the
receiver. A player’s strategy includes both a sender-strategy and a receiver-strategy. It is well
known that the ESS of a symmetrized game correspond exactly to the strict Nash equilibria of the
original game. In the discussion that follows we consider only the symmetrized Lewis signaling
game.
8 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
games present no evolutionary mysteries. The following quote represents this stance
fairly well:
Honest signaling ... would be expected if the signaler and receiver
have identical interests in an evolutionary sense... Communica-
tion between two such individuals would be akin to communication
between two cells or two organs within an individual and one in
general would not find reliability puzzling... (Searcy and Nowicki,
2005, 20, emphasis added)
While Maynard Smith suggested that cases with more than one ESS – like the
Lewis signaling game – represent a situation where one must consider “change as
well as constancy”, Searcy and Nowicki appear to disagree. Although no explicit
argument is given for Searcy and Nowicki’s conclusion that signaling “would be
expected” in this context, we understand why such a conclusion might seem ap-
pealing. If sender a receiver have identical interests, there appears to be no reason
why they should not be able to communicate. While there is more than one ESS
in the Lewis signaling game, they all feature perfect communication. If one is only
interested in the property of signaling successfully without regard for how that sig-
naling is achieved, then one expects signaling to evolve. We will see shortly that
this argument is invalid because, even in a common interest game like this one,
evolution need not take populations to an ESS.
3.3. Dynamic analysis. A more careful analysis of the dynamics of evolution in
these games does not bear out Searcy and Nowicki’s optimistic view of common
interest signaling. First one should note that there are a variety of other Nash
equilibria where less than perfect signaling is possible. In all these games there
are equilibria known as pooling equilibria, where no information is transmitted.
Suppose that the sender ignores the state of the world and sends the same signal
regardless of state, and suppose that the receiver ignores the signal and takes the
action which is most likely to be best (or chooses some action randomly among
those who are most likely to be best). This set of strategies is an equilibrium; no
player can do better by switching, but it is not an ESS.
When N > 2 there can also be states where some information is communicated,
but less than perfect communication is achieved. Consider the strategies pictured
in figure 4. Here xand yrepresent probabilities which lie strictly between 0 and 1.
This strategy is an equilibrium; neither player can gain by switching. However, it
features less than perfect communication. While state 3 is communicated perfectly,
states 1 and 2 are pooled onto signal 1.
These equilibria are not ESSs, not because they can be invaded by a mutant, but
instead because certain mutants will not be eliminated by evolution. Therefore in
METHODOLOGY IN BIOLOGICAL GAME THEORY 9
Figure 4. A partial pooling equilibrium
order to determine the evolutionary significance of total pooling and partial pooling
equilibria we must turn to a model which explicitly considers how strategies change
over time.
We will first consider perhaps the simplest model of evolution in games, the
replicator dynamics (Taylor and Jonker, 1978; Hofbauer and Sigmund, 1998). This
set of differential equations requires that individual strategies increase in frequency
in a population only when they do better than the population average. In this
model population proportions are treated as real numbers, and so the population
is presumed to be infinitely divisible. Also, a strategy receives, as its payoff, the
average it would receive against the population. As a result the replicator dynamics
represents individuals as interacting at random. Using this underlying model for
evolution we can return to the two questions we began with.
First, what populations are stable utilizing this model of evolution? A general
fact about this dynamics is that all ESSs are asymptotically stable. If the popu-
lation is at an ESS and there is a small perturbation in the strategy frequencies,
the population will bounce back to the ESS. In other words, if the population’s
strategy frequencies start close to an ESS, the population will not only stay nearby
but also converge to it.
However, other population states are Lyapunov stable. This means that small
mutations do not snowball into large scale changes in the population; populations
10 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Figure 5. Basins of attraction for total pooling equilibria in the
two-state, two-signal, two-act signaling game. The x-axis repre-
sents the probability of state 1.
starting close to a Liapunov stable population state will stay nearby.5Except in
the special case where there are two states of the world, signals, and acts and the
two states are equally probable, several of the pooling or partial pooling equilibria
are Lyapunov stable (Huttegger, 2007; Pawlowitsch, 2008).
Second, what populations are likely to evolve? In the special case discussed
above where there are two states, signals, and acts and the states are equally
probable, essentially every initial starting population evolves to a signaling system
(Huttegger, 2007).6In this case, Searcy and Nowicki were right; signaling should
be expected. However, in every signaling game where there are more signals or the
states are not equally likely there is some significant set of initial populations that
evolve to imperfect or no communication.
Figure 5 illustrates the situation for a 2 ×2×2 signaling game. Here you can
see that signaling is assured when the states are equally likely. However, signaling
is far from guaranteed when one state is 9 times more likely than the other.7
Huttegger (2007), Pawlowitsch (2008), and Huttegger et al. (2010) have shown
that the partial pooling equilibria have positive basins of attraction as well. Already
when there are three states (which are equiprobable), signals, and acts, five percent
of the initial starting proportions evolved to a state with partial communication.
The replicator dynamics does not include any method for representing mutation,
where new strategies can be constantly introduced into the population. However,
5This is a weaker concept of stability than asymptotic stability. Liapunov stable population states
are nonetheless significant as approximations.
6Formally, the set of states that evolve to a signaling system comprise a set of measure 1.
7This figure was generated by simulation using the related discrete time replicator dynamics.
METHODOLOGY IN BIOLOGICAL GAME THEORY 11
Figure 6. An end state for the spatial Lewis signaling game (from
Zollman 2005).
one can modify this dynamics slightly in order to account for this possibility. One
such modification is known as the selection-mutation dynamics (Hofbauer, 1985).
Many things are possible under this dynamics. It might be the case that all initial
states evolve to the signaling systems for certain amounts of mutation, but it need
not be the case for others (Hofbauer and Huttegger, 2008).
One might also want to relax some of the other assumptions of the replicator
dynamics. For instance, if one relaxes the assumption that the population is ef-
fectively infinite, one can use the Moran process (Moran, 1962). Again we get a
variegated picture, where sometimes these non-signaling equilibria are avoided but
not always (Huttegger et al., 2010; Pawlowitsch, 2007).
It is also important to consider a situation where individuals interact non-
randomly. Perhaps they are constrained by physical space to only interact with
people that are near them. Signaling games in this type of situation have been
considered by Wagner (2009) and Zollman (2005). Zollman (2005) shows when
individuals are mapped unto a plane and constrained to interact with others who
are near them, new stable states emerge. Every simulation resulted in a state like
the one pictured in Figure 6. Here there are two co-existing signaling systems (one
pictured in white the other in black). This state is not an ESS, but is stable (in
a weak sense) and, more to the point, is what will result from evolution. Wagner
(2009) confirmed that these types of non-equilibrium states are possible even if one
modifies the underlying structure of interaction to different types of social networks.
We believe that the results from evolutionary dynamics, taken together, indicate
two important things. Firstly, the ESS analysis is incomplete. Considering the
12 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Lewis signaling game only in terms of ESS will suggest that “signaling ought to
be expected.” An explicitly dynamical model of the process casts doubt on this
strong conclusion. This is especially true for the replicator dynamics, which is
strongly related to the concept of ESS in that both the replicator dynamics and
ESS make similar assumptions about the population. What this illustrates is that
the equilibrium methodology does not live up to its goal of being dynamics agnostic.
Different dynamics yield different outcomes, some but not all of which coincide with
the equilibrium analysis.
Secondly, considering evolutionary dynamics reveals a much more interesting and
complicated picture of the relationship between the game, the process of evolution,
and the expected outcomes which warrants investigation in order to develop a
deeper understanding of the process by which signaling is to emerge.
For a variety of reasons the Lewis signaling game has not been extensively studied
by biologists. Perhaps this is because of the simple facts about ESS or perhaps for
some other reason such as the lack of conflicting interests and the resulting supposed
triviality of signaling reliability. As a result, it is hard to say exactly how important
the failure of the ESS methodology is in this case. We will now turn to a game
that has been extensively studied and that (in some cases) has a unique strict
Nash equilibrium and, therefore, a unique ESS. Again we will show how the static
analysis has been misleading, and we will show why an explicitly dynamic analysis
will make one doubt the general conclusion that results from finding a particular
type of equilibrium in the game.
4. The Sir Philip Sidney game
The Lewis signaling game features common interest – in every state the sender
and the receiver have the same ordering over the potential actions. This situation
has not been extensively studied by ecologists. Instead they have focused on the
apparently mysterious case where signaling exists, but where there are not perfectly
overlapping interest between the sender and the receiver.
In situations where the sender an receiver no longer have common interest, there
is a mystery about stability. Consider for example a canonical case – signaling
between potential mates. Individuals of one sex (here we will say males) might
differ in quality, and the other sex (here females) would prefer to mate with those
of higher quality.8In such a situation high quality males would be selected to make
their quality conspicuous. However, low quality males would also be selected to
display whatever characteristic the high quality males use to signal their quality.
But, if both high and low quality males appear the same in some respect, females
8This is the standard story which begs many questions, such as the exact nature of quality.
METHODOLOGY IN BIOLOGICAL GAME THEORY 13
would be selected to ignore this trait, and the signal would cease to serve any
evolutionary function.
Zahavi (1975) noted that despite this evolutionary story there appears to be
a large number of cases where males successfully signal their quality to females.
Zahavi developed the “handicap principle” where he suggested that only those
signals which involved costs would be stable. This notion was formalized by Grafen
(1990) and then simplified to a particular case by Maynard Smith (1991).
Maynard Smith considered a situation different from sexual signaling where in-
dividuals nonetheless have a conflict of interest. He considered the situation of a
child begging for a resource from its parent. The child could be in one of two states,
needy or healthy. While both needy and healthy children benefited from receiving
the resource, the needy child benefited more. The child could communicate with
the parent by sending a costly signal, which the parent can observe (the parent
cannot directly observe the state of the child). Once observing the signal, the par-
ent decides whether or not to transfer the resource, and reduce its own individual
fitness to benefit its child.
4.1. Static analysis. If the two players are unrelated, the dominant strategy for
the parent is to keep the resource. After all, what does it gain from reducing its own
fitness? This remains true if the parent and the child are related to a low degree.
But, once they are related to a sufficiently high degree, the parent wishes to donate
the resource to the needy child but not to the healthy one, because the gain to the
needy child is sufficiently high to warrant reducing her own fitness, whereas the
gain for the healthy one is not. Both the healthy and the needy child would prefer
to secure the resource, however. So, in this case there is a conflict of interest when
the child is healthy (the child wants a resource that the parent would prefer not to
give). When the parent and the child are related to sufficiently high degree, this
game is similar to the common interest signaling game discussed in Section 3. The
child only wants the resource if it is needy, and the parent only wants to transfer
the resource if the child is needy.
As indicated before, the central mystery for ecologists has been the middle case
where interests diverge. Why, when the child has an incentive to lie, would honest
signaling persist? Maynard Smith showed that whenever one is in this situation
one can impose a cost on the child for sending the signal which is sufficiently high
that only the needy child is willing to pay the cost in order to secure the resource.
When the game has partial conflict of interest and a single signal with sufficiently
high cost, the unique ESS of the game is one where only the needy child signals
and where the parent transfers the resource only if she observes the child signaling.
14 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Maynard Smith’s game has been taken to illustrate how cost (or handicaps) can
explain the stability of signaling in the face of conflicts of interest. Signaling would
not be an equilibrium, and would thus be unstable, if there were no cost. But when
there is sufficiently high cost it is the only strict equilibrium.
4.2. Other equilibria. Bergstrom and Lachmann (1997) first identified a poten-
tial problem with the explanation for signaling offered by Maynard Smith. They
compare the signaling equilibrium to total pooling states. Like the Lewis signaling
game, total pooling states are stable but in a weaker sense than ESS. However,
they can have the property of being Pareto superior to the signaling state – i.e.
the total pooling state is better for both the parent and child than the state of
signaling. It would be odd, Bergstrom and Lachmann claim, for evolution to lead
from a (weakly) stable superior state to an inferior one.
Huttegger and Zollman (2010) also show that another state of interest exists in
this game, a hybrid equilibrium. When the cost of the signal is too low to support
a signaling equilibrium, but nonetheless above a certain threshold, there exists an
equilibrium where some communication occurs. In this equilibrium the needy child
always sends the signal and the healthy child sometimes sends it, and the parent
sometimes transfers the resource when it receives the signal but not always. This
is an equilibrium of the game. Neither the parent nor the chick could do better by
switching. But it is not an ESS.
In his initial paper, Maynard Smith was only concerned with offering an expla-
nation of stability, and does not consider other equilibria. In later work (Maynard
Smith and Harper, 2003) he seems sensitive to the existence of pooling equilibria
and recognizes that one must offer an explanation for why the signaling equilibrium
is arrived at rather than another. Even those who prefer an equilibrium based anal-
ysis should be concerned. By expanding our purview to consider equilibria beyond
ESS, we have multiplied the number of possibilities. If one were to simply stop here,
very little could be said about the evolvability of different types of signaling. One
simply cannot answer that question now without considering dynamics explicitly.
4.3. Dynamic analysis. When one turns to a dynamic analysis, one finds similar
concerns that plagued the Lewis signaling game. In the Sir Philip Sidney game,
pooling equilibria remain a problem. One can begin at the “best case” for signaling,
when the parent and the child are clones. In such a case, a gain for the parent is a
gain for the child and vice versa. In this situation there is no conflict of interest. The
game is much like the Lewis signaling game. Just like in the Lewis signaling game
there are two ESS where the parent and the chick perfectly coordinate, and like in
the Lewis signaling game the dynamic analysis reveals similar concerns. Figure 7
(from Huttegger and Zollman (2010)) shows the basins of attraction for signaling
METHODOLOGY IN BIOLOGICAL GAME THEORY 15
Figure 7. Basins of attraction for the signaling ESS in the Sir
Philip Sidney game when the parent and the child are clones
for various amounts of cost. Again one sees that pooling equilibria pose a problem,
especially when it is either very likely or very unlikely that the chick is needy.
What about the cases of primary interest to biologists, where there is conflict
of interest between the parent and the child? Here too, Huttegger and Zollman
find problems. In these situations, where there is a unique strict Nash equilibrium
which features signaling, they find that signaling is less likely to evolve than pooling
equilibria! In some cases they studied the probability of a random population
evolving to signaling was less than 20%.
This presents a significant problem for the equilibrium methodology. In order
to see why, let us revisit the explanation it attempts to offer. The question, which
began much of biological investigation into signaling, is one about stability. Why
do we observe many different instances of signaling in the face of conflict of in-
terest, where there are apparently strong evolutionary forces that should drive us
away from that state? The proffered explanation from Zahavi, Grafen, and May-
nard Smith centers around cost: this state is stable because the signal carries a
sufficiently high cost that it is not “profitable” for the signalers to lie.
In this very limited respect, the static analysis succeeds. That signaling is an
ESS in the Sir Philip Sidney game does demonstrate that it is stable under most
evolutionary dynamics. So, if our interest is the very narrow question of stability,
16 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
then we have a legitimate answer. However, this explanation comes at a price. The
very same model explains the stability of signaling and indicates that signaling is
unlikely to evolve. An evolutionary mystery has been answered by substituting
another. This new mystery might lead one to question the adequacy of the answer
to the stability question, for if a behavior is made stable only by making it difficult
to evolve, it seems unlikely that this is the correct explanation for its stability in
the first place.
By restricting themselves only to an equilibrium based approach, those who
have analyzed the Sir Philip Sidney game have obscured this fact. Thus, even if
the equilibrium method succeeds at being general, this generality comes at a cost.
One might produce explanations that, from a wider perspective, seem implausible.
5. Static and Dynamic Approaches
As we have mentioned repeatedly throughout the paper, the main advantage
of the equilibrium methodology seems to be its claim of generality. We should
expect equilibria like Maynard Smith’s ESSs to be observable across a wide range
of distinct ecological circumstances. When is this claim of generality justified?
There are empirical aspects to this question. In this paper, we have focused on its
theoretical aspects, however. We think that the generality claim of an equilibrium
methodology also has implications for somewhat more specific models that include
dynamical details of evolutionary processes. More precisely, if an equilibrium is
claimed to be a very general outcome of evolution, then it should also be a significant
state in many dynamic models of evolution. Otherwise, we have reason to doubt
that our explanation of a real world state in terms of an equilibrium is correct. We
have shown argued that in two biologically relevant situations this generality does
not obtain.
In addition to the cases we discuss here, there are other situations that might
cause one to worry about the generality of equilibrium based approaches. Some
of the problems we discuss here are endemic of games that feature a non-trivial
extensive form (Huttegger, 2010). Other important games have similar situations,
like the widely studied ultimatum game which seeks to model situations of very
simple economic exchange (Zollman, 2008). Furthermore, Wagner (2011) has re-
cently provided another example which illustrates a problem with the equilibrium
based approach. Wagner studies a zero-sum signaling game, where the sender and
the receiver never agree about which action is best to perform in a given state of the
world. One would expect when interests are so radically divergent that communica-
tion would never emerge, and this is exactly what the equilibrium based approach
indicates. However, he shows in the replicator dynamics that one observes chaotic
behavior, where some partial communication will emerge only to be later destroyed.
METHODOLOGY IN BIOLOGICAL GAME THEORY 17
These concerns have shown that the ESS methodology does not always achieve
its primary aim: generality. But, even if it were to achieve this, we believe that
there are other concerns. In the Sir Philip Sidney game, we have shown how the
equilibrium methodology’s focus on the question of stability obscured other relevant
considerations. The purported explanation for the stability of signaling in the face
of partial conflict of interest succeeded, but only by introducing another mystery:
how could such a behavior have evolved? If we explain the stability of a behavior
only by introducing a model that also suggests this behavior is very difficult to
evolve, one might want to question the adequacy of the explanation for stability as
well. One would not have realized that such a mystery had been introduced without
considering an explicitly dynamic model. This is an independent reason, beyond
concerns of generality, to eschew any methodology which focuses exclusively on the
question of stability.
Although we have been critical of ESS methodology or, more general, of a static
equilibrium methodology, we would like to emphasize that we don’t think that ESS
and other equilibrium concepts are useless. Quite to the contrary, we think that they
are indispensable tools that allow us to get a basic understanding of evolutionary
processes without getting tangled up in subtle dynamical considerations right in
the beginning of an investigation. What we deny is that one can conclude that a
state is a significant evolutionary outcome from the fact that it has been shown to
be an equilibrium. We propose that a result like this one is an ingredient in an
evolutionary explanation that needs to be supplemented by other results, empirical
as well as theoretical. Other theoretical results are particularly important, first of
all in order to guide empirical research, and secondly to explore the equilibrium
in terms of more specific dynamical assumptions. We thus argue for a pluralistic
approach to study evolutionary outcomes that takes advantage of the plethora of
methods that are available in evolutionary and mathematical biology.
References
Bergstrom, C. T. and M. Lachmann (1997). Signalling among relatives. I. Is costly
signalling too costly? Philosophical Transactions of the royal Society of London
B 352, 609–617.
Bergstrom, C. T. and P. Godfrey-Smith (1998). On The Evolution of Behavioral
Heterogeneity in Individuals and Populations. Biology and Philosophy 13, 205–
231.
Fisher, R. A. (1915). The evolution of sexual preference. Eugenics Review 7,
184–192.
Grafen, A. (1990). Biological signals as handicaps. Journal of Theoretical Biol-
ogy 144, 517–546.
18 SIMON M. HUTTEGGER AND KEVIN J.S. ZOLLMAN
Hofbauer, J. (1985). The selection mutation equation. Journal of Mathematical
Biology 23, 41–53.
Hofbauer, J. and S. Huttegger (2008). Feasibility of communication in binary sig-
naling games. Journal of Theoretical Biology 254 (4), 843–849.
Hofbauer, J. and K. Sigmund (1998). Evolutionary Games and Population Dynam-
ics. Cambridge: Cambridge University Press.
Huttegger, S. (2007, January). Evolution and explanation of meaning. Philosophy
of Science 74 (1), 1–27.
Huttegger, S. (2010). Generic Properties of Evolutionary Games and Adaptationism
Journal of Philosophy 107(2), 80–102.
Huttegger, S., B. Skyrms, R. Smead, and K. Zollman (2010). Evolutionary dy-
namics of Lewis signaling games: Signaling systems vs. partial pooling. Syn-
these 172 (1), 177–191.
Huttegger, S. and K. J. Zollman (2009). Signaling games: The dynamics of evolu-
tion and learning. In A. Benz (Ed.), Language, Games, and Evolution.
Huttegger, S. and K. J. Zollman (2010). Dynamic stability and basins of attraction
in the Sir Philip Sidney game. Proceedings of the Royal Society of London B 277,
1915–1922.
Kerr, B., M. A. Riley, M. W. Feldman, and B. J. M. Bohannan (2002, 11 July).
Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors.
Nature 418, 171–173.
Lewis, D. (1969). Convention: A Philosophical Study. Cambridge: Harvard Uni-
versity Press.
Maynard Smith, J. (1982). Evolution and the Theory of Games. Cambridge: Cam-
bridge University Press.
Maynard Smith, J. (1991). Honest signaling, the Philip Sidney game. Animal
Behavior 42, 1034–1035.
Maynard Smith, J. and D. Harper (2003). Animal signals. Oxford: Oxford Univer-
sity Press.
Maynard Smith, J. and G. Price (1973). The logic of animal conflict. Nature 146,
15–18.
Moran, P. (1962). The Stastical Process of Evolutionary Theory. Oxford: Clarendon
Press.
Nowak, M. (1990, January). An evolutionary stable strategy may be inaccessible.
Journal of Theoretical Biology 142 (2), 237–241.
Pawlowitsch, C. (2007). Finite populations choose an efficient language. Journal
of Theoretical Biology 249, 606–617.
Pawlowitsch, C. (2008). Why does evolution not always lead to an optimal signaling
system. Games and Economic Behavior 63, 203–226.
METHODOLOGY IN BIOLOGICAL GAME THEORY 19
Searcy, W. A. and S. Nowicki (2005). The Evolution of Animal Communication.
Princeton: Princeton University Press.
Sinervo, B. and M. Lively (1996, March 21). The rock-paper-scissors game and the
evolution of alternative male strategies. Nature 380, 240–243.
Taylor, P. and L. Jonker (1978). Evolutionarily stable strategies and game dynam-
ics. Mathematical Biosciences 40, 145–156.
Wagner, E. (2009). Communication and structured correlation. Forthcoming in
Erkenntnis.
Wagner, E. (2011). Deterministic Chaos and the Evolution of Meaning. British
Journal for the Philosophy of Science forthcoming.
W¨arneryd, K. (1993). Cheap talk, coordination, and evolutionary stability. Games
and Economic Behavior 5, 532–546.
Zahavi, A. (1975). Mate selection – a selection for a handicap. Journal of Theoretical
Biology 53, 205–214.
Zollman, K. J. (2005). Talking to neighbors: The evolution of regional meaning.
Philosophy of Science 72, 69–85.
Zollman, K. J. (2008). Explaining Fairness in Complex Environments. Politics,
Philosophy, and Economics 7(1), 81 – 97.
