Content uploaded by Kevin J S Zollman

Author content

All content in this area was uploaded by Kevin J S Zollman on Jan 17, 2014

Content may be subject to copyright.

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.