76 •
Chapter 2
mous or time homogeneous. Thus, the system abstracts from historically varying influences on the equations, such as the state of knowledge, technology, institutional facts taken as given, or the weather. Of
course, if we understood the dynamics of these time-varying influences,
we could include them as state variables in the dynamical system.
Whether the time-homogeneous nature of the replicator dynamic is a
problem or not depends on the question at hand; for many problems,
abstracting from, say, climate change is reasonable and for some it is
not. The interpretation of the emergence of individual property rights in
chapter 11 is a case in which variations in weather make a significant
difference. If the selection processes described by the replicator dynamic
are slow relative to the changes in the underlying technologies and other
exogenous data defining the underlying game, the dynamical system
may never reach the neighborhood of the stationary values of p (as
these will be continually displaced by exogenous changes.)
A third problem with the replicator dynamic is suggested by its name:
it cannot be used to study innovation. To study genuine novelty (as
opposed to differential replication of existing traits), I need to introduce
the complementary concept of an evolutionarily stable strategy.
Evolutionary Stability and Societal Outcomes
Under what conditions can a population be “invaded” by a new trait?
Concrete examples of such an invasion include the rapid spread of the
practice of having small rather than large families in many countries
over the past century. Or think of late feudal European society, “invaded” by a small number of Italian and other merchants using entirely
new business practices such as double-entry bookkeeping and the community responsibility system of contractual enforcement (Greif 2002,
Padgett 2002). The invaders prospered and eventually transformed the
feudal order. Other examples include corrupt business practices invading a community of honest traders, or the deferential forms of address
of a linguistic community being invaded by familiar pronouns.
While the replicator dynamic is a convenient analytical tool, a trait
absent from a population in period t cannot be copied in period t 1.
Recall that the stationarity condition for p is satisfied at p 1 and
p 0, irrespective of the payoffs that might accrue to the absent strategy, were it present. These values of p are always stationary in the replicator dynamic but may not be Nash equilibria and may not be asymptotically stable: small perturbations around p 0 and p 1 may not
be self-correcting. It is not difficult to extend replicator dynamic models
to take account of both innovations and chance; we will return to these
Spontaneous Order
•
77
stochastic evolutionary models in the closing chapters. Here, rather
than explicitly incorporating chance into the replicator equation, we
will introduce a handy shortcut for getting innovation into the picture,
the notion of evolutionary stability.
Not surprisingly, biologists pioneered the modeling of innovation.
Their interest in whether a small number of mutants could proliferate in
a large population motivated the key concept of an evolutionarily stable
strategy. The basic idea is that a population all playing an evolutionary
stable strategy will repel an invasion of individuals playing some other
strategy. Consider a large (strictly, infinite) population in which individuals are randomly paired to interact (along the lines of the model immediately above.) Suppose, as above, that we are considering two behavioral traits, x and y. Trait y is evolutionarily stable against x if there
exists some positive fraction of the population, p⬃, such that if the fraction of the population playing x is less than p⬃, then the incumbent
strategy (y) will produce more replicas than x and hence will eliminate
the entrant. I will introduce a case shortly in which you will see that the
“invasion barrier” p⬃ ∈ (0, 1) is an unstable interior equilibrium and it
defines the boundary of the basin of attraction of p 0 and p 1
mentioned above.
To see what evolutionary stability entails, we want to know what will
happen in a large population composed entirely of y’s if a small number
of x’s are introduced. Using eq. (2.6), this means evaluating ⌬p at
p ε where ε is arbitrarily small. We know that ⌬p will have the sign
of
bx(ε) by(ε) 兵ε␲(x, x) (1 ε)␲(x, y)其 兵ε␲(y, x) (1 ε)␲(y, y)其
A behavioral trait y is an evolutionarily stable strategy (ESS) with respect to some other strategy x if and only if bx(ε) by(ε) ⬍ 0, which
for arbitrarily small ε is the case when
␲(y, y) ⬎ ␲(x, y)
(2.9)
or when
␲(y, y) ␲(x, y)
and
␲(y, x) ⬎ ␲(x, x).
Thus, an ESS is a best response to itself (at least weakly, and if it is a
weak best response to itself then the other strategy is not a best response to itself). As small perturbations of p around an ESS are (by the
above reasoning) self-correcting, we know that every ESS is a symmetric
Nash equilibrium that is asymptotically stable in the replicator dynamic. Where the mutant may be a weak best response to itself (that is,
the last inequality in eq. (2.9) is not strict, but instead ␲(y,x) ⱖ ␲(x,x))
78 •
Chapter 2
then y may be neutrally stable: the invader may not be eliminated, but it
will not proliferate as a result of payoff monotonic updating, either.5 Of
course, such a neutrally stable state (NSS) can be invaded through a
process of drift (that is, further exogenously generated innovations) and
this has important implications in some applications (see, e.g., chapter
11). The NSS and the ESS are thus increasingly stringent evolutionary
refinements of Nash equilibria. Every ESS is an NSS and every NSS is a
Nash equilibrium; but of course the converse is not true.
The converse of evolutionary stability is the capacity to invade, which
Axelrod and Hamilton (1981) termed initial viability.6 If x is initially
viable against y then y is not an ESS. Notice that y’s status as an ESS
with respect to x says nothing about its status with respect to some
other trait k or two mutants k and x occurring simultaneously.
We often want to know if a mixed population (that is, one for which
p ∈ (0, 1)) is invadeable by a rare mutant. We can do this by noting
that a population, all of which adopts the same mixed strategy, is for
this reason homogeneous in strategies even though it is behaviorally
heterogeneous in the sense that at any given moment different individuals take different actions. Representing the polymorphic population as
one in which all individuals adopt a mixed strategy (playing x and y
with probability p* and (1 p*) respectively), we can refer to this
mixed strategy as an interior (or mixed) ESS with respect to some other
strategy k if, should a small number of k’s be introduced, they would be
eliminated. For p* to be an ESS, it must be stationary and asymptotically stable in the replicator dynamic; were this not the case, the
expected payoffs to the strategies making up the mixed population
(called the support of the mixed strategy) would be unequal in the
neighborhood of p*, so the payoff to one of these strategies in the support would exceed the payoff to the mixed strategy and a mutant bearing this pure strategy could invade.
Just as the replicator model is uninformative about dynamics on the
“edges” of a population (that is, for p 0 or p 1), the concepts of
initial viability and evolutionary stability are unilluminating about the
dynamic governing p when it is interior. It is generally useful to combine
the two approaches, asking of the stationary extreme values of p
whether it is an ESS, that is, asymptotically stable.
The Hawk Dove Game illustrates these concepts. As everyone knows,
Hawks are hawkish, and Doves are peace loving. The game is commonly applied to culturally or genetically transmitted human behavioral
traits such as aggression and sharing, but it was initially developed to
5
6
Thus every NSS is Lyapunov stable.
In biological usage, “viable” means capable of living and developing normally.
Spontaneous Order
•
79
Table 2.2
Hawk Dove Game (row player’s payoffs)
Hawk
Dove
Hawk
Dove
a (v-c)兾2
c0
bv
d v兾2
Note: fitness (number of offspring produced)
is equal to ␾ plus the game payoffs.
study contests among other animals. Here is the game. Doves, when
they meet, share a prize, while when Hawks meet, they fight over the
prize, inflicting costs on one another; and when a Hawk meets a Dove,
the Hawk takes the prize. A similar account maintaining the feathered
metaphor but actually applying to automobiles, makes it a “chicken”
game in which the “hang tough” drivers never swerve, so when they
meet, they crash, but when they meet a “chicken” (one who swerves),
they garner (presumably psychological) benefits, while the swerver is
humiliated. The prize to be divided is v, the cost of losing a fight is c,
and the probability of a Hawk winning a contest against another Hawk
(they are identical) is 1⁄2. Doves divide the prize equally and without
cost. Thus, the payoff matrix is as shown in table 2.2, from which it is
readily seen that as long as c ⬎ v neither H nor D is an ESS. (A handy
way to search for ESS’s in large matrices of row payoffs is to ask: is the
entry in the main diagonal the largest entry in the column? If it is, that
column represents an ESS.)
Members of this population are randomly paired, so letting bh(p) and
bd(p) be the expected payoff to being a Hawk and a Dove, respectively,
in a population in which the fraction of hawks is p, the expected payoffs illustrated in figure 2.2 are:
bh(p) pa (1 p)b
bd(p) pc (1 p)d
(2.10)
To illustrate the use of the replicator equation in a fitness-based evolutionary process, assume that at the end of a period, each member of the
population produces a number of exact replicas (excluding mutations)
equal to ␸ plus the payoff to the game, so the payoffs are in units of
offspring surviving to reproductive age, that is, fitness (␸ is called the
“baseline fitness”). The assumption that a single member (rather than a
pair) produces offspring simplifies the modeling; this clonal or asexual
reproductive assumption is a simple (and often useful) alternative to the
more realistic modeling of replicator processes based on sexual re-
80 •
Chapter 2
v
bh
1/2 v
bd
0
0
1/2(v − c)
p*
0
Hawks as a fraction of the population, p
1
Figure 2.2 Frequency-dependent payoffs in the Hawk Dove Game. The number
of replicas is equal to the payoffs plus a constant.
production. Normalizing the total population to unity, we can write
next year’s population frequency of hawks, p as
p′ =
p(bh + ␸)
pbh + (1 − p)bd + ␸
(2.11)
Read the numerator to say, “There were p Hawks in the population this
year and each of them made bh ␸ offspring, giving us p(bh ␸)
Hawks next year.” The denominator gives us the total number of Hawks
and Doves combined, next year. Given the normalization of population
size to unity, the total replicas made is also equal to the average or b.
We are interested in ⌬p, so subtracting p from both sides of eq. (2.11)
we have
⌬p ≡ p′ − p =
p(bh + ␸) p{p(bh + ␸) + (1 − p)(bd + ␸)}
−
b−
b−
(2.12)
which, with a little rearranging, and using the values in the payoff matrix to express (bh bd) as 1⁄2(v pc), gives us
b⌬p p(1 p)(bh bd) p(1 p)1⁄2(v pc),
(2.12)
which is exactly the replicator dynamic equation already derived (by
different routes) for the model of residential segregation and the general
case of cultural trait updating presented in previous sections.
Stationary interior values of p are those for which bh(p) bd(p), so