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. Dawid Mathematical Social Sciences 37 1999 265 –280
population strategy given s . Thus, the evolution of the population strategy is given by
t
the following dynamical system
As
t
]] s
5 as 1 1 2 a diags
1
t 11 t
t
9
s As
t t
We call this dynamical system replicator dynamics with inertia, since the case a 50
corresponds to the discrete time replicator dynamics see e.g. Weibull, 1995. The replicator dynamics has been analyzed in great detail in the game theory literature see
e.g. Hofbauer and Sigmund, 1988; Weissing, 1991; Weibull, 1995, however it is mainly motivated by biological rather than by economic models. Here we provide an
interpretation of the replicator dynamics with inertia as an imitation dynamics in the spirit of word of mouth learning see Ellison and Fudenberg, 1995; Dawid, 1999. Note
however that a certain level of inertia always has to be present if the dynamics is interpreted according to the model presented above. Cabrales and Sobel 1992 consider
the dynamics Eq. 1 in a more technical context without giving a specific economic interpretation of the model and without restricting the range of
a. Of course the dynamics can also be motivated by a model where all agents have information about the
payoffs of all other agents in the population and with probability 12 a choose some
other agent to imitate, where again the probability to be imitated is proportional to the past success.
It is well known that any Nash equilibrium of the symmetric game with payoff matrix
A is a fixed point of both the discrete time replicator dynamics and also its continuous time pendant
~s 5 diags[As 2 1
s9As]. 2
However, there are fixed points which are no equilibria; for example all vertices of the simplex. As already pointed out in the introduction every evolutionary stable strategy is
locally asymptotically stable with respect to the continuous time replicator dynamics. The discrete time replicator dynamics however may diverge from an equilibrium even if
it is evolutionary stable because the equilibrium may be ‘overshot’ by such a wide margin that the trajectory departs more and more from the equilibrium Weissing, 1991.
Since the notion of overshooting is a central point of the present paper we would like to make precise what we mean by this term. We say that a discrete time dynamic shows
overshooting near an equilibrium if the fixed point is unstable with respect to the discrete time dynamics but stable with respect to Eq. 2. Note however that even if the
discrete time dynamics is stable the convergence speed varies with changing levels of inertia. In particular there might be oscillating converging paths where the oscillations
could be dampened by increasing the level of inertia. On the other hand, a very high level of inertia causes tiny approach steps and accordinlgy a slow convergence. It is the
aim of this paper to derive characterizations of the crucial level of inertia which avoids instability due to overshooting.
3. Local stability analysis
In this section we first show how the spectrum of the discrete time replicator dynamics with inertia is connected to the spectrum of the continuous time replicator
H . Dawid Mathematical Social Sciences 37 1999 265 –280
269
dynamics. In particular, we prove that the stability of an equilibrium with respect to the continuous time dynamics guarantees that this equilibrium is stable with respect to the
discrete time dynamics with inertia if the level of inertia is sufficiently high. Afterwards, we derive conditions which guarantee that the continuous time dynamics is locally
attracted by a quasi strict equilibrium with a support of at most three pure strategies.
The first Lemma establishes that the eigenvalues of the linearization of the replicator dynamics with inertia are closely related to the eigenvalues of the continuous time
dynamics.
Lemma 1 Let
m be a symmetric Nash equilibrium of the game G. Then m is a fixed point of Eq. 1 and the eigenvalues
l of the Jacobian of Eq. 1 at m are given by
i
1 2 a
]] l 5 1 1
n ,
i i
v
n
where v is the game value at the equilibrium m,m and
hn j is the spectrum of the
i i 51
Jacobian of the continuous time replicator dynamics at m. Furthermore, the external
eigenvalue of the discrete time dynamics corresponds to the external eigenvalue of the continuous time dynamics
.
Proof. Note first that Eq. 1 can be written as
1 2 a
]]
9
s 5 fs 5 s 1
diags As 2 1
s As
t 11 t
t t
t t
t
9
s As
t t
9
Further, diags As 2 1
s As is the right hand side of the continuous time replicator
t t
t t
dynamics. Let us denote the Jacobian of this expression at
m by Jm. Calculating the
differential at m yields
diag
m Am 2
1
m 9Am
1 2 a
1 2 a
]]] ]]
D m 5 I 1
J m9Am 2
5 I 1 J.
f 2
v
50
m9Am
Since the eigenvectors of both dynamics coincide it is obvious that l is the external
i
eigenvalue of the linearization of Eq. 1 if and only if m is the external eigenvalue of
i
the linearization of Eq. 2. Thus, we get the lemma. h
Lemma 1 says that every eigenvalue of the discrete time dynamics with inertia lies
n
i
]
somewhere on the straight line between 1 and 1 1 where
n is an eigenvalue of the
i v
continuous time dynamics. In particular this shows the well known fact that the discrete time dynamics even with inertia can never be locally stable unless the continuous time
dynamics is stable. On the other hand it is obvious from Lemma 1 that if all n have
i
negative real parts there exists a value a such that all eigenvalues of the discrete time
dynamics with inertia a a lie within the unit circle. We state this fact in Corrollary 1.
Corollary 1 Assume the symmetric Nash equilibrium
m is a hyperbolic locally asymtotically fixed point of the continuous time replicator dynamics Eq
. 2. Then there is a level of inertia
a such that m is locally asymptotically stable with respect to any discrete time replicator dynamics with inertia
a a. This corrollary shows that there is some critical level of inertia such that the dynamic
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. Dawid Mathematical Social Sciences 37 1999 265 –280
behavior for lower levels of inertia qualitatively looks like the discrete time replicator dynamic, whereas it resembles the behavior of the continuous time replicator dynamics
for larger inertia. Thus, it is important for the determination of the stability of the discrete time replicator dynamics with inertia to derive condtitions guaranteeing the
stability of an equilibrium with respect to the continuous time replicator dynamics. In order to derive such conditions we have to determine the spectrum of the linearization of
the continuous time replicator dynamics. It is impossible to give a closed form analytical expression for the spectrum, but in Lemma 2 we reduce the problem to an eigenvalue
problem of a simpler matrix M . We denote by A the submatrix of A consisting of all
m m
rows and columns corresponding to the pure strategies in the support of m and by
C m
ˆ m [ D
the vector consisting only of the positive elements of m.
Lemma 2 Let
m be a symmetric Nash equilibrium with support Cm and v be the equilibrium payoff
. The spectrum of Jm is given by Y 5 2 v J P\ 0 ,
3 h
j h j
where 2 v is the external eigenvalue and J is the spectrum of ˆ
ˆ M: 5 diag
m I 2 1
m 9A
m
and
9
P 5
he Am 2 vj.
j j [
⁄ C m
Proof: Calculating J
m gives
J m 5 diagAm 2
1
m9Am 1 diagmA 2
1
m9A 1 A9.
Since the simplex is invariant in the replicator dynamics there is a single external or transversal eigenvalue where the corresponding eigenvector is no element of the tangent
n n
space t5 hx[R uo
x 50 j. It is easy to see that the external eigenvalue is given by
i 51 i
2 v, where the corresponding right-eigenvector is m. If we reorder the strategies in a
way such that the first C m strategies are in the support of m we realize that the first
C m rows of the first matrix in the sum are zero, whereas the last n2Cm rows of the
second matrix are zero. Accordingly, J
m has the form J
J
1 2
J
m 5 ,
S D
D ˆ
9
where J J 5diag
m A2
1
m9A1A9 and D5diage Am 2v
. Thus, all
1 2
j j [
⁄ C
m
elements of P are eigenvalues of J
m. Furthermore this implies that the remaining
3
eigenvalues of J
m coincide with the eigenvalues of J . We have
1
ˆ ˆ
ˆ J 5 diag
m A 2
1
m 9A 2 v
11
9 5 M 2 v diag m
11 9.
1 m
m m
3
These facts have already been observed by several researchers in this field, see e.g. Lemma 45 in Bomze and ¨
Potscher 1989.
H . Dawid Mathematical Social Sciences 37 1999 265 –280
271
Note further that also the simplex containing all mixed strategies with support C m is
invariant in the replicator dynamics which implies that all eigenspaces of J with
1 C
m
ˆ ˆ
exception of the external eigenvector lie in the tangent space t5
hx [D uo
i [C m
ˆ x 50
j. However, for any vector x [ t we have
i
ˆ ˆ
J x 5 M x
1 m
which implies that the remaining eigenvalues of J
m coincide with the eigenvalues of
M . Noting that m is a right-eigenvector of M for eigenvalue 0 establishes the claim of
m
the lemma. h
Taking into account Lemma 2 we only have to calculate the eigenvalues of M to
m
determine the stability of an equilibrium with respect to the continuous time replicator dynamics and to find the right level of inertia which will avoid overshooting in the
discrete time dynamics. In cases where the support of the equilibrium contains not more than three pure strategies the related calculations can be carried out without specifying
the payoff matrix A. We will provide these results in the remaining part of this section. To avoid the case where the equilibrium is a non-hyperbolic fixed point we restrict our
attention to quasi-strict Nash equilibria see e.g. van Damme, 1991.
Every quasi-strict Nash equilibrium in pure strategies is strict and accordingly locally asymptotically stable with respect to both the continuous time and the discrete time
replicator dynamics. If the support of m contains two pure strategies we easily get the
following characterization.
Proposition 1 Let
m be a quasi-strict Nash equilibrium with Cm5 hi ,i j. Then m is
1 2
locally asymptotically stable with respect to the dynamics Eq . 2 if
C1 a 2 a
2 a 1 a
, 0.
i i i i
i i i i
1 1 1 2
2 1 2 2
Proof: Since
m is a quasi-strict equilibrium we have
e A
m , v ; j [
⁄ Cm
j
and accordingly all elements of P in Eq. 3 are negative. Considering the elements J we know that one of them is zero and accordingly the other one is given by tr M .
m
Simple calculations show
trM 5
m m a 2 a
2 a 1 a
m i
i i i
i i i i
i i
1 2
1 1 1 2
2 1 2 2
and the positivity of m and m implies the proposition. h
i i
1 2
To formulate the stability conditions for an equilibrium with the support of three pure strategies we introduce the following notation. Let [a]
denote the 232 submatrix of A
ijkl
consisting of rows i and j and columns k and l: a
a
ik il
[a] 5
.
S D
ijkl
a a
jk jl
We define
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. Dawid Mathematical Social Sciences 37 1999 265 –280
n n
i 1j 1k 1l
SA 5
O O O O
21 det[a]
ijkl i 51 j .i k 51 l .k
and ˆ
D 5 diag m A .
m m
Proposition 2 Let
m be a symmetric quasi-strict Nash equilibrium of the game with support C
m5 hi , i , i j. Then m is locally asymptotically stable with respect to the
1 2
3
continuous time replicator dynamics if and only if the following two conditions are satisfied
:
C2 v . trD
m
C3 SA . 0.
m
Proof: According to Lemma 2 the eigenvalues of J
m are given by Eq. 3. Just like in the proof of proposition 1 we conclude that all elements of P are negative which leaves
us with showing that all elements of J but the external eigenvalue 0 are negative. The elements of J are given by the solutions of the equation
3 2
l 2 trM l 1 WM l 2 detM 5 0,
m m
m
where WM is the sum of the major subdeterminants of M . Since we have shown that
m m
0 is an eigenvalue of M we have detM 50. This leaves us with showing that the real
m m
parts of the two solutions of
2
l 2 trM l 1 WM 5 0
m m
are negative. It is well known that this holds if and only if WM .0 and trM,0.
m
Tiresome but straightforward calculations establish further that
trM 5 trD 2 v, WM 5 m m m SA .
m m
m i
i i
m
1 2
3
If at least one the two inequalities holds the other way round at least one of the two eigenvalues has a positive real part and accordingly
m is unstable. If either C2 or C3 hold as equality
m , m , m is a non-hyperbolic fixed point of the replicator dynamics
i i
i
1 2
3
on the simplex spanned by hi ,i ,i j. It follows from the classification in Bomze 1983
1 2
3
that no non-hyperbolic interior fixed point can be asymptotically stable with respect to the replicator dynamics for three or less pure strategies.
h Another static criterion for dynamic stability of the continuous time replicator
dynamic based on the definiteness of a matrix derived from the payoff matrix and the
4
support of the equilibrium was given by Haigh 1975. However, our approach enables us also to conveniently express the actual eigenvalues of the linearization of Eq. 2
4
Actually, Haigh’s criterion implies m to be an ESS which due to a standard result implies that m is
asymptotically stable with respect to Eq. 2.
H . Dawid Mathematical Social Sciences 37 1999 265 –280
273
which also allows us to actually compute the level of inertia needed to avoid overshooting. Measuring the level of inertia from real data seems to be impossible which
might suggest that calculating the exact values of the crucial level of inertia is not of high relevance. However, since the derivation of the actual crucial level of inertia yields,
as a corrollary, a stability criterion for the discrete time replicator dynamics and also allows some insights into the qualitative behavior of the dynamics with inertia we
present the derivation of the crucial level of inertia for equilibria with a support of three
5
pure strategies.
Proposition 3 Let
m be a symmetric quasi-strict Nash equilibrium with support C
m5 hi , i , i j. If C2 and C3 hold then there is a level of inertia
1 2
3
v
2 trD
m
]]]]] a 5 max 0,1 2 v
S D
m m m SA
i i
i m
1 2
3
such that m is locally asymptotically stable with respect to the dynamics Eq. 1 for
every a .a. If a ,a the equilibrium m is unstable.
Proof: We have to show that all eigenvalues
l of the linearization of Eq. 1 at m have
i
a modulus smaller than one. Let l be an arbitrary relevant eigenvalue. Then we know
i
from Lemma 1 that 1 2
a ]]
l 5 1 1 n ,
i i
v where
n is either an element of P or J\
h0j. If n [P then
i i
9 9
1 2
ae Am 2 v e A
m
j j
]]]]]] ]]
l 5 1 1 5
a 1 1 2 a .
i
v v
Since e A
m ,v we have l[0,1 for any a [[0,1.
j
Considering the two eigenvalues in J\
h0j we distinguish between the case where the
eigenvalues of M are real and the case where they are complex. Assume first that they
m
]]]]
are real i.e. trD v 22 m m m SA . Denote by n the smaller of the two real
m 1
2 3
m i
œ
eigenvalues. We have
trD 2 v
]]]]]]]]] 1
m 2
]]] ]
n 5 2
trD 2 v 2 4 m m m SA .
i m
i i
i m
œ
1 2
3
2 2
This yields
trD 2 v ,
n , 0
m i
and accordingly
trD
m
]] l . a 1 1 2 a
i
v
5
The crucial level of inertia at equilibria with the support of two pure strategies can be easily calculated using the proof of proposition 1.
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. Dawid Mathematical Social Sciences 37 1999 265 –280
which implies by C2 that l [[0,1 for all a [[0,1. Obviously, we also have l [[0,1
i j
for the eigenvalue corresponding to the larger of the two real relevant eigenvalues of M . Thus, in such a case we should have
a50 and indeed using the condition that the
m
eigenvalues are real and C2 we get
2 2
v
2 trD 2v 2 2v trD
v 2 trD
m m
m
]]]]] ]]]]]
]]]]] 1 2 v
, 1 2 , 1 2
0.
m m m SA
4
m m m SA
4
m m m SA
i i
i m
i i
i m
i i
i m
1 2
3 1
2 3
1 2
3
]]]] If the relevant eigenvalues are complex i.e. trD .v 22
m m m SA we have to
m 1
2 3
m
œ
1 2 a
]
check that the modulus of the eigenvalue l 511
n is smaller than 1, where n and
i i
i v
¯ n are the two relevant eigenvalues of M . We have
i m
2
1 2 a
1 2 a
2 2
]] ]]]
⇔ 1 1 2
Re n 1
ul u , 1u n u , 1
i 2
i i
v v
2
1 2 a
1 2 a
]] ]]]
⇔
trD 2 v 1 m m m SA , 0
m 2
i i
i m
1 2
3
v v
v
2 trD
m
]]]]] ⇔
1 2 v ,
a.
m m m SA
i i
i m
1 2
3
Putting together the two cases establishes the proposition. h
Since the ‘traditional’ discrete time replicator dynamics corresponds to the case where a 50 we get as an immediate corrollary the following characterization of equilibria with
the support of three strategies which are locally asymptotically stable with respect to the discrete time replicator dynamics
Corollary 2 Under the assumptions of proposition 3 an equilibrium m is stable with
respect to the discrete time replicator dynamics without inertia if
m m m SA . vv 2 trD .
4
i i
i m
m
1 2
3
If the inequality holds the other way round m is unstable.
It is further interesting to notice that the reasoning in the proof of proposition 3 shows that in the case of an equilibrium with the support of three pure strategies overshooting
can never appear if all eigenvalues are real in this case all eigenvalues are positive. With other words, if the equilibrium is approached linearly rather than in spirals the
discrete dynamics never overshoots the equilibrium. However, such a situation might occur if the support of the equilibrium consists only of two pure strategies.
In the case of multiple equilibria one equilibrium becoming unstable can of course imply convergence of the population distribution towards another equilibrium of the
game. In such a case the level of inertia present in the population becomes a crucial factor of equilibrium selection. We will present a game where the level of inertia indeed
selects the equilibrium to be learned in the next section.
H . Dawid Mathematical Social Sciences 37 1999 265 –280
275
4. An example
