H . Dawid Mathematical Social Sciences 37 1999 265 –280
277
our reasoning that for a ,a this equilibrium is almost a global attractor. Only
3 2
trajectories with either s 50 or s 5
m do not eventually converge towards m . Since
0,4 1
2 13
]
the equilibrium m with equilibrium payoff v5
for the agents Pareto dominates m
3
with v 50.2 a higher level of inertia in the population can for a large set of initial population distributions significanlty increase the long run payoff of all agents in the
population.
5. Monotone selection dynamics with inertia
In the previous sections we always considered the discrete time replicator dynamics with inertia motivated by an imitation model where the agents imitate another agent with
a probability proportional to the relative payoff of this agents current strategy. Let us now consider the more general case where the probability that an agent currently using
strategy i imitates another agent currently using j is given by xp s, where p is
ij ij
non-negative and
9 9
9 9
p s . p s ⇒
e As , e As and p s . p s
⇔
e As . e As 5
ij kj
i k
ij ik
j k
hold for all i, j,k [I. This means that the probability to imitate an agent increases with his payoff and either decreases or is constant with the own payoff. The corresponding
population dynamics reads
s 5 s 1
xs
O
s p s 2 p s . 6
i,t 11 i,t
i,t j,t
ji t
ij t
j [I
Similar dynamics in continuous time have recently been analyzed in Hofbauer 1995. Note also that the replicator dynamics with inertia is a special case of this type of
e As
9
j
]
dynamics with p s5 . We call this kind of dynamics a smooth monotone imitation
ij
s 9As
n
dynamics with inertia a 512x if all p s are continuously differentiable on D , satisfy
ij
Eq. 5 and the scaling condition 1
]
E O O
s s p s ds 5 1 7
S D
i j ij
D
i [I j [I
n
s [ D
n
holds. D denotes the volume of D . Contrary to the case of the replicator dynamics the
probability to stick with the old strategy given by 12 x
o s p s depends on both the
j[I j
ij
currently used strategy i and the population state s in this more general formulation. The scaling condition Eq. 7 guarantees that
a 512x is the average measured over the
n
strategy space D probability that an agent does not imitate a new strategy. Of course x
again has to be sufficiently small to satisfy x max
p s1. It is easy to see that
n
i, j [I,s [ D
ij
any Nash equilibrium of the game is a rest point of a monotone imitation dynamics with inertia and that every interior fixed point has to be a Nash equilibrium.
278 H
. Dawid Mathematical Social Sciences 37 1999 265 –280
The monotone imitation dynamics Eq. 6 are a special case of a monotone selection
8
dynamics s
5 diags 1 diags gs , 8
t 11 t
t t
n n
n
where g: D ∞R is continuously differentiable and satisfies s9gs50;s[D and
9 9
g s . g s ⇔
e As . e As. 9
i j
i j
Cressman 1997 showed in a continuous time framework that the stability properties of an interior rest point with respect to a smooth monotone selection dynamics are
basically the same as with respect to the continuous time replicator dynamics. Using this result it is easy to show that the conditions C2 and C3 ensure that there is a crucial
level of inertia for every smooth monotone imitation dynamics with inertia. Again, a quasi-strict equilibrium with the support of at most three pure strategies is locally
asymptotically stable with respect to these dynamics if the level of inertia is larger than this crucial level. We state this formally in the next proposition.
Proposition 4 Let
m be a symmetric quasi-strict Nash equilibrium with support C
m5 hi ,i ,i j and assume that C2 and C3 hold. Then for every smooth imitation
1 2
3
dynamics with inertia there exists a crucial level of inertia a,1 such that m is locally
asymptotically stable with respect to this dynamics for any level of inertia larger than a.
Proof: We assume here that
m is an interior fixed point of the dynamics with three pure strategies. The extension to quasi-strict equilibria with the support of three pure
strategies is the same as in the proof of Eq. 3. The Jacobian of the dynamics Eq. 6 at m is
V5 I 1 xK,
where K is the linearization of the continuous time dynamics
~s 5 s
O
s p s 2 p s. 10
i i
j ji
ij j [I
It is easy to see that Eq. 10 is a smooth monotone selection dynamics as defined in Cressman 1997 and in the same paper it is shown that the Jacobian of any smooth
selection dynamics at an interior hyperbolic fixed point is a positive multiple of the Jacobian of the continuous time replicator dynamics at this fixed point. This means that
there exists a scalar c .0 such that
K 5 cJ,
8
Nachbar 1990 calls a dynamics satisfying Eq. 9 relative monotone.
H . Dawid Mathematical Social Sciences 37 1999 265 –280
279
where J is the linearization of Eq. 2 at
m. Denoting by l , l the two relevant
1 2
eigenvalues of V we have l 5 1 1 xcn , i 5 1,2,
i i
where
n are again the two relevant eigenvalues of J. Since we know from proposition
i
Eq. 2 that C2 and C3 imply that the real parts of both n are negative it is obvious
i
that there exists a x.0 such that the moduli of both l are smaller than one if x ,x.
i
h Note that the actual value for the crucial level of inertia depends on the functions p
ij
and accordingly we can not give a general expression here. However, it is interesting to notice that the conditions C2 and C3 are very general conditions in the sense that
they ensure stability of any monotone dynamics as long as the level of inertia is sufficiently large. Finally, we would like to mention that also imitation rules of the form
9 9
p s5
Ce As2e As where C is non-negative and increasing e.g. Hofbauer 1995
ij j
i
fit into our framework and similar results hold there too.
6. Conclusions
