Prove the analogue of Proposition 10.7 for the RD and the notion of dominance in mixed strategies. That is, show that for any path x (·) induced by the RD that starts at interior initial conditions,...

of dominance in mixed strategies. That is, show that for any path
x(·) induced by the

RD that starts at interior initial conditions, limt→∞
xkq
(t) = 0 for any pure strategy

skq
that is dominated in the sense of Definition 2.1.

Hint:
Suppose that, say, population 1 has the dominated strategy
s1q
and let
σ1 be

a mixed strategy that dominates it. Define then the real function
φ
:
_
n−1 → R

given by
φ(x1) =__nr=1
σ1r
log
x1r_− log
x1q
and determine its evolution alongany path of the system by computing its time derivative.