  We introduce a modification of Perron's method, where semi-solutions are
considered in a carefully defined asymptotic sense. With this definition, we
can show, in a rather elementary way, that in a zero-sum game or a control
problem (with or without model uncertainty), the value function over all
strategies coincides with the value function over Markov strategies discretized
in time. Therefore, there are always discretized Markov $\varepsilon$-optimal
strategies, (uniform with respect to the bounded initial condition). With a
minor modification, the method produces a value and approximate saddle points
for an asymmetric game of feedback strategies vs. counter-strategies.
