  The paper is devoted to dynamic games. We consider a general enough
framework, which is not limited to e.g. differential games and could
accommodate both discrete and continuous time. Assuming common dynamics, we
study two game families with total payoffs that are defined either as the
Ces\`{a}ro average (long run average game family) or Abel average (discounting
game family) of the running costs. We study a robust strategy that would
provide a near-optimal total payoff for all sufficiently small discounts and
for all sufficiently large planning horizons. Assuming merely the Dynamic
Programming Principle, we prove the following Tauberian theorem: if a strategy
is uniformly optimal for one of the families (when discount goes to zero for
discounting games, when planning horizon goes to infinity in long run average
games) and its value functions converge uniformly, then, for the other family,
this strategy is also uniformly optimal and its value functions converge
uniformly to the same limit.
