  Admissible strategies, i.e. those that are not dominated by any other
strategy, are a typical rationality notion in game theory. In many classes of
games this is justified by results showing that any strategy is admissible or
dominated by an admissible strategy. However, in games played on finite graphs
with quantitative objectives (as used for reactive synthesis), this is not the
case.
  We consider increasing chains of strategies instead to recover a satisfactory
rationality notion based on dominance in such games. We start with some
order-theoretic considerations establishing sufficient criteria for this to
work. We then turn our attention to generalised safety/reachability games as a
particular application. We propose the notion of maximal uniform chain as the
desired dominance-based rationality concept in these games. Decidability of
some fundamental questions about uniform chains is established.
