  We study turn-based quantitative games of infinite duration opposing two
antagonistic players and played over graphs. This model is widely accepted as
providing the adequate framework for formalizing the synthesis question for
reactive systems. This important application motivates the question of strategy
complexity: which valuations (or payoff functions) admit optimal positional
strategies (without memory)? Valuations for which both players have optimal
positional strategies have been characterized by Gimbert and Zielonka for
finite graphs and by Colcombet and Niwi\'nski for infinite graphs. However, for
reactive synthesis, existence of optimal positional strategies for the opponent
(which models an antagonistic environment) is irrelevant. Despite this fact,
not much is known about valuations for which the protagonist admits optimal
positional strategies, regardless of the opponent. In this work, we
characterize valuations which admit such strategies over infinite game graphs.
Our characterization uses the vocabulary of universal graphs, which has also
proved useful in understanding recent breakthrough results regarding the
complexity of parity games. More precisely, we show that a valuation admitting
universal graphs which are monotone and well-ordered is positional over all
game graphs, and -- more surprisingly -- that the converse is also true for
valuations admitting neutral colors. We prove the applicability and elegance of
the framework by unifying a number of known positionality results, proving new
ones, and establishing closure under lexicographical products. Finally, we
discuss a class of prefix-independent positional objectives which is closed
under countable unions.
