  This paper presents a case study for the application of semiring semantics
for fixed-point formulae to the analysis of strategies in B\"uchi games.
Semiring semantics generalizes the classical Boolean semantics by permitting
multiple truth values from certain semirings. Evaluating the fixed-point
formula that defines the winning region in a given game in an appropriate
semiring of polynomials provides not only the Boolean information on who wins,
but also tells us how they win and which strategies they might use. This is
well-understood for reachability games, where the winning region is definable
as a least fixed point. The case of B\"uchi games is of special interest, not
only due to their practical importance, but also because it is the simplest
case where the fixed-point definition involves a genuine alternation of a
greatest and a least fixed point. We show that, in a precise sense, semiring
semantics provide information about all absorption-dominant strategies --
strategies that win with minimal effort, and we discuss how these relate to
positional and the more general persistent strategies. This information enables
applications such as game synthesis or determining minimal modifications to the
game needed to change its outcome. Lastly, we discuss limitations of our
approach and present questions that cannot be immediately answered by semiring
semantics.
