Discounting successive payments is a well-known procedure used in the
theory of stochastic games since a seminal paper of Shapley. Recently
de Alfaro, Henzinger and Majumdar observed that discounting can be
pertinent in the context of much more recent theory of stochastic
parity games which were proposed as a tool for verification of
probabilistic systems. The particular discounting of de Alfaro et al.
is in fact very close to the original ideas of Shapley but suffers
from some needless restrictions. Dropping these constraints results
in a more elegant theory that includes new interesting generalizations
of parity and mean payoff games as limit cases. We shall present
selected results concerning these new games as well as some open
problems.
