  Bewley and Kohlberg (1976) and Mertens and Neyman (1981) have proved,
respectively, the existence of the asymptotic value and the uniform value in
zero-sum stochastic games with finite state space and finite action sets. In
their work, the total payoff in a stochastic game is defined either as a Cesaro
mean or an Abel mean of the stage payoffs. This paper presents two findings:
first, we generalize the result of Bewley and Kohlberg to a more general class
of payoff evaluations and we prove with a counterexample that this result is
tight. We also investigate the particular case of absorbing games. Second, for
the uniform approach of Mertens and Neyman, we provide another counterexample
to demonstrate that there is no natural way to generalize the result of Mertens
and Neyman to a wider class of payoff evaluations.
