**MSC:**- 90D15 Stochastic games, See also {93E05}

every player $k$ there

is

a function $r^k:S\rightarrow {\bf R}$ from the state space $S$ to the

real numbers

such that for every $\epsilon>0$ there is an $\epsilon$

equilibrium such that with probability

at least $1-\epsilon$ no state $s$ is reached

where

the future expected payoff for any player $k$

differs from $r^k(s)$ by more than $\epsilon$.

We demonstrate an example of a recursive two-person non-zero-sum

stochastic game

with only three non-absorbing states and limit average payoffs

that is not valued, (but

does have $\epsilon$ equilibria for every positive

$\epsilon$). In this respect

two-person non-zero-sum stochastic games are

very different from their

zero-sum varieties. N. Vieille proved that

all such games with finitely

many states have an $\epsilon$ equilibrium

for every positive $\epsilon$, and our example shows that any proof of this

result

must be qualitatively

different from the existing proofs for zero-sum games.

To show that our example is not valued

we need that the existence of

$\epsilon$ equilibria for all positive $\epsilon$

implies

a ``perfection'' property. Should there exist a stochastic game

without an $\epsilon$ equilibrium for some $\epsilon >0$,

this perfection property

may be useful for demonstrating this fact. Furthermore our

example sows some doubt concerning the existence of

$\epsilon$ equilibria for two-person non-zero-sum stochastic games

with countably many states.