A stochastic mean-payoff game is a mean-payoff game with additional chance vertices. At each such graph vertex, a prescribed discrete distribution selects the next outgoing graph edge.
Writing 
for the reward of graph edge 
, the value at a starting graph
vertex 
is
![sup_(sigma)inf_(tau)E_(v,sigma,tau)[liminf_(T->infty)1/Tsum_(t=0)^(T-1)w(e_t)],](/images/equations/StochasticMean-PayoffGame/NumberedEquation1.svg) |
where 
and 
range over the players' history-dependent strategies,
which may use private randomness. The expectation
value is taken after the pathwise infimum limit.
