Thresholds for sensitive optimality and Blackwell optimality in stochastic games

We investigate refinements of the mean-payoff criterion in two-player zero-sum perfect-information stochastic games. A strategy is Blackwell optimal if it is optimal in the discounted game for all discount factors sufficiently close to $1$. The notion of $d$-sensitive optimality interpolates between mean-payoff optimality (corresponding to the case $d=-1$) and Blackwell optimality ($d=\infty$). The Blackwell threshold ${\alpha }_{\mathsf{bw}}\in 0,1$ is the discount factor above which all optimal strategies in the discounted game are guaranteed to be Blackwell optimal. The $d$-sensitive threshold ${\alpha }_{\mathsf{d}}\in$0,1 is defined analogously.