We study functions $f : [0, 1]^d \rightarrow [0, 1]^d$ that are both monotone and contracting, and we consider the problem of finding an $\varepsilon$-approximate fixed point of $f$. We show that the problem lies in the complexity class UEOPL. We give an algorithm that finds an $\varepsilon$-approximate fixed point of a three-dimensional monotone contraction using $O(\log (1/\varepsilon))$ queries to $f$. We also give a decomposition theorem that allows us to use this result to obtain an algorithm that finds an $\varepsilon$-approximate fixed point of a $d$-dimensional monotone contraction using $O((c \cdot \log (1/\varepsilon))^{\lceil d / 3 \rceil})$ queries to $f$ for some constant $c$. Moreover, each step of both of our algorithms takes time that is polynomial in the representation of $f$. These results are strictly better than the best-known results for functions that are only monotone, or only contracting. All of our results also apply to Shapley stochastic games, which are known to be reducible to the monotone contraction problem. Thus we put Shapley games in UEOPL, and we give a faster algorithm for approximating the value of a Shapley game.

翻译：我们研究定义在$[0, 1]^d \rightarrow [0, 1]^d$上且同时满足单调性与收缩性的函数$f$，并考虑寻找$f$的$\varepsilon$-近似不动点问题。我们证明该问题属于复杂度类UEOPL。我们提出一种算法，通过$O(\log (1/\varepsilon))$次对$f$的查询，即可找到三维单调收缩函数的$\varepsilon$-近似不动点。我们还给出一个分解定理，利用该定理可将上述结果推广至$d$维情形，得到一种通过$O((c \cdot \log (1/\varepsilon))^{\lceil d / 3 \rceil})$次查询（其中$c$为常数）寻找$d$维单调收缩函数$\varepsilon$-近似不动点的算法。此外，我们两种算法的每一步均在$f$的表示规模下具有多项式时间复杂度。这些结果严格优于目前仅满足单调性或仅满足收缩性的函数的最佳已知结果。我们所有结论均可应用于Shapley随机博弈——该问题已知可归约为单调收缩问题。由此我们将Shapley博弈纳入UEOPL复杂度类，并为逼近Shapley博弈值提供了更高效的算法。