Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings.

翻译：对称策略改进是由Schewe等人（ICALP 2015）提出的一种算法，可用于求解有向图上的双人博弈（如平局博弈和均值支付博弈）。与众所周知的传统策略改进算法不同，该算法同时迭代双方的策略。对称版本能快速解决已知的策略改进最坏情况实例，但其最坏情况复杂度此前尚不明确。我们提出了一类针对对称策略改进的最坏情况实例，在该类实例上，对称版本同样需要指数级步骤。值得注意的是，这些实例对任何改进规则均表现出该行为，这与传统策略改进中通常针对特定改进规则手工构造困难实例的情况形成鲜明对比。我们提出了一种广义对称策略迭代版本，其更弱地依赖于双方策略的相互影响。然而，该广义版本同样存在相同缺陷。