Direct reciprocity is a wide-spread mechanism for evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well known approach is given by reactive strategies, which respond to the co-player's previous move. Here we extend reactive strategies to longer memories. A reactive-$n$ strategy takes into account the sequence of the last $n$ moves of the co-player. A reactive-$n$ counting strategy records how often the co-player has cooperated during the last $n$ rounds. We derive an algorithm to identify all partner strategies among reactive-$n$ strategies. We give explicit conditions for all partner strategies among reactive-2, reactive-3 strategies, and reactive-$n$ counting strategies. Partner strategies are those that ensure mutual cooperation without exploitation. We perform evolutionary simulations and find that longer memory increases the average cooperation rate for reactive-$n$ strategies but not for reactive counting strategies. Paying attention to the sequence of moves is necessary for reaping the advantages of longer memory.

翻译：直接互惠是合作演化中一种广泛存在的机制。在重复博弈中，参与者可以根据先前的结果调节自身行为。一个著名的研究方法是反应策略，即基于对手上一步的行动做出回应。本文我们将反应策略扩展到更长的记忆范围：反应-n策略会考虑对手最近n个动作的序列，而反应-n计数策略则记录对手在最近n轮中合作的次数。我们推导出了一个算法，用于识别所有属于反应-n策略中的伙伴策略。我们给出了反应-2策略、反应-3策略以及反应-n计数策略中所有伙伴策略的显式条件。伙伴策略是指那些能确保相互合作且不被剥削的策略。通过演化模拟，我们发现：更长的记忆能提高反应-n策略的平均合作率，但对反应计数策略无效。只有关注动作序列，才能充分利用更长记忆带来的优势。