Stochastic games have become a prevalent framework for studying long-term multi-agent interactions, especially in the context of multi-agent reinforcement learning. In this work, we comprehensively investigate the concept of constant-memory strategies in stochastic games. We first establish some results on best responses and Nash equilibria for behavioral constant-memory strategies, followed by a discussion on the computational hardness of best responding to mixed constant-memory strategies. Those theoretic insights are later verified on several sequential decision-making testbeds, including the Iterated Prisoner's Dilemma, the Iterated Traveler's Dilemma, and the Pursuit domain. This work aims to enhance the understanding of theoretical issues in single-agent planning under multi-agent systems, and uncover the connection between decision models in single-agent and multi-agent contexts. The codebase and the full version of this paper is available at github.com/Fernadoo/Const-Mem.