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In the real world, people/entities usually find matches independently and autonomously, such as finding jobs, partners, roommates, etc. It is possible that this search for matches starts with no initial knowledge of the environment. We propose the use of a multi-agent reinforcement learning (MARL) paradigm for a spatially formulated decentralized two-sided matching market with independent and autonomous agents. Having autonomous agents acting independently makes our environment very dynamic and uncertain. Moreover, agents lack the knowledge of preferences of other agents and have to explore the environment and interact with other agents to discover their own preferences through noisy rewards. We think such a setting better approximates the real world and we study the usefulness of our MARL approach for it. Along with conventional stable matching case where agents have strictly ordered preferences, we check the applicability of our approach for stable matching with incomplete lists and ties. We investigate our results for stability, level of instability (for unstable results), and fairness. Our MARL approach mostly yields stable and fair outcomes.