We study rational synthesis for stochastic multi-agent systems with unknown dynamics, where each agent is equipped with its own probabilistic temporal objective specified using Probabilistic Linear Temporal Logic on Finite Traces ($\mathsf{PLTL}_f}$). We impose solution concepts (e.g., Nash equilibrium, subgame-perfect equilibrium) as constraints on the joint strategy profile, using payoffs derived from each agent's probability of satisfying its temporal objective. In particular, we formulate per-agent rational synthesis as finding a strategy profile -- if one exists -- such that (i) the designated agent's probabilistic temporal constraint is satisfied, and (ii) the profile represents a stable solution under a chosen solution concept. We give conditional convergence guarantees for tabular Nash-Q learning, certify satisfaction with simultaneous finite-sample confidence bounds, and assess strategic stability with approximate best-response checking. Our approach operates in a stochastic \emph{Multi-Agent Reinforcement Learning} (MARL) environment, where model-free learning algorithms attempt to attain extremal satisfaction probabilities for temporal objectives automatically compiled into reward automata. Our approach is empirically validated using 8 general-sum concurrent stochastic games.

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