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This paper considers a dynamic process of n-person social choice problems under q-majority where a status-quo policy is challenged by an opposing policy drawn randomly in each period. The opposing policy becomes the next status-quo if it receives at least q votes. We characterize stochastically stable policies under a boundedly rational choice rule of voters. Under the best response rule with mutations, a Condorcet winner is stochastically stable for all q-quota rules, and uniquely so if q is greater than the minmax quota. Under the logit choice rule, the Borda winner is stochastically stable under the unanimity rule. Our evolutionary approach provides a dynamic foundation of the mini-max policies in multidimensional choice problems with Euclidean preferences.