This paper proposes a novel game-theoretical autonomous decision-making framework to address a task allocation problem for a swarm of multiple agents. We consider cooperation of self-interested agents, and show that our proposed decentralized algorithm guarantees convergence of agents with social inhibition to a Nash stable partition (i.e., social agreement) within polynomial time. The algorithm is simple and executable based on local interactions with neighbor agents under a strongly connected communication network and even in asynchronous environments. We analytically present a mathematical formulation for computing the lower bound of suboptimality of the outcome, and additionally show that at least 50% of suboptimality can be guaranteed if social utilities are nondecreasing functions with respect to the number of coworking agents. The results of numerical experiments confirm that the proposed framework is scalable, fast adaptable against dynamical environments, and robust even in a realistic situation.