The automation of agents that bargain within markets is a challenging open problem. In these settings a seller agent can take advantage from the competition among several buyer agents and vice versa. In this paper we focus on one-sided competition situations wherein there are one seller and several buyers (the analysis of the reverse situations is analogous). We provide two original contributions. Firstly, we extend state-of-the-art models to satisfactorily capture competition among buyers. More precisely, we enrich the alternating-offers protocol by allowing the seller agent to exploit the outside option, namely, the option to leave the negotiation she is carrying on and choose a different buyer with whom to negotiate. Secondly, we provide a preliminary game theoretical analysis both to develop an efficient solving algorithm and to show the effectiveness of the employment of the outside option in bargaining.