This article focuses on dyadic negotiations in which negotiators have asymmetric best alternatives to the negotiated agreement (BATNAs). The article sets out to contribute to this domain, arguing that it is important to consider negotiators’ knowledge of their opponents’ BATNAs. The study uses a job negotiation simulation and examined the effects of the knowledge of opponents' BATNAs on agreement efficiency (indexed by joint outcome) and negotiators' abilities to claim values (bargaining strength). In a 2 x 2 experiment, findings indicated that strong negotiators’ knowledge of opponents’ BATNAs increases their bargaining strength but hinders their efficiency; weak negotiators’ knowledge alone reduces their bargaining strength but increases efficiency; and the detrimental impact of strong negotiators’ knowledge on efficiency outweighs the benefit of weak negotiators’ knowledge. The impact of knowledge on efficiency depends on which party has access to it. Paradoxically, weakness leads to efficiency and strength can lead to “winning” an impoverished prize.