A conventional biometric authentication system is often designed to distinguish genuine accesses from zero-effort impostor attacks. However, when operating in an adversarial environment, the system has to be robust against presentation attacks such as spoofing. An effective solution to reduce the impact of spoofing attack is to consider both the matching score and liveness score when making the accept/reject decision. In this paper, we consider the joint decision space of matching and liveness scores in the presence of both spoofing attack and zero-effort attack, with application to signature verification. Our investigation aims to understand how decision thresholds in the above space should be optimized. This leads to two dichotomies of methods, namely brute-force approach versus probabilistic approach; and single threshold versus double-threshold approach. This view leads to three novel methods that have never been reported. Based on the experimental results carried out on an off-line signature database, the novel methods turn out to outperform simpler methods with only matching score.