Leung et al. (2010) [Leung JY-T, Pinedo M, Wan G (2010) Competitive two-agent scheduling and its applications. Oper. Res. 58:458–469] considered a two-agent nonpreemptive single-machine scheduling problem. Agent A is responsible for n1 jobs with due dates 𝑑1,…,𝑑𝑛 and has as the objective the minimization of the total tardiness of the n1 jobs. Agent B is responsible for n2 jobs and has as the objective the minimization of the total completion time of the n2 jobs. The problem is to find a schedule for the 𝑛1+𝑛2 jobs that minimizes the objective of agent A (with regard to his n1 jobs) while keeping the objective of agent B (with regard to his n2 jobs) below or at a fixed level Q. Leung et al. (2010) [Leung JY-T, Pinedo M, Wan G (2010) Competitive two-agent scheduling and its applications. Oper. Res. 58:458–469], in their theorem 3, showed that this problem can be solved through dynamic programming in pseudopolynomial time. However, in the proof of their theorem and in their dynamic programming formulation, there is an error that requires some changes in their proof.