Arguments are statements used to persuade someone or in support of a claim. However, these are not perfect and the opponents can exploit them to build their own arguments. In this paper, we propose a new litigation success function that (i) considers the intrinsic and immutable strength of the arguments presented by the plaintiff and defendant, (ii) solves the limitations of the existing literature in dealing with the English fee-shifting system, and (iii) is flexible and tractable in analytical terms. As a robustness check, the proposed litigation success function confirms several important results in the literature, but also brings new insights on how argumentation strength affects the individuals' efforts and the decision to file and contest lawsuits in different fee-shifting systems. In this context, we also show how to introduce the worldwide-accepted presumption of innocence in favor of the defendant and the possibility of settlement. Keywords: Litigation success function, Contests success function, Argumentation strength, Fee-shifting systems. JEL classification: C72, D72, D74, K41.