Litigants' Agreement to Special Rules of Adjudication

The mishna (Sanhedrin 24a) discusses a case where the litigants agree that someone who should be invalid to testify or judge in their case, e.g., a sinner or a relative of one of the litigants, will be able to testify. Rabbi Meir says that the litigants can back out of the agreement; Rabbanan say that they cannot. The gemara concludes that they are arguing about a case where one wants to back out of the agreement after the g'mar din (the end of the court case, when the ruling is rendered), and the halacha is that he cannot back out. However, prior to that, either side can back out of the agreement to allow an invalid witness or dayan. If a kinyan (act of finalization) was made, no one can subsequently back out.
Rishonim dispute how a kinyan prevents the sides from backing out of their agreement. The Nimukei Yosef (Sanhedrin 4b in the pages of the Rif) posits that the kinyan is not of a directly monetary nature, but comes to concretize the procedural agreement. The Ra'avan says that the litigants must make a kinyan on the money in dispute to be paid according to the decision.
Rishonim explain that not only regarding the viability of witnesses and dayanim but in regard to a variety of rules of adjudication, the sides can alter the regular rules. These include beginning deliberations at night, using translators, etc. These changes are not limited to technical issues or matters of a specific Torah preference but apply as well to matters of serious content that could easily affect the case’s outcome. For example, a sinner as a witness is suspected of lying for financial gain, yet he can testify if so agreed. The use of translators also can hinder the dayanim’s ability to understand and interrogate the litigants. (One area where agreement is invalid is about going to adjudicate under the auspices of a secular court.) The reason that these agreements are accepted is that since the goal is to resolve arguments between the parties, there is little reason to prevent people from using a framework with which they are satisfied.
Regarding present day practice, the signing of an arbitration agreement is deemed to be a form of kinyan to the agreements between the sides based on the rule of situmta (whatever is considered binding by society becomes halachically so). That is why special rules that a given beit din employs that are referred to in the arbitration agreement are binding. It is common now, for example, to allow adjudication to take place at night. Regarding matters of content, many batei din receive authority to extract money even when there is a (minority) opinion that payment should not be made. Sometimes the law of the land is accepted relatively broadly. It is possible to validate a ruling, even if it ends up being based on a mistake, or to allow the dayanim to rule either based on strict law or based on compromise. Examples of rulings where this was employed can be found in editions 26-27 of Halacha P'suka.