When two [litigants] appear before you for trial, one weak [in influence], the other powerful [in influence], so long as you have not heard their pleas or [even after you have heard their pleas, — [so long as] you are in doubt which way the judgment will incline, you may say to them, 'I am not bound to try your case,' lest the one of powerful [influence] be pronounced guilty and persecute the Judge. However, if you have [already] heard their pleas, and you know which way the judgment will incline, you may not say [to them], 'I am not bound to try your case.' And if he was appointed [to serve as Judge] for the public, he is obliged to try their case. Gloss: So too, if they were both powerful [litigants] he is required to try their case. There is, however, an [authority] who writes that at present it is nevertheless the adopted practice not to interfere with violators [of the law] because there is danger in [this] matter lest he [the violator] get him into trouble with the government.
It is a religious duty to say to the litigants at the outset, 'Do you wish [to resort to] law or [to submit to] arbitration?' If they consented [to submit to] arbitration, they bring about a settlement between them. And just as [the Judge] is cautioned not to pervert judgment, so too, is he cautioned not to direct the arbitration [in favour of] one party more than the other. Every Court of Law that always brings about a settlement is considered praiseworthy. When does this apply? — Before the conclusion of the trial, — even though one has heard their pleas and knows which way the judgment will incline, — it is a religious duty to attempt a settlement; but after the Judge concluded the trial and made the pronouncement, 'So-and-so, you are innocent, So-and-so, you are guilty,' he is not permitted to attempt a settlement between them. However, another person who is not a Judge is permitted to attempt a settlement between them, [provided] that this is not [attempted] in a Court sitting that is appointed for the purpose [of holding] legal [sessions]. If the Court of Law imposed an oath upon one of them, the Court of Law is permitted to bring about a settlement between them [even after the conclusion of the trial], so that [the one who is bound to take an oath] should be exempt from the responsibility thereof. The Court of Law cannot compel one to fall in with [the principle of equity, i.e., actting] beyond the requirements of the line of justice, although that would appear to them to be the proper thing [to do]. However, some differ with [this opinion].
...
§ The Sages taught in a baraita (Tosefta 1:2–8): Just as judgment is performed by three judges, so too, mediation is performed by three judges.
Once the verdict of the judgment has been issued, it is not permitted for you to mediate a dispute.
...