Two opinions what does the judge’s discretion, שׁוּדָא דְּדַיָּינֵי, mean.

 TB Kiddushin 74 discusses the credibility of different types of people when they testify. After a court case is decided, when is the judge credibility accepted and when is it rejected?

“The Gemara cites the continuation of the baraita: Similarly, a judge is deemed credible to say: I found this person victorious in a civil case, and I found this one obligated to pay. In what case is this statement said? When the litigants are still standing before him. But if the litigants are not standing before him but have left, he is not deemed credible. The Gemara asks: And let us see who holds the writ of a favorable verdict. Why is there a need to rely on the statement of the judge?

The Gemara answers: No, it is necessary to teach this halakha in a case where their writs of a favorable verdict have been torn up and cannot be examined. The Gemara asks: If so, then let him return and judge them again, and presumably the same verdict will be issued. The Gemara answers: It was a case of the judges’ discretion [שׁוּדָא דְּדַיָּינֵי -shuda dedayyanei]. In certain cases, the verdict depends on the decision of the judges based solely on their sense of which litigant deserves to win. There is no guarantee that they will make the same decision the second time around.” (Sefaria.org translation)

Rashi and his grandson Rebeinu Tam disagree what is the definition of the judge’s discretion, [שׁוּדָא דְּדַיָּינֵי -shuda dedayyanei]. Rashi ד"ה בְּשׁוּדָא דְּדַיָּינֵי says

judge’s discretion comes into play when logic or law cannot be applied. It’s his discretion to decide the case as he wishes. He gives the example of two men with the same name and are neighbors and relatives. The judge says it appears to me the case favors this one and not the other one. (See TB Ketubot 85b)

Tosefot ד"ה שׁוּדָא דְּדַיָּינֵי cite several problems with this definition and then quotes Rebeinu Tam. He says that the judge has the power and therefore the discretion to decide which side is more credible. To prove his point he cites the halakhic principle that the court has the power to make property ownerless, הפקר בית דין הפקר.