There is really only one rule of pesaq that depends on who said what -- acharei rabbim lehatos (lean toward the majority).
Within a court (such as the Sanhedrin), this was a law of bitul berov -- nullification by majority. Meaning, after the votes are counted in the court, the opinion is that of the court, and it is the number of judges in the court that gives the opinion its weight. Not the number of judges who voted for the winning opinion.
Tosafos (Eiruvin 13b) say that the heavenly voice that came down to tell us "אלו ואלו דברי אלהים חיים הן, והלכה כבית הלל-- these and those are the words of the 'Living' G-d, but the halakhah is like Beis Hillel" actually endorsed the majority. It is not that we are ruling according to Divine Revelation rather than halachic process (which Eiruvin 6b tells us we should not do). So, why did we need a heavenly voice? Tosafos explain, "דבית שמאי הוו חריפי טפי -- for Beis Shammai were sharper."
Another example where sharpness doesn't win is Rabbi Meir (also Eiruvin 13b).
אמר רבי אחא בר חנינא גלוי וידוע לפני מי שאמר והיה העולם שאין בדורו של רבי מאיר כמותו ומפני מה לא קבעו הלכה כמותו שלא יכלו חביריו לעמוד על סוף דעתו שהוא אומר על טמא טהור ומראה לו פנים על טהור טמא ומראה לו פנים
Rabbh Acha bar Chanina said: It is revealed and known before the "One Who Spoke and the World Was" that in Rabbi Meir's general, there was no one like him/ And why didn't they establish the halakhah to be like him? Because his peers couldn't get at his conclusions. [Rabbi Meir] would say that the impure is pure, and prove the point. And he would say that the pure is impure and prove the point.
Rabbi Meir would explain both sides of a dispute so thoroughly and his peers so couldn't keep up that they couldn't figure out what he ruled!
Although when his student Rebbe did know Rabbi Meir's ruling, "סתם משנה כר' מאיר -- an unattributed mishnah is like Rabbi Meir" and "הלכה כסתם נשנה -- the halakhah follows the unattributed mishnah" over a mishnah where the ruling has a citation.
And that is more typical. Usually the person with a the sharper line of reasoning gains followers, and so his position ends up winning. Like the way Abayei so impressed his peers in the discussion of those 6 topics, or Rav Nachman when discussing monetary laws. Not because sharpness wins -- as we saw with Batei Hillel veShammai, but because sharpness usually causes popularity.
Notice the portrayal of Beis Shammai in the gemara I quoted was that it was more of an "eilitist" school. Smaller, and only accepting the sharpest students.
This relates to what Maurice Mizrahi wrote in his answer, drawing from another part of the same discussion in the gemara. Beis Hillel merited gaining following not only because they were more numerous themselves, but also because they showed respect for all opinions. People had a choice between listening to a Shammaite promote his own view, or a Hillelite present both sides of the argument fairly. And that humility won them followers.
This is what the gemara means in part of his quote:
ללמדך שכל המשפיל עצמו הקדוש ברוך הוא מגביהו וכל המגביה עצמו הקדוש ברוך הוא משפילו כל המחזר על הגדולה גדולה בורחת ממנו וכל הבורח מן הגדולה גדולה מחזרת אחריו
This teaches you that God raises up those who humbles themselves, and humbles those who exalt themselves; that greatness flees those who seek it and follows those who flee it...
(Which I guess means you should believe Maurice Mizrahi's answer. Because unlike Beis Hillel, I gave my own answer first!)