Gemara

If one says that his ox gored Ploni, or Ploni's ox, he pays due to his admission;

The rule is, anyone who pays more than he damaged does not pay due to his admission.

(Rav Papa): Half damage is considered like Mamon (not a fine), since animals are prone to do damage unless guarded;

One should pay full damage for failure to guard his animal. The Torah had mercy on him and makes him pay only half, since the animal was not yet established to gore.

(Rav Huna brei d'Rav Yehoshua): Half damage is considered a fine, since animals, even unguarded, are not prone to do damage;

One should not pay at all. The Torah levies a fine of half-damage to encourage people to guard their animals.

(Mishnah): If he says that his ox gored Ploni or Ploni's ox, he pays due to his own admission.

Suggestion: The Mishnah discusses a Tam. (This shows that half-damage is Mamon!)

Rejection: No, it discusses a Mu'ad.

(Mishnah): The rule is, anyone who pays more than he damaged does not pay due to his own admission.

Inference: One who pays less than he damaged does pay due to his admission!

Rejection: No, one who pays like he damaged pays due to his admission.

Question: If one who pays less than he damaged does not pay due to his own admission, it should say that one who does not pay like he damaged does not pay due to his own admission. This includes one who pays more and one who pays less!

This refutes the opinion that half-damage is a fine. The Halachah is, it is a fine.

Question: How can the Halachah follow an opinion that was refuted?

Answer: It was refuted because the Mishnah did not say 'One who does not pay like he damaged...' (but this is not a refutation);

We could not teach this, for it is not always true! A tradition from Sinai teaches that half-damage of pebbles (if an animal set an object in motion, and the object damaged) is Mamon.

Since the Halachah is that half-damage is a fine, if a dog killed and ate sheep, or a cat ate large chickens, this is unusual (any unusual damage pays half, like goring), so we do not force one to pay for it in Bavel. (We do not collect fines in Chutz la'Aretz.)

Bava Kama 84b (Rava): (Becoming) Mu'ad does not apply in Bavel.

Rishonim

Rif (Bava Kama 6a): The Halachah is that half-damage is a fine, except for half-damage of pebbles. We have a tradition that pebbles are Mamon. Since half-damage is a fine, if a dog killed and ate sheep, or a cat ate large chickens, this is unusual, so we do not collect for it in Bavel.

Ran (DH Gemara): Since half-damage is a fine, if he admits he is exempt.

Rambam (Hilchos Nizkei Mamon 2:7): Any half payment is a fine, except for damage of pebbles, which is a tradition (from Sinai).

Rambam (Hilchos Sanhedrin 5:9): Judges in Chutz la'Aretz collect only common matters in which a party loses money. They do not collect for an animal that damaged another or wounded another. They do not collect half-damage, except for damage of pebbles, which is Mamon, not a fine.

Rosh (Bava Kama 1:20): The Halachah is that half-damage is a fine. Therefore, anything unusual, even if the damager benefits, is not Shen (eating), rather, Keren (goring). It pays half-damage. Even though one pays half-damage for pebbles, this is Mamon, because it is normal. Since half-damage is a fine, if a dog killed and ate sheep, or a cat ate large chickens, this is unusual, so we do not collect for it in Bavel.

Or Some'ach (Nizkei Mamon 2:9): It is unusual for a dog to kill sheep, so paying for the loss due to death is a fine (and we do not collect it in Bavel). However, it is normal to eat the meat, so one pays fully for this (even in Bavel).

Poskim

Shulchan Aruch (CM 1:1): Judges in Chutz la'Aretz collect only common matters in which a party loses money. They do not collect for an animal that damaged another or wounded another. They do not collect half-damage, except for damage of pebbles, which is Mamon, not a fine.

Tur: If an animal damaged an animal through Keren (goring) or its derivatives, we do not collect this in Bavel. If it is Tam, we hold that half-damage is a fine, which we do not judge in Chutz la'Aretz. We do not make an animal Mu'ad in Bavel. Even if it became Mu'ad in Eretz Yisrael and went to Bavel, we do not judge it, for it is not common.

Beis Yosef (DH u'Mah she'Chosav v'Chen, based on Rashi Bava Kama 84b DH Ein): Since we do not collect for a Tam that gored, it cannot become Mu'ad.

Rejection (Tosfos Rid Bava Kama 84b DH v'Ha): We do not collect from a Tam of an Cheresh, lunatic or child, yet we testify in front of an Apotropos in order to make it Mu'ad. We could do similarly for a Tam in Bavel! Rather, testimony to make a Mu'ad must be in front of judges with Semichah, which we lack in Bavel.

SMA (4): If an animal damaged anything in an unusual way, we do not collect for it in Bavel.