Gemara

17a (Mishnah): If a Dalil (something tied to a chicken's leg) broke Kelim, it pays half-damage.

19b (Rav Huna): It became tied by itself. If one tied it, he pays full damage.

If it became tied by itself, we cannot say that the Dalil's owner pays. If the Dalil was hidden, he is blameless! If it was open, he was negligent! (He should pay full damage.) Rather, the owner of the chicken pays.

Question: He doesn't pay full damage due to "when a man will open a pit", not when an ox will open a pit (create an obstacle). If so, he should be totally exempt!

Answer: Rather, the chicken threw the Dalil. (This is Tzeroros.) Rav Huna teaches about a Hefker Dalil. If it became attached to a chicken by itself and damaged, the chicken's owner is exempt. If one tied it, he pays full damage.

53a (Mishnah): If an animal fell forwards into a pit, due to the noise of the digging, he (the owner of the pit) is liable.

Question: Why is the pit's owner liable? The one digging caused the damage!

Answer (Rav Simi bar Ashi): The Beraisa is like R. Noson, who says that since the pit damaged, what cannot be collected from the other damager (the digger is exempt, for it is only Gerama) is collected from the pit.

(Beraisa): If Reuven's ox pushed Shimon's ox into Levi's pit, Reuven is liable, and Levi is exempt;

R. Noson says, Reuven and Levi each pay half.

Contradiction (Beraisa - R. Noson): Reuven pays quarter-damage, and Levi pays three fourths.

Answer: A Mu'ad pays half the damage. A Tam pays quarter-damage.

Question: If R. Noson holds that the ox and pit both did all the damage, even a Tam should pay half! If he holds that each did half the damage, a Tam should pay a fourth, the pit should pay a half, and Shimon should lose a fourth!

Rava: R. Noson was a judge. He descended to the depth of judgment (to reach the correct Halachah)! Really, each did all the damage. When a Tam damages alone, it pays half-damage. Now that it had a partner, it only pays half of that.

Answer #2: Really, each did half the damage. Shimon can tell Levi 'my ox died in your pit. What I cannot collect from Reuven, I can collect from you.'

(Rava): If Reuven put a rock on the mouth of Levi's pit, and Shimon's ox tripped on it and fell in the pit, R. Noson and Chachamim argue about this.

Suggestion: Chachamim exempt Levi only when Reuven's ox pushed the ox in, for even without the pit, Reuven's ox would have killed it. Here, without the pit, the rock would not have damaged, so they agree that Levi pays half the damage!

Rejection: This is not so. If not for the rock, the pit would not have damaged.

(Rava): If Reuven's ox and Shimon pushed (a person) into Levi's pit, all are liable.

Rishonim

Rif and Rosh (5:12): Presumably, the Halachah follows R. Noson, for our Mishnah is like him.

Rosh: Also, Rava's law of a person and ox that pushed into a pit is like R. Noson.

Rambam (Hilchos Nizkei Mamon 2:10): If a string was tied to Levi's chicken's leg and a Kli was caught on the string and rolled and broke, he pays half damage. This is if a man tied it. If it became tied by itself, Levi is exempt. If the string was not Hefker, rather, of David, David pays half-damage, for this is a rolling pit.

Rambam (11): If David hid away his string and the chicken stepped on it and took it out and it became tied to its foot, even David is exempt, for this is Ones.

Rebuttal (Ra'avad): The Rambam understands that when it was David's string, he and Levi each pay half, for a pit never pays half. This is wrong. If it damaged after came to rest, David pays full damage, for it is his pit. Levi does not pay for his chicken's pit. If it damaged while rolling, David is exempt and Levi pays half, for it is Tzeroros of his chicken's impetus.

Gra (Likutim in Frankel Rambam): The Rambam is incomprehensible. Rather, Levi pays full damage, half-damage, or is exempt, if it was dragged normally, thrown, or damaged after it rested, respectively. This is if no one tied it and the string was Hefker or its owner hid it away. If David left his string exposed, or he put a string (even another's) in an exposed place or tied it to the chicken, David pays full damage, half-damage, or is exempt, if it damaged normally, was thrown, or damaged after it rested, respectively. This refers to a Hefker chicken. If it was Levi's chicken, if it damaged normally, each pays half. If it was thrown, Levi pays a quarter, for it is Tzeroros and both of them damaged. David pays three quarters, like R. Noson, since both of them were negligent. If it damaged after it rested, both are exempt.

Rambam (12:18): If one was digging a pit and an animal fell forwards into the pit due to the noise of the digging, he is liable.

Magid Mishneh: The Rambam rules like R. Noson because was a judge and descended to the depth of judgment. Also, the Mishnah is like him.

Rambam (19): If Reuven's ox pushed an animal into Levi's pit, if it was Mu'ad, Reuven and Levi each pay half the damage. If it was Tam, Reuven pays quarter-damage from his ox, and Levi pays three quarters from his best property

Rosh (2:2): If David left his Dalil in Reshus ha'Rabim and it became tied to Levi's chicken and if it damaged through walking normally, each of them pays half. If the chicken threw it, Levi pays a quarter, for (Tzeroros normally pays a half, and) he has a partner, and David pays three quarters, like R. Noson, who says that what cannot be collected from him is collected from the other.

Poskim

Shulchan Aruch (390:10): If a string was tied to Levi's chicken's leg and a Kli got caught on the string and rolled and broke, he pays half damage.

Rema: If it was David's string and a Hefker chicken, he is fully liable whether the chicken damaged through dragging or throwing, since he did not hide away the string. If Levi's chicken damaged through dragging, both of them pay together. If it damaged through throwing, Levi pays a quarter and David pays three quarters.

Beis Yosef (DH veha'Rambam): The Rif omitted this, for it is uncommon.

SMA (27): David pays the rest, for he would have paid full damage if the chicken were Hefker.

Rema: If Yosef tied it to the chicken, he is liable in every case, even if David owns the string. The same applies if David hid it and Yosef put it in an open place.

SMA (29): He could have that this is even if Levi owns the chicken. It seems that this was no bigger Chidush, for Yosef became fully responsible.

Dagul me'Revavah: The SMA's words apply when Yosef tied it, but if he only put it in an open place, we must say that the chicken is Hefker. If it is Levi's, Yosef is no more liable than one who left his string exposed; also Levi pays.

Shulchan Aruch (CM 410:31): If David was digging in Levi's pit to widen it and Shimon's animal fell into the pit due to the noise of the digging, David is exempt, for it is Gerama. Since Shimon cannot collect from David, he collects from Levi.

Shulchan Aruch (32): If Reuven's ox pushed Shimon's animal into Levi's pit, if it was Mu'ad, Reuven and Levi each pay half. If it was Tam, Reuven pays a quarter from his ox, and Levi pays three quarters from his best property. Shimon can tell Levi 'my carcass is in your pit.'