We do find half-damages in cases of damage by eating. Tosafot on Bava Kamma 17b, s.v. “digging in a trash heap,” writes: “And even though he taught pebbles as a case of foot, he thereby also teaches pebbles as a case of tooth, where it acts for its own benefit.”

If so, the question returns to its place: in such a case, can the injured party claim the second half from the damager on the basis of benefit?

In pebbles of tooth there is benefit from what the animal is doing (digging in the trash heap), but what was damaged by the pebbles does not benefit it.

What about a dog that eats sheep? That is atypical tooth, meaning it benefits, but since that is not its usual way, it pays half-damages.

And in this case the question with which I opened this thread comes back, no?

In this case the injured party would collect half-damages as damages, and two-thirds of the second half on the basis of benefit.
Right or not?

By logic it seems reasonable to me, and also in terms of: when it cannot be collected from this basis, it is collected from the other basis.

I also wanted to ask you something else in this connection. There is the rule regarding one who robs his fellow and then another comes and eats it: if he wants, he collects from this one, and if he wants, he collects from that one. That is, if the victim collects from the robber, then the eater is no longer obligated to pay, even though he benefited, since the victim has already gotten his money.
And seemingly one could say: after all, the robber paid his debt on the basis of robbery, meaning because he caused his fellow a monetary loss. But the eater owes money not because he caused him a loss, but because he received benefit. So why should the robber’s payment exempt the one who benefited from payment? After all, in the two cases it is not the same basis of liability.

(I don’t know the topics well enough to answer on my own…)

Why two-thirds? I didn’t understand. It does indeed seem that he pays the damage and makes up the difference up to the amount of the benefit.
As for the question about robbery too, it seems that the law is like that: the eater is obligated to pay for the benefit, and regarding the damage one can collect from either of them. But just as with one person he does not pay damage and benefit simultaneously, so too with two people there is no logic that they should pay double. The injured party is not supposed to receive more than the amount he lost.
However, now the robber who paid can sue the eater, who benefited from what was his. For if the robber paid, then the eater benefited from the robber, not from the victim of the robbery, since the item is considered his insofar as he paid for it. Therefore there is room to obligate him to pay the robber for the benefit. According to this, in the end the eater is always obligated to pay, because he benefited. And the robber can use the benefit payments to restore the robbery. And by the rule of Rabbi Nathan’s lien, the victim of the robbery can sue the eater directly. According to this, the intent of the Talmud when it said that one can collect from either of them is that if the eater fled abroad and it is impossible to collect from him, then one collects from the robber.