Commentary for Bava Kamma 3:8

He pays the balance, that he should evaluate how much this one is obligated for what he damaged and how much the second is obligated also, and if the damages are equivalent, this one and this one go out (with nothing)...

במותר חצי נזק – they estimate what the damages of this one are greater than the damage of that one, and in that surplus, the person who caused greater damage pays the one-half assessment.

If two oxen which were accounted harmless hurt one another, the owner pays half-damages for that one which suffered the greater hurt.
If both were attested dangers full damages are payable for that one which suffered the greater hurt.
If one was accounted harmless and the other was an attested danger, that which was an attested danger as against that which was accounted harmless must pay full damages for the greater hurt that the other has suffered, while that which was accounted harmless, as against that which was an attested danger, pays only half damages for the greater hurt that the other has suffered.
So, too, if two men hurt one another, full damages are payable for that one which suffered the greater hurt.
If a man and an ox which was accounted harmless hurt one another, the man as against the ox accounted harmless must pay full damages for the greater hurt that the other has suffered, while the ox accounted harmless, as against the man, pays only half damages for the greater hurt that the other suffered. Rabbi Akiva says: “Even if an ox accounted harmless hurt a man, full damages must be paid for that one which suffered the greater hurt.
This mishnah returns to deal with a subject that we dealt with in the last mishnah of chapter one and in the fourth and fifth mishnayoth of chapter 2 and that is the goring ox. Remember, there are two types of goring oxen, one that is a previously attested danger (muad), who has been testified against. The second type is innocent (tam) meaning he is not a previously attested danger. When a muad damages the owner will pay full damages from the best of his land. When a tam damages the owner will pay half damages from the value of the tam itself. This means that the upper limit of liability will be the value of the damaging animal.
We will continue to deal with the goring ox for the next two and a half chapters. I will not be explaining the concepts of tam and muad every time we encounter them. I will try to reference places where I did explain them. If you are wondering why the mishnah is so fascinated by the goring ox, it is due to the fact that the Torah mentions the ox quite frequently (Exodus 21:28-32, 35-36). It is indeed the paradigm for the damaging animal.
The rendering of this mishnah into sensible English is very difficult since the mishnah speaks in shorthand. However, the explanation should make more sense to you.
The first section deals with two harmless oxen that gore each other. In general the owner of each is obligated to pay half damages to the other. Here we figure out what was the greater damage and the owner of the less injured ox will pay half of that amount. A table will help.
Value of ox before injury
Value after injury
Damages
Amount owed
100 (tam)
30
70
35
50 (tam)
30
20
10
In this case the owner of the ox worth fifty will pay twenty-five to the owner of the ox worth 100.
In the second case both of the animals were muad and will therefore pay full damages. Our table now looks like this:
Value of ox before injury
Value after injury
Damages
Amount owed
100 (muad)
30
70
70
50 (muad)
30
20
20
In this case the owner of the ox worth fifty will pay fifty to the owner of the ox worth 100.
In the third case one ox was muad and one was tam. The muad will owe half damages and the tam full damages. According to our example in this case the animal worth 100 was a muad and therefore will owe full damages for the animal worth 50. The animal worth 50 is a tam and will therefore pay half damages for the animal worth 100. Our table now looks like this:
Value of ox before injury
Value after injury
Damages
Amount owed
100 (muad)
30
70
35
50 (tam)
30
20
30
In this case the owner of the ox worth fifty will pay 5 to the owner of the ox worth 100.
The fourth case of the mishnah deals with human beings who injure one another. Since a human being is always a muad (see chapter one mishnah four), this is similar to case number two.
The fifth and final case deals with a human being (who is always muad) and a harmless ox (tam) who injure one another. This case is similar to case number three. We will nevertheless bring a new table.
Value before injury
Value after injury
Damages
Amount owed
1000-- human (muad)
500
500
250
50--ox (tam)
20
30
30
In this case the owner of the ox will pay the human 220. Rabbi Akiva disagrees. According to him an ox that injures a human being always pays full damages as if it was a muad. Therefore in the previous scenario the owner of the ox will pay 470 to the human.

And if the damages of one of them were more, that it was appropriate for one to pay more than what is appropriate for the other to pay, he who damaged more pays the difference. And it is not neccesary to give an example because this is clear.

מועד בתם משלם במותר – meaning to say, if he (i.e., the forewarned animal) damaged the innocuous animal to a greater extent than what the innocuous animal did.

And that the pasuk says ' if it gores a minor man or woman, like this judgement should be done to it. It's a disagreement between Rabbi Akiva and the sages in the explanation of this pasuk, the sages say like the judgement of one ox to another ox so too the judgement [of an ox] to a person, meaning to say that the unaccustomed is obligated in half damages and the accustomed full damages. and this has already reached us that all agree thata man is always accustomed. And Rabbi Akiva says like this judgement refers to the accustomed cow, because the pasuk comes to speak about this matter, thus it will be in general the law of an ox that damages a man whther it is accustomed of unaccustomed

אדם בתם משלם במותר נזק שלם – for a person is forever forewarned.

And the halakah is not like Rabbi Akiva

ותם באדם – he (i.e., the owner of the innocuous animal) pays the surplus of one-half damages, as it is written (Exodus 21:31): “So, too, if it gores a minor, male or female, [the owner] shall be dealt with according to the same rule.” Just as the judgment of an ox harming another ox, so too the judgement of an ox harming a human being. Just as an ox goring an ox – [the owner of] the innocuous ox who gored another ox pays half-damages, so too, with an ox which goes a person – [the owner of] the innocuous ox pays one-half damages and the [owner of the] forewarned ox pays full damages.

ר"ע אומר אף תם שחבל באדם משלם במותר נזק שלם – as he expounds “[shall be dealt with according to] the same rule” for a law that this verse removes from it. For it speaks about a forewarned ox. And this is what the Biblical verse said “"כמשפט הזה /the same rule – refer to the forewarned ox that [its owner] pays full damages. [The words] "יעשה לו" /shall be dealt with – means for every ox that gores a person, even if it is an innocuous ox. But the Halakha is not according to Rabbi Akiva.