The gemara says that if a man and an ox would together push another animal into a bor, they would equally split the damages between the the man who pushed, the ox who pushed and the bor. Tosafos (here and on 6a) point out that we must be speaking about the person pushing accidentally, because if he intentionally pushed, why should the ba'al ha'bor be chayev! Just as if one throws his friends talis into a fire, or pushes his friends animal into a bor, the thrower and pusher are fully responsible, not the ba'al ha'eish or the ba'al ha'bor, here also the pusher should be more chayev than the ba'al ha'bor.
It is not clear from Tosafos what exactly the din would be if the person pushed with intent, together with an ox, into the bor of a third person. The ketzos Ha'choshen (410:2) explains the paradox. From the perspective of the person who pushed intentionally, the ox who pushed is his partner, but the bor is not. However, from the perspective of the ox who pushed, both the person and the bor are it's partners in this damage. So, if the person will pay half, the ox half, and the ba'al ha'bor exempt, the ox can claim that the ba'al ha'bor is his partner. If the person pays 2/3 of the damage and the ox 1/3, the person can claim that from his perspective the ox was his half partner in this damage. Therefore, the ketzos says that the person pays 1/2 the damage, and the shor and bor split the other half. Why? Because if the person pushed alone he would pay full, now that he has the ox as a partner he pays half. The remaining half the shor should pay, but the shor has the bor as a partner so they each pay 1/4. Based on this, the case in the gemara can be speaking when the person does it with intent, and all 3 participate in paying, but not equally. However, Tosafos implies that when the person does it with intent, the owner of the bor is completely exempt.
In the hagahos on the ketzos, they quote from R' Shimon Shkopf - From the perspective of the ox, there is no difference if the person was intentional or not, so the ox has 2 partners. From the perspective of the person who did it with intent, he only has one partner. Therefore, the ox pays 1/3, the person pays 1/2, and the remaining 1/6 is payed by the ba'al ha'bor.
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