“The victim and the perpetrator share in the [payment] of damages.” (Bava Kamma 14b) Being a victim of an accident is never pleasant even if one is fully compensated for any losses. One can never repay the time, upset and stress such causes. And almost always one is not fully compensated neither for actual losses and surely not for the time spend dealing with the fallout. This is unavoidable and a situation the halacha well understood and to a certain extent even perpetuated.
One who has the misfortune of having themselves or their property attacked by an animal that is a tam can only collect half damages. In that sense its better to be attacked by a muad, an animal that has a history of attack as that would allow the victim to collect nezek shalem, full damages. Furthermore those half damages are limited to a maximum of the value of the animal itself. If an animal causes $1,000 of damages but the animal is worth only $100 the owner would pay only $100 and not even the half damages of $500. And the victim stands to lose even more due to the inevitable delay from the time of incident until monies are actually collected. For example say a tam attacks another animal killing it in the process. The amount one can collect is (half) the difference between the value of the animal while it was alive and its value as a carcass. However if the value of the carcass deceases from the time of the attack until it is actually sold no adjustment is made even though the actual loss is now greater.   
We have previously defined a tam as an animal that has yet to gore more than three times. There is a most interesting argument regarding the time frame in which an animal changes its classification from tam to muad, a regularly goring animal that must pay full damages. Rabbi Meir argues that once a warning was issued three times even on the same day and the animal continues to gore it becomes a muad. Rabbi Yehuda argues that the three attacks take place on different days. Three attacks on one day is a bad day but does not turn one into a muad
Yet once a muad does not mean always a muad. Just as an animal can go from tam to muad, it can go from muad to tam. Rabbi Yehuda’s view is just as it becomes muad if it gores on three different days it can return to it tam status if it goes three days without goring[1]. Rabbi Meir who does not concern himself with a specific time frame says that an animal returns to its tam status when children can safely play with the animal.
As noted above before the animal can be declared a muad a warning must be issued. There is fascinating debate (Bava Kamma 24a) regarding who is exactly is being warned. Is it the animal so to speak or the owner who needs to be warned? In other words do we need three separate gorings to demonstrate that the animal is a wild one and full damages must be paid from here on in? Or is it the action or inaction of the owner that is the cause of the animal goring? Perhaps full damages cannot be paid unless and until the owner is given a chance (or two or three) to ensure their animal won't do so again. According to this view if three sets of witnesses were to testify at the same time that the ox had be gored it would still be a tam as there was no opportunity for corrective action on the part of the owner. 
A similar view is expressed by Rav Meir regarding the ox of a child that gores. The Mishna (Bava Kamma 39a) explains that in such a situation the court will appoint a guardian to watch over the animal. When the child becomes an adult they are given  the responsibility to care for his possessions. Rabbi Meir claims that the animal regains its status as a tam affording the child turned adult an opportunity to properly guard the animal before having to pay full damages.
In assessing the proper amount of damages to be paid there are multiple factors at work making it a complex exercise. It is fascinating to see how important a factor is that of the relationship between an animal and it owner.
[1] This is somewhat akin to the notion of kbolo kach polto that a vessel can become koshered if the same method of use is applied as to what made it non-kosher in the first place. Thus a pot used for cooking needs to be koshered by heating water in the pot, an oven by heating it to the highest temperature for which it was used and a bbq by applying a direct flame on the grill.