Tiger (male) + Lion (female) =
.
(sterile)
Lion (male) + Tiger (female) =
.
(fertile)
Lion (male) + Ligon (female) =
.
(fertile)


.


This could get real crazy...
.


Doc

Too far, I'm afraid. Look at all the "designer" dogs:
.
.

Can't imagine a good reason to mix wild species that are becoming rare enough as it is.

Not the same, or for the same reasons, IMO -- canines are not endangered by any means (just the contrary) and they're not interbreeding big cats for pets.

My WAG for a reason to mix the big cats is that as their populations decline (reducing the respective gene pools) inbreeding becomes a factor in precipitating extinction -- not enough variety to maintain genetic integrity*. By mixing in just a small amount of "new blood" -- e.g. breeding a liliger (1/4 tiger) into the lion gene pool may strengthen the lion lines tremendously (and vice versa with tigers if they can overcome the sterility issue with tigons). Once the initial "strengthening" of gene pools is accomplished (possibly with the help of the laboratory) nature can then be allowed to continue on its course.
Doc