The odds are 50% for being the robot's owner, if it's really so closely contested they're both willing to gamble their lives on it... not a lot of room for someone knowingly lying about it. The odds are also 50% for picking the poisoned glass, and that is not dependent on ownership. These are not the same 50% chances - so, the chances work out among four options (owner is or isn't poisoned, not-owner is poisoned or not). On the other hand, three of the four are nonlethal options, a 25% chance of being fatal (both not-owner and picked poison) so that would make sense if it was only about Joe being wrong about his chances, not overestimating. Also only a 25% chance of getting the robot to act (both owner and picked poison), so not terribly useful, either.
So, perhaps we can assume
both drinks are poisoned. But that only brings it back down to 50% odds, owner or not, since the glass picked is now irrelevant. However, Jack goes first - this is given. Perhaps the extra odds against his survival have to do with the potency of the poison? If the poison is very quick, the extra time could nibble away the odds of survival even if an antidote is given. Or it could increase Joe's chance of surviving (as in, he can take an antidote based on what he sees happening to Jack), but increasing Jack's odds don't decrease Joe's odds.
hmm, what if
they're different poisons, and the robot only has the antidote to one? then only the owner will survive, reversing that previous 75% odds to against Joe (must be both owner and pick the right glass to live). On the other hand, there's 25% chance against the owner surviving anyway in this scenario, so maybe that's a bit extreme. On the other hand, the question does say they can't live with the possibility of the other taking the credit, so I think this is going to be my answer. That, or Jack's poison is so unique only Jack's robot will have the antidote - so if Joe is not the owner, he definitely dies (50% chance), if he is the owner, he might die anyway (some extra % chance the robot will not have the correct antidote). Jack certainly lives, since owner or not he has the right antidote.
Both glasses are poisoned, and the robot is openly given "the antidote"...which only counters one of two poisons, in scenario one, or is outright false, if scenario two. The first possibility is with each glass having different poisons. This would lead to 75% odds against either of them surviving, since to live they must be both the owner and have picked the right glass. Or, possibility two is that perhaps they are both poisoned with some incredibly proprietary poison of Jack's manufacture, because if the robot is his, it has the right antidote... but if it belongs to Joe, there is some extra % chance it won't have or be able to get the real antidote in time.