Logically, the second player should accept any non-zero offer. In actual experiments people tend to reject offers less than about 20%. The exact percentage varies with culture. In some cultures it is considered an insult if the second player is not offered more than 50%. All cultures have a sense of fairness. People will reject what is clearly the logical thing to do if they think they're being treated too unfairly.
Versions of the Ultimatum Game designed for experiments with non-human primates show similar results. When one primate sees another consistently rewarded with more food or a more favored food for the same effort or less they eventually reject their lesser gift or even throw it at the experimenter. An innate sense of fairness, of justice is apparently a primate birthright, an evolved trait going back millions of years. Push primates, including humans, too far and they will rebel, even if it is not in their best interest.
Republicans who ignore today's Gilded Age income divergence are ignoring something more visceral than self-interest or even greed. At some point the non-rich will decide that the split is too unfair and reject it, possibly violently. This isn't just an issue of political philosophy or even logic. It's biological fact. Republicans ignore it at their peril. Unfortunately, ignoring or even attacking science that contradicts their dogma is something Republicans excel at.