Very nice video and good explanation. A question that came to my mind was though: Aside from the influence of "fairness", does B know about the game and the consequences for A in case of B rejecting the offer? If so, why would B ever settle for something less than 9 dollars. In this situation A has to choose between taking 1 dollar or nothing. In case the game rules are known by everyone, this would leave B with all the power given both try to maximize their income.
Yes, unless the researcher is trying to understand the role of information asymmetry in particular, in general the rules are known to all players of the game in economic experiments. This allows us transparently to identify peoples' responses in a way that is compatible with the incentives involved and to compare the different institutional settings offered by different "rules of the game" in different contexts. For example, the $1 received by the receiver is likely to be subjectively experienced differently if they known the proposer had $2 total, $5 total, $10, $20 and so on. In the first case of receiving $1 of a total of $2 is likely to be perceived as fairer than $1 of $10.
It depends on what the person's underlying preferences are and what their beliefs are about what their counterpart's preferences are. Remember that "rationality" is basically just a statement about someone behaving consistently with a particular set of goals. Rationality and self-interest are not the same thing. A person could be rational and self-interested or self-regarding, or they could be rational and other-regarding in some way. But even if one person is rational and self-regarding, that does not mean they have a belief that the other player is self-regarding. If a rational and self-regarding player offered zero, then they would also not be offering an incentive for their rational and self-regarding partner to accept the offer as they would be indifferent between zero from rejecting and zero from accepting; so even with self-regarding players, one would expect an offer of the lowest feasible amount greater than zero.
