The behaviour described above could be motivated by the so-called altruistic punishment.
The ultimatum game is a two-player game, in which the first player plays the role of the offeror while the second is the respondent. The first one is endowed with a certain amount of money (for example, 100 $) and has to make an offer about how to split it between the two of them. If the respondent accepts the offer, each player would receive the amount of money that had been proposed by the offeror. If the respondent does not accept, both will earn 0.
A respondent will accept any offer that maximizes his utility. If utility meant exactly the same as money earnings, the respondent would accept any offer in which he receives at least 1 $, as he would be left better off than before (when he had 0$). But this is not true, as in many cases the respondent would choose to punish the other player by refusing his offer so that both earn 0$, if he considers the other has done an unfair distribution. This behaviour is known as the altruistic punishment, as although the offeror is punished, he learns a lesson from it.
If the offeror had expected that possibility he would go for more egalitarian distributions, so that he makes sure the respondent does not punish him and both manage to earn some money. This is why in the end many proposers offer half of the money in the game.
