Costly punishment in the ultimatum game evokes moral concern, in particular when framed as payoff reduction

The ultimatum game is a common economic experiment in which some participants reject another's unfair offer of how to split some money, even though it leaves them both worse off. This costly behavior can be seen as enforcement of a fairness norm and has been labeled “altruistic punishment”, suggesting that it is a moral thing to do. But is this behavior viewed as moral by participants? Is it viewed as punishment? And are the payoff consequences...