Ultimatum game: regret or fairness?

In the ultimatum game, the challenge is to explain why responders reject non-zero offers thereby defying classical rationality. Fairness and related notions have been the main explanations so far. We explain this rejection behavior via the following principle: if the responder regrets less about losing the offer than the proposer regrets not offering the best option, the offer is rejected. This principle qualifies as a rational punishing behavior and it replaces the experimentally falsified classical rationality (the subgame perfect Nash equilibrium) that leads to accepting any non-zero offer. The principle is implemented via the transitive regret theory for probabilistic lotteries. The expected utility implementation is a limiting case of this. We show that several experimental results normally prescribed to fairness and intent-recognition can be given an alternative explanation via rational punishment; e.g. the comparison between "fair" and "superfair", the behavior under raising the stakes etc. Hence we also propose experiments that can distinguish these two scenarios (fairness versus regret-based punishment). They assume different utilities for the proposer and responder. We focus on the mini-ultimatum version of the game and also show how it can emerge from a more general setup of the game.