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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.
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