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.

翻译：在最后通牒博弈中，解释为何回应者会拒绝非零提议从而违背经典理性是一个关键挑战。公平及相关概念一直是主要的解释方向。我们通过以下原则解释这种拒绝行为：如果回应者对失去该提议的遗憾程度小于提议者未提供最优选项的遗憾程度，则提议被拒绝。这一原则被视为一种理性惩罚行为，并取代了经实验证伪的经典理性（子博弈完美纳什均衡），后者导致任何非零提议均被接受。该原则基于概率彩票的传递性遗憾理论实现，期望效用实现是其特例。我们证明，通常归因于公平和意图识别的多项实验结果（例如"公平"与"超公平"的对比、提高赌注时的行为反应等）可通过理性惩罚获得替代性解释。因此，我们提出能够区分这两种场景（公平与基于遗憾的惩罚）的实验设计，其假设提议者与回应者具有不同的效用函数。我们聚焦于迷你最后通牒博弈变体，并展示了其如何从更一般的博弈设定中衍生。