We study the impact on payoff distribution of varying the probability (opportunity) that a player has of becoming the proposer in an ultimatum game (UG). Subjects' assignment to roles within the UG was randomised before the interactions. Subjects played 20 rounds anonymously and with random rematching at each round. We compare the outcomes of four different settings that differed according to the distribution of opportunities between the pair of players in each round, and across the whole 20 rounds. The results clearly point to the existence of a discontinuity in the origin of the opportunity spectrum.Allowing a player a 1% probability of becoming the proposer brings about significantly lower offers and higher acceptance rates with respect to the benchmark case where a player has no such a chance. As such probability is raised to 20% and 50%, this same trend continues, but the effects are generally no longer significant with respect to the 1% setting. In one case the monotonic pattern is violated. We conclude that subjects in our experiment appear to be motivated mostly by the purely symbolic aspect of opportunity rather than by the actual fairness in the allocation of opportunities
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