We study a model in which a group of agents make a sequence of collective decisions on whether to remain in the current state of the system or switch to an alternative state, as proposed by one of them. Examples for instantiations of this model include the step-wise refinement of a bill of law by means of amendments to be voted on, as well as resource allocation problems, where agents successively alter the current allocation by means of a sequence of deals. We specifically focus on cases where the majority rule is used to make each of the collective decisions, as well as variations of the majority rule where different quotas need to be met to get a proposal accepted. In addition, we allow for cases in which the same proposal may be made more than once. As this can lead to infinite sequences, we investigate the effects of introducing a deadline bounding the number of proposals that can be made. We use both analytical and experimental means to characterise situations in which we can expect to see a convergence effect, in the sense that the expected payoff of each agent will become independent from the initial state of the system, as long as the deadline is chosen large enough.
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