Fairly Allocating (Contiguous) Dynamic Indivisible Items with Few Adjustments

We study the problem of dynamically allocating indivisible items to a group of agents in a fair manner. We assume that the items are goods and the valuation functions are additive without specification. Due to the negative results to achieve fairness, we allow adjustments to make fairness attainable with the objective to minimize the number of adjustments. We obtain positive results to achieve EF1 for (1) two agents with mixed manna, (2) restricted additive or general identical valuations, and (3) the default setting. We further impose the contiguity constraint on the items and require that each agent obtains a consecutive block of items. We obtain both positive and negative results to achieve either EF1 or proportionality with an additive approximate factor. In particular, we establish matching lower and upper bounds to achieve approximate proportionality for identical valuations. Our results exhibit the large discrepancy between the identical model and nonidentical model in both contiguous and noncontiguous settings. All our positive results are computationally efficient.