Most Competitive Mechanisms in Online Fair Division

This paper combines two key ingredients for online algorithms - competitive analysis (e.g. the competitive ratio) and advice complexity (e.g. the number of advice bits needed to improve online decisions) - in the context of a simple online fair division model where items arrive one by one and are allocated to agents via some mechanism. We consider four such online mechanisms: the popular Ranking matching mechanism adapted from online bipartite matching and the Like, Balanced Like and Maximum Like allocation mechanisms firstly introduced for online fair division problems. Our first contribution is that we perform a competitive analysis of these mechanisms with respect to the expected size of the matching, the utilitarian welfare, and the egalitarian welfare. We also suppose that an oracle can give a number of advice bits to the mechanisms. Our second contribution is to give several impossibility results; e.g. no mechanism can achieve the egalitarian outcome of the optimal offline mechanism supposing they receive partial advice from the oracle. Our third contribution is that we quantify the competitive performance of these four mechanisms w.r.t. the number of oracle requests they can make. We thus present a most competitive mechanism for each objective.