We study a dynamic social choice problem in which an alternative is chosen at each round according to the reported valuations of a set of agents. In the interests of obtaining a solution that is both efficient and fair, we aim to maximize the long-term Nash welfare, which is the product of all agents' utilities. We present and analyze two greedy algorithms for this problem, including the classic Proportional Fair (PF) algorithm. We analyze several versions of the algorithms and how they relate, and provide an axiomatization of PF. Finally, we evaluate the algorithms on data gathered from a computer systems application.

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