Nash Social Welfare in Selfish and Online Load Balancing

In load-balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load-balancing problems are selfish load balancing (a.k.a. load-balancing games), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and online load balancing, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both problems under the objective of minimizing the Nash Social Welfare, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in loadbalancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal, as it matches the performance of any possible online algorithm.