In the online load balancing problem on related machines, we have a set of jobs (with different sizes) arriving online, and we need to assign each job to a machine immediately upon its arrival, so as to minimize the makespan, i.e., the maximum completion time. In classic mechanism design problems, we assume that the jobs are controlled by selfish agents, with the sizes being their private information. Each job (agent) aims at minimizing its own cost, which is its completion time plus the payment charged by the mechanism. Truthful mechanisms guaranteeing that every job minimizes its cost by reporting its true size have been well-studied [Aspnes et al. JACM 1997, Feldman et al. EC 2017]. In this paper, we study truthful online load balancing mechanisms that are well-behaved [Epstein et al., MOR 2016]. Well-behavior is important as it guarantees fairness between machines, and implies truthfulness in some cases when machines are controlled by selfish agents. Unfortunately, existing truthful online load balancing mechanisms are not well-behaved. We first show that to guarantee producing a well-behaved schedule, any online algorithm (even non-truthful) has a competitive ratio at least $Omega(sqrt{m})$, where m is the number of machines. Then we propose a mechanism that guarantees truthfulness of the online jobs, and produces a schedule that is almost well-behaved. We show that our algorithm has a competitive ratio of $O(log m)$. Moreover, for the case when the sizes of online jobs are bounded, the competitive ratio of our algorithm improves to $O(1)$. Interestingly, we show several cases for which our mechanism is actually truthful against selfish machines.
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