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Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines

We study problems of scheduling jobs on related machines so as to minimize the makespan in the setting where machines are strategic agents. In this problem, each job $j$ has a length $l_{j}$ and each machine $i$ has a private speed $t_{i}$. The running time of job $j$ on machine $i$ is $t_{i}l_{j}$. We seek a mechanism that obtains speed bids of machines and then assign jobs and payments to machines so that the machines have incentive to report true speeds and the allocation and payments are also envy-free. We show that 1. A deterministic envy-free, truthful, individually rational, and anonymous mechanism cannot approximate the makespan strictly better than $2-1/m$, where $m$ is the number of machines. This result contrasts with prior work giving a deterministic PTAS for envy-free anonymous assignment and a distinct deterministic PTAS for truthful anonymous mechanism. 2. For two machines of different speeds, the unique deterministic scalable allocation of any envy-free, truthful, individually rational, and anonymous mechanism is to allocate all jobs to the quickest machine. This allocation is the same as that of the VCG mechanism, yielding a 2-approximation to the minimum makespan. 3. No payments can make any of the prior published monotone and locally efficient allocations that yield better than an $m$-approximation for $\qcmax$ \cite{aas, at,ck10, dddr, kovacs} a truthful, envy-free, individually rational, and anonymous mechanism.