We present an extension of the overtaking criterion, called fixed-step overtaking social welfare relation (SWR), and its leximin counterpart, called fixed-step W-leximin SWR, for the evaluation of infinite utility streams. Our SWRs satisfy Fixed-step Anonymity. First, we characterize them by using the consistency, called Weak Fixed-step Indifference Consistency, which links indifference for infinite utility streams to indifference for their truncated vectors that are taken periodically. Second, we characterize another utilitarian extension, called S-overtaking SWR, by replacing the consistency with Fixed-step Anonymity. The S-overtaking SWR is a subrelation of the fixed-step overtaking SWR. Meanwhile, the leximin counterpart of the S-overtaking SWR coincides with the fixed-step W-leximin SWR, and we obtain its alternative characterization using Fixed-step Anonymity. Finally, we generalize the impossibility of combining Fixed-step Anonymity with the catching-up SWR by showing the Pareto and strong consistency axioms satisfied by the catching-up SWR come in conflict with Fixed-step Anonymity.
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