The final inspiral of double neutron star and neutron-star-black-hole binaries are likely to be detected by advanced networks of ground-based gravitational wave (GW) interferometers. Maximizing the science returns from such a discovery will require the identification of an electromagnetic counterpart. Here we critically evaluate and compare several possible counterparts, including short-duration gamma-ray bursts (SGRBs), "orphan" optical and radio afterglows, and day-long optical transients powered by the radioactive decay of heavy nuclei synthesized in the merger ejecta ("kilonovae"). We assess the promise of each counterpart in terms of four "Cardinal Virtues": detectability, high fraction, identifiability, and positional accuracy. Taking into account the search strategy for typical error regions of tens of square degrees, we conclude that SGRBs are the most useful to confirm the cosmic origin of a few GW events, and to test the association with neutron star mergers. However, for the more ambitious goal of localizing and obtaining redshifts for a large sample of GW events, kilonovae are instead preferred. Off-axis optical afterglows are detectable for at most tens of percent of events, while radio afterglows are promising only for energetic relativistic ejecta in a high-density medium. Our main recommendations are: (1) an all-sky gamma-ray satellite is essential for temporal coincidence detections, and for GW searches of gamma-ray-triggered events; (2) the Large Synoptic Survey Telescope should adopt a one-day cadence follow-up strategy, ideally with 0.5 hr per pointing to cover GW error regions; and (3) radio searches should focus on the relativistic case, which requires observations for a few months.
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