The oceanic asthenosphere is observed to have high electrical conductivity, which is highly anisotropic in some locations. In the directions parallel and normal to the plate motion, the conductivity is of the order of 10(-1) and 10(-2) S m(-1), respectively, which cannot be explained by the conductivity of anhydrous olivine. But because hydrogen can be incorporated in olivine at mantle pressures, this observation has been attributed to olivine hydration, which might cause anisotropically high conductivity by proton migration. To examine this hypothesis, here we report the effect of water on electrical conductivity and its anisotropy for hydrogen-doped and undoped olivine at 500-1,500 K and 3 GPa. The hydrous olivine has much higher conductivity and lower activation energy than anhydrous olivine in the investigated temperature range. Nevertheless, extrapolation of the experimental results suggests that conductivity of hydrous olivine at the top of the asthenosphere should be nearly isotropic and only of the order of 10(-2) S m(-1). Our data indicate that the hydration of olivine cannot account for the geophysical observations, which instead may be explained by the presence of partial melt elongated in the direction of plate motion.