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We study the impact of competitive personalized pricing in a Hotelling duopoly model where consumers can purchase from both firms. We show that the impact crucially depends on the magnitude of the additional utility from consuming the second product. Compared with uniform pricing, personalized pricing benefits both consumers and firms when the additional utility is moderate; but it harms consumers while benefiting firms when the additional utility is large. These results contrast with the existing research on competitive personalized pricing, which assumes that consumers purchase one product only.