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In many markets, firms can price discriminate between their own customers and their rivals' customers, charging one price to consumers who prefer their own product and another price to consumers who prefer a rival's product. We find that when demand is symmetric, charging a lower price to a rival's customers is always optimal. When demand is asymmetric, however, it may be more profitable to charge a lower price to one's own customers. Surprisingly, price discrimination can lead to lower prices to all consumers, not only to the group that is more elastic, but also to the less elastic group.