Marketing research has traditionally focused on centralized brand-extension strategies where a brand expands its product offerings by controlling the design, production, marketing and sales of new products `in-house'. However, luxury brands frequently use `brand licensing' as a decentralized brand-extension strategy under which a brand licenses its brand name to an `external licensee' that designs, produces and sells the new product. Licensing is a time-efficient and cost-effective brand-extension strategy for luxury brands to reach out to their aspirational, low-end consumers (`followers') who value a brand more when more high-end consumers (`snobs') purchase the brand's primary product (i.e., `positive popularity effect'). On the other hand, over-licensing might dilute the brand for snobs who value brand exclusivity (i.e., `negative popularity effect'). We develop a game-theoretic model to study luxury brand licensing in the presence of these two countervailing forces. First, in the monopoly setting (a benchmark), we find that the monopoly brand should license only when the negative popularity effect is not too high, and it should prefer `royalty licensing' over `fixed-fee licensing' when the negative popularity effect is intermediate. Second, to explicate our analysis, we study the duopoly setting under fixed-fee contracts. In contrast to the monopoly setting, we find that fixed-fee licensing can `soften' price competition between brands so that licensing is `always' profitable for both brands under competition. Interestingly, in equilibrium under fixed-fee contracts, competing brands face a prisoner's dilemma and both brands prefer not to license, even though both would be better off if they could commit to fixed-fee licensing. Finally, we expand our analysis of the duopoly model by incorporating royalty licensing in addition to fixed-fee licensing. We find that, in contrast to fixed-fee licensing, royalty licensing can `intensify' price competition so that both brands have to lower their prices. Consequently, when the positive popularity effect is sufficiently strong, fixed-fee licensing `dominates' royalty licensing. We also show that, under competition, luxury brands should adopt royalty contracts only when the licensing market is large, and positive and negative popularity effects are small enough.