Barbie needed to keep winning the hearts of the youngest children buying a fashion doll for the first time (or persuading their parents to do so) is an example of:

Type 1 Rivalry: Competing For Potential Customers

Explanation:

Rivalry in the terms of market and customers have a very different approach. Growth in sales and share of market are the two factors which play an active role in the growth of competition in the market. Rivalries among the sellers brings them to the constant war to win the market and the customers. They adopt many strategies and build plans to develop and retain their businesses in the market. In Type 1 rivalry, the sellers are in constant war to allure new customers. These customers are new customers who have never purchased any of such product before.  

In the given excerpt, the competitive strategy adopted by Barbie is type 1 rivalry in which it is in constant competition to bring new customers.

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